This is a multi-part message in MIME format. ---------------0605101120308 Content-Type: text/plain; name="06-151.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-151.keywords" Green's function, discrete Laplacian, stationary phase method, Fourier transform, convexity, Molchanov-Vainberg Laplacian ---------------0605101120308 Content-Type: application/postscript; name="Mol25.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Mol25.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.95a Copyright 2005 Radical Eye Software %%Title: Mol25.dvi %%Pages: 44 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMR17 CMR12 CMSY8 CMBX9 CMR9 CMMI9 CMSY9 TeX-cmex9 %%+ MSBM7 CMMI5 CMR6 CMMI6 CMSY6 MSBM10 CMBX12 CMR10 CMTI10 CMR7 CMMI10 %%+ CMSY10 CMEX10 CMMI7 CMSY7 CMR8 CMBX10 CMR5 CMSY5 MSAM10 CMTI8 CMMI8 %%+ MSAM7 TeX-cmex7 CMTI9 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o Mol25.ps Mol25.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2006.05.09:1312 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 39 /quoteright put dup 45 /hyphen put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 58 /colon put dup 65 /A put dup 66 /B put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 76 /L put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 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cleartomark %%EndFont %%BeginFont: TeX-cmex7 %!PS-AdobeFont-1.0: TeX-cmex7 001.001 % Filtered by type1fix.pl 0.05 %%EndComments 13 dict dup begin /FontInfo 16 dict dup begin /Copyright (see\040copyright\040of\040original\040TeX\040font) def /FamilyName (TeX\040cmex7) def /FullName (TeX\040cmex7\040Regular) def /ItalicAngle 0 def /Notice (converted\040after\040April\0402001) def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Regular) def /isFixedPitch false def /version (001.001) def end readonly def /FontName /TeX-cmex7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 80 /P put readonly def /FontBBox {-14 -2954 1627 771} readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSAM7 %!PS-AdobeFont-1.1: MSAM7 2.1 %%CreationDate: 1992 Oct 17 08:30:29 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 22 /harpoonupright put dup 54 /lessorequalslant put dup 62 /greaterorequalslant put readonly def /FontBBox{0 -576 1569 1076}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 13 /gamma put dup 14 /delta put dup 27 /sigma put dup 33 /omega put dup 59 /comma put dup 65 /A put dup 66 /B put dup 69 /E put dup 73 /I put dup 76 /L put dup 85 /U put dup 101 /e put dup 116 /t put dup 120 /x put dup 121 /y put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 82 /R put dup 84 /T put dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTI8 %!PS-AdobeFont-1.1: CMTI8 1.0 %%CreationDate: 1991 Aug 18 21:07:42 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 110 /n put dup 111 /o put dup 114 /r put dup 117 /u put dup 121 /y put readonly def /FontBBox{-35 -250 1190 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put dup 22 /harpoonupright put dup 54 /lessorequalslant put dup 62 /greaterorequalslant put readonly def /FontBBox{8 -463 1331 1003}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 48 /prime put readonly def /FontBBox{21 -944 1448 791}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 61 /equal put readonly def /FontBBox{-341 -250 1304 965}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 70 /F put dup 74 /J put dup 76 /L put dup 77 /M put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put dup 122 /z put dup 123 /endash put readonly def /FontBBox{-301 -250 1164 946}readonly def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 8 /Phi put dup 9 /Psi put dup 11 /ff put dup 12 /fi put dup 35 /numbersign put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 67 /C put dup 69 /E put dup 70 /F put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 124 /emdash put readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 84 /T put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 1D432B0D064B56C598F4338C319309181D78E1629A31ECA5DD8536379B03C383 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cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 33 /arrowright put dup 35 /arrowdown put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 68 /D put dup 86 /V put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 107 /bardbl put dup 114 /nabla put readonly def /FontBBox{-15 -951 1252 782}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 24 /xi put dup 25 /pi put dup 27 /sigma put dup 30 /phi put dup 31 /chi put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 62 /greater put dup 64 /partialdiff put dup 66 /B put dup 67 /C put dup 70 /F put dup 77 /M put dup 78 /N put dup 81 /Q put dup 85 /U put dup 86 /V put dup 88 /X put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 80 /summationtext put dup 82 /integraltext put dup 83 /uniontext put dup 84 /intersectiontext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 98 /hatwide put dup 104 /bracketleftBig put dup 105 /bracketrightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 14 /openbullet put dup 15 /bullet put dup 17 /equivalence put dup 18 /reflexsubset put dup 24 /similar put dup 26 /propersubset put dup 33 /arrowright put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 55 /mapsto put dup 68 /D put dup 70 /F put dup 82 /R put dup 83 /S put dup 85 /U put dup 86 /V put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 24 /xi put dup 25 /pi put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 85 /U put dup 86 /V put dup 88 /X put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 8 /Phi put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 59 /semicolon put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 99 /c put dup 100 /d put dup 105 /i put dup 110 /n put dup 111 /o put dup 115 /s put dup 116 /t put dup 120 /x put dup 126 /tilde put readonly def /FontBBox{-27 -250 1122 750}readonly def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 33 /exclam put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 79 /O put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put readonly def /FontBBox{-163 -250 1146 969}readonly def currentdict end currentfile eexec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19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 3 /Lambda put dup 6 /Sigma put dup 8 /Phi put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 19 /acute put dup 20 /caron put dup 33 /exclam put dup 34 /quotedblright put dup 35 /numbersign put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 126 /tilde put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 3 /asteriskmath put dup 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 78 /N put dup 100 /d put dup 109 /m put dup 110 /n put dup 120 /x put dup 121 /y put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 61 /equal put dup 99 /c put dup 105 /i put readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 20 /kappa put dup 34 /epsilon put dup 78 /N put dup 100 /d put dup 104 /h put dup 106 /j put dup 107 /k put dup 108 /l put dup 110 /n put dup 116 /t put dup 120 /x put readonly def /FontBBox{37 -250 1349 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633 89DE8BB201D5DB4627484A80A431B6AFDBBBF23D4157D4AFE17E6B1C853DD417 25F84CD55402AB88AB7EEFDEDBF2C2C731BD25567C53B474CCF739188A930039 098A197F9C4BE7594D79442B2C8A67447DE44698321145D7689B91EF235EA80E B600AA8E238064F154284096C4C2554EFE8DDF13AFF8D3CE30E0999375C0FEE6 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def /version (001.001) def end readonly def /FontName /TeX-cmex9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 90 /Z put readonly def /FontBBox {-26 -2961 1503 773} readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633 89DE8BAE4F093966CD2192CE95EB0F323A6BABFDACCFCF27D91F7869A0E46CA5 9AAF6905783E8AC1F3F9875A76F97187738432F8D14E61574CB292FFB9740871 66839799D8CAF6E0DFE00012EE6D46A2B3655F29705BE37FD5EDA1C765AA2CF5 C5AD37207ED1EE9DB82FF31A33307FFA16911406557336AF92F50B603C7BD336 73EC060F68318378A6F599DDADA5A21504CADBA1E1F4B1A22962BA1BB39ADC7B E8CC92F196549457877C9636A8A7EFAC1C3745644C0FD151C70B9FAD69B02C1F FE5ED071CA1CF3D4A70909B6A3986687D8FB87391E0C665A4EBD2993161FA81B 2B8F7221294CCB11AE65A31E8903DCC3AC1A47765E880ACDDC88509418B04F0F 2D84FD3007D411EEE2DFF237D99A10430524F07480302DFD698B571A023B08A1 36F84F09BCECBE34ECC1CFD251EBBF338CFF7C197D9B33CBD9ACA7171370527C CFA0F7FD5DE8B62790D7DF23F004AEDA717F35B5E2B260A8AAFDE31164AFA47C 686735EC47454F42AC5037D97B410DC373EB0CEFE3C41E243EDA86FF582AB53E 7B56D103AD816F2D7DA35DE239DE30CA5645A377E77A1980B984C195E605841B 36C82FF23D95B5FF770C3AB37C2D657FD1731E91FA0446C316C68F62626B900D 8574C26A5150F80628D1AABF76459D28E0B2493D9F16B3EA6B965C802BFEF566 834D9939CEAF52A614F6123279A923D266DF0024DC5A11D6953BF8C1C79F27D1 95B9ABDFBB49E7972BD265540371848C1AEAD41AB6D62C4226DBE6CCEB5B3F67 FA688263AC3F9D884A72E847ECB99779C6F01836813C0F8E7C6041260E9A8DE9 0BF4BF49871A978AEF81110A121B271BFB64299CADC25EB53A697A8611054A29 421CE6C381A282B15F669367C183FDDF6D913B8C3251DFECFBB6BBBF973E41B2 72F2D3821B038514DAD7D612CFD357089BBD948D821D31735A7C6CD6F34F3CE0 53ED120DE2D44D59E8C39F7A766D5E0CD891201217349547337C8C0F21735057 404BCA98F3B6BDF2924693C8AD227DCECD01418D5C806031D1297FCD87C9B73F CA27EC0E8FFCD74B560390D3912582D47771F6EFED668CD697E7862ADAF75DB7 EECDD38E2CD4CE87F0C333A4675743360C25BED1891227B8503592948A85CD94 CA28C887D4BB3369C475E9FF55F3317B9888C360698F15D14160BDCFF9C4EE1F E38810FEB06578C18D24B1BB97A7328E8ECC11548E1A189AD3C84E63002049A0 743647C711527B57C286B971A857D442F748A9F0F260C04367F1 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY9 %!PS-AdobeFont-1.1: CMSY9 1.0 %%CreationDate: 1991 Aug 15 07:22:27 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 106 /bar put readonly def /FontBBox{-30 -958 1146 777}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI9 %!PS-AdobeFont-1.1: CMMI9 1.100 %%CreationDate: 1996 Jul 23 07:53:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 59 /comma put dup 71 /G put dup 100 /d put dup 101 /e put dup 109 /m put dup 110 /n put dup 120 /x put dup 122 /z put readonly def /FontBBox{-29 -250 1075 750}readonly def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 8 /Phi put dup 11 /ff put dup 12 /fi put dup 19 /acute put dup 20 /caron put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 74 /J put dup 76 /L put dup 77 /M put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-39 -250 1036 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 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cleartomark %%EndFont %%BeginFont: CMBX9 %!PS-AdobeFont-1.1: CMBX9 1.0 %%CreationDate: 1991 Aug 20 16:36:25 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 65 /A put dup 97 /a put dup 98 /b put dup 99 /c put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-58 -250 1195 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 50 /element put dup 54 /negationslash put dup 102 /braceleft put dup 103 /braceright put dup 107 /bardbl put dup 110 /backslash put dup 114 /nabla put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 19 /acute put dup 44 /comma put dup 45 /hyphen put dup 48 /zero put dup 50 /two put dup 51 /three put dup 53 /five put dup 54 /six put dup 56 /eight put dup 65 /A put dup 67 /C put dup 68 /D put dup 71 /G put dup 72 /H put dup 75 /K put dup 77 /M put dup 79 /O put dup 80 /P put dup 81 /Q put dup 83 /S put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 121 /y put readonly def 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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cleartomark %%EndFont TeXDict begin 39139632 55387786 1000 600 600 (Mol25.dvi) @start /Fa 128[39 3[39 1[37 1[51 35 41 25 31 32 35 39 39 43 63 20 2[24 39 35 24 35 39 35 35 39 10[57 1[55 43 56 59 52 59 1[69 48 2[30 57 59 50 52 58 55 54 57 6[24 12[27 24 32[47 11[{}46 74.7198 /CMTI9 rf /Fb 135[59 2[62 44 44 2[62 56 62 93 31 2[31 1[56 1[51 62 50 1[54 11[86 14[70 74 2[80 85 6[31 58[{}21 99.6264 /CMBX12 rf /Fc 175[70 80[{}1 58.1154 /TeX-cmex7 rf /Fd 233[29 22[{}1 66.4176 /MSAM7 rf /Fe 193[52 7[52 54[{}2 58.1154 /MSAM7 rf /Ff 134[35 40 3[25 14[33 15[48 8[48 2[31 3[52 2[53 53 5[20 25[44 5[40 12[31 36 13[{}15 66.4176 /CMMI8 rf /Fg 171[44 1[48 82[{}2 66.4176 /MSBM7 rf /Fh 134[35 3[38 2[30 2[36 40 1[18 2[22 36 33 1[33 36 33 33 36 97[{}14 66.4176 /CMTI8 rf /Fi 193[65 7[65 31[35 18[65 3[{}4 83.022 /MSAM10 rf /Fj 207[18 47[45{}2 41.511 /CMSY5 rf /Fk 194[43 7[28 28 28 28 28 28 4[43 1[22 22 40[{}10 41.511 /CMR5 rf /Fl 132[48 42 50 1[69 50 53 37 38 39 1[53 48 53 80 27 50 1[27 53 48 29 44 53 42 53 46 10[72 1[66 53 72 72 65 2[91 57 1[49 3[60 2[69 68 72 6[27 48 48 48 48 48 48 48 48 48 48 48[{}48 83.022 /CMBX10 rf /Fm 131[71 1[31 37 37 51 37 39 27 28 28 37 39 35 39 59 20 37 1[20 39 35 22 31 39 31 39 35 3[20 1[20 4[53 1[51 39 52 1[48 55 53 65 44 2[25 2[46 48 1[51 1[53 3[55 1[20 20 8[35 35 1[20 24 20 55 1[27 27 4[59 22[39 41 1[55 51 6[59 44{}60 66.4176 /CMR8 rf /Fn 165[55 5[55 1[60 3[60 10[60 67[{}5 83.022 /MSBM10 rf /Fo 141[55 6[34 20 2[34 34 15[42 17[52 17[45 66 19 12[34 1[66 26[52 2[34 52 20 52{}17 58.1154 /CMSY7 rf /Fp 133[32 1[38 48 33 39 25 31 31 3[41 59 21 35 27 1[39 1[32 31 35 30 29 36 8[54 1[39 45 3[52 2[53 63 6[42 2[48 50 1[35 1[52 1[52 20 20 18[44 4[31 42 1[41 40 2[38 1[39 30 3[39 1[32 34 2[30 35 38 43 11[{}49 58.1154 /CMMI7 rf /Fq 165[39 5[39 1[42 14[42 67[{}4 58.1154 /MSBM7 rf /Fr 150[39 39 5[46 6[92 46 106 120 3[69 69 39 1[88 46[66 66 5[62 6[61 61 50 50 3[28 8[35 35 2[{}21 83.022 /CMEX10 rf /Fs 141[69 3[42 2[42 23 32 32 42 42 9[55 55 4[51 52 1[50 70 11[60 1[64 12[0 0 2[55 55 83 7[83 83 6[83 6[65 1[65 5[65 65 1[42 42 7[65 2[42 65 23 65{}35 83.022 /CMSY10 rf /Ft 133[39 41 47 59 40 48 30 39 37 3[50 73 25 43 34 29 48 40 41 39 43 36 36 44 8[69 1[48 57 2[63 66 53 63 67 81 57 1[46 36 1[65 53 61 69 59 63 1[44 1[65 42 65 23 23 18[54 4[39 52 54 52 49 1[36 47 1[47 36 2[48 48 1[39 41 2[37 43 47 53 11[{}65 83.022 /CMMI10 rf /Fu 129[33 5[35 3[26 26 3[33 37 4[19 4[37 30 5[19 1[19 29[51 1[19 1[33 33 33 33 33 33 33 33 33 33 4[51 1[26 26 31[47 6[55 41{}29 58.1154 /CMR7 rf /Fv 131[85 42 34 40 39 55 38 45 28 34 35 38 42 42 47 68 21 38 25 25 42 38 25 38 42 38 38 42 10[62 62 59 47 3[64 1[74 52 2[32 62 64 54 1[63 59 58 62 5[25 25 6[42 42 42 2[25 30 25 2[34 34 25 5[25 18[73 1[47 51 11[{}58 83.022 /CMTI10 rf /Fw 129[42 1[83 42 37 44 44 60 44 46 32 33 33 44 46 42 46 69 23 44 25 23 46 42 25 37 46 37 46 42 3[23 42 23 1[62 1[85 62 62 60 46 61 1[57 65 62 76 52 65 43 30 62 65 54 57 63 60 59 62 3[65 1[23 23 42 42 42 42 42 42 42 42 42 42 42 23 28 23 65 42 32 32 23 3[69 42 23 12[42 42 4[69 1[46 48 2[60 1[60 2[58 1[69 52{}90 83.022 /CMR10 rf /Fx 133[60 71 1[97 71 75 52 53 55 1[75 67 75 112 37 2[37 75 67 41 61 75 60 1[65 12[94 75 100 1[92 101 105 1[81 2[50 1[106 85 1[103 2[102 10[67 67 67 67 67 67 3[45 5[37 39[{}41 119.552 /CMBX12 rf /Fy 165[50 22[54 67[{}2 74.7198 /MSBM10 rf /Fz 207[18 44[32 1[19 48{}4 49.8132 /CMSY6 rf /FA 134[32 35 9[38 54 8[32 21[48 66[40 11[{}7 49.8132 /CMMI6 rf /FB 150[18 5[27 37[47 3[30 30 30 30 30 30 30 30 30 30 4[47 1[24 24 40[{}16 49.8132 /CMR6 rf /FC 135[33 3[23 5[37 1[20 31 24 1[34 3[31 21[45 43[28 13[34 7[33 38 11[{}13 41.511 /CMMI5 rf /FD 171[33 84[{}1 49.8132 /MSBM7 rf /FE 165[43 90[{}1 74.7198 /TeX-cmex9 rf /FF 149[21 55[51 77 15[77 32[60{}5 74.7198 /CMSY9 rf /FG 133[36 1[44 9[46 68 7[36 40 28[60 11[21 59[{}8 74.7198 /CMMI9 rf /FH 128[38 3[38 34 41 41 55 41 43 30 30 30 41 43 38 43 64 21 41 1[21 43 38 23 34 43 34 43 38 3[21 1[21 1[58 1[79 58 58 55 43 1[60 52 2[70 48 1[39 1[58 60 50 52 59 1[54 58 3[60 2[21 38 38 38 38 38 38 38 38 38 38 1[21 26 21 60 1[30 30 21 18[38 38 6[43 45 2[55 6[64 1[{}72 74.7198 /CMR9 rf /FI 139[34 35 36 14[39 49 43 31[67 65[{}7 74.7198 /CMBX9 rf /FJ 141[59 3[35 2[35 3[35 35 47[0 3[47 46[35 55 20 55{}11 66.4176 /CMSY8 rf /FK 134[51 2[51 54 38 38 38 1[54 49 54 81 27 51 1[27 54 1[30 43 54 43 54 49 11[73 1[54 1[76 66 76 1[89 1[76 2[73 77 2[75 70 1[73 8[49 1[49 49 1[49 49 1[49 2[33 27 24[49 19[{}41 99.6264 /CMR12 rf /FL 133[58 4[73 51 52 51 1[73 66 73 1[36 2[36 2[40 58 73 58 1[66 20[83 4[104 86 30[36 39[{}19 143.462 /CMR17 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 end %%EndSetup %%Page: 1 1 TeXDict begin 1 0 bop 673 872 a FL(Green's)43 b(F)-11 b(unctions)44 b(of)e(Generalized)j(Laplacians)1599 1112 y FK(Philipp)s(e)34 b(P)m(oulin)2295 1076 y FJ(\003)1038 1345 y FK(Departmen)m(t)f(of)f(Mathematics)i(and)f(Statistics)1567 1461 y(McGill)g(Univ)m(ersit)m(y)1382 1577 y(805,)f(rue)h(Sherbro)s(ok) m(e)g(Ouest)1144 1694 y(Mon)m(tr)m(\023)-46 b(eal,)33 b(Qu)m(\023)-46 b(eb)s(ec,)34 b(H3A-2K6,)d(Canada)1778 2073 y FI(Abstract)837 2196 y FH(The)38 b(Green's)g(function)g(of)g(an) n(y)f(discretization,)42 b(\001,)f(of)d(the)f(Laplacian)h(in)722 2288 y(dimension)26 b FG(d)g FH(is)g(giv)n(en)g(b)n(y)1418 2493 y 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Fw(=)k Ft(O)r Fw(\()p Ft(r)2239 1492 y Fo(\000)p Fp(N)6 b Fo(\000)p Fu(1)2441 1526 y Fw(\))p Ft(:)515 1792 y Fw(If)40 b(the)f(function)h Ft(f)48 b Fw(also)39 b(dep)r(ends)h(on)f(a)g(parameter,)i Ft(t)p Fw(,)h(w)n(e)d(sa)n(y)g(that)g(the)h(previous)515 1892 y(estimates/asymptotics)25 b(are)h Fv(uniform)34 b Fw(in)27 b Ft(t)g Fw(if)h(the)f(constan)n(ts)f Ft(C)33 b Fw(ma)n(y)27 b(b)r(e)g(c)n(hosen)f(inde-)515 1992 y(p)r(enden)n(tly)i (of)f Ft(t)p Fw(.)515 2266 y Fx(2)134 b(Oscillatory)39 b(In)l(tegral)g(without)f(Stationary)h(Phase)716 2416 y(P)l(oin)l(t)515 2597 y Fw(The)h(follo)n(wing)e(theorems)i(are)e (stated)i(in)g(the)h(w)n(a)n(y)d(they)i(are)f(used)h(when)g(studying) 515 2697 y(F)-7 b(ourier)22 b(transforms)g(o)n(v)n(er)g(surfaces.)34 b(They)24 b(establish)f(the)h(existence)f(of)g(neigh)n(b)r(orho)r(o)r (ds)515 2797 y(on)30 b(whic)n(h)g(a)g(certain)f(phenomenon)h(o)r (ccurs,)g(giv)n(en)g(a)f(\014xed)i(phase.)44 b(The)30 b(giv)n(en)g(phase,)515 2896 y Ft(')p Fw(\()p Ft(x;)14 b(t)p Fw(\),)29 b(is)e(supp)r(osed)h(to)f(b)r(e)h(smo)r(oth)f(in)h(\()p Ft(x;)14 b(t)p Fw(\))24 b Fs(2)g Fn(R)2186 2866 y Fp(d)2243 2896 y Fs(\002)18 b Fn(R)2386 2866 y Fp(m)2449 2896 y Fw(.)515 3096 y Fl(Lemma)71 b Fv(L)l(et)35 b Ft(d)f Fw(=)f(1)p Fv(.)56 b(Supp)l(ose)36 b Ft(@)1706 3108 y Fp(x)1747 3096 y Ft(')e Fs(6)p Fw(=)g(0)h Fv(at)g(a)h(given)h Fw(\()p Ft(x)2496 3108 y Fu(0)2534 3096 y Ft(;)14 b(t)2601 3108 y Fu(0)2638 3096 y Fw(\))34 b Fs(2)g Fn(R)22 b Fs(\002)h Fn(R)3023 3065 y Fp(m)3086 3096 y Fv(.)56 b(Then,)515 3195 y(ther)l(e)33 b(exists)f(an)i(arbitr)l(arily)g(smal)t(l)g(cub)l (e,)g Ft(U)c Fs(\002)20 b Ft(B)t Fv(,)35 b(c)l(ontaining)e Fw(\()p Ft(x)2698 3207 y Fu(0)2736 3195 y Ft(;)14 b(t)2803 3207 y Fu(0)2840 3195 y Fw(\))34 b Fv(such)f(that)g(the)515 3295 y(fol)t(lowing)27 b(holds:)38 b(if)26 b(the)f(amplitude,)i Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))p Fv(,)27 b(is)e(smo)l(oth)h(in)f (the)g(neighb)l(orho)l(o)l(d)i(of)f Fn(R)8 b Fs(\002)p 3312 3228 68 4 v 8 w Ft(B)515 3394 y Fv(and)30 b(vanishes)h(on)f Ft(U)1192 3364 y Fu(c)1244 3394 y Fs(\002)p 1327 3328 V 18 w Ft(B)t Fv(,)g(then)1599 3577 y Fs(j)p Ft(I)1658 3589 y Fp(f)1702 3577 y Fw(\()p Ft(r)n(;)14 b(t)p Fw(\))p Fs(j)24 b Fw(=)e Ft(O)r Fw(\()p Ft(r)2138 3543 y Fo(\0001)2262 3577 y Fw(\))515 3760 y Fv(uniformly)30 b(in)g Ft(t)23 b Fs(2)p 1127 3693 V 24 w Ft(B)t Fv(.)1233 3730 y Fu(4)515 4059 y Fl(Pro)s(of:)94 b Fw(By)38 b(con)n(tin)n(uit)n(y)-7 b(,)40 b Ft(@)1474 4071 y Fp(x)1515 4059 y Ft(')p Fw(\()p Ft(x;)14 b(t)p Fw(\))42 b Fs(6)p Fw(=)d(0)f(on)f(a)g(certain)h(cub)r(e) g Ft(U)2737 4029 y Fo(0)2785 4059 y Fs(\002)25 b Ft(B)2942 4029 y Fo(0)3003 4059 y Fw(con)n(taining)515 4158 y(\()p Ft(x)594 4170 y Fu(0)632 4158 y Ft(;)14 b(t)699 4170 y Fu(0)736 4158 y Fw(\).)36 b(The)26 b(op)r(erator)e Ft(D)h Fw(=)1629 4126 y Fu(1)p 1521 4140 251 4 v 1521 4187 a Fp(@)1556 4195 y FC(x)1593 4187 y Fp(')p Fu(\()p Fp(x;t)p Fu(\))1796 4158 y Fs(\016)14 b Ft(@)1896 4170 y Fp(x)1963 4158 y Fw(is)26 b(th)n(us)f(w)n(ell)h(de\014ned)g(on)f (this)h(cub)r(e,)h(where)515 4283 y(it)g(satis\014es)f Ft(D)r Fw(e)1010 4253 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(x;t)p Fu(\))1267 4283 y Fw(=)d(i)p Ft(r)r Fw(e)1454 4253 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(x;t)p Fu(\))1715 4283 y Fw(and)k Ft(D)1947 4253 y Fu(t)2000 4283 y Fw(=)c Fs(\000)p Ft(@)2197 4295 y Fp(x)2255 4283 y Fs(\016)2433 4251 y Fu(1)p 2324 4265 V 2324 4312 a Fp(@)2359 4320 y FC(x)2396 4312 y Fp(')p Fu(\()p Fp(x;t)p Fu(\))2584 4283 y Fw(.)37 b(Let)27 b Ft(U)f Fs(\002)17 b Ft(B)27 b Fs(3)d Fw(\()p Ft(x)3205 4295 y Fu(0)3243 4283 y Ft(;)14 b(t)3310 4295 y Fu(0)3347 4283 y Fw(\))515 4397 y(b)r(e)25 b(a)f(cub)r(e)h(whose)e (closure)h(is)g(in)h Ft(U)1640 4367 y Fo(0)1675 4397 y Fs(\002)12 b Ft(B)1819 4367 y Fo(0)1842 4397 y Fw(.)36 b(If)25 b Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))25 b(satis\014es)f(the) h(asserted)e(prop)r(erties,)p 515 4471 1146 4 v 607 4525 a FB(4)642 4548 y Ff(U)697 4525 y FB(c)747 4548 y Fm(denotes)e(the)e (complemen)n(tary)i(of)e Ff(U)25 b Fm(with)20 b(resp)r(ect)g(to)f(a)h (set)f(determined)h(b)n(y)g(the)f(con)n(text;)k(here,)515 4627 y Ff(U)570 4604 y FB(c)621 4627 y Fm(=)c Fg(R)d FJ(n)g Ff(U)7 b Fm(.)1926 5255 y Fw(4)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 515 523 a Fw(then)28 b(for)f(an)n(y)g Ft(N)32 b Fi(>)22 b Fw(0)27 b(and)h Ft(t)23 b Fs(2)p 1536 456 68 4 v 23 w Ft(B)1105 737 y Fs(j)p Ft(I)1164 749 y Fp(f)1208 737 y Fw(\()p Ft(r)n(;)14 b(t)p Fw(\))p Fs(j)83 b Fw(=)1668 681 y(1)p 1638 718 103 4 v 1638 794 a Ft(r)1677 770 y Fp(N)1764 617 y Fr(\014)1764 667 y(\014)1764 716 y(\014)1764 766 y(\014)1792 624 y(Z)1838 813 y Fp(U)1894 737 y Fw(\()p Ft(D)1997 703 y Fp(N)2060 737 y Fw(e)2097 703 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(x;t)p Fu(\))2331 737 y Fw(\))p Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))19 b(d)p Ft(x)2703 617 y Fr(\014)2703 667 y(\014)2703 716 y(\014)2703 766 y(\014)1480 970 y Fw(=)1668 913 y(1)p 1638 950 V 1638 1026 a Ft(r)1677 1003 y Fp(N)1764 849 y Fr(\014)1764 899 y(\014)1764 949 y(\014)1764 999 y(\014)1792 857 y(Z)1838 1045 y Fp(U)1908 970 y Fw(e)1945 935 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(x;t)p Fu(\))2178 970 y Fw(\()p Ft(D)2281 935 y Fu(t)2311 970 y Fw(\))2343 935 y Fp(N)2407 970 y Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))19 b(d)p Ft(x)2747 849 y Fr(\014)2747 899 y(\014)2747 949 y(\014)2747 999 y(\014)1480 1195 y Fi(6)1638 1138 y Ft(C)1697 1150 y Fp(N)p 1638 1175 123 4 v 1648 1251 a Ft(r)1687 1227 y Fp(N)1770 1195 y Ft(;)515 1402 y Fw(where)27 b(the)h(constan)n(t)f Ft(C)1292 1414 y Fp(N)1383 1402 y Fw(do)r(es)g(not)g(dep)r(end)i(on)e Ft(t)c Fs(2)p 2253 1336 68 4 v 24 w Ft(B)t Fw(.)971 b Fi(\003)639 1602 y Fw(The)28 b(m)n(ultidimensional)f(analogue)f(follo)n (ws:)515 1770 y Fl(Theorem)31 b(1)41 b Fv(Supp)l(ose)35 b Fs(r)1399 1782 y Fp(x)1441 1770 y Ft(')p Fw(\()p Ft(x)1574 1782 y Fu(0)1612 1770 y Ft(;)14 b(t)1679 1782 y Fu(0)1717 1770 y Fw(\))31 b Fs(6)p Fw(=)g(0)j Fv(for)i(a)e(given)h Fw(\()p Ft(x)2466 1782 y Fu(0)2504 1770 y Ft(;)14 b(t)2571 1782 y Fu(0)2609 1770 y Fw(\))31 b Fs(2)h Fn(R)2819 1740 y Fp(d)2880 1770 y Fs(\002)21 b Fn(R)3026 1740 y Fp(m)3089 1770 y Fv(.)53 b(Then,)515 1870 y(ther)l(e)33 b(exists)f(an)i(arbitr)l (arily)g(smal)t(l)g(cub)l(e,)g Ft(U)c Fs(\002)20 b Ft(B)t Fv(,)35 b(c)l(ontaining)e Fw(\()p Ft(x)2698 1882 y Fu(0)2736 1870 y Ft(;)14 b(t)2803 1882 y Fu(0)2840 1870 y Fw(\))34 b Fv(such)f(that)g(the)515 1970 y(fol)t(lowing)25 b(holds:)36 b(if)23 b(the)g(amplitude,)j Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))p Fv(,)24 b(is)f(smo)l(oth)g(in)g(the)f(neighb)l(orho)l(o)l (d)j(of)e Fn(R)3203 1940 y Fp(d)3245 1970 y Fs(\002)p 3313 1903 V 3 w Ft(B)515 2069 y Fv(and)30 b(vanishes)h(on)f Ft(U)1192 2039 y Fu(c)1244 2069 y Fs(\002)p 1327 2003 V 18 w Ft(B)t Fv(,)g(then)1599 2238 y Fs(j)p Ft(I)1658 2250 y Fp(f)1702 2238 y Fw(\()p Ft(r)n(;)14 b(t)p Fw(\))p Fs(j)24 b Fw(=)e Ft(O)r Fw(\()p Ft(r)2138 2204 y Fo(\0001)2262 2238 y Fw(\))515 2407 y Fv(uniformly)30 b(in)g Ft(t)23 b Fs(2)p 1127 2340 V 24 w Ft(B)t Fv(.)515 2576 y Fl(Pro)s(of:)86 b Fw(By)34 b(assumption,)j Ft(@)1513 2595 y Fp(x)1551 2578 y Fk(\()p FC(k)q Fk(\))1636 2576 y Ft(')p Fw(\()p Ft(x)1769 2588 y Fu(0)1807 2576 y Ft(;)14 b(t)1874 2588 y Fu(0)1911 2576 y Fw(\))36 b Fs(6)p Fw(=)f(0)f(for)g(a)h(certain)f(1)h Fi(6)g Ft(k)j Fi(6)d Ft(d)p Fw(,)i(sa)n(y)-7 b(,)36 b(for)515 2675 y Ft(k)26 b Fw(=)e(1.)37 b(In)n(terpreting)27 b(\()p Ft(x)1316 2645 y Fu(\(2\))1406 2675 y Ft(;)14 b(:)g(:)g(:)g(;)g(x)1638 2645 y Fu(\()p Fp(d)p Fu(\))1729 2675 y Ft(;)g(t)p Fw(\))28 b(as)g(a)f(parameter,)g(there)h(exists)f(an)h(arbitrarily)515 2791 y(small)22 b(cub)r(e,)j Ft(U)1000 2803 y Fu(1)1046 2791 y Fs(\002)9 b Fw(\()p Ft(U)1209 2803 y Fu(2)1256 2791 y Fs(\002)g(\001)14 b(\001)g(\001)9 b(\002)g Ft(U)1567 2803 y Fp(d)1614 2791 y Fs(\002)g Ft(B)t Fw(\),)25 b(con)n(taining)d (\()p Ft(x)2313 2748 y Fu(\(1\))2313 2813 y(0)2403 2791 y Fw(;)14 b Ft(x)2487 2748 y Fu(\(2\))2487 2813 y(0)2576 2791 y Ft(;)g(:)g(:)g(:)g(;)g(x)2808 2748 y Fu(\()p Fp(d)p Fu(\))2808 2813 y(0)2899 2791 y Ft(;)g(t)2966 2803 y Fu(0)3003 2791 y Fw(\))23 b(on)g(whic)n(h)515 2891 y(the)32 b(previous)g(lemma)g(applies.)50 b(Hence,)34 b(if)f Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))33 b(satis\014es)e(the)i(asserted)e (prop)r(erties,)515 2990 y(then)1372 2999 y Fr(\014)1372 3049 y(\014)1372 3098 y(\014)1372 3148 y(\014)1399 3006 y(Z)1446 3195 y Fp(U)1491 3203 y Fk(1)1541 3119 y Fw(e)1578 3085 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(x;t)p Fu(\))1812 3119 y Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))19 b(d)p Ft(x)2152 3085 y Fu(\(1\))2241 2999 y Fr(\014)2241 3049 y(\014)2241 3098 y(\014)2241 3148 y(\014)2292 3119 y Fi(6)2390 3063 y Ft(C)2449 3075 y Fp(N)p 2390 3100 123 4 v 2399 3176 a Ft(r)2438 3152 y Fp(N)515 3327 y Fw(uniformly)27 b(in)h(\()p Ft(x)1069 3297 y Fu(\(2\))1159 3327 y Ft(;)14 b(:)g(:)g(:)g(;)g(x)1391 3297 y Fu(\()p Fp(d)p Fu(\))1481 3327 y Ft(;)g(t)p Fw(\))24 b Fs(2)p 1682 3260 94 4 v 23 w Ft(U)1739 3339 y Fu(2)1794 3327 y Fs(\002)18 b Ft(:)c(:)g(:)19 b Fs(\002)p 2076 3260 96 4 v 18 w Ft(U)2133 3339 y Fp(d)2190 3327 y Fs(\002)p 2273 3260 68 4 v 18 w Ft(B)t Fw(.)37 b(In)28 b(particular,)766 3545 y Fs(j)p Ft(I)825 3557 y Fp(f)869 3545 y Fw(\()p Ft(r)n(;)14 b(t)p Fw(\))p Fs(j)84 b Fw(=)1289 3425 y Fr(\014)1289 3475 y(\014)1289 3525 y(\014)1289 3574 y(\014)1317 3432 y(Z)1363 3621 y Fp(U)1408 3629 y Fk(2)1441 3621 y Fo(\002)p Fp(:::)o Fo(\002)p Fp(U)1649 3630 y FC(d)1702 3432 y Fr(Z)1748 3621 y Fp(U)1793 3629 y Fk(1)1844 3545 y Fw(e)1881 3511 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(x;t)p Fu(\))2114 3545 y Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))19 b(d)p Ft(x)2454 3511 y Fu(\(1\))2576 3545 y Fw(d)p Ft(x)2669 3511 y Fu(\(2\))2772 3545 y Ft(:)14 b(:)g(:)33 b Fw(d)p Ft(x)2995 3511 y Fu(\()p Fp(d)p Fu(\))3086 3425 y Fr(\014)3086 3475 y(\014)3086 3525 y(\014)3086 3574 y(\014)1142 3776 y Fi(6)82 b Fs(j)p Ft(U)1369 3788 y Fu(2)1425 3776 y Fs(\002)18 b Ft(:)c(:)g(:)k Fs(\002)g Ft(U)1763 3788 y Fp(d)1802 3776 y Fs(j)1835 3720 y Ft(C)1894 3732 y Fp(N)p 1835 3757 123 4 v 1844 3833 a Ft(r)1883 3809 y Fp(N)515 3984 y Fw(uniformly)27 b(in)h Ft(t)23 b Fs(2)p 1121 3917 68 4 v 23 w Ft(B)5 b Fw(.)2102 b Fi(\003)515 4283 y Fl(Sc)m(holium)74 b Fv(A)37 b(similar)i(r)l(esult)d(holds)j (when)f(c)l(onsidering)h(oscil)t(latory)h(inte)l(gr)l(als)d(over)515 4382 y(the)30 b(torus,)1561 4407 y Fr(Z)1607 4595 y Fq(T)1646 4579 y FC(d)1699 4520 y Fw(e)1736 4485 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(x;t)p Fu(\))1969 4520 y Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))19 b(d)p Ft(x;)515 4707 y Fv(without)41 b(stationary)g (phase)h(p)l(oint.)72 b(One)40 b(then)h(assumes)g Ft(f)9 b Fw(\()p Fs(\001)p Ft(;)14 b(t)p Fw(\))40 b Fv(is)i(p)l(erio)l(dic)g (for)g(ev-)515 4807 y(ery)30 b Ft(t)g Fv(\(inste)l(ad)g(of)h(b)l(eing)f (c)l(omp)l(actly)h(supp)l(orte)l(d\).)40 b(The)30 b(same)h(pr)l(o)l(of) g(works,)g(due)f(to)g(the)515 4907 y(absenc)l(e)25 b(of)h(b)l(oundary)f (term)g(when)g(inte)l(gr)l(ating)g(by)g(p)l(arts)g(a)g(smo)l(oth)g(p)l (erio)l(dic)i(function.)1926 5255 y Fw(5)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 515 523 a Fl(Cauc)m(h)m(y)42 b(Principal)f(V)-8 b(alue)83 b Fw(W)-7 b(e)36 b(no)n(w)f(turn)g(our)g(atten)n(tion)g(to)g (Cauc)n(h)n(y)g(principal)515 623 y(v)-5 b(alues)30 b(of)h(oscillatory) e(in)n(tegrals)h(without)h(stationary)f(phase)g(p)r(oin)n(t.)47 b(T)-7 b(o)31 b(this)g(end,)h(w)n(e)515 722 y(no)n(w)i(consider)g(a)g (parameter)g(\()p Ft(e;)14 b(t)p Fw(\))35 b Fs(2)h Fn(R)23 b Fs(\002)g Fn(R)2034 692 y Fp(m)2132 722 y Fw(and)35 b(a)f(phase,)j Ft(')p Fw(\()p Ft(x)p Fw(;)14 b Ft(e;)g(t)p Fw(\),)37 b(smo)r(oth)e(in)515 822 y(\()p Ft(x)p Fw(;)14 b Ft(e;)g(t)p Fw(\))23 b Fs(2)h Fn(R)931 792 y Fp(d)988 822 y Fs(\002)18 b Fw(\()p Fn(R)h Fs(\002)f Fn(R)1325 792 y Fp(m)1388 822 y Fw(\).)37 b(W)-7 b(e)28 b(th)n(us)f(let)1306 1039 y Ft(I)1342 1051 y Fp(f)1385 1039 y Fw(\()p Ft(r)r Fw(;)14 b Ft(e;)g(t)p Fw(\))24 b(=)1743 926 y Fr(Z)1789 1114 y Fq(R)1831 1098 y FC(d)1884 1039 y Fw(e)1921 1004 y Fu(i)p Fp(')p Fu(\()p Fp(x)p Fu(;)p Fp(e;t)p Fu(\))2172 1039 y Ft(f)9 b Fw(\()p Ft(x)p Fw(;)14 b Ft(e;)g(t)p Fw(\))k(d)p Ft(x)515 1255 y Fw(and)27 b(study)758 1467 y(p)p Ft(:)p Fw(v)q Ft(:)909 1354 y Fr(Z)955 1543 y Fo(j)p Fp(\021)r Fo(\000)p Fp(e)p Fo(j)p Fp(<\016)1297 1411 y Fw(1)p 1226 1448 185 4 v 1226 1524 a Ft(\021)22 b Fs(\000)c Ft(e)1420 1467 y(I)1456 1479 y Fp(f)1500 1467 y Fw(\()p Ft(r)r Fw(;)c Ft(\021)s(;)g(t)p Fw(\))20 b(d)p Ft(\021)26 b Fw(=)d(lim)1980 1521 y Fp(")p Fo(#)p Fu(0)2101 1354 y Fr(Z)2147 1543 y Fp("<)p Fo(j)p Fp(\021)r Fo(\000)p Fp(e)p Fo(j)p Fp(<\016)2573 1411 y Fw(1)p 2501 1448 V 2501 1524 a Ft(\021)f Fs(\000)c Ft(e)2696 1467 y(I)2732 1479 y Fp(f)2775 1467 y Fw(\()p Ft(r)r Fw(;)c Ft(\021)s(;)g(t)p Fw(\))20 b(d)p Ft(\021)515 1698 y Fw(for)27 b(a)g(giv)n(en)g Ft(\016)f(>)d Fw(0.)36 b(Our)27 b(results)g(are)g(based)g(on)g(the)h (follo)n(wing)e(elemen)n(tary)h(estimate:)515 1897 y Fl(Lemma)51 b Fv(Supp)l(ose)26 b Ft(f)9 b Fw(\()p Ft(h)p Fw(\))26 b Fv(is)g(a)g(c)l(ontinuously)f(di\013er)l(entiable)j (function)d(in)h(the)g(neighb)l(or-)515 1997 y(ho)l(o)l(d)31 b(of)f Fw([)p Fs(\000)p Ft(\016)s(;)14 b(\016)s Fw(])p Fv(,)30 b(wher)l(e)h Ft(\016)26 b(>)c Fw(0)30 b Fv(is)g(given.)39 b(Then,)31 b(for)f(any)h Ft(")22 b Fs(2)i Fw(]0)p Ft(;)14 b(\016)s Fw([)1299 2100 y Fr(\014)1299 2150 y(\014)1299 2200 y(\014)1299 2250 y(\014)1299 2300 y(\014)1326 2133 y(Z)1372 2321 y Fp(")p Fe(6)p Fo(j)p Fp(h)p Fo(j)p Fe(6)p Fp(\016)1646 2189 y Ft(f)9 b Fw(\()p Ft(h)p Fw(\))p 1646 2226 162 4 v 1703 2303 a Ft(h)1837 2246 y Fw(d)p Ft(h)1931 2100 y Fr(\014)1931 2150 y(\014)1931 2200 y(\014)1931 2250 y(\014)1931 2300 y(\014)1981 2246 y Fi(6)23 b Fw(2)p Ft(\016)d Fw(max)2164 2303 y Fo(j)p Fp(h)p Fo(j)p Fe(6)p Fp(\016)2341 2246 y Fs(j)p Ft(f)2414 2211 y Fo(0)2437 2246 y Fw(\()p Ft(h)p Fw(\))p Fs(j)p Ft(:)515 2490 y Fv(In)29 b(p)l(articular,)1304 2514 y Fr(\014)1304 2564 y(\014)1304 2614 y(\014)1304 2664 y(\014)1304 2714 y(\014)1332 2660 y Fw(p)p Ft(:)p Fw(v)q Ft(:)1483 2547 y Fr(Z)1566 2567 y Fp(\016)1529 2735 y Fo(\000)p Fp(\016)1641 2604 y Ft(f)9 b Fw(\()p Ft(h)p Fw(\))p 1641 2641 V 1698 2717 a Ft(h)1831 2660 y Fw(d)p Ft(h)1925 2514 y Fr(\014)1925 2564 y(\014)1925 2614 y(\014)1925 2664 y(\014)1925 2714 y(\014)1976 2660 y Fi(6)23 b Fw(2)p Ft(\016)d Fw(max)2159 2717 y Fo(j)p Fp(h)p Fo(j)p Fe(6)p Fp(\016)2335 2660 y Fs(j)p Ft(f)2408 2625 y Fo(0)2431 2660 y Fw(\()p Ft(h)p Fw(\))p Fs(j)p Ft(:)639 3016 y Fw(The)28 b(pro)r(of,)f(based)g(on)g (the)h(Mean)g(V)-7 b(alue)27 b(Theorem,)g(is)h(left)g(to)f(the)h (reader.)515 3190 y Fl(Theorem)j(2)41 b Fv(Supp)l(ose)30 b Fs(r)1394 3202 y Fp(x)1436 3190 y Ft(')p Fw(\()p Ft(x)1569 3202 y Fu(0)1608 3190 y Ft(;)14 b Fw(0)p Ft(;)g(t)1754 3202 y Fu(0)1790 3190 y Fw(\))24 b Fs(6)p Fw(=)e(0)29 b Fv(for)i(a)f(given)g Fw(\()p Ft(x)2504 3202 y Fu(0)2542 3190 y Ft(;)14 b(t)2609 3202 y Fu(0)2647 3190 y Fw(\))23 b Fs(2)g Fn(R)2840 3160 y Fp(d)2897 3190 y Fs(\002)18 b Fn(R)3040 3160 y Fp(m)3103 3190 y Fv(.)39 b(Then,)515 3290 y(ther)l(e)34 b(exists)g(an)g(arbitr)l(arily)i(smal)t(l)f(cub)l (e,)h Ft(U)31 b Fs(\002)21 b Ft(B)t Fv(,)36 b(c)l(ontaining)f Fw(\()p Ft(x)2711 3302 y Fu(0)2749 3290 y Ft(;)14 b(t)2816 3302 y Fu(0)2853 3290 y Fw(\))34 b Fv(and)h(an)f(arbi-)515 3389 y(tr)l(arily)d(smal)t(l)h Ft(\016)d(>)24 b Fw(0)31 b Fv(such)g(that)g(the)g(fol)t(lowing)i(holds:)43 b(if)32 b(the)f(amplitude,)i Ft(f)9 b Fw(\()p Ft(x;)14 b(h;)g(t)p Fw(\))p Fv(,)32 b(is)515 3489 y(smo)l(oth)26 b(in)g(the)f(neighb)l (orho)l(o)l(d)k(of)d Fn(R)1667 3459 y Fp(d)1715 3489 y Fs(\002)10 b Fw([)o Fs(\000)p Ft(\016)s(;)k(\016)s Fw(])c Fs(\002)p 2102 3422 68 4 v 10 w Ft(B)28 b Fv(and)e(vanishes)h (on)f Ft(U)2858 3459 y Fu(c)2901 3489 y Fs(\002)10 b Fw([)o Fs(\000)p Ft(\016)s(;)k(\016)s Fw(])9 b Fs(\002)p 3287 3422 V 10 w Ft(B)t Fv(,)515 3589 y(then)1336 3597 y Fr(\014)1336 3647 y(\014)1336 3696 y(\014)1336 3746 y(\014)1336 3796 y(\014)1364 3742 y Fw(p)p Ft(:)p Fw(v)q Ft(:)1515 3629 y Fr(Z)1598 3650 y Fp(\016)1561 3818 y Fo(\000)p Fp(\016)1673 3686 y Ft(I)1709 3698 y Fp(f)1753 3686 y Fw(\()p Ft(r)n(;)14 b(h;)g(t)p Fw(\))p 1673 3723 331 4 v 1815 3799 a Ft(h)2033 3742 y Fw(d)p Ft(h)2127 3597 y Fr(\014)2127 3647 y(\014)2127 3696 y(\014)2127 3746 y(\014)2127 3796 y(\014)2177 3742 y Fw(=)23 b Ft(O)r Fw(\()p Ft(r)2401 3708 y Fo(\0001)2525 3742 y Fw(\))515 3972 y Fv(uniformly)30 b(in)g Ft(t)23 b Fs(2)p 1127 3906 68 4 v 24 w Ft(B)t Fv(.)515 4147 y Fl(Pro)s(of:)52 b Fw(By)19 b(the)h(lemma)f(it)h(su\016ces)g(to)f(estimate)h Ft(@)2154 4159 y Fp(h)2197 4147 y Ft(I)2233 4159 y Fp(f)2276 4147 y Fw(\()p Ft(r)n(;)14 b(h;)g(t)p Fw(\).)35 b(In)20 b(fact,)h(the)f(dominated)515 4246 y(con)n(v)n(ergence)25 b(theorem)i(implies)706 4458 y Ft(@)750 4470 y Fp(h)793 4458 y Ft(I)829 4470 y Fp(f)873 4458 y Fw(\()p Ft(r)n(;)14 b(h;)g(t)p Fw(\))23 b(=)1235 4345 y Fr(Z)1281 4534 y Fq(R)1323 4517 y FC(d)1376 4458 y Fw(e)1413 4424 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(x;h;t)p Fu(\))1705 4458 y Fw(\()p Ft(@)1781 4470 y Fp(h)1824 4458 y Ft(f)9 b Fw(\()p Ft(x;)14 b(h;)g(t)p Fw(\))19 b(+)f(i)p Ft(r)r(f)9 b Fw(\()p Ft(x;)14 b(h;)g(t)p Fw(\))p Ft(@)2658 4470 y Fp(h)2702 4458 y Ft(')p Fw(\()p Ft(x;)g(h;)g(t)p Fw(\)\))20 b(d)p Ft(x:)515 4675 y Fw(Hence,)31 b(b)n(y)f(Theorem)g(1)g(there)g(exist)h(an)f (arbitrarily)f(small)h(cub)r(e,)i Ft(U)d Fs(\002)20 b Ft(B)t Fw(,)31 b(con)n(taining)515 4774 y(\()p Ft(x)594 4786 y Fu(0)632 4774 y Ft(;)14 b(t)699 4786 y Fu(0)736 4774 y Fw(\))28 b(and)f(an)h(arbitrarily)d(small)i Ft(\016)g(>)22 b Fw(0)27 b(suc)n(h)h(that)g(for)f Ft(f)9 b Fw(\()p Ft(x;)14 b(h;)g(t)p Fw(\))27 b(as)g(stipulated)1588 4991 y Fs(j)p Ft(@)1655 5003 y Fp(h)1699 4991 y Ft(I)1735 5003 y Fp(f)1778 4991 y Fw(\()p Ft(r)n(;)14 b(h;)g(t)p Fw(\))p Fs(j)24 b Fi(6)2173 4935 y Ft(C)2232 4947 y Fp(N)p 2173 4972 123 4 v 2183 5048 a Ft(r)2222 5024 y Fp(N)1926 5255 y Fw(6)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 515 523 a Fw(uniformly)27 b(in)h(\()p Ft(h;)14 b(t)p Fw(\))23 b Fs(2)h Fw([)p Fs(\000)p Ft(\016)r(;)14 b(\016)s Fw(])19 b Fs(\002)p 1600 456 68 4 v 18 w Ft(B)t Fw(.)37 b(The)28 b(result)f(follo)n(ws.)916 b Fi(\003)639 722 y Fw(Applying)28 b(the)g(ab)r(o)n(v)n(e)f(to)g(the)h(phase)41 b(~)-56 b Ft(')q Fw(\()p Ft(x;)14 b(h)p Fw(;)g Ft(e;)g(t)p Fw(\))23 b(=)g Ft(')p Fw(\()p Ft(x;)14 b(h)19 b Fw(+)f Ft(e;)c(t)p Fw(\))28 b(and)f(the)h(ampli-)515 822 y(tude)722 800 y(~)704 822 y Ft(f)8 b Fw(\()p Ft(x;)14 b(h)p Fw(;)g Ft(e;)g(t)p Fw(\))24 b(=)e Ft(f)9 b Fw(\()p Ft(x;)14 b(h)19 b Fw(+)f Ft(e;)c(t)p Fw(\),)27 b(one)h(obtains:)515 1021 y Fl(Corollary)58 b Fv(Supp)l(ose)29 b Fs(r)1350 1033 y Fp(x)1392 1021 y Ft(')p Fw(\()p Ft(x)1525 1033 y Fu(0)1563 1021 y Ft(;)14 b(e)1639 1033 y Fu(0)1676 1021 y Ft(;)g(t)1743 1033 y Fu(0)1780 1021 y Fw(\))24 b Fs(6)p Fw(=)e(0)28 b Fv(for)i(a)f(given)g Fw(\()p Ft(x)2490 1033 y Fu(0)2528 1021 y Ft(;)14 b(e)2604 1033 y Fu(0)2641 1021 y Ft(;)g(t)2708 1033 y Fu(0)2745 1021 y Fw(\))23 b Fs(2)h Fn(R)2939 991 y Fp(d)2993 1021 y Fs(\002)16 b Fn(R)g Fs(\002)g Fn(R)3291 991 y Fp(m)3354 1021 y Fv(.)515 1121 y(Then,)35 b(ther)l(e)f(exist)f(an)h(arbitr)l(arily)h(smal)t(l)g (cub)l(e,)g Ft(U)30 b Fs(\002)21 b Ft(B)t Fv(,)35 b(c)l(ontaining)f Fw(\()p Ft(x)2917 1133 y Fu(0)2955 1121 y Ft(;)14 b(t)3022 1133 y Fu(0)3059 1121 y Fw(\))34 b Fv(and)g(an)515 1220 y(arbitr)l(arily)29 b(smal)t(l)g Ft(\016)d(>)c Fw(0)27 b Fv(such)h(that)g(the)g(fol)t(lowing)i(holds:)39 b(if)28 b(the)g(amplitude,)h Ft(f)9 b Fw(\()p Ft(x;)14 b(e;)g(t)p Fw(\))p Fv(,)515 1320 y(is)28 b(smo)l(oth)f(in)h(the)g(neighb)l(orho)l (o)l(d)i(of)e Fn(R)1763 1290 y Fp(d)1815 1320 y Fs(\002)14 b Fw([)o Ft(e)1955 1332 y Fu(0)2011 1320 y Fs(\000)k Ft(\016)s(;)c(e)2210 1332 y Fu(0)2265 1320 y Fw(+)k Ft(\016)s Fw(])c Fs(\002)p 2504 1253 V 14 w Ft(B)30 b Fv(and)e(vanishes)h(on)f Ft(U)3268 1290 y Fu(c)3314 1320 y Fs(\002)515 1420 y Fw([)p Ft(e)577 1432 y Fu(0)632 1420 y Fs(\000)18 b Ft(\016)s(;)c(e)831 1432 y Fu(0)886 1420 y Fw(+)k Ft(\016)t Fw(])g Fs(\002)p 1134 1353 V 18 w Ft(B)t Fv(,)30 b(then)1261 1530 y Fr(\014)1261 1580 y(\014)1261 1630 y(\014)1261 1680 y(\014)1261 1729 y(\014)1288 1675 y Fw(p)p Ft(:)p Fw(v)q Ft(:)1439 1562 y Fr(Z)1485 1751 y Fo(j)p Fp(\021)r Fo(\000)p Fp(e)p Fo(j)p Fp(<\016)1757 1619 y Ft(I)1793 1631 y Fp(f)1836 1619 y Fw(\()p Ft(r)n(;)14 b(\021)s(;)g(t)p Fw(\))p 1757 1656 328 4 v 1828 1732 a Ft(\021)22 b Fs(\000)c Ft(e)2112 1675 y Fw(d)p Ft(\021)2202 1530 y Fr(\014)2202 1580 y(\014)2202 1630 y(\014)2202 1680 y(\014)2202 1729 y(\014)2253 1675 y Fw(=)23 b Ft(O)r Fw(\()p Ft(r)2477 1641 y Fo(\0001)2601 1675 y Fw(\))515 1942 y Fv(uniformly)30 b(in)g Fw(\()p Ft(e;)14 b(t)p Fw(\))23 b Fs(2)h Fw([)p Ft(e)1329 1954 y Fu(0)1384 1942 y Fs(\000)18 b Ft(\016)s(;)c(e)1583 1954 y Fu(0)1639 1942 y Fw(+)k Ft(\016)s Fw(])g Fs(\002)p 1886 1876 68 4 v 18 w Ft(B)t Fv(.)515 2316 y Fx(3)134 b(Oscillatory)34 b(In)l(tegral)f(with)f(Non)f(Degenerate)j(Sta-)716 2466 y(tionary)46 b(Phase)f(P)l(oin)l(ts)515 2647 y Fl(Quadratic)38 b(Phase)83 b Fw(W)-7 b(e)32 b(no)n(w)f(in)n(v)n(estigate)g(the)h(deca)n (y)f(of)h(oscillatory)e(in)n(tegrals)h(ad-)515 2747 y(mitting)g(non)f (degenerate)f(stationary)g(phase)h(p)r(oin)n(ts.)46 b(Via)30 b(the)h Fv(Morse)i(lemma)p Fw(,)f(whic)n(h)515 2847 y(is)38 b(pro)n(v)n(en)f(b)r(elo)n(w,)k(our)d(study)h(reduces)f(to)g (oscillatory)f(in)n(tegrals)g(whose)h(phases)g(are)515 2946 y(canonical,)26 b(not)i(degenerate)e(quadratic)h(forms:)1340 3205 y Ft(Q)p Fw(\()p Ft(x)p Fw(\))d(=)1673 3101 y Fp(s)1628 3126 y Fr(X)1631 3303 y Fp(j)s Fu(=1)1748 3205 y Fw(\()p Ft(x)1827 3170 y Fu(\()p Fp(j)s Fu(\))1915 3205 y Fw(\))1947 3170 y Fu(2)2003 3205 y Fs(\000)2170 3101 y Fp(d)2127 3126 y Fr(X)2086 3304 y Fp(k)q Fu(=)p Fp(s)p Fu(+1)2289 3205 y Fw(\()p Ft(x)2368 3170 y Fu(\()p Fp(k)q Fu(\))2461 3205 y Fw(\))2493 3170 y Fu(2)2531 3205 y Ft(;)515 3469 y Fw(where)j(0)22 b Fi(6)h Ft(s)g Fi(6)g Ft(d)p Fw(.)639 3569 y(Exceptionally)-7 b(,)28 b(in)h(the)f(next)h(three)f(lemmas)g (the)h(v)-5 b(ariable)28 b Ft(r)f Fs(2)e Fn(R)j Fw(is)g(allo)n(w)n(ed)f (to)i(b)r(e)515 3669 y(negativ)n(e.)35 b(Moreo)n(v)n(er)25 b Ft(d)f Fw(=)e(1)27 b(\(so)h Ft(x)g Fw(v)-5 b(aries)26 b(in)i Fn(R)p Fw(\).)515 3868 y Fl(Lemma)60 b Fv(Supp)l(ose)31 b Fs(j)p Ft(r)r Fs(j)25 b Ft(>)e Fw(1)p Fv(.)40 b(F)-6 b(or)31 b(al)t(l)g Ft(l)26 b Fs(2)e Fn(N)30 b Fv(ther)l(e)h(exist)f(c)l (onstants)f Ft(c)2802 3825 y Fu(\()p Fp(l)p Fu(\))2802 3891 y Fp(j)2904 3868 y Fs(2)24 b Fn(C)30 b Fv(indep)l(en-)515 3968 y(dent)f(of)i Fs(j)p Ft(r)r Fs(j)g Fv(such)e(that)1178 4102 y Fr(Z)1261 4123 y Fo(1)1224 4291 y(\0001)1360 4215 y Fw(e)1397 4181 y Fo(\000)p Fu(\(1)p Fo(\000)p Fu(i)p Fp(r)r Fu(\))p Fp(x)1676 4156 y Fk(2)1711 4215 y Ft(x)1758 4181 y Fp(l)1802 4215 y Fw(d)p Ft(x)24 b Fw(=)f Fs(j)p Ft(r)r Fs(j)2092 4181 y Fo(\000)2154 4155 y FC(l)p Fk(+1)p 2154 4168 91 3 v 2186 4201 a(2)2300 4111 y Fo(1)2273 4136 y Fr(X)2276 4313 y Fp(j)s Fu(=0)2407 4215 y Ft(c)2443 4172 y Fu(\()p Fp(l)p Fu(\))2443 4238 y Fp(j)2520 4215 y Fs(j)p Ft(r)r Fs(j)2605 4181 y Fo(\000)p Fp(j)2693 4215 y Ft(:)515 4622 y Fl(Pro)s(of:)60 b Fw(Let)26 b(us)f(denote)g(b)n (y)g(\()p Fs(\001)p Fw(\))1560 4569 y Fk(1)p 1561 4578 29 3 v 1561 4612 a(2)1629 4622 y Fw(the)g(branc)n(h)g(of)g(the)h (square)e(ro)r(ot)g(whose)h(singular)f(cut)515 4730 y(is)j(the)h(p)r (ositiv)n(e)f(imaginary)f(axis.)36 b(Let)28 b Ft(z)f Fw(=)22 b(\(1)c Fs(\000)g Fw(i)p Ft(r)r Fw(\))2220 4678 y Fk(1)p 2222 4687 V 2222 4720 a(2)2264 4730 y Ft(x)p Fw(.)37 b(Then,)1007 4848 y Fr(Z)1053 5036 y Fq(R)1113 4961 y Fw(e)1150 4927 y Fo(\000)p Fu(\(1)p Fo(\000)p Fu(i)p Fp(r)r Fu(\))p Fp(x)1429 4901 y Fk(2)1464 4961 y Ft(x)1511 4927 y Fp(l)1556 4961 y Fw(d)p Ft(x)23 b Fw(=)1760 4844 y Fr(\022)1821 4848 y(Z)1867 5036 y Fp(\015)1924 4961 y Fw(e)1961 4927 y Fo(\000)p Fp(z)2047 4901 y Fk(2)2083 4961 y Ft(z)2126 4927 y Fp(l)2169 4961 y Fw(d)p Ft(z)2258 4844 y Fr(\023)2333 4961 y Fw([\(1)18 b Fs(\000)g Fw(i)p Ft(r)r Fw(\))2635 4904 y Fk(1)p 2636 4913 V 2636 4947 a(2)2678 4961 y Fw(])2701 4927 y Fo(\000)p Fp(l)p Fo(\000)p Fu(1)2864 4961 y Ft(;)386 b Fw(\(1\))1926 5255 y(7)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 515 523 a Fw(where)27 b Ft(\015)32 b Fw(is)c(the)g(orien)n(ted)e(path)i(\(1)18 b Fs(\000)g Fw(i)p Ft(r)r Fw(\))1850 471 y Fk(1)p 1851 480 29 3 v 1851 513 a(2)1894 523 y Fn(R)p Fw(.)37 b(Observ)n(e)26 b(that)1413 784 y(\(1)19 b Fs(\000)f Fw(i)p Ft(r)r Fw(\))1693 727 y Fk(1)p 1694 736 V 1694 770 a(2)1759 784 y Fw(=)23 b Ft(\033)1894 796 y Fp(r)1931 784 y Fs(j)p Ft(r)r Fs(j)2026 727 y Fk(1)p 2027 736 V 2027 770 a(2)2083 667 y Fr(\022)2154 728 y Fw(1)p 2154 765 42 4 v 2155 841 a Ft(r)2224 784 y Fs(\000)18 b Fw(i)2330 667 y Fr(\023)2401 662 y Fk(1)p 2401 671 29 3 v 2401 704 a(2)2457 784 y Ft(;)793 b Fw(\(2\))515 1012 y(where)1488 1141 y Ft(\033)1535 1153 y Fp(r)1595 1141 y Fw(=)1683 1024 y Fr(\032)1787 1090 y Fw(1)110 b(if)h Ft(r)26 b(>)c Fw(1)1787 1190 y(i)129 b(if)111 b Ft(r)26 b(<)c Fs(\000)p Fw(1)515 1340 y(\(so)27 b Ft(\033)696 1352 y Fp(r)761 1340 y Fw(do)r(es)g(not)h(dep)r(end)g(on)f Fs(j)p Ft(r)r Fs(j)p Fw(\).)38 b(Moreo)n(v)n(er)25 b(the)j(T)-7 b(a)n(ylor)26 b(expansion)1460 1589 y([\()p Ft(w)21 b Fs(\000)d Fw(i\))1743 1532 y Fk(1)p 1744 1541 V 1744 1575 a(2)1786 1589 y Fw(])1809 1555 y Fo(\000)p Fp(l)p Fo(\000)p Fu(1)1995 1589 y Fw(=)2109 1485 y Fo(1)2082 1510 y Fr(X)2085 1687 y Fp(j)s Fu(=0)2216 1589 y Ft(a)2260 1546 y Fu(\()p Fp(l)p Fu(\))2260 1612 y Fp(j)2337 1589 y Ft(w)2398 1555 y Fp(j)3273 1589 y Fw(\(3\))515 1877 y(is)40 b(v)-5 b(alid)40 b(for)f(all)h Fs(j)p Ft(w)r Fs(j)45 b Ft(<)f Fw(1,)e(b)n(y)e(c)n(hoice)g(of)g(the)g(branc)n(h)f(of) h Ft(z)2556 1825 y Fk(1)p 2556 1834 V 2556 1867 a(2)2598 1877 y Fw(.)74 b(Th)n(us,)44 b(substituting)515 1977 y Ft(w)25 b Fw(=)e(sgn)o(\()p Ft(r)r Fw(\))941 1944 y Fu(1)p 922 1958 73 4 v 922 2005 a Fo(j)p Fp(r)r Fo(j)1031 1977 y Fw(in)28 b(the)g(ab)r(o)n(v)n(e,)f(the)h(result)f(follo)n(ws)f (from)i(\(1\),)g(\(2\),)f(and)h(\(3\).)337 b Fi(\003)515 2276 y Fl(Lemma)59 b Fv(Supp)l(ose)30 b Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))31 b Fv(is)f(smo)l(oth)g(in)g(the)g(neighb)l(orho) l(o)l(d)i(of)f Fn(R)18 b Fs(\002)p 2819 2209 68 4 v 18 w Ft(B)34 b Fv(and)d(vanishes)515 2375 y(on)g Ft(U)701 2345 y Fu(c)753 2375 y Fs(\002)p 837 2309 V 19 w Ft(B)t Fv(,)g(wher)l(e)h Ft(U)i Fs(\032)24 b Fn(R)31 b Fv(and)g Ft(B)e Fs(\032)c Fn(R)1871 2345 y Fp(m)1964 2375 y Fv(ar)l(e)31 b(cub)l(es.)42 b(F)-6 b(or)31 b(any)g Ft(l)26 b Fs(2)g Fn(N)k Fv(ther)l(e)h(exists)g(a)515 2475 y Ft(C)574 2487 y Fp(l)623 2475 y Ft(>)22 b Fw(0)29 b Fv(indep)l(endent)i(of)f Fs(j)p Ft(r)r Fs(j)h Fv(satisfying)1316 2587 y Fr(\014)1316 2637 y(\014)1316 2687 y(\014)1316 2736 y(\014)1344 2594 y(Z)1427 2615 y Fo(1)1390 2783 y(\0001)1526 2707 y Fw(e)1563 2673 y Fu(i)p Fp(r)r(x)1653 2648 y Fk(2)1688 2707 y Ft(x)1735 2673 y Fp(l)1761 2707 y Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))19 b(d)p Ft(x)2101 2587 y Fr(\014)2101 2637 y(\014)2101 2687 y(\014)2101 2736 y(\014)2153 2707 y Fi(6)j Ft(C)2299 2719 y Fp(l)2325 2707 y Fs(j)p Ft(r)r Fs(j)2410 2673 y Fo(\000)2473 2647 y FC(l)p Fk(+1)p 2473 2660 91 3 v 2504 2693 a(2)515 2955 y Fv(uniformly)30 b(in)g Ft(t)23 b Fs(2)p 1127 2888 68 4 v 24 w Ft(B)t Fv(.)515 3154 y Fl(Pro)s(of:)78 b Fw(Let)33 b Ft(\037)p Fw(\()p Ft(x)p Fw(\))h(b)r(e)e(a)g(compactly)g(supp)r(orted)h(smo)r(oth)f(function)h (on)f Fn(R)h Fw(suc)n(h)f(that)515 3253 y(0)22 b Fi(6)h Ft(\037)p Fw(\()p Ft(x)p Fw(\))h Fi(6)f Fw(1)k(ev)n(erywhere)f(and)1442 3486 y Ft(\037)p Fw(\()p Ft(x)p Fw(\))e(=)1716 3369 y Fr(\032)1820 3435 y Fw(0)110 b(if)i Fs(j)p Ft(x)p Fs(j)23 b Fi(>)g Fw(2)1820 3535 y(1)110 b(if)i Fs(j)p Ft(x)p Fs(j)23 b Fi(6)g Fw(1)p Ft(:)515 3738 y Fw(F)-7 b(or)27 b(an)g(arbitrarily)f(\014xed)h Ft(")c(>)g Fw(0,)1621 3671 y Fr(R)1660 3768 y Fq(R)1720 3738 y Fw(e)1757 3708 y Fu(i)p Fp(r)r(x)1847 3683 y Fk(2)1883 3738 y Ft(x)1930 3708 y Fp(l)1956 3738 y Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))19 b(d)p Ft(x)k Fw(=)g Fv(I)31 b Fw(+)18 b Fv(II)c Ft(;)27 b Fw(where)1148 3972 y Fv(I)97 b Fw(=)1424 3859 y Fr(Z)1470 4047 y Fo(j)p Fp(x)p Fo(j)p Fp(<)p Fu(2)p Fp(")1682 3972 y Fw(e)1719 3938 y Fu(i)p Fp(r)r(x)1809 3913 y Fk(2)1844 3972 y Ft(x)1891 3938 y Fp(l)1917 3972 y Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))p Ft(\037)2211 3880 y Fr(\020)2271 3916 y Ft(x)p 2271 3953 48 4 v 2275 4029 a(")2328 3880 y Fr(\021)2410 3972 y Fw(d)p Ft(x;)1116 4208 y Fv(II)97 b Fw(=)1424 4095 y Fr(Z)1470 4284 y Fo(j)p Fp(x)p Fo(j)p Fp(>")1649 4208 y Fw(e)1686 4174 y Fu(i)p Fp(r)r(x)1776 4149 y Fk(2)1811 4208 y Ft(x)1858 4174 y Fp(l)1884 4208 y Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))2126 4116 y Fr(\020)2190 4208 y Fw(1)k Fs(\000)g Ft(\037)2399 4116 y Fr(\020)2458 4152 y Ft(x)p 2458 4189 V 2462 4265 a(")2515 4116 y Fr(\021)c(\021)2661 4208 y Fw(d)p Ft(x:)639 4452 y Fw(Concerning)21 b Fv(I)13 b Fw(,)24 b(there)e(exist)g(constan)n (ts)f(\(generically)f(denoted)j(b)n(y)e Fv(Const)h Fw(\))h(indep)r(en-) 515 4552 y(den)n(t)28 b(of)f Ft(t)c Fs(2)p 928 4485 68 4 v 24 w Ft(B)t Fw(,)k(but)i(dep)r(ending)f(on)f Ft(l)r Fw(,)g(satisfying)1260 4789 y Fs(j)p Fv(I)13 b Fs(j)24 b Fi(6)e Fv(Const)1708 4676 y Fr(Z)1791 4697 y Fu(2)p Fp(")1754 4865 y Fo(\000)p Fu(2)p Fp(")1888 4789 y Fs(j)p Ft(x)p Fs(j)1981 4755 y Fp(l)2026 4789 y Fw(d)p Ft(x)i Fw(=)e Fv(Const)g Ft(")2501 4755 y Fp(l)p Fu(+1)2610 4789 y Ft(:)1926 5255 y Fw(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 639 523 a Fw(Concerning)29 b Fv(II)14 b Fw(,)31 b(let)g Ft(D)e Fw(=)1538 490 y Fu(1)p 1536 504 38 4 v 1536 552 a Fp(x)1604 523 y Fs(\016)19 b Ft(@)1709 535 y Fp(x)1751 523 y Fw(,)32 b(whic)n(h)e(is)g(w)n(ell)g(de\014ned)h (on)f(the)h(supp)r(ort)f(of)h(the)515 642 y(in)n(tegrand.)36 b(Since)1171 609 y Fu(1)p 1145 623 85 4 v 1145 671 a(2i)p Fp(r)1240 642 y Ft(D)29 b Fw(\014xes)f(e)1563 612 y Fu(i)p Fp(r)r(x)1653 587 y Fk(2)1688 642 y Fw(,)914 898 y Fv(II)36 b Fw(=)1102 781 y Fr(\022)1204 842 y Fw(1)p 1173 879 105 4 v 1173 955 a(2i)p Ft(r)1287 781 y Fr(\023)1348 798 y Fp(N)1425 785 y Fr(Z)1471 973 y Fo(j)p Fp(x)p Fo(j)p Fp(>")1650 898 y Fw(e)1687 864 y Fu(i)p Fp(r)r(x)1777 838 y Fk(2)1812 898 y Fw(\()p Ft(D)1915 864 y Fu(t)1946 898 y Fw(\))1978 864 y Fp(N)2041 898 y Ft(x)2088 864 y Fp(l)2114 898 y Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))2356 806 y Fr(\020)2406 898 y Fw(1)k Fs(\000)g Ft(\037)2615 806 y Fr(\020)2674 842 y Ft(x)p 2674 879 48 4 v 2678 955 a(")2731 806 y Fr(\021\021)2863 898 y Fw(d)p Ft(x:)515 1138 y Fw(In)28 b(general,)e(if)i Ft(g)s Fw(\()p Ft(x)p Fw(\))g(is)g(a)f(compactly)g(supp)r(orted)g(smo)r(oth)h(function,)g (then)1397 1367 y Ft(D)1468 1332 y Fu(t)1498 1367 y Ft(x)1545 1332 y Fp(l)1571 1367 y Ft(g)s Fw(\()p Ft(x)p Fw(\))c(=)1837 1250 y Fr(\022)1898 1367 y Fs(\000)p Ft(@)2007 1379 y Fp(x)2067 1367 y Fs(\016)2140 1311 y Fw(1)p 2137 1348 V 2137 1424 a Ft(x)2194 1250 y Fr(\023)2269 1367 y Ft(x)2316 1332 y Fp(l)2342 1367 y Ft(g)s Fw(\()p Ft(x)p Fw(\))515 1604 y(is)h(of)g(the)g(form)g Ft(x)1069 1574 y Fp(l)p Fo(\000)p Fu(2)1180 1604 y Ft(h)p Fw(\()p Ft(x)p Fw(\),)h(where)f Ft(h)p Fw(\()p Ft(x)p Fw(\))g(is)g(also)f(a)h(compactly)f(supp)r(orted) h(smo)r(oth)g(func-)515 1703 y(tion.)39 b(It)29 b(follo)n(ws)e(b)n(y)h (induction)h(that)f(there)h(exist)f(constan)n(ts)f(indep)r(enden)n(t)i (of)g Ft(t)f Fw(and)g Ft(r)r Fw(,)515 1803 y(but)g(dep)r(ending)g(on)f Ft(N)37 b Fw(and)27 b Ft(l)r Fw(,)g(whic)n(h)h(satisfy)1247 2020 y Fs(j)p Fv(II)13 b Fs(j)83 b Fi(6)g Fv(Const)22 b Fs(j)p Ft(r)r Fs(j)1918 1986 y Fo(\000)p Fp(N)2047 1907 y Fr(Z)2093 2096 y Fo(j)p Fp(x)p Fo(j)p Fp(>")2272 2020 y Fs(j)p Ft(x)p Fs(j)2365 1986 y Fp(l)p Fo(\000)p Fu(2)p Fp(N)2553 2020 y Fw(d)p Ft(x)1453 2218 y Fw(=)83 b Fv(Const)22 b Fs(j)p Ft(r)r Fs(j)1918 2184 y Fo(\000)p Fp(N)2033 2218 y Ft(")2072 2184 y Fp(l)p Fo(\000)p Fu(2)p Fp(N)6 b Fu(+1)2326 2218 y Ft(;)515 2397 y Fw(pro)n(vided)26 b(that)i Ft(N)k(>)1233 2364 y Fp(l)p Fu(+1)p 1233 2378 106 4 v 1269 2426 a(2)1348 2397 y Fw(.)639 2497 y(In)c(total,)1237 2596 y Fs(j)p Fv(I)k Fw(+)18 b Fv(II)13 b Fs(j)23 b Fi(6)g Fv(Const)e Fw(\()p Ft(")1920 2562 y Fp(l)p Fu(+1)2048 2596 y Fw(+)d Fs(j)p Ft(r)r Fs(j)2216 2562 y Fo(\000)p Fp(N)2332 2596 y Ft(")2371 2562 y Fp(l)p Fo(\000)p Fu(2)p Fp(N)6 b Fu(+1)2624 2596 y Fw(\))515 2752 y(for)27 b(suc)n(h)g(an)g Ft(N)9 b Fw(.)37 b(Cho)r(osing)27 b Ft(")22 b Fw(=)h Fs(j)p Ft(r)r Fs(j)1677 2722 y Fo(\000)1740 2700 y Fk(1)p 1740 2709 29 3 v 1740 2742 a(2)1810 2752 y Fw(then)28 b(completes)f(the)h(pro)r(of.)576 b Fi(\003)515 3051 y Fl(Lemma)74 b Fv(Consider)38 b(a)g(function,)h Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))p Fv(,)40 b(smo)l(oth)e(in)f(the)g (neighb)l(orho)l(o)l(d)j(of)e Fn(R)23 b Fs(\002)p 3312 2984 68 4 v 24 w Ft(B)515 3151 y Fv(\(wher)l(e)31 b Ft(B)e Fs(\032)c Fn(R)1026 3120 y Fp(m)1120 3151 y Fv(is)31 b(a)h(cub)l(e\))e(and)i(vanishing)g(when)f Fs(j)p Ft(x)p Fs(j)26 b Ft(<)f(")30 b Fv(\(wher)l(e)i Ft(")24 b(>)h Fw(0)p Fv(\).)42 b(Supp)l(ose)515 3250 y(that)29 b(for)i(al)t(l)g Ft(N)h Fi(>)22 b Fw(0)30 b Fv(ther)l(e)f(exist)h(a)g Ft(D)1735 3262 y Fp(N)1821 3250 y Ft(>)22 b Fw(0)30 b Fv(and)g(an)g Ft(\013)2313 3262 y Fp(N)2399 3250 y Ft(>)22 b Fw(0)30 b Fv(such)f(that)1432 3429 y Fs(j)p Ft(@)1504 3395 y Fp(N)1499 3450 y(x)1566 3429 y Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))p Fs(j)24 b Fi(6)f Ft(D)1998 3441 y Fp(N)2060 3429 y Fw(\(1)c(+)f Fs(j)p Ft(x)p Fs(j)2329 3395 y Fp(\013)2372 3403 y FC(N)2430 3429 y Fw(\))515 3608 y Fv(uniformly)30 b(in)g Ft(t)23 b Fs(2)p 1127 3542 V 24 w Ft(B)t Fv(.)39 b(Then,)1334 3717 y Fr(Z)1417 3737 y Fo(1)1381 3905 y(\0001)1517 3830 y Fw(e)1554 3795 y Fu(\(i)p Fp(r)r Fo(\000)p Fu(1\))p Fp(x)1781 3770 y Fk(2)1816 3830 y Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))19 b(d)p Ft(x)24 b Fw(=)e Ft(O)r Fw(\()p Ft(r)2403 3795 y Fo(\0001)2527 3830 y Fw(\))515 4073 y Fv(uniformly)30 b(in)g Ft(t)23 b Fs(2)p 1127 4006 V 24 w Ft(B)t Fv(.)515 4272 y Fl(Pro)s(of:)104 b Fw(The)41 b(deriv)-5 b(ation)40 b Ft(D)48 b Fw(=)1704 4240 y Fu(1)p 1701 4254 38 4 v 1701 4301 a Fp(x)1776 4272 y Fs(\016)27 b Ft(@)1889 4284 y Fp(x)1972 4272 y Fw(is)41 b(w)n(ell)g(de\014ned)g(on)g(the)g(supp)r(ort)g(of)g(the)515 4391 y(in)n(tegrand.)36 b(Notice)27 b(that)1395 4359 y Fu(1)p 1369 4373 85 4 v 1369 4420 a(2i)p Fp(r)1463 4391 y Ft(D)j Fw(\014xes)d(e)1786 4361 y Fu(i)p Fp(r)r(x)1876 4336 y Fk(2)1940 4391 y Fw(and)g Ft(D)2172 4361 y Fu(t)2225 4391 y Fw(=)c Fs(\000)p Ft(@)2422 4403 y Fp(x)2482 4391 y Fs(\016)2554 4359 y Fu(1)p 2552 4373 38 4 v 2552 4420 a Fp(x)2599 4391 y Fw(.)37 b(Then,)680 4532 y Fr(\014)680 4582 y(\014)680 4632 y(\014)680 4682 y(\014)707 4540 y(Z)754 4729 y Fq(R)813 4653 y Fw(e)850 4619 y Fu(\(i)p Fp(r)r Fo(\000)p Fu(1\))p Fp(x)1077 4594 y Fk(2)1113 4653 y Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))19 b(d)p Ft(x)1453 4532 y Fr(\014)1453 4582 y(\014)1453 4632 y(\014)1453 4682 y(\014)1504 4653 y Fw(=)1592 4536 y Fr(\022)1706 4597 y Fw(1)p 1663 4634 128 4 v 1663 4710 a(2)p Fs(j)p Ft(r)r Fs(j)1800 4536 y Fr(\023)1861 4553 y Fp(N)1938 4508 y Fr(\014)1938 4557 y(\014)1938 4607 y(\014)1938 4657 y(\014)1938 4707 y(\014)1966 4540 y(Z)2012 4729 y Fo(j)p Fp(x)p Fo(j)p Fe(>)p Fp(")2190 4653 y Fw(e)2227 4619 y Fu(i)p Fp(r)r(x)2317 4594 y Fk(2)2353 4653 y Fw(\()p Ft(D)2456 4619 y Fu(t)2486 4653 y Fw(\))2518 4619 y Fp(N)2581 4653 y Fw(\(e)2650 4619 y Fo(\000)p Fp(x)2740 4594 y Fk(2)2777 4653 y Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\)\))19 b(d)p Ft(x)3149 4508 y Fr(\014)3149 4557 y(\014)3149 4607 y(\014)3149 4657 y(\014)3149 4707 y(\014)3191 4653 y Ft(:)515 4907 y Fw(Our)j(assumption)g(on)h(the)g(deriv)-5 b(ativ)n(es)22 b(of)h Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))23 b(mak)n(es)f(the)h(in)n(tegral)f(on)h(the)g(righ)n(t)f(side) 515 5006 y(of)34 b(the)g(previous)f(equation)h(uniformly)g(b)r(ounded)g (in)g Ft(t)g Fs(2)p 2435 4940 68 4 v 35 w Ft(B)t Fw(,)i(where)d Ft(N)43 b Fw(is)34 b(arbitrarily)1926 5255 y(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 515 523 a Fw(\014xed.)37 b(The)27 b(result)h(follo)n(ws.)1895 b Fi(\003)639 722 y Fw(In)28 b(dimension)g(one)f(w)n(e)g(are)g(in)n(terested)g(in)h(the)g(phase)f Ft(x)2448 692 y Fu(2)2485 722 y Fw(,)h(so)f(let)1414 947 y Ft(I)1450 959 y Fp(f)1494 947 y Fw(\()p Ft(r)n(;)14 b(t)p Fw(\))24 b(=)1771 834 y Fr(Z)1854 855 y Fo(1)1817 1023 y(\0001)1953 947 y Fw(e)1990 913 y Fu(i)p Fp(r)r(x)2080 888 y Fk(2)2116 947 y Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))19 b(d)p Ft(x:)515 1179 y Fw(According)24 b(to)h(our)f(con)n(v)n (en)n(tion)g Ft(r)h(>)e Fw(0.)36 b(Ho)n(w)n(ev)n(er,)23 b(since)i(w)n(e)g(are)f(also)g(in)n(terested)g(in)i(the)515 1278 y(phase)e Fs(\000)p Ft(x)855 1248 y Fu(2)892 1278 y Fw(,)i(w)n(e)e(allo)n(w)g Ft(r)k Fw(to)d(b)r(e)g(negativ)n(e.)35 b(W)-7 b(e)25 b(are)f(ready)g(to)g(compute)h(the)h(asymptotic)515 1378 y(expansion)g(of)i Ft(I)1031 1390 y Fp(f)1074 1378 y Fw(\()p Ft(r)n(;)14 b(t)p Fw(\))29 b(\(resp.)36 b Ft(I)1545 1390 y Fp(f)1589 1378 y Fw(\()p Fs(\000)p Ft(r)n(;)14 b(t)p Fw(\)\))28 b(when)g Fs(j)p Ft(r)r Fs(j)c(!)f(1)p Fw(:)515 1561 y Fl(Theorem)31 b(3)41 b Fv(Consider)27 b(an)e(amplitude,)j Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))p Fv(,)27 b(smo)l(oth)f(in)f(the)g(neighb)l(orho)l(o)l(d)j(of)e Fn(R)9 b Fs(\002)p 515 1594 68 4 v 515 1660 a Ft(B)34 b Fv(and)c(vanishing)i(on)e Ft(U)1328 1630 y Fu(c)1380 1660 y Fs(\002)p 1463 1594 V 18 w Ft(B)t Fv(,)h(wher)l(e)g Ft(U)h Fs(\032)23 b Fn(R)30 b Fv(and)h Ft(B)c Fs(\032)d Fn(R)2489 1630 y Fp(m)2582 1660 y Fv(ar)l(e)30 b(cub)l(es.)39 b(Ther)l(e)31 b(exist)515 1760 y(c)l(onstants,)e Ft(a)949 1772 y Fp(j)984 1760 y Fw(\()p Ft(t)p Fw(\))p Fv(,)i(dep)l(ending)g (smo)l(othly)g(on)e Ft(t)23 b Fs(2)p 2114 1693 V 24 w Ft(B)34 b Fv(such)29 b(that)1198 1899 y Fr(Z)1281 1920 y Fo(1)1244 2088 y(\0001)1380 2012 y Fw(e)1417 1978 y Fu(i)p Fp(r)r(x)1507 1953 y Fk(2)1543 2012 y Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))19 b(d)p Ft(x)24 b Fs(\030)e(j)p Ft(r)r Fs(j)2079 1978 y Fo(\000)2142 1956 y Fk(1)p 2142 1965 29 3 v 2142 1998 a(2)2225 1908 y Fo(1)2198 1933 y Fr(X)2201 2110 y Fp(j)s Fu(=0)2332 2012 y Ft(a)2376 2024 y Fp(j)2411 2012 y Fw(\()p Ft(t)p Fw(\))p Fs(j)p Ft(r)r Fs(j)2590 1978 y Fo(\000)2654 1951 y FC(j)p 2653 1965 V 2653 1998 a Fk(2)515 2294 y Fv(uniformly)33 b(in)g Ft(t)c Fs(2)p 1144 2228 68 4 v 29 w Ft(B)37 b Fv(when)c Ft(r)e Fs(!)e(1)p Fv(.)48 b(The)33 b(same)h(r)l(esult)e(holds)i(when)f Ft(r)e Fs(!)e(\0001)p Fv(,)k(with)515 2394 y(di\013er)l(ent)d(c)l (onstants)f Ft(a)1254 2406 y Fp(j)1288 2394 y Fw(\()p Ft(t)p Fw(\))p Fv(.)515 2577 y Fl(Pro)s(of:)80 b Fw(Let)33 b Ft(\037)p Fw(\()p Ft(x)p Fw(\))h(b)r(e)f(a)f(smo)r(oth,)i(b)r (ounded,)h(compactly)d(supp)r(orted)h(function)g(suc)n(h)515 2676 y(that)28 b Ft(\037)p Fw(\()p Ft(x)p Fw(\))c(=)e(1)27 b(on)h(an)f(in)n(terv)-5 b(al)27 b(con)n(taining)p 1973 2610 66 4 v 27 w Ft(U)g Fs([)19 b(f)p Fw(0)p Fs(g)p Fw(.)35 b(Then,)28 b(for)f Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))28 b(as)f(stated)1200 2901 y Ft(I)1236 2913 y Fp(f)1280 2901 y Fw(\()p Ft(r)n(;)14 b(t)p Fw(\))23 b(=)1557 2788 y Fr(Z)1603 2977 y Fq(R)1663 2901 y Fw(e)1700 2867 y Fu(\(i)p Fp(r)r Fo(\000)p Fu(1\))p Fp(x)1927 2842 y Fk(2)1963 2901 y Fw(\()14 b(e)2046 2867 y Fp(x)2084 2842 y Fk(2)2120 2901 y Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))g(\))p Ft(\037)p Fw(\()p Ft(x)p Fw(\))20 b(d)p Ft(x:)515 3131 y Fw(By)h(T)-7 b(a)n(ylor's)20 b(theorem)h(\(in)h(dimension)g(1\),)h(for)e(an)n(y)g (\014xed)h Ft(N)30 b Fw(there)22 b(exists)f(a)h(p)r(olynomial,)1602 3394 y Ft(P)1655 3406 y Fp(t)1684 3394 y Fw(\()p Ft(x)p Fw(\))i(=)1937 3291 y Fp(N)1907 3316 y Fr(X)1914 3494 y Fp(l)p Fu(=0)2041 3394 y Ft(b)2077 3406 y Fp(l)2102 3394 y Fw(\()p Ft(t)p Fw(\))p Ft(x)2243 3360 y Fp(l)2269 3394 y Ft(;)515 3667 y Fw(and)32 b(a)g(smo)r(oth)g(remainder,)g Ft(R)1537 3679 y Fp(t)1567 3667 y Fw(\()p Ft(x)p Fw(\),)i(b)r(oth)f (dep)r(ending)g(smo)r(othly)f(on)g(\()p Ft(x;)14 b(t)p Fw(\))32 b Fs(2)f Fn(R)22 b Fs(\002)p 3289 3601 68 4 v 21 w Ft(B)t Fw(,)515 3767 y(suc)n(h)27 b(that)1363 3867 y(e)1400 3832 y Fp(x)1438 3807 y Fk(2)1474 3867 y Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))23 b(=)g Ft(P)1866 3879 y Fp(t)1896 3867 y Fw(\()p Ft(x)p Fw(\))c(+)f Ft(x)2156 3832 y Fp(N)6 b Fu(+1)2304 3867 y Ft(R)2367 3879 y Fp(t)2396 3867 y Fw(\()p Ft(x)p Fw(\))p Ft(:)515 4016 y Fw(Th)n(us,)27 b Ft(I)784 4028 y Fp(f)828 4016 y Fw(\()p Ft(r)n(;)14 b(t)p Fw(\))28 b(decomp)r(oses)f(in)n(to)g Fv(I)k Fw(+)19 b Fv(II)31 b Fw(+)18 b Fv(III)13 b Fw(,)28 b(where)1282 4283 y Fv(I)97 b Fw(=)1589 4179 y Fp(N)1558 4204 y Fr(X)1565 4383 y Fp(l)p Fu(=0)1692 4283 y Ft(b)1728 4295 y Fp(l)1753 4283 y Fw(\()p Ft(t)p Fw(\))1861 4170 y Fr(Z)1908 4358 y Fq(R)1968 4283 y Fw(e)2005 4249 y Fu(\(i)p Fp(r)r Fo(\000)p Fu(1\))p Fp(x)2232 4223 y Fk(2)2267 4283 y Ft(x)2314 4249 y Fp(l)2358 4283 y Fw(d)p Ft(x;)1250 4537 y Fv(II)g Fw(=)1558 4424 y Fr(Z)1604 4613 y Fq(R)1664 4537 y Fw(e)1701 4503 y Fu(\(i)p Fp(r)r Fo(\000)p Fu(1\))p Fp(x)1928 4478 y Fk(2)1964 4537 y Ft(x)2011 4503 y Fp(N)6 b Fu(+1)2158 4537 y Ft(R)2221 4549 y Fp(t)2250 4537 y Fw(\()p Ft(x)p Fw(\))p Ft(\037)p Fw(\()p Ft(x)p Fw(\))21 b(d)p Ft(x;)1218 4759 y Fv(III)97 b Fw(=)1558 4646 y Fr(Z)1604 4835 y Fq(R)1664 4759 y Fw(e)1701 4725 y Fu(\(i)p Fp(r)r Fo(\000)p Fu(1\))p Fp(x)1928 4700 y Fk(2)1964 4759 y Ft(P)2017 4771 y Fp(t)2046 4759 y Fw(\()p Ft(x)p Fw(\)\()p Ft(\037)p Fw(\()p Ft(x)p Fw(\))21 b Fs(\000)d Fw(1\))g(d)p Ft(x:)1905 5255 y Fw(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 639 527 a Fw(Concerning)32 b Fv(I)46 b Fw(the)34 b(\014rst)f(lemma)g(giv)n(es)e(the)j(existence)f (of)g Ft(c)2612 484 y Fu(\()p Fp(l)p Fu(\))2612 550 y Fp(j)2721 527 y Fs(2)f Fn(C)i Fw(suc)n(h)e(that)i(for)515 626 y(an)n(y)27 b Fs(j)p Ft(r)r Fs(j)d Ft(>)e Fw(1)603 888 y Fv(I)97 b Fw(=)910 785 y Fp(N)879 810 y Fr(X)886 988 y Fp(l)p Fu(=0)1013 888 y Ft(b)1049 900 y Fp(l)1074 888 y Fw(\()p Ft(t)p Fw(\))p Fs(j)p Ft(r)r Fs(j)1253 854 y Fo(\000)1316 828 y FC(l)p Fk(+1)p 1316 841 91 3 v 1348 874 a(2)1462 785 y Fo(1)1435 810 y Fr(X)1438 986 y Fp(j)s Fu(=0)1569 888 y Ft(c)1605 845 y Fu(\()p Fp(l)p Fu(\))1605 912 y Fp(j)1682 888 y Fs(j)p Ft(r)r Fs(j)1767 854 y Fo(\000)p Fp(j)732 1191 y Fw(=)82 b Fs(j)p Ft(r)r Fs(j)964 1157 y Fo(\000)1027 1134 y Fk(1)p 1027 1143 29 3 v 1027 1177 a(2)1114 1087 y Fp(N)1083 1112 y Fr(X)1090 1291 y Fp(l)p Fu(=0)1217 1191 y Ft(b)1253 1203 y Fp(l)1278 1191 y Fw(\()p Ft(t)p Fw(\))p Fs(j)p Ft(r)r Fs(j)1457 1157 y Fo(\000)1524 1134 y FC(l)p 1520 1143 V 1520 1177 a Fk(2)1625 1112 y Fr(X)1577 1305 y Fp(j)s Fe(6)1669 1279 y FC(N)5 b Fj(\000)p FC(l)p 1669 1292 115 3 v 1712 1325 a Fk(2)1807 1191 y Ft(c)1843 1148 y Fu(\()p Fp(l)p Fu(\))1843 1214 y Fp(j)1920 1191 y Fs(j)p Ft(r)r Fs(j)2005 1157 y Fo(\000)p Fp(j)2112 1191 y Fw(+)2225 1087 y Fp(N)2195 1112 y Fr(X)2202 1291 y Fp(l)p Fu(=0)2328 1191 y Ft(b)2364 1203 y Fp(l)2389 1191 y Fw(\()p Ft(t)p Fw(\))p Fs(j)p Ft(r)r Fs(j)2568 1157 y Fo(\000)2632 1131 y FC(l)p Fk(+1)p 2632 1144 91 3 v 2663 1177 a(2)2799 1112 y Fr(X)2751 1305 y Fp(j)s(>)2843 1279 y FC(N)g Fj(\000)p FC(l)p 2843 1292 115 3 v 2886 1325 a Fk(2)2981 1191 y Ft(c)3017 1148 y Fu(\()p Fp(l)p Fu(\))3017 1214 y Fp(j)3094 1191 y Fs(j)p Ft(r)r Fs(j)3179 1157 y Fo(\000)p Fp(j)3267 1191 y Ft(:)515 1480 y Fw(Notice)27 b(that)1173 1537 y Fp(N)1142 1562 y Fr(X)1149 1740 y Fp(l)p Fu(=0)1276 1640 y Ft(b)1312 1652 y Fp(l)1337 1640 y Fw(\()p Ft(t)p Fw(\))p Fs(j)p Ft(r)r Fs(j)1516 1606 y Fo(\000)1579 1581 y FC(l)p Fk(+1)p 1579 1594 91 3 v 1611 1626 a(2)1746 1562 y Fr(X)1698 1754 y Fp(j)s(>)1791 1729 y FC(N)5 b Fj(\000)p FC(l)p 1791 1742 115 3 v 1834 1774 a Fk(2)1929 1640 y Ft(c)1965 1597 y Fu(\()p Fp(l)p Fu(\))1965 1664 y Fp(j)2042 1640 y Fs(j)p Ft(r)r Fs(j)2127 1606 y Fo(\000)p Fp(j)2238 1640 y Fw(=)22 b Ft(O)r Fw(\()p Fs(j)p Ft(r)r Fs(j)2507 1606 y Fo(\000)2571 1584 y FC(N)p 2571 1593 50 3 v 2582 1626 a Fk(2)2630 1606 y Fo(\000)p Fu(1)2719 1640 y Fw(\))515 1913 y(uniformly)h(in)h Ft(t)f Fs(2)p 1114 1846 68 4 v 24 w Ft(B)28 b Fw(when)c Ft(r)h Fs(!)f(1)f Fw(\(resp.)36 b Ft(r)25 b Fs(!)f(\0001)p Fw(\).)35 b(Consequen)n(tly)-7 b(,)24 b(w)n(e)f(ha)n(v)n(e)g(found)515 2013 y(co)r(e\016cien)n(ts,)k Ft(a)1002 2025 y Fp(k)1043 2013 y Fw(\()p Ft(t)p Fw(\),)h(smo)r(oth)f (in)h Ft(t)23 b Fs(2)p 1710 1946 V 24 w Ft(B)t Fw(,)28 b(satisfying)1249 2275 y Fv(I)36 b Fw(=)23 b Fs(j)p Ft(r)r Fs(j)1490 2241 y Fo(\000)1553 2218 y Fk(1)p 1553 2227 29 3 v 1553 2261 a(2)1640 2171 y Fp(N)1609 2196 y Fr(X)1609 2375 y Fp(k)q Fu(=0)1744 2275 y Ft(a)1788 2287 y Fp(k)1828 2275 y Fw(\()p Ft(t)p Fw(\))p Fs(j)p Ft(r)r Fs(j)2007 2241 y Fo(\000)2071 2218 y FC(k)p 2071 2227 33 3 v 2073 2261 a Fk(2)2135 2275 y Fw(+)18 b Ft(O)r Fw(\()p Fs(j)p Ft(r)r Fs(j)2400 2241 y Fo(\000)2464 2218 y FC(N)p 2464 2227 50 3 v 2474 2261 a Fk(2)2523 2241 y Fo(\000)p Fu(1)2612 2275 y Fw(\))515 2546 y(uniformly)27 b(in)h Ft(t)23 b Fs(2)p 1121 2479 68 4 v 23 w Ft(B)5 b Fw(.)639 2645 y(Moreo)n(v)n(er,) 25 b(the)j(last)f(t)n(w)n(o)g(lemmas)h(giv)n(e)e(resp)r(ectiv)n(ely) 1239 2836 y Fv(II)36 b Fw(=)22 b Ft(O)r Fw(\()p Fs(j)p Ft(r)r Fs(j)1608 2802 y Fo(\000)1672 2779 y FC(N)p 1672 2788 50 3 v 1682 2822 a Fk(2)1731 2802 y Fo(\000)p Fu(1)1820 2836 y Fw(\))56 b(and)f Fv(III)36 b Fw(=)23 b Ft(O)r Fw(\()p Fs(j)p Ft(r)r Fs(j)2499 2802 y Fo(\0001)2623 2836 y Fw(\))515 3017 y(uniformly)k(in)h Ft(t)23 b Fs(2)p 1121 2950 68 4 v 23 w Ft(B)5 b Fw(.)37 b(In)27 b(total,)1149 3274 y Ft(I)1185 3286 y Fp(f)1228 3274 y Fw(\()p Ft(r)n(;)14 b(t)p Fw(\))24 b(=)f Fs(j)p Ft(r)r Fs(j)1591 3240 y Fo(\000)1653 3218 y Fk(1)p 1653 3227 29 3 v 1653 3260 a(2)1740 3170 y Fp(N)1710 3195 y Fr(X)1709 3374 y Fp(k)q Fu(=0)1844 3274 y Ft(a)1888 3286 y Fp(k)1929 3274 y Fw(\()p Ft(t)p Fw(\))p Fs(j)p Ft(r)r Fs(j)2108 3240 y Fo(\000)2171 3218 y FC(k)p 2171 3227 33 3 v 2173 3260 a Fk(2)2236 3274 y Fw(+)18 b Ft(O)r Fw(\()p Fs(j)p Ft(r)r Fs(j)2501 3240 y Fo(\000)2564 3218 y FC(N)p 2564 3227 50 3 v 2575 3260 a Fk(2)2624 3240 y Fo(\000)p Fu(1)2713 3274 y Fw(\))515 3545 y(uniformly)27 b(in)h Ft(t)23 b Fs(2)p 1121 3478 68 4 v 23 w Ft(B)32 b Fw(when)c Ft(r)e Fs(!)d(1)28 b Fw(\(resp.)36 b Ft(r)26 b Fs(!)d(\0001)p Fw(\),)28 b(as)f(desired.)581 b Fi(\003)639 3744 y Fw(Our)28 b(treatmen)n(t)h(of)g(a)f(parameter)f Ft(t)f Fs(2)f Fn(R)1958 3714 y Fp(m)2050 3744 y Fw(p)r(ermits)k(to)g (deduce)g(from)f(the)h(previous)515 3844 y(theorem)22 b(its)h(m)n(ultidimensional)g(analogue)e(b)n(y)h(induction.)36 b(Indeed,)24 b(let)f(us)g(consider)f(the)515 3944 y(phase)1340 4121 y Ft(Q)p Fw(\()p Ft(x)p Fw(\))i(=)1673 4017 y Fp(s)1628 4042 y Fr(X)1631 4218 y Fp(j)s Fu(=1)1748 4121 y Fw(\()p Ft(x)1827 4086 y Fu(\()p Fp(j)s Fu(\))1915 4121 y Fw(\))1947 4086 y Fu(2)2003 4121 y Fs(\000)2170 4017 y Fp(d)2127 4042 y Fr(X)2086 4220 y Fp(k)q Fu(=)p Fp(s)p Fu(+1)2289 4121 y Fw(\()p Ft(x)2368 4086 y Fu(\()p Fp(k)q Fu(\))2461 4121 y Fw(\))2493 4086 y Fu(2)2531 4121 y Ft(;)515 4366 y Fw(where)j Ft(x)c Fs(2)h Fn(R)964 4335 y Fp(d)1030 4366 y Fw(and)k(0)22 b Fi(6)h Ft(s)g Fi(6)f Ft(d)p Fw(.)37 b(Returning)28 b(to)f(the)h(con)n(v)n(en)n(tion)e(that)i Ft(r)e(>)d Fw(0,)k(w)n(e)g(let)1399 4584 y Ft(I)1435 4596 y Fp(f)1479 4584 y Fw(\()p Ft(r)n(;)14 b(t)p Fw(\))23 b(=)1756 4471 y Fr(Z)1802 4660 y Fq(R)1844 4643 y FC(d)1897 4584 y Fw(e)1934 4550 y Fu(i)p Fp(r)r(Q)p Fu(\()p Fp(x)p Fu(\))2131 4584 y Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))19 b(d)p Ft(x:)515 4807 y Fw(Then,)515 5006 y Fl(Corollary)53 b Fv(Consider)28 b(an)e(amplitude,)i Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))p Fv(,)28 b(smo)l(oth)e(in)h(the)f(neighb)l(orho)l (o)l(d)i(of)f Fn(R)3265 4976 y Fp(d)3314 5006 y Fs(\002)1905 5255 y Fw(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 515 456 68 4 v 515 523 a Ft(B)32 b Fv(and)c(vanishing)h(on)f Ft(U)1319 493 y Fu(c)1366 523 y Fs(\002)p 1444 456 V 13 w Ft(B)5 b Fv(,)28 b(wher)l(e)g Ft(U)k Fs(\032)23 b Fn(R)2034 493 y Fp(d)2100 523 y Fv(and)28 b Ft(B)g Fs(\032)22 b Fn(R)2497 493 y Fp(m)2588 523 y Fv(ar)l(e)28 b(cub)l(es.)38 b(Ther)l(e)28 b(exist)515 623 y(c)l(onstants)h Ft(a)924 635 y Fp(j)959 623 y Fw(\()p Ft(t)p Fw(\))h Fv(dep)l(ending)h(smo)l(othly)g(on)e Ft(t)23 b Fs(2)p 2063 556 V 24 w Ft(B)34 b Fv(such)c(that)1228 756 y Fr(Z)1274 945 y Fq(R)1316 928 y FC(d)1369 869 y Fw(e)1406 835 y Fu(i)p Fp(r)r(Q)p Fu(\()p Fp(x)p Fu(\))1603 869 y Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))19 b(d)p Ft(x)24 b Fs(\030)e Ft(r)2093 835 y Fo(\000)2156 812 y FC(d)p 2156 821 31 3 v 2157 855 a Fk(2)2242 765 y Fo(1)2215 790 y Fr(X)2217 967 y Fp(j)s Fu(=0)2348 869 y Ft(a)2392 881 y Fp(j)2427 869 y Fw(\()p Ft(t)p Fw(\))p Ft(r)2560 835 y Fo(\000)2624 808 y FC(j)p 2623 822 29 3 v 2623 855 a Fk(2)515 1145 y Fv(uniformly)30 b(in)g Ft(t)23 b Fs(2)p 1127 1079 68 4 v 24 w Ft(B)t Fv(,)30 b(when)g Ft(r)c Fs(!)d(1)p Fv(.)515 1345 y Fl(Pro)s(of:)59 b Fw(Supp)r(ose)24 b(the)g(result)g(holds)g(for)f(a)h(certain)g Ft(d)11 b Fs(\000)g Fw(1.)36 b(Let)24 b Ft(x)2622 1357 y Fu(1)2683 1345 y Fw(=)e(\()p Ft(x)2849 1314 y Fu(\(1\))2939 1345 y Ft(;)14 b(:)g(:)g(:)g(;)g(x)3171 1314 y Fu(\()p Fp(d)p Fo(\000)p Fu(1\))3347 1345 y Fw(\))515 1444 y(and)27 b Ft(Q)742 1456 y Fu(1)779 1444 y Fw(\()p Ft(x)858 1456 y Fu(1)896 1444 y Fw(\))h(b)r(e)g(de\014ned)g(b)n(y)1494 1621 y Ft(Q)p Fw(\()p Ft(x)p Fw(\))c(=)f Ft(Q)1849 1633 y Fu(1)1886 1621 y Fw(\()p Ft(x)1965 1633 y Fu(1)2003 1621 y Fw(\))18 b Fs(\006)g Fw(\()p Ft(x)2215 1587 y Fu(\()p Fp(d)p Fu(\))2307 1621 y Fw(\))2339 1587 y Fu(2)2376 1621 y Ft(:)515 1798 y Fw(Recall)24 b(that)h Ft(U)33 b Fw(ma)n(y)24 b(b)r(e)h(written)g Ft(U)1661 1810 y Fu(1)1710 1798 y Fs(\002)12 b(\001)i(\001)g(\001)f(\002)f Ft(U)2031 1810 y Fp(d)2069 1798 y Fw(.)36 b(Then,)25 b(b)n(y)g(the)g(inductiv)n (e)f(h)n(yp)r(othesis,)515 1897 y(for)j(an)g(arbitrary)f Ft(N)31 b Fi(>)23 b Fw(0)570 2112 y Ft(I)606 2124 y Fp(f)650 2112 y Fw(\()p Ft(r)n(;)14 b(t)p Fw(\))83 b(=)1047 1999 y Fr(Z)1093 2187 y Fp(U)1138 2196 y FC(d)1191 2112 y Fw(e)1228 2078 y Fo(\006)p Fu(i)p Fp(r)r Fu(\()p Fp(x)1396 2053 y Fk(\()p FC(d)p Fk(\))1475 2078 y Fu(\))1501 2053 y Fk(2)1551 1999 y Fr(Z)1597 2187 y Fq(R)1639 2171 y FC(d)p Fj(\000)p Fk(1)1765 2112 y Fw(e)1802 2078 y Fu(i)p Fp(r)r(Q)1906 2086 y Fk(1)1938 2078 y Fu(\()p Fp(x)2002 2086 y Fk(1)2034 2078 y Fu(\))2064 2112 y Ft(f)9 b Fw(\()p Ft(x)2193 2124 y Fu(1)2231 2112 y Fw(;)14 b Ft(x)2315 2078 y Fu(\()p Fp(d)p Fu(\))2406 2112 y Ft(;)g(t)p Fw(\))k(d)p Ft(x)2616 2124 y Fu(1)2673 2112 y Fw(d)p Ft(x)2766 2078 y Fu(\()p Fp(d)p Fu(\))899 2381 y Fw(=)1047 2268 y Fr(Z)1093 2457 y Fp(U)1138 2466 y FC(d)1191 2381 y Fw(e)1228 2347 y Fo(\006)p Fu(i)p Fp(r)r Fu(\()p Fp(x)1396 2322 y Fk(\()p FC(d)p Fk(\))1475 2347 y Fu(\))1501 2322 y Fk(2)1551 2239 y Fr( )1617 2381 y Ft(r)1656 2347 y Fo(\000)1718 2321 y FC(d)p Fj(\000)p Fk(1)p 1718 2334 104 3 v 1756 2367 a(2)1850 2278 y Fp(N)6 b Fu(+1)1862 2302 y Fr(X)1861 2481 y Fp(k)q Fu(=0)2007 2381 y Ft(b)2043 2393 y Fp(k)2083 2381 y Fw(\()p Ft(x)2162 2347 y Fu(\()p Fp(d)p Fu(\))2254 2381 y Ft(;)14 b(t)p Fw(\))p Ft(r)2392 2347 y Fo(\000)2454 2325 y FC(k)p 2454 2334 33 3 v 2456 2367 a Fk(2)2519 2381 y Fw(+)k Ft(O)r Fw(\()p Ft(r)2738 2347 y Fo(\000)2802 2321 y FC(d)p Fk(+)p FC(N)5 b Fk(+1)p 2802 2334 194 3 v 2884 2367 a(2)3009 2381 y Fw(\))3041 2239 y Fr(!)3140 2381 y Fw(d)p Ft(x)3233 2347 y Fu(\()p Fp(d)p Fu(\))515 2649 y Fw(uniformly)22 b(in)i(\()p Ft(x)1060 2618 y Fu(\()p Fp(d)p Fu(\))1151 2649 y Ft(;)14 b(t)p Fw(\))23 b Fs(2)p 1352 2582 96 4 v 24 w Ft(U)1409 2661 y Fp(d)1456 2649 y Fs(\002)p 1530 2582 68 4 v 9 w Ft(B)t Fw(.)35 b(The)23 b(follo)n(wing)f(estimates)h(then)g(hold)g(when)g Ft(r)j Fs(!)d(1)p Fw(,)515 2748 y(uniformly)k(in)h Ft(t)23 b Fs(2)p 1121 2682 V 23 w Ft(B)5 b Fw(:)686 3002 y Ft(I)722 3014 y Fp(f)766 3002 y Fw(\()p Ft(r)n(;)14 b(t)p Fw(\))83 b(=)1163 2898 y Fp(N)6 b Fu(+1)1174 2923 y Fr(X)1174 3101 y Fp(k)q Fu(=0)1320 3002 y Ft(r)1359 2967 y Fo(\000)1421 2942 y FC(d)p Fk(+)p FC(k)q Fj(\000)p Fk(1)p 1421 2955 179 3 v 1496 2987 a(2)1628 2889 y Fr(Z)1674 3077 y Fp(U)1719 3086 y FC(d)1772 3002 y Fw(e)1809 2967 y Fo(\006)p Fu(i)p Fp(r)r Fu(\()p Fp(x)1977 2942 y Fk(\()p FC(d)p Fk(\))2056 2967 y Fu(\))2082 2942 y Fk(2)2118 3002 y Ft(b)2154 3014 y Fp(k)2195 3002 y Fw(\()p Ft(x)2274 2967 y Fu(\()p Fp(d)p Fu(\))2365 3002 y Ft(;)14 b(t)p Fw(\))19 b(d)p Ft(x)2576 2967 y Fu(\()p Fp(d)p Fu(\))2685 3002 y Fw(+)f Ft(O)r Fw(\()p Ft(r)2904 2967 y Fo(\000)2968 2942 y FC(d)p Fk(+)p FC(N)5 b Fk(+1)p 2968 2955 194 3 v 3050 2987 a(2)3175 3002 y Fw(\))1015 3295 y(=)1163 3191 y Fp(N)h Fu(+1)1174 3216 y Fr(X)1174 3395 y Fp(k)q Fu(=0)1320 3295 y Ft(r)1359 3260 y Fo(\000)1421 3235 y FC(d)p Fk(+)p FC(k)q Fj(\000)p Fk(1)p 1421 3248 179 3 v 1496 3280 a(2)1628 3153 y Fr( )1694 3295 y Ft(r)1733 3260 y Fo(\000)1795 3238 y Fk(1)p 1795 3247 29 3 v 1795 3280 a(2)1882 3191 y Fp(N)1851 3216 y Fr(X)1859 3395 y Fp(l)p Fu(=0)1985 3295 y Ft(c)2021 3307 y Fp(k)q(;l)2103 3295 y Fw(\()p Ft(t)p Fw(\))p Ft(r)2236 3260 y Fo(\000)2303 3238 y FC(l)p 2299 3247 V 2299 3280 a Fk(2)2342 3153 y Fr(!)2426 3295 y Fw(+)18 b Ft(O)r Fw(\()p Ft(r)2645 3260 y Fo(\000)2708 3235 y FC(d)p Fk(+)p FC(N)5 b Fk(+1)p 2708 3248 194 3 v 2791 3280 a(2)2916 3295 y Fw(\))1015 3588 y(=)83 b Ft(r)1202 3554 y Fo(\000)1264 3531 y FC(d)p 1264 3540 31 3 v 1265 3574 a Fk(2)1354 3484 y Fp(N)1323 3509 y Fr(X)1326 3686 y Fp(j)s Fu(=0)1457 3588 y Ft(a)1501 3600 y Fp(j)1536 3588 y Fw(\()p Ft(t)p Fw(\))p Ft(r)1669 3554 y Fo(\000)1733 3527 y FC(j)p 1732 3541 29 3 v 1732 3574 a Fk(2)1793 3588 y Fw(+)18 b Ft(O)r Fw(\()p Ft(r)2012 3554 y Fo(\000)2075 3528 y FC(d)p Fk(+)p FC(N)5 b Fk(+1)p 2075 3541 194 3 v 2158 3574 a(2)2283 3588 y Fw(\))p Ft(:)515 3849 y Fw(Since)28 b Ft(N)36 b Fw(is)27 b(arbitrary)-7 b(,)26 b(this)i(completes)f(the)h(pro)r(of.) 1125 b Fi(\003)515 4163 y Fl(Morse)25 b(Lemma)83 b Fw(The)22 b(in)n(v)n(estigation)f(of)h(oscillatory)f(in)n(tegrals)g(with)i(non)f (degenerate)515 4263 y(stationary)k(phase)h(p)r(oin)n(ts)g(reduces)g (to)h(the)f(ab)r(o)n(v)n(e)g(case)f(b)n(y)i(means)f(of)g Fv(Morse's)k(lemma)p Fw(,)515 4362 y(whic)n(h)c(w)n(e)g(no)n(w)g(pro)n (v)n(e.)639 4462 y(Let)g(the)f(an)n(ticipated)g(phase,)g Ft(\036)p Fw(\()p Ft(h;)14 b(t)p Fw(\),)28 b(b)r(e)e(a)g(smo)r(oth)g (function)h(in)f(\()p Ft(h;)14 b(t)p Fw(\))24 b Fs(2)f Fn(R)3121 4432 y Fp(d)3176 4462 y Fs(\002)15 b Fn(R)3316 4432 y Fp(m)515 4561 y Fw(satisfying)1036 4661 y Ft(\036)p Fw(\(0)p Ft(;)f(t)p Fw(\))23 b(=)g(0)p Ft(;)69 b Fs(r)1572 4673 y Fp(h)1615 4661 y Ft(\036)p Fw(\(0)p Ft(;)14 b(t)p Fw(\))23 b(=)g(0)55 b(and)69 b(det)14 b(D)2440 4627 y Fu(2)2440 4682 y Fp(h)2483 4661 y Ft(\036)p Fw(\(0)p Ft(;)g(t)p Fw(\))24 b Fs(6)p Fw(=)e(0)515 4807 y(for)27 b Fv(al)t(l)38 b Ft(t)24 b Fs(2)h Ft(B)t Fw(,)j(where)g Ft(B)g Fs(\032)c Fn(R)1501 4777 y Fp(m)1592 4807 y Fw(is)k(a)g(giv)n (en)f(cub)r(e)i(con)n(taining)e(a)g(giv)n(en)h Ft(t)2881 4819 y Fu(0)2942 4807 y Fs(2)c Fn(R)3081 4777 y Fp(m)3144 4807 y Fw(.)39 b(Suc)n(h)515 4907 y(a)26 b(function)h(can)f(b)r(e)h (expressed)e Fv(like)34 b Fw(a)26 b(quadratic)f(form,)h(but)h(with)g (co)r(e\016cien)n(ts)f(v)-5 b(arying)515 5006 y(smo)r(othly)27 b(in)h(\()p Ft(h;)14 b(t)p Fw(\):)1905 5255 y(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 515 623 a Fl(Lemma)50 b Fv(Under)25 b(the)g(ab)l(ove)h(cir)l(cumstanc)l(es)f(ther)l(e)g(exist)g(functions,) i Ft(\036)2824 635 y Fp(j)s(k)2896 623 y Fw(\()p Ft(h;)14 b(t)p Fw(\))p Fv(,)27 b(smo)l(oth)515 722 y(in)j Fw(\()p Ft(h;)14 b(t)p Fw(\))23 b Fs(2)g Fn(R)957 692 y Fp(d)1014 722 y Fs(\002)18 b Fn(R)1157 692 y Fp(m)1250 722 y Fv(and)30 b(satisfying)1344 982 y Ft(\036)p Fw(\()p Ft(h;)14 b(t)p Fw(\))23 b(=)1726 878 y Fp(d)1683 903 y Fr(X)1686 1079 y Fp(j)s Fu(=1)1860 878 y Fp(d)1817 903 y Fr(X)1817 1081 y Fp(k)q Fu(=1)1951 982 y Ft(\036)2000 994 y Fp(j)s(k)2073 982 y Fw(\()p Ft(h;)14 b(t)p Fw(\))p Ft(h)2300 947 y Fu(\()p Fp(j)s Fu(\))2387 982 y Ft(h)2435 947 y Fu(\()p Fp(k)q Fu(\))2527 982 y Ft(;)515 1247 y Fv(wher)l(e)30 b Ft(\036)798 1259 y Fp(j)s(k)870 1247 y Fw(\()p Ft(h;)14 b(t)p Fw(\))24 b(=)e Ft(\036)1209 1259 y Fp(k)q(j)1281 1247 y Fw(\()p Ft(h;)14 b(t)p Fw(\))p Fv(.)515 1447 y Fl(Pro)s(of:)68 b Fw(Using)28 b(our)g(h)n(yp)r(otheses)h(on)f Ft(\036)p Fw(\()p Ft(h;)14 b(t)p Fw(\),)30 b(the)f(fundamen)n(tal)g (theorem)g(of)g(calculus)515 1546 y(and)e(in)n(tegration)f(b)n(y)i (parts)f(giv)n(e)f(for)h(an)n(y)g Ft(t)c Fs(2)h Ft(B)941 1778 y(\036)p Fw(\()p Ft(h;)14 b(t)p Fw(\))24 b(=)1281 1665 y Fr(Z)1364 1686 y Fu(1)1327 1854 y(0)1415 1778 y Ft(@)1459 1790 y Fp(s)1494 1778 y Fw(\()p Ft(\036)p Fw(\()p Ft(sh;)14 b(t)p 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Fw(matrix,)i(denoted)e(b)n(y)f(\010\()p Ft(h;)14 b(t)p Fw(\).)35 b(The)20 b(\()p Ft(j;)14 b(k)s Fw(\)-th)21 b(elemen)n(t)515 2605 y(of)27 b(\010\()p Ft(h;)14 b(t)p Fw(\))28 b(is)g(then)g(giv)n(en)f(b)n(y)g(the)h (function)g Ft(\036)1998 2617 y Fp(j)s(k)2070 2605 y Fw(\()p Ft(h;)14 b(t)p Fw(\),)28 b(so)f(the)h(previous)f(lemma)g(giv)n (es)1583 2783 y Ft(\036)p Fw(\()p Ft(h;)14 b(t)p Fw(\))24 b(=)f Ft(h)1971 2748 y Fu(t)2000 2783 y Fw(\010\()p Ft(h;)14 b(t)p Fw(\))p Ft(h:)515 2960 y Fw(F)-7 b(or)31 b(an)n(y)g(giv)n(en)g Ft(t)f Fs(2)h Ft(B)36 b Fw(the)c(rank)f(of)h(\010\(0)p Ft(;)14 b(t)p Fw(\))32 b(is)g(read)f(through)g(the)h(Hessian)f(of)h Ft(\036)p Fw(\()p Ft(h;)14 b(t)p Fw(\))515 3060 y(using)27 b(the)h(follo)n(wing,)f(straigh)n(tforw)n(ard)d(relation,)j(whic)n(h)g (is)h(left)g(to)g(the)g(reader:)515 3259 y Fl(Lemma)59 b Fv(Under)30 b(the)f(ab)l(ove)i(cir)l(cumstanc)l(es,)f(for)40 b Fw(an)n(y)29 b Ft(t)23 b Fs(2)g Ft(B)t Fv(,)30 b Fw(D)2671 3229 y Fu(2)2671 3282 y Fp(h)2715 3259 y Ft(\036)p Fw(\(0)p Ft(;)14 b(t)p Fw(\))23 b(=)g(2\010\(0)p Ft(;)14 b(t)p Fw(\))p Ft(:)639 3458 y Fw(In)27 b(order)e(to)h(p)r(erform)g (Lagrange's)e(algorithm)i(one)g(uses)g(the)h(follo)n(wing)e(elemen)n (tary)515 3558 y(line/column)i(op)r(erations:)639 3720 y Fs(\017)41 b Fw(Giv)n(en)30 b(a)f(scalar)f Ft(c)f Fs(6)p Fw(=)f(0,)k(to)f(m)n(ultiply)h(the)g Ft(j)5 b Fw(-th)30 b(ro)n(w)f(and)g(then)h(the)g Ft(j)5 b Fw(-th)30 b(column)722 3819 y(b)n(y)e Ft(c)p Fw(,)f(whic)n(h)h(is)f(denoted)h(b)n(y)f Ft(C)6 b(L)1796 3831 y Fp(j)1831 3819 y Fw(\()p Ft(c)p Fw(\);)639 3983 y Fs(\017)41 b Fw(Giv)n(en)23 b(a)g(scalar)f Ft(c)h Fs(2)h Fn(C)p Fw(,)g(to)f(add)g Ft(c)h Fw(times)f(the)h Ft(k)s Fw(-th)g(ro)n(w)e(to)h(the)h Ft(j)5 b Fw(-th)23 b(ro)n(w)f(and)i(then)722 4083 y Ft(c)h Fw(times)g(the)f Ft(k)s Fw(-th)h(column)f(to)h(the)g Ft(j)5 b Fw(-th)24 b(column,)h(whic)n(h)g(is)f(denoted)h(b)n(y)f Ft(C)6 b(L)3184 4095 y Fp(j)s(k)3255 4083 y Fw(\()p Ft(c)p Fw(\);)639 4247 y Fs(\017)41 b Fw(T)-7 b(o)29 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b(diagonal)f(matrix)i(ha)n(ving)e Fs(\006)p Fw(1)h(as)h(its)g(\014rst)f(t)n(w)n(o)g(blo)r(c)n(ks)g(and)h (a)f(square)g(\()p Ft(d)20 b Fs(\000)e Fw(2\))h Fs(\002)515 2026 y Fw(\()p Ft(d)i Fs(\000)g Fw(2\))31 b(matrix)g(as)f(its)i(third)f (blo)r(c)n(k.)48 b(All)32 b(the)f(op)r(erations)f(used)i(for)f(this)g (second)g(step)515 2126 y(are)26 b(represen)n(ted)h(b)n(y)g(matrices)g (ha)n(ving)f(smo)r(oth)i(elemen)n(ts)f(in)h(\()p Ft(h;)14 b(t)p Fw(\))23 b Fs(2)h Ft(V)2869 2138 y Fu(2)2925 2126 y Fs(\002)18 b Ft(B)3071 2138 y Fu(2)3108 2126 y Fw(,)28 b(where)1238 2308 y(\(0)p Ft(;)14 b(t)1379 2320 y Fu(0)1416 2308 y Fw(\))23 b Fs(2)g Ft(V)1597 2320 y Fu(2)1654 2308 y Fs(\002)18 b Ft(B)1800 2320 y Fu(2)1860 2308 y Fs(\022)k Ft(V)1995 2320 y Fu(1)2052 2308 y Fs(\002)c Ft(B)2198 2320 y Fu(1)2258 2308 y Fs(\022)23 b Ft(V)2394 2320 y Fu(0)2450 2308 y Fs(\002)18 b Ft(B)2596 2320 y Fu(0)2633 2308 y Ft(:)639 2491 y Fw(So)38 b(on)g(and)g(so)g(forth)g(one)g (transforms)e(\010\()p Ft(h;)14 b(t)p Fw(\))39 b(in)n(to)f(a)g (diagonal)e(matrix)i(ha)n(ving)515 2591 y(elemen)n(ts)i Fs(\006)p Fw(1)g(only)-7 b(.)75 b(All)41 b(the)g(required)e(op)r (erations)h(are)f(smo)r(oth)h(\(in)h(the)g(previous)515 2690 y(sense\))29 b(for)f(\()p Ft(h;)14 b(t)p Fw(\))29 b(v)-5 b(arying)28 b(in)i(a)e(cub)r(e)i Ft(V)1810 2702 y Fp(d)1868 2690 y Fs(\002)19 b Ft(B)2015 2702 y Fp(d)2083 2690 y Fw(con)n(taining)28 b(\(0)p Ft(;)14 b(t)2629 2702 y Fu(0)2666 2690 y Fw(\).)41 b(One)29 b(then)g(applies)515 2790 y(p)r(erm)n(utations)24 b Ft(C)6 b(L)1142 2802 y Fp(j)s(k)1214 2790 y Fw(,)25 b(so)g(the)g(resulting)g(matrix)f(b)r (ecomes)h(diag\(1)p Ft(;)14 b(:)g(:)g(:)g(;)g Fw(1)p Ft(;)g Fs(\000)p Fw(1)p Ft(;)g(:)g(:)g(:)e(;)i Fs(\000)p Fw(1\),)515 2889 y(where)38 b(the)h(elemen)n(t)g(1)g(is)f(rep)r(eated,) k(sa)n(y)-7 b(,)40 b Ft(s)f Fw(times.)71 b(Since)39 b Ft(s)g Fw(do)r(es)f(not)h(dep)r(end)g(on)515 2989 y(\()p Ft(h;)14 b(t)p Fw(\))23 b Fs(2)g Ft(V)843 3001 y Fp(d)898 2989 y Fs(\002)15 b Ft(B)1041 3001 y Fp(d)1080 2989 y Fw(,)26 b(the)h(previous)e(lemma)h(and)f(the)i(Sylv)n(ester)e(Inertia)g (Theorem)h(p)r(ermit)515 3089 y(to)h(reco)n(v)n(er)e Ft(s)j Fw(from)f(the)h(signature)f(of)g(D)1824 3059 y Fu(2)1824 3112 y Fp(h)1868 3089 y Ft(\036)p Fw(\(0)p Ft(;)14 b(t)2058 3101 y Fu(0)2095 3089 y Fw(\),)28 b(whic)n(h)g(is)f (then)h(\()p Ft(s;)14 b(d)19 b Fs(\000)f Ft(s;)c Fw(0\).)639 3188 y(In)28 b(summary)-7 b(,)27 b(w)n(e)g(ha)n(v)n(e)g(pro)n(v)n(en:) 515 3388 y Fl(Lemma)52 b Fv(Under)26 b(the)g(ab)l(ove)i(cir)l(cumstanc) l(es)d(ther)l(e)i(exist)f(a)g(cub)l(e,)h Ft(V)2724 3400 y Fp(d)2774 3388 y Fs(\002)11 b Ft(B)2913 3400 y Fp(d)2951 3388 y Fv(,)27 b(c)l(ontaining)515 3487 y Fw(\(0)p Ft(;)14 b(t)656 3499 y Fu(0)693 3487 y Fw(\))36 b Fv(and)h(line)l(ar)g(map,)i Ft(Q)p Fw(\()p Ft(h;)14 b(t)p Fw(\))p Fv(,)39 b(whose)e(matrix)f (elements)g(ar)l(e)h(smo)l(oth)g(in)f Fw(\()p Ft(h;)14 b(t)p Fw(\))36 b Fs(2)515 3587 y Ft(V)563 3599 y Fp(d)620 3587 y Fs(\002)18 b Ft(B)766 3599 y Fp(d)835 3587 y Fv(such)30 b(that)1041 3770 y Ft(Q)p Fw(\()p Ft(h;)14 b(t)p Fw(\))1286 3728 y Fu(t)1316 3770 y Fw(\010\()p Ft(h;)g(t)p Fw(\))p Ft(Q)p Fw(\()p Ft(h;)g(t)p Fw(\))23 b(=)g(diag\(1)p Ft(;)14 b(:)g(:)g(:)g(;)g Fw(1)p Ft(;)g Fs(\000)p Fw(1)p Ft(;)g(:)g(:)g(:)e(;)i Fs(\000)p Fw(1\))p Ft(:)515 3952 y Fv(In)25 b(the)i(ab)l(ove)g(the)f (element)g Fw(1)f Fv(is)h(r)l(ep)l(e)l(ate)l(d)h Ft(s)f Fv(times,)h(wher)l(e)g Fw(\()p Ft(s;)14 b(d)c Fs(\000)g Ft(s;)k Fw(0\))26 b Fv(is)g(the)g(signatur)l(e)515 4052 y(of)k Fw(D)675 4022 y Fu(2)675 4075 y Fp(h)719 4052 y Ft(\036)p Fw(\(0)p Ft(;)14 b(t)909 4064 y Fu(0)946 4052 y Fw(\))p Fv(.)639 4251 y Fw(It)40 b(is)f(th)n(us)g(tempting)h(to) f(consider)f(the)i Fv(non)g(line)l(ar)49 b Fw(mapping)39 b Ft(h)j Fs(7!)g Ft(Q)p Fw(\()p Ft(h;)14 b(t)p Fw(\))3242 4221 y Fo(\000)p Fu(1)3331 4251 y Ft(h)515 4351 y Fw(de\014ned)20 b(on)f Ft(V)948 4363 y Fp(d)1007 4351 y Fw(as)h(a)f(p)r(oten)n(tial)h (c)n(hange)e(of)i(v)-5 b(ariables)19 b(giv)n(en)g(a)g(\014xed)h Ft(t)j Fs(2)g Ft(B)2853 4363 y Fp(d)2892 4351 y Fw(.)34 b(Considering)515 4450 y Ft(t)23 b Fs(2)g Ft(B)709 4462 y Fp(d)772 4450 y Fw(as)g(a)g(parameter,)g Fs(f)p Ft(Q)p Fw(\()p Ft(h;)14 b(t)p Fw(\))1639 4420 y Fo(\000)p Fu(1)1727 4450 y Ft(h)p Fs(g)1817 4473 y Fp(t)p Fo(2)p Fp(B)1937 4482 y FC(d)1999 4450 y Fw(is)24 b(indeed)g(a)f(family)h(of)f(smo)r (oth)h(mappings)515 4550 y(dep)r(ending)k(smo)r(othly)g(on)g Ft(t)h Fw(in)f(the)h(strong)e(sense)h(that)h(\()p Ft(h;)14 b(t)p Fw(\))24 b Fs(7!)h Ft(Q)p Fw(\()p Ft(h;)14 b(t)p Fw(\))2897 4520 y Fo(\000)p Fu(1)2986 4550 y Ft(h)28 b Fw(is)g(join)n(tly)515 4650 y(smo)r(oth)35 b(on)h Ft(V)988 4662 y Fp(d)1051 4650 y Fs(\002)23 b Ft(B)1202 4662 y Fp(d)1241 4650 y Fw(.)61 b(Moreo)n(v)n(er,)35 b(all)h(these)g(mappings) f(map)h(0)f(to)h(0.)60 b(They)36 b(really)515 4749 y(consist)25 b(of)h(in)n(v)n(ertible)g(c)n(hanges)f(of)h(v)-5 b(ariables)25 b(when)h(restricting)f(suitably)h(the)h(ranges)d(of)515 4849 y Ft(h)j Fw(and)h Ft(t)p Fw(,)f(as)g(sho)n(wn)g(b)r(elo)n(w:)1905 5255 y(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 515 523 a Fl(Lemma)55 b Fv(Under)27 b(the)h(ab)l(ove)h(cir)l(cumstanc)l(es)f(ther)l(e)g(exists)f(a)h(cub)l (e,)g Ft(V)2792 493 y Fo(0)2830 523 y Fs(\002)13 b Ft(B)2975 493 y Fo(0)2999 523 y Fv(,)28 b(such)g(that)1438 704 y Fw(\(0)p Ft(;)14 b(t)1579 716 y Fu(0)1616 704 y Fw(\))23 b Fs(2)h Ft(V)1817 669 y Fo(0)1858 704 y Fs(\002)18 b Ft(B)2008 669 y Fo(0)2055 704 y Fs(\032)k Ft(V)2190 716 y Fp(d)2248 704 y Fs(\002)c Ft(B)2394 716 y Fp(d)2433 704 y Ft(;)515 884 y Fv(on)30 b(which)1536 984 y Fw(\()p Ft(h;)14 b(t)p Fw(\))23 b Fs(7!)g Fw(\()p Ft(Q)p Fw(\()p Ft(h;)14 b(t)p Fw(\))2121 950 y Fo(\000)p Fu(1)2211 984 y Ft(h;)g(t)p Fw(\))515 1132 y Fv(is)38 b(a)h(smo)l(oth)f(di\013e)l (omorphism.)67 b(In)37 b(addition,)43 b(this)38 b(di\013e)l(omorphism)j (maps)e Ft(V)3152 1102 y Fo(0)3199 1132 y Fs(\002)25 b Ft(B)3356 1102 y Fo(0)515 1232 y Fv(onto)30 b(an)f(op)l(en)i(b)l (ounde)l(d)f(set,)g(henc)l(e)g(c)l(ontaine)l(d)g(in)g(a)g(c)l(ertain)g (cub)l(e)2751 1211 y Fw(~)2727 1232 y Ft(V)2793 1208 y Fo(0)2835 1232 y Fs(\002)18 b Ft(B)2985 1201 y Fo(0)3009 1232 y Fv(.)515 1431 y Fl(Pro)s(of:)58 b Fw(Let)23 b Ft(F)12 b Fw(\()p Ft(h;)i(t)p Fw(\))24 b(=)e(\()p Ft(Q)p Fw(\()p Ft(h;)14 b(t)p Fw(\))1608 1401 y Fo(\000)p Fu(1)1698 1431 y Ft(h;)g(t)p Fw(\))23 b(for)g(\()p Ft(h;)14 b(t)p Fw(\))24 b Fs(2)f Ft(V)2320 1443 y Fp(d)2370 1431 y Fs(\002)10 b Ft(B)2508 1443 y Fp(d)2546 1431 y Fw(.)36 b(A)24 b(direct)f (computation)515 1530 y(sho)n(ws)j(that)1329 1630 y(det)14 b(D)1521 1645 y Fu(\()p Fp(h;t)p Fu(\))1662 1630 y Ft(F)e Fw(\(0)p Ft(;)i(t)1868 1642 y Fu(0)1904 1630 y Fw(\))24 b(=)e(det)14 b Ft(Q)p Fw(\(0)p Ft(;)g(t)2383 1642 y Fu(0)2420 1630 y Fw(\))2452 1596 y Fo(\000)p Fu(1)2541 1630 y Ft(;)515 1778 y Fw(whic)n(h)27 b(is)g(not)g(zero.)36 b(Th)n(us,)27 b Ft(F)12 b Fw(\()p Ft(h;)i(t)p Fw(\))27 b(is)h(a)e(lo)r(cal)h (di\013eomorphism)g(in)g(a)g(neigh)n(b)r(orho)r(o)r(d)f(of)515 1878 y(\(0)p Ft(;)14 b(t)656 1890 y Fu(0)693 1878 y Fw(\).)36 b(Cho)r(osing)25 b(a)g(cub)r(e,)i Ft(V)1496 1848 y Fo(0)1534 1878 y Fs(\002)14 b Ft(B)1680 1848 y Fo(0)1727 1878 y Fs(3)23 b Fw(\(0)p Ft(;)14 b(t)1946 1890 y Fu(0)1983 1878 y Fw(\),)27 b(whose)d(closure)h(is)g(included)i(in)e(this)h(last) 515 1978 y(neigh)n(b)r(orho)r(o)r(d)g(\(and)i(in)g Ft(V)1375 1990 y Fp(d)1432 1978 y Fs(\002)18 b Ft(B)1578 1990 y Fp(d)1617 1978 y Fw(\))28 b(then)g(yields)f(the)h(result.)845 b Fi(\003)639 2177 y Fw(The)32 b(previous)e(di\013eomorphism)h(maps)g Ft(V)2024 2147 y Fo(0)2068 2177 y Fs(\002)20 b Ft(B)2220 2147 y Fo(0)2275 2177 y Fw(on)n(to)31 b(a)f(b)r(ounded)i(op)r(en)g(set) 3238 2156 y(~)3219 2177 y Fs(D)f(\022)539 2255 y Fw(~)515 2276 y Ft(V)582 2252 y Fo(0)610 2276 y Fs(\002)5 b Ft(B)747 2246 y Fo(0)770 2276 y Fw(.)35 b(W)-7 b(e)21 b(w)n(an)n(t)f(to)h(\014x) g Ft(t)i Fs(2)g Ft(B)1564 2246 y Fo(0)1588 2276 y Fw(,)f(so)e(let)1867 2255 y(~)1841 2276 y Fs(D)1905 2288 y Fp(t)1958 2276 y Fw(=)i Fs(f)p Ft(h)h Fs(2)g Fn(R)2296 2246 y Fp(d)2358 2276 y Fw(;)37 b(\()p Ft(h;)14 b(t)p Fw(\))23 b Fs(2)2718 2255 y Fw(~)2699 2276 y Fs(D)r(g)p Ft(:)d Fw(Then,)3111 2255 y(~)3085 2276 y Fs(D)3149 2288 y Fp(t)3201 2276 y Fs(\022)3313 2255 y Fw(~)3289 2276 y Ft(V)3356 2252 y Fo(0)515 2376 y Fw(is)27 b(an)h(op)r(en)f(set)h(in)g Fn(R)1201 2346 y Fp(d)1239 2376 y Fw(.)37 b(Let)28 b(us)g(consider)e (the)i(restriction)1467 2557 y Ft(V)1534 2522 y Fo(0)1581 2557 y Fs(!)1713 2536 y Fw(~)1687 2557 y Fs(D)1751 2569 y Fp(t)1780 2557 y Ft(;)41 b(h)23 b Fs(7!)g Ft(Q)p Fw(\()p Ft(h;)14 b(t)p Fw(\))2266 2522 y Fo(\000)p Fu(1)2356 2557 y Ft(h;)515 2737 y Fw(whic)n(h)29 b(is)g(also)f(a)g(smo)r(oth)h (di\013eomorphism.)41 b(It)29 b(ma)n(y)g(b)r(e)g(used)g(as)f(a)h(smo)r (oth)g(in)n(v)n(ertible)515 2837 y(c)n(hange)d(of)i(v)-5 b(ariables)26 b(b)n(y)h(setting)1678 2995 y(~)1677 3017 y Ft(h)c Fw(=)f Ft(Q)p Fw(\()p Ft(h;)14 b(t)p Fw(\))2080 2983 y Fo(\000)p Fu(1)2169 3017 y Ft(h)515 3198 y Fw(for)27 b Ft(h)c Fs(2)h Ft(V)859 3168 y Fo(0)882 3198 y Fw(.)38 b(Notice)28 b(that)g(0)f(is)h(then)g(mapp)r(ed)g(to)g(0.)37 b(Let)28 b Ft(P)2448 3210 y Fp(t)2506 3198 y Fw(b)r(e)g(the)g(in)n(v)n (erse)e(c)n(hange)h(of)515 3298 y(v)-5 b(ariables,)26 b(so)1770 3397 y Ft(h)d Fw(=)g Ft(P)1982 3409 y Fp(t)2011 3397 y Fw(\()2044 3375 y(~)2043 3397 y Ft(h)p Fw(\))515 3545 y(for)647 3523 y(~)646 3545 y Ft(h)30 b Fs(2)835 3524 y Fw(~)809 3545 y Fs(D)873 3557 y Fp(t)902 3545 y Fw(.)50 b(Then,)1221 3523 y(~)1220 3545 y Ft(h)30 b Fw(=)f Ft(P)1457 3510 y Fo(\000)p Fu(1)1445 3566 y Fp(t)1546 3545 y Fw(\()p Ft(h)p Fw(\))h(=)g Ft(Q)p Fw(\()p Ft(h;)14 b(t)p Fw(\))2028 3515 y Fo(\000)p Fu(1)2117 3545 y Ft(h)p Fw(,)33 b(whic)n(h)e(implies)h Ft(h)e Fw(=)g Ft(Q)p Fw(\()p Ft(h;)14 b(t)p Fw(\))3167 3523 y(~)3166 3545 y Ft(h)31 b Fw(and)515 3645 y(hence)1180 3826 y Ft(\036)p Fw(\()p Ft(P)1314 3838 y Fp(t)1345 3826 y Fw(\()1378 3804 y(~)1377 3826 y Ft(h)p Fw(\))p Ft(;)14 b(t)p Fw(\))83 b(=)g Ft(\036)p Fw(\()p Ft(h;)14 b(t)p Fw(\))1639 3950 y(=)83 b Ft(\036)p Fw(\()p Ft(Q)p Fw(\()p Ft(h;)14 b(t)p Fw(\))2114 3928 y(~)2113 3950 y Ft(h;)g(t)p Fw(\))1639 4075 y(=)1788 4053 y(~)1787 4075 y Ft(h)1835 4040 y Fu(t)1865 4075 y Ft(Q)p Fw(\()p Ft(h;)g(t)p Fw(\))2110 4033 y Fu(t)2140 4075 y Fw(\010\()p Ft(h;)g(t)p Fw(\))p Ft(Q)p Fw(\()p Ft(h;)g(t)p Fw(\))2625 4053 y(~)2624 4075 y Ft(h)1639 4281 y Fw(=)1831 4177 y Fp(s)1787 4202 y Fr(X)1789 4379 y Fp(j)s Fu(=1)1907 4281 y Fw(\()1940 4259 y(~)1939 4281 y Ft(h)1987 4247 y Fu(\()p Fp(j)s Fu(\))2074 4281 y Fw(\))2106 4247 y Fu(2)2162 4281 y Fs(\000)2329 4177 y Fp(d)2286 4202 y Fr(X)2245 4381 y Fp(k)q Fu(=)p Fp(s)p Fu(+1)2448 4281 y Fw(\()2481 4259 y(~)2480 4281 y Ft(h)2528 4247 y Fu(\()p Fp(k)q Fu(\))2621 4281 y Fw(\))2653 4247 y Fu(2)2690 4281 y Ft(:)515 4545 y Fw(W)-7 b(e)28 b(ha)n(v)n(e)e(th)n(us) i(pro)n(v)n(en)e(the)i Fv(Morse)j(lemma)j Fw(with)28 b(sp)r(ecial)g(care)e(of)i(the)g(parameter)e Ft(t)p Fw(:)515 4726 y Fl(Theorem)31 b(4)41 b Fv(Given)f(a)f(cub)l(e)g Ft(B)44 b Fs(\032)39 b Fn(R)1807 4696 y Fp(m)1909 4726 y Fv(c)l(ontaining)h(a)f(\014xe)l(d)f Ft(t)2635 4738 y Fu(0)2673 4726 y Fv(,)j(supp)l(ose)f Ft(\036)p Fw(\()p Ft(h;)14 b(t)p Fw(\))40 b Fv(is)515 4826 y(smo)l(oth)30 b(in)g Fw(\()p Ft(h;)14 b(t)p Fw(\))23 b Fs(2)h Fn(R)1240 4796 y Fp(d)1297 4826 y Fs(\002)18 b Ft(B)33 b Fv(and)e(satis\014es) 1012 5006 y Ft(\036)p Fw(\(0)p Ft(;)14 b(t)p Fw(\))24 b(=)e(0)p Ft(;)73 b Fs(r)1552 5018 y Fp(h)1595 5006 y Ft(\036)p Fw(\(0)p Ft(;)14 b(t)p Fw(\))24 b(=)f(0)p Ft(;)72 b Fv(and)i Fw(det)14 b(D)2463 4972 y Fu(2)2463 5027 y Fp(h)2507 5006 y Ft(\036)p Fw(\(0)p Ft(;)g(t)p Fw(\))23 b Fs(6)p Fw(=)g(0)1905 5255 y(15)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 515 523 a Fv(for)44 b Fw(all)34 b Ft(t)e Fs(2)f Ft(B)t Fv(.)53 b(Then,)37 b(ther)l(e)e(exists)f(a)h (cub)l(e)f Ft(V)2101 493 y Fo(0)2146 523 y Fs(\002)21 b Ft(B)2299 493 y Fo(0)2354 523 y Fs(\032)31 b Fn(R)2510 493 y Fp(d)2571 523 y Fs(\002)21 b Ft(B)39 b Fv(c)l(ontaining)c Fw(\(0)p Ft(;)14 b(t)3310 535 y Fu(0)3347 523 y Fw(\))515 623 y Fv(such)30 b(that)g(the)g(fol)t(lowing)j(holds:)41 b(for)30 b(al)t(l)i Ft(t)23 b Fs(2)h Ft(B)2063 593 y Fo(0)2117 623 y Fv(ther)l(e)30 b(exists)g(an)g(invertible)h(change)g (of)515 722 y(variables)1735 800 y Fw(~)1734 822 y Ft(h)23 b Fw(=)g Ft(P)1958 786 y Fo(\000)p Fu(1)1946 842 y Fp(t)2047 822 y Fw(\()p Ft(h)p Fw(\))515 971 y Fv(on)30 b Ft(V)700 941 y Fo(0)724 971 y Fv(,)g(smo)l(oth)g(and)g(with)h(smo)l(oth)f (inverse,)h(mapping)g Fw(0)e Fv(to)h Fw(0)p Fv(,)f(which)j(satis\014es) 1240 1231 y Ft(\036)p Fw(\()p Ft(P)1374 1243 y Fp(t)1405 1231 y Fw(\()1438 1209 y(~)1437 1231 y Ft(h)p Fw(\))p Ft(;)14 b(t)p Fw(\))23 b(=)1771 1127 y Fp(s)1727 1152 y Fr(X)1729 1329 y Fp(j)s Fu(=1)1847 1231 y Fw(\()1880 1209 y(~)1879 1231 y Ft(h)1927 1197 y Fu(\()p Fp(j)s Fu(\))2014 1231 y Fw(\))2046 1197 y Fu(2)2102 1231 y Fs(\000)2269 1127 y Fp(d)2226 1152 y Fr(X)2185 1331 y Fp(k)q Fu(=)p Fp(s)p Fu(+1)2388 1231 y Fw(\()2421 1209 y(~)2420 1231 y Ft(h)2468 1197 y Fu(\()p Fp(k)q Fu(\))2561 1231 y Fw(\))2593 1197 y Fu(2)2630 1231 y Ft(:)515 1517 y Fv(The)38 b(r)l(esulting)e(family)j(of)e(changes)h(of)g(variables,)j Fs(f)p Ft(P)2295 1529 y Fp(t)2324 1517 y Fw(\()2357 1496 y(~)2356 1517 y Ft(h)q Fw(\))p Fs(g)2479 1540 y Fp(t)p Fo(2)p Fp(B)2602 1524 y Fj(0)2628 1517 y Fv(,)e(dep)l(ends)f(di\013e)l (omor-)515 1629 y(phic)l(al)t(ly)32 b(on)e Ft(t)23 b Fs(2)g Ft(B)1135 1599 y Fo(0)1189 1629 y Fv(in)29 b(the)h(fol)t(lowing) i(sense:)39 b(setting)2328 1609 y Fw(~)2302 1629 y Fs(D)2366 1641 y Fp(t)2419 1629 y Fw(=)22 b Ft(P)2571 1594 y Fo(\000)p Fu(1)2559 1650 y Fp(t)2660 1629 y Fw(\()p Ft(V)2760 1599 y Fo(0)2783 1629 y Fw(\))p Fv(,)1644 1808 y Fw(~)1625 1829 y Fs(D)j Fw(=)1828 1750 y Fr([)1802 1928 y Fp(t)p Fo(2)p Fp(B)1925 1911 y Fj(0)1987 1808 y Fw(~)1961 1829 y Fs(D)2025 1841 y Fp(t)2073 1829 y Fs(\002)18 b(f)p Ft(t)p Fs(g)515 2108 y Fv(is)30 b(an)g(op)l(en)g(set)f(in)h Fn(R)1209 2078 y Fp(d)1266 2108 y Fs(\002)18 b Fn(R)1409 2078 y Fp(m)1502 2108 y Fv(\(c)l(ontaine)l(d)30 b(in)g(a)g(cub)l(e)2278 2087 y Fw(~)2266 2108 y Ft(V)2333 2078 y Fo(0)2374 2108 y Fs(\002)18 b Ft(B)2524 2078 y Fo(0)2548 2108 y Fv(\))30 b(on)f(which)1362 2270 y Fw(~)1343 2291 y Fs(D)c(!)e Ft(V)1605 2257 y Fo(0)1647 2291 y Fs(\002)18 b Ft(B)1797 2257 y Fo(0)1820 2291 y Ft(;)73 b Fw(\()1949 2269 y(~)1948 2291 y Ft(h)q(;)14 b(t)p Fw(\))23 b Fs(7!)g Fw(\()p Ft(P)2310 2303 y Fp(t)2340 2291 y Fw(\()2373 2269 y(~)2372 2291 y Ft(h)p Fw(\))p Ft(;)14 b(t)p Fw(\))515 2474 y Fv(is)30 b(a)g(di\013e)l(omorphism.)515 2690 y Fl(Con)m(tin)m(uation)83 b Fw(The)31 b(corollary)e(of)j(Theorem)f(3)g(joined)g(with)h(the)g (Morse)f(lemma)g(\014-)515 2789 y(nally)c(yield:)515 2972 y Fl(Theorem)k(5)41 b Fv(Supp)l(ose)31 b Ft(')p Fw(\()p Ft(x;)14 b(t)p Fw(\))30 b Fv(is)h(smo)l(oth)f(in)g Fw(\()p Ft(x;)14 b(t)p Fw(\))24 b Fs(2)f Fn(R)2402 2942 y Fp(d)2459 2972 y Fs(\002)18 b Fn(R)2602 2942 y Fp(m)2695 2972 y Fv(and)30 b(satis\014es)1171 3154 y Fs(r)1240 3166 y Fp(x)1282 3154 y Ft(')p Fw(\()p Ft(x)1415 3166 y Fu(0)1453 3154 y Ft(;)14 b(t)1520 3166 y Fu(0)1557 3154 y Fw(\))23 b(=)g(0)59 b Fv(and)74 b Fw(det)14 b(D)2198 3120 y Fu(2)2198 3175 y Fp(x)2240 3154 y Ft(')p Fw(\()p Ft(x)2373 3166 y Fu(0)2411 3154 y Ft(;)g(t)2478 3166 y Fu(0)2516 3154 y Fw(\))23 b Fs(6)p Fw(=)g(0)p Ft(:)515 3337 y Fv(Then,)42 b(ther)l(e)c(exists)h(an)f(arbitr)l(arily)j(smal)t (l)e(cub)l(e,)i Ft(U)34 b Fs(\002)24 b Ft(B)t Fv(,)42 b(c)l(ontaining)d Fw(\()p Ft(x)3007 3349 y Fu(0)3045 3337 y Ft(;)14 b(t)3112 3349 y Fu(0)3149 3337 y Fw(\))39 b Fv(such)515 3437 y(that)29 b(the)g(fol)t(lowing)j(holds:)39 b(if)30 b Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))30 b Fv(is)f(smo)l(oth)h(in)f(the)g(neighb)l(orho)l(o)l(d)j(of)e Fn(R)3014 3407 y Fp(d)3070 3437 y Fs(\002)p 3151 3370 68 4 v 16 w Ft(B)j Fv(and)515 3536 y(vanishes)e(on)f Ft(U)1031 3506 y Fu(c)1082 3536 y Fs(\002)p 1165 3470 V 18 w Ft(B)5 b Fv(,)30 b(then)1350 3781 y Ft(I)1386 3793 y Fp(f)1429 3781 y Fw(\()p Ft(r)n(;)14 b(t)p Fw(\))24 b Fs(\030)f Fw(e)1744 3747 y Fu(i)p Fp(r)r(\022)r Fu(\()p Fp(t)p Fu(\))1909 3781 y Ft(r)1948 3747 y Fo(\000)2011 3724 y FC(d)p 2011 3733 31 3 v 2012 3767 a Fk(2)2097 3677 y Fo(1)2070 3702 y Fr(X)2072 3879 y Fp(j)s Fu(=0)2203 3781 y Ft(a)2247 3793 y Fp(j)2282 3781 y Fw(\()p Ft(t)p Fw(\))p Ft(r)2415 3747 y Fo(\000)2479 3720 y FC(j)p 2479 3734 29 3 v 2479 3767 a Fk(2)2521 3781 y Ft(;)515 4052 y Fv(wher)l(e)29 b Ft(\022)r Fw(\()p Ft(t)p Fw(\))g Fv(is)g(r)l(e)l(al) g(value)l(d,)h(the)f Ft(a)1615 4064 y Fp(j)1649 4052 y Fw(\()p Ft(t)p Fw(\))p Fv('s)h(ar)l(e)f(c)l(omplex)g(value)l(d,)h (and)f(al)t(l)g(these)g(functions)515 4152 y(ar)l(e)h(smo)l(oth)g(in)g Ft(t)23 b Fs(2)p 1171 4085 68 4 v 23 w Ft(B)t Fv(.)39 b(Mor)l(e)l(over,)32 b(these)e(estimates)g(ar)l(e)g(uniform)g(in)g Ft(t)23 b Fs(2)p 2959 4085 V 23 w Ft(B)t Fv(.)515 4334 y Fl(Pro)s(of:)64 b Fw(By)27 b(the)h(Implicit)g(F)-7 b(unction)28 b(Theorem)f(there)h(exists)f(a)g(smo)r(oth)g(function)1471 4517 y Ft(B)1534 4529 y Fu(0)1594 4517 y Fs(\032)c Fn(R)1742 4483 y Fp(m)1828 4517 y Fs(!)g Fn(R)1994 4483 y Fp(d)2033 4517 y Ft(;)69 b(t)23 b Fs(7!)g Fw(x\()p Ft(t)p Fw(\))515 4700 y(de\014ned)38 b(on)g(a)f(cub)r(e)i Ft(B)1286 4712 y Fu(0)1364 4700 y Fs(3)h Ft(t)1489 4712 y Fu(0)1565 4700 y Fw(suc)n(h)d(that)i(\()p Fs(r)2054 4712 y Fp(x)2096 4700 y Ft(')p Fw(\)\(x\()p Ft(t)p Fw(\))p Ft(;)14 b(t)p Fw(\))42 b(=)e(0)d(for)h(all)g Ft(t)i Fs(2)h Ft(B)3170 4712 y Fu(0)3245 4700 y Fw(and)515 4799 y(x\()p Ft(t)621 4811 y Fu(0)658 4799 y Fw(\))23 b(=)g Ft(x)848 4811 y Fu(0)886 4799 y Fw(.)37 b(Without)28 b(loss)f(of)g(generalit)n(y)f(w)n (e)i(also)e(supp)r(ose)1592 4982 y(det)14 b(D)1784 4948 y Fu(2)1784 5002 y Fp(x)1826 4982 y Ft(')p Fw(\(x\()p Ft(t)p Fw(\))p Ft(;)g(t)p Fw(\))24 b Fs(6)p Fw(=)f(0)1905 5255 y(16)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 515 523 a Fw(for)27 b(all)g Ft(t)c Fs(2)h Ft(B)952 535 y Fu(0)989 523 y Fw(.)37 b(Let)1316 695 y Ft(\036)p Fw(\()p Ft(h;)14 b(t)p Fw(\))24 b(=)f Ft(')p Fw(\(x\()p Ft(t)p Fw(\))c(+)f Ft(h;)c(t)p Fw(\))k Fs(\000)g Ft(')p Fw(\(x\()p Ft(t)p Fw(\))p Ft(;)c(t)p Fw(\))p Ft(;)515 867 y Fw(whic)n(h)32 b(is)g(smo)r(oth)g(in)g(\()p Ft(h;)14 b(t)p Fw(\))31 b Fs(2)g Fn(R)1600 836 y Fp(d)1660 867 y Fs(\002)21 b Ft(B)1809 879 y Fu(0)1847 867 y Fw(.)50 b(Then,)34 b Ft(\036)p Fw(\()p Ft(h;)14 b(t)p Fw(\))33 b(satis\014es)e(the)i(h)n(yp)r(otheses)e(of)515 966 y(Morse's)19 b(lemma.)35 b(Hence,)22 b(there)f(exists)g(a)f(cub)r(e,)j Ft(V)2145 936 y Fo(0)2173 966 y Fs(\002)5 b Ft(B)2310 936 y Fo(0)2333 966 y Fw(,)23 b(con)n(taining)d(\(0)p Ft(;)14 b(t)2917 978 y Fu(0)2954 966 y Fw(\))21 b(and)g(whose)515 1066 y(closure)26 b(is)i(in)g Fn(R)1033 1036 y Fp(d)1090 1066 y Fs(\002)18 b Ft(B)1236 1078 y Fu(0)1273 1066 y Fw(,)28 b(and)f(a)g(family)h(of)f(di\013eomorphisms)1260 1238 y Ft(P)1325 1202 y Fo(\000)p Fu(1)1313 1258 y Fp(t)1423 1238 y Fw(:)37 b Ft(V)1550 1203 y Fo(0)1596 1238 y Fs(!)1728 1217 y Fw(~)1702 1238 y Fs(D)1766 1250 y Fp(t)1819 1238 y Fs(\022)1919 1217 y Fw(~)1906 1238 y Ft(V)1973 1203 y Fo(0)2107 1238 y Fw(\(where)28 b Ft(t)23 b Fs(2)g Ft(B)2578 1207 y Fo(0)2601 1238 y Fw(\))515 1409 y(suc)n(h)k(that,)h(letting)1169 1387 y(~)1168 1409 y Ft(h)23 b Fw(=)f Ft(P)1391 1374 y Fo(\000)p Fu(1)1379 1430 y Fp(t)1480 1409 y Fw(\()p Ft(h)p Fw(\),)28 b(one)g(obtains)1240 1663 y Ft(\036)p Fw(\()p Ft(P)1374 1675 y Fp(t)1405 1663 y Fw(\()1438 1641 y(~)1437 1663 y Ft(h)p Fw(\))p Ft(;)14 b(t)p Fw(\))23 b(=)1771 1559 y Fp(s)1727 1584 y Fr(X)1729 1761 y Fp(j)s Fu(=1)1847 1663 y Fw(\()1880 1641 y(~)1879 1663 y Ft(h)1927 1629 y Fu(\()p Fp(j)s Fu(\))2014 1663 y Fw(\))2046 1629 y Fu(2)2102 1663 y Fs(\000)2269 1559 y Fp(d)2226 1584 y Fr(X)2185 1763 y Fp(k)q Fu(=)p Fp(s)p Fu(+1)2388 1663 y Fw(\()2421 1641 y(~)2420 1663 y Ft(h)2468 1629 y Fu(\()p Fp(k)q Fu(\))2561 1663 y Fw(\))2593 1629 y Fu(2)2630 1663 y Ft(:)515 1923 y Fw(Notice)j(that)g(0)c(=)h Ft(x)1151 1935 y Fu(0)1204 1923 y Fs(\000)14 b Fw(x\()p Ft(t)1389 1935 y Fu(0)1427 1923 y Fw(\))23 b Fs(2)h Ft(V)1627 1893 y Fo(0)1651 1923 y Fw(.)36 b(Consequen)n(tly)-7 b(,)26 b(there)f(exists)h(an)f(arbitrarily)f(small)515 2023 y(cub)r(e,)k Ft(U)f Fs(\002)18 b Ft(B)27 b Fs(3)d Fw(\()p Ft(x)1149 2035 y Fu(0)1187 2023 y Ft(;)14 b(t)1254 2035 y Fu(0)1291 2023 y Fw(\),)28 b(suc)n(h)f(that)1546 2195 y(\()p Ft(x)19 b Fs(\000)f Fw(x\()p Ft(t)p Fw(\))p Ft(;)c(t)p Fw(\))24 b Fs(2)f Ft(V)2133 2161 y Fo(0)2175 2195 y Fs(\002)18 b Ft(B)2325 2161 y Fo(0)515 2367 y Fw(for)31 b(all)g(\()p Ft(x;)14 b(t)p Fw(\))31 b Fs(2)p 1059 2300 66 4 v 30 w Ft(U)f Fs(\002)p 1231 2300 68 4 v 20 w Ft(B)5 b Fw(.)49 b(In)31 b(other)g(w)n(ords,)h Ft(U)d Fs(\002)21 b Ft(B)36 b Fw(is)31 b(mapp)r(ed)h(on)n(to)f(a)g(region)g(whose)515 2466 y(closure)26 b(is)i(in)g Ft(V)1039 2436 y Fo(0)1081 2466 y Fs(\002)18 b Ft(B)1231 2436 y Fo(0)1282 2466 y Fw(via)27 b(the)h(c)n(hange)f(of)g(v)-5 b(ariables)27 b(\()p Ft(h;)14 b(t)p Fw(\))23 b(=)g(\()p Ft(x)c Fs(\000)f Fw(x\()p Ft(t)p Fw(\))p Ft(;)c(t)p Fw(\).)639 2566 y(Let)42 b(us)g(consider)e(an)h(amplitude,)46 b Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\),)46 b(satisfying)41 b(the)h(asserted)e(prop)r (erties.)515 2665 y(Since)28 b(for)f(a)g(\014xed)g Ft(t)c Fs(2)p 1260 2599 V 24 w Ft(B)32 b Fw(the)c(in)n(tegrand)e(in)1392 2875 y Ft(I)1428 2887 y Fp(f)1472 2875 y Fw(\()p Ft(r)n(;)14 b(t)p Fw(\))24 b(=)1749 2762 y Fr(Z)1795 2951 y Fq(R)1837 2934 y FC(d)1890 2875 y Fw(e)1927 2841 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(x;t)p Fu(\))2161 2875 y Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))19 b(d)p Ft(x)515 3089 y Fw(is)27 b(supp)r(orted)h(in)g Ft(U)9 b Fw(,)27 b(it)h(follo)n(ws)f(that)g(the)h(righ)n(t-hand)f(side) g(in)515 3298 y Ft(I)551 3310 y Fp(f)594 3298 y Fw(\()p Ft(r)n(;)14 b(t)p Fw(\))24 b(=)872 3185 y Fr(Z)918 3374 y Fq(R)960 3357 y FC(d)1013 3298 y Fw(e)1050 3264 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(h)p Fu(+x\()p Fp(t)p Fu(\))p Fp(;t)p Fu(\))1447 3298 y Ft(f)9 b Fw(\()p Ft(h)p Fw(+x\()p Ft(t)p Fw(\))p Ft(;)14 b(t)p Fw(\)d)p Ft(h)24 b Fw(=)f(e)2122 3264 y Fu(i)p Fp(r)r(')p Fu(\(x\()p Fp(t)p Fu(\))p Fp(;t)p Fu(\))2443 3185 y Fr(Z)2489 3374 y Fq(R)2531 3357 y FC(d)2584 3298 y Fw(e)2621 3264 y Fu(i)p Fp(r)r(\036)p Fu(\()p Fp(h;t)p Fu(\))2852 3298 y Ft(f)9 b Fw(\()p Ft(h)p Fw(+x\()p Ft(t)p Fw(\))p Ft(;)14 b(t)p Fw(\)d)p Ft(h)515 3532 y Fw(is)21 b(supp)r(orted)h(in)g Ft(V)1135 3502 y Fo(0)1159 3532 y Fw(.)35 b(Then,)23 b(b)n(y)e(Morse's)g(lemma)h(the)g(c)n(hange)f (of)h(v)-5 b(ariables)2955 3510 y(~)2954 3532 y Ft(h)23 b Fw(=)g Ft(P)3178 3496 y Fo(\000)p Fu(1)3166 3552 y Fp(t)3267 3532 y Fw(\()p Ft(h)p Fw(\))515 3631 y(is)k(a)n(v)-5 b(ailable,)27 b(yielding)930 3841 y Ft(I)966 3853 y Fp(f)1010 3841 y Fw(\()p Ft(r)n(;)14 b(t)p Fw(\))23 b(=)g(e)1324 3807 y Fu(i)p Fp(r)r(')p Fu(\(x\()p Fp(t)p Fu(\))p Fp(;t)p Fu(\))1646 3728 y Fr(Z)1715 3902 y Fu(~)1692 3917 y Fo(D)1744 3925 y FC(t)1789 3841 y Fw(e)1826 3807 y Fu(i)p Fp(r)r(Q)p Fu(\()1957 3791 y(~)1956 3807 y Fp(h)o Fu(\))2024 3841 y Ft(f)9 b Fw(\()p 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Ft(B)k Fw(and)c(v)-5 b(anishes)26 b(for)516 4685 y(~)515 4707 y Ft(h)32 b(=)-51 b Fs(2)677 4686 y Fw(~)664 4707 y Ft(V)731 4677 y Fo(0)754 4707 y Fw(,)28 b(the)g(corollary)d(of)j(Theorem)e(3)i (then)g(completes)f(the)h(pro)r(of.)579 b Fi(\003)639 4907 y Fw(Finally)-7 b(,)27 b(the)f(follo)n(wing)g(result)f(is)i(an)e (in)n(teresting)h(application)g(of)g(our)f(treatmen)n(t)h(of)515 5006 y(a)h(parameter:)1905 5255 y(17)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 515 523 a Fl(Theorem)31 b(6)41 b Fv(Supp)l(ose)31 b Ft(')p Fw(\()p Ft(x;)14 b(t)p Fw(\))30 b Fv(is)h(smo)l(oth)f(in)g Fw(\()p Ft(x;)14 b(t)p Fw(\))24 b Fs(2)f Fn(R)2402 493 y Fp(d)2459 523 y Fs(\002)18 b Fn(R)2602 493 y Fp(m)2695 523 y Fv(and)30 b(satis\014es)1150 704 y Fs(r)1219 716 y Fp(x)1261 704 y Ft(')p Fw(\()p Ft(x)1394 716 y Fu(0)1432 704 y Ft(;)14 b(t)1499 716 y Fu(0)1536 704 y Fw(\))24 b(=)e(0)59 b Fv(and)h Fw(rank)13 b(D)2212 669 y Fu(2)2212 724 y Fp(x)2254 704 y Ft(')p Fw(\()p Ft(x)2387 716 y Fu(0)2426 704 y 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FB(5)642 5006 y Fm(A)23 b(smo)r(oth)h(path,)h Ff(\015)t Fm(\()p Ff(t)p Fm(\),)g(is)d Fh(r)l(e)l(gular)32 b Fm(if)23 b Ff(\015)1733 4983 y Fz(0)1756 5006 y Fm(\()p Ff(t)p Fm(\))e FJ(6)p Fm(=)e(0)24 b(for)f(an)n(y)h Ff(t)p Fm(.)1905 5255 y Fw(18)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 515 623 a Fl(Lemma)63 b Fv(Supp)l(ose)32 b Ft(f)9 b Fw(\()p Ft(h)p Fw(\))32 b Fv(is)g(smo)l(oth)g(in)g Ft(h)26 b Fs(2)h Fn(R)p Fv(,)32 b(c)l(omp)l(actly)h(supp)l(orte)l(d,)g (and)g(analytic)515 722 y(at)c Fw(0)p Fv(.)39 b(Then,)1201 865 y Fw(p)p Ft(:)p Fw(v)q Ft(:)1352 752 y Fr(Z)1435 773 y Fo(1)1398 941 y(\0001)1534 865 y Fw(e)1571 831 y Fu(i)p Fp(r)r(h)1675 809 y Ft(f)9 b Fw(\()p Ft(h)p Fw(\))p 1675 846 162 4 v 1732 922 a Ft(h)1866 865 y Fw(d)p Ft(h)23 b Fw(=)f Ft(\031)s Fw(i)p Ft(f)9 b Fw(\(0\))19 b(+)f Ft(O)r Fw(\()p Ft(r)2537 831 y Fo(\0001)2661 865 y Fw(\))515 1063 y Fv(when)30 b Ft(r)c Fs(!)d(1)p Fv(.)515 1263 y Fl(Pro)s(of:)89 b Fw(F)-7 b(or)35 b Ft(")i(>)f Fw(0,)i(let)e Ft(C)1487 1275 y Fp(")1559 1263 y Fw(b)r(e)g(a)g(smo)r (oth)f(regular)f(path)j(starting)e(at)g Fs(\000)p Fw(2)p Ft(")p Fw(,)i(going)515 1362 y(through)27 b([)p Fs(\000)p Fw(2)p Ft(")n(;)14 b Fs(\000)p Ft(")p Fw(],)27 b(then)i(a)n(v)n(oiding) c(the)j(origin,)f(but)h(sta)n(ying)e(in)1220 1545 y Fs(f)p Ft(z)g Fs(2)e Fn(C)f Fw(;)37 b Fs(j)p Fw(Re)14 b(\()p Ft(z)t Fw(\))p Fs(j)22 b Ft(<)h(")k Fw(and)h(0)22 b Ft(<)h Fw(Im)14 b Ft(z)26 b(<)d(")p Fs(g)p Ft(;)515 1728 y Fw(and)k(\014nally) h(going)e(through)h([)p Ft(";)14 b Fw(2)p Ft(")o Fw(].)37 b(Since)1443 1958 y(e)1480 1924 y Fu(i)p Fp(r)r(z)1580 1902 y Ft(f)9 b Fw(\()p Ft(z)t Fw(\))p 1580 1939 157 4 v 1637 2015 a Ft(z)1769 1958 y Fw(=)1867 1902 y Ft(f)g Fw(\(0\))p 1867 1939 156 4 v 1923 2015 a Ft(z)2051 1958 y Fw(+)46 b Fv(analytic)515 2164 y Fw(in)28 b(a)f(punctured)h(neigh)n (b)r(orho)r(o)r(d)e(of)i(the)g(origin,)1392 2390 y(lim)1400 2444 y Fp(")p Fo(#)p Fu(0)1521 2277 y Fr(Z)1567 2466 y Fp(C)1615 2474 y FC(")1665 2390 y Fw(e)1702 2356 y Fu(i)p 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3010 a Ft(z)2815 2953 y Fw(d)p Ft(z)2086 3192 y Fw(=)2234 3079 y Fr(Z)2280 3267 y Fp(\015)2336 3192 y Fw(e)2373 3157 y Fu(i)p Fp(r)r(z)2473 3135 y Ft(f)g Fw(\()p Ft(z)t Fw(\))p 2473 3173 V 2530 3249 a Ft(z)2658 3192 y Fw(d)p Ft(z)t(;)515 3428 y Fw(where)25 b Ft(\015)j Fw(=)22 b Ft(\015)954 3440 y Fp(")985 3448 y Fk(0)1048 3428 y Fw(for)j(a)h(\014xed,)g(but)g (small)g(enough)f Ft(")2152 3440 y Fu(0)2212 3428 y Ft(>)d Fw(0.)36 b(The)26 b(result)g(follo)n(ws)e(from)i(the)515 3527 y(statemen)n(t)h(preceding)g(the)h(lemma.)1634 b Fi(\003)639 3726 y Fw(The)28 b(analyticit)n(y)e(assumption)h(ma)n(y)g (b)r(e)h(remo)n(v)n(ed)e(in)i(the)f(follo)n(wing)g(w)n(a)n(y|where)e(a) 515 3826 y(parameter)h Ft(t)d Fs(2)g Fn(R)1103 3796 y Fp(m)1194 3826 y Fw(is)k(also)g(in)n(tro)r(duced)g(for)g(later)g(purp)r (ose:)515 4009 y Fl(Theorem)k(7)41 b Fv(Given)d Ft(\016)i(>)c Fw(0)h Fv(and)g(a)h(cub)l(e)f Ft(B)k Fs(\032)36 b Fn(R)2221 3979 y Fp(m)2284 4009 y Fv(,)j(c)l(onsider)g(a)e(function,)j Ft(f)9 b Fw(\()p 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b Fi(6)h Ft(\037)p Fw(\()p Ft(x)p Fw(\))i Fi(6)e Fw(1)j(on)g Fn(R)h Fw(and)f Ft(\037)p Fw(\()p Ft(x)p Fw(\))e Fs(\021)f Fw(1)j(in)h(a)f(neigh)n(b)r(orho)r(o)r (d)g(of)g(0.)42 b(The)30 b(considered)1905 5255 y(19)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 515 523 a Fw(principal)27 b(v)-5 b(alue)27 b(then)h(decomp)r(oses)f(in)n(to)g Fv(I)32 b Fw(+)18 b Fv(II)32 b Fw(+)18 b Fv(III)13 b Fw(,)28 b(where)1185 735 y Fv(I)96 b Fw(=)1460 622 y Fr(Z)1506 810 y Fq(R)1566 735 y Fw(e)1603 700 y Fu(i)p Fp(r)r(h)1708 678 y Ft(f)9 b Fw(\()p Ft(h;)14 b(t)p Fw(\))p 1708 715 229 4 v 1798 791 a Ft(h)1946 735 y Fw(\(1)19 b Fs(\000)f Ft(\037)p Fw(\()p Ft(h)p Fw(\)\))h(d)p Ft(h;)1153 962 y Fv(II)96 b Fw(=)1460 849 y Fr(Z)1506 1037 y Fq(R)1566 962 y Fw(e)1603 928 y Fu(i)p Fp(r)r(h)1708 906 y Ft(f)9 b Fw(\()p Ft(h;)14 b(t)p Fw(\))k Fs(\000)g Ft(f)9 b Fw(\(0)p Ft(;)14 b(t)p Fw(\))p 1708 943 553 4 v 1960 1019 a Ft(h)2270 962 y(\037)p Fw(\()p Ft(h)p Fw(\))19 b(d)p Ft(h;)69 b Fw(and)1120 1189 y Fv(III)97 b Fw(=)82 b Ft(f)9 b Fw(\(0)p Ft(;)14 b(t)p Fw(\)p)p Ft(:)p Fw(v)q Ft(:)1834 1076 y Fr(Z)1880 1265 y Fq(R)1940 1189 y Fw(e)1977 1155 y Fu(i)p Fp(r)r(h)2081 1133 y Ft(\037)p Fw(\()p Ft(h)p Fw(\))p 2081 1170 165 4 v 2139 1246 a Ft(h)2274 1189 y Fw(d)p Ft(h:)515 1408 y Fw(The)30 b(lemma)f(of)h(Theorem)f(1)h(implies)g Fv(I)40 b Fw(=)26 b Ft(O)r Fw(\()p Ft(r)2062 1378 y Fo(\0001)2186 1408 y Fw(\))k(and)g Fv(II)40 b Fw(=)26 b Ft(O)r Fw(\()p Ft(r)2743 1378 y Fo(\0001)2867 1408 y Fw(\))k(uniformly)g(in)515 1507 y Ft(t)h Fs(2)p 663 1441 68 4 v 32 w Ft(B)t Fw(.)753 1477 y Fu(6)842 1507 y Fw(Since)i Ft(\037)p Fw(\()p Ft(x)p Fw(\))g(is)g(analytic)f(at)g(0)g(and)h Ft(\037)p Fw(\(0\))e(=)g(1,)j (the)f(result)f(follo)n(ws)f(from)i(the)515 1607 y(lemma.)2536 b Fi(\003)639 1806 y Fw(W)-7 b(e)31 b(no)n(w)f(turn)h(our)e(atten)n (tion)i(to)f(a)g(phase,)h Ft(')p Fw(\()p Ft(h;)14 b(t)p Fw(\),)31 b(smo)r(oth)g(in)f(\()p Ft(h;)14 b(t)p Fw(\))29 b Fs(2)f Fn(R)20 b Fs(\002)g Fn(R)3293 1776 y Fp(m)3356 1806 y Fw(,)515 1906 y(suc)n(h)38 b(that)h Ft(@)948 1918 y Fp(h)991 1906 y Ft(')p Fw(\(0)p Ft(;)14 b(t)1186 1918 y Fu(0)1224 1906 y Fw(\))42 b Fs(6)p Fw(=)f(0.)70 b(Our)38 b(study)h(reduces)f(to)h(the)g(previous)f(theorem)g(b)n(y)g(a)515 2005 y(c)n(hange)28 b(of)h(v)-5 b(ariables,)29 b(regarding)e(whic)n(h)i (the)h(follo)n(wing)e(elemen)n(tary)h(result)g(is)g(helpful:)515 2105 y Fv(Given)h(a)f(cub)l(e)g Ft(B)f Fs(\032)22 b Fn(R)1250 2075 y Fp(m)1313 2105 y Fv(,)30 b(supp)l(ose)g Ft(f)9 b Fw(\()p Ft(h;)14 b(t)p Fw(\))29 b Fv(is)h(smo)l(oth)f(in)h(a)g (neighb)l(orho)l(o)l(d)h(of)f Fs(f)p Fw(0)p Fs(g)16 b(\002)p 3286 2038 V 17 w Ft(B)5 b Fv(,)515 2205 y(wher)l(e)30 b(it)g(satis\014es)1392 2304 y Ft(f)9 b Fw(\(0)p Ft(;)14 b(t)p Fw(\))22 b(=)h(0)59 b Fv(and)h Ft(@)2061 2316 y Fp(h)2104 2304 y Ft(f)9 b Fw(\(0)p Ft(;)14 b(t)p Fw(\))23 b Fs(6)p Fw(=)f(0)p Ft(:)515 2458 y Fv(Then,)773 2418 y Fp(h@)847 2427 y FC(h)886 2418 y Fp(f)7 b Fu(\()p Fp(h;t)p Fu(\))p 773 2439 288 4 v 829 2487 a Fp(f)g Fu(\()p 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Fv(and)c(vanishes)h(on)f Fw(])p Fs(\000)p Ft(\016)s(;)14 b(\016)s Fw([)2977 2924 y Fu(c)3025 2966 y Fs(\002)p 3103 2899 V 13 w Ft(B)5 b Fv(,)28 b(then)944 3182 y Fw(p)p Ft(:)p Fw(v)q Ft(:)1095 3069 y Fr(Z)1141 3258 y Fq(R)1201 3182 y Fw(e)1238 3148 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(h;t)p Fu(\))1482 3126 y Ft(f)9 b Fw(\()p Ft(h;)14 b(t)p Fw(\))p 1482 3163 229 4 v 1573 3239 a Ft(h)1740 3182 y Fw(d)p Ft(h)23 b Fw(=)f Ft(\033)s(\031)s Fw(i)p Ft(f)9 b Fw(\(0)p Ft(;)14 b(t)p Fw(\)e)2327 3148 y Fu(i)p Fp(r)r(')p Fu(\(0)p Fp(;t)p Fu(\))2575 3182 y Fw(+)k Ft(O)r Fw(\()p Ft(r)2794 3148 y Fo(\0001)2918 3182 y Fw(\))515 3408 y Fv(uniformly)30 b(in)g Ft(t)23 b Fs(2)p 1127 3341 68 4 v 24 w Ft(B)t Fv(,)30 b(wher)l(e)g Ft(\033)d Fw(=)c(sgn)o(\()p Ft(@)1841 3420 y Fp(h)1884 3408 y Ft(')p Fw(\(0)p Ft(;)14 b(t)2079 3420 y Fu(0)2117 3408 y Fw(\)\))p Ft(:)515 3607 y Fl(Pro)s(of:)97 b Fw(By)39 b(con)n(tin)n(uit)n(y)f(there)h(exists)f(a)h(cub)r(e,)j(])p Fs(\000)p Ft(\016)s(;)14 b(\016)s Fw([)26 b Fs(\002)f 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1552 a Ft( )2246 1564 y Fp(t)2275 1552 y Fw(\()2308 1531 y(~)2307 1552 y Ft(h)q Fw(\))2422 1484 y Ft(:)515 1707 y Fw(Notice)44 b(that)g Ft(g)s Fw(\(0)p Ft(;)14 b(t)p Fw(\))49 b(=)h Ft(f)9 b Fw(\(0)p Ft(;)14 b(t)p Fw(\).)85 b(Moreo)n(v)n(er,)45 b Ft(g)s Fw(\()2183 1685 y(~)2182 1707 y Ft(h;)14 b(t)p Fw(\))44 b(v)-5 b(anishes)44 b(when)g Ft(t)50 b Fs(2)p 3134 1640 68 4 v 50 w Ft(B)e Fw(and)516 1785 y(~)515 1806 y Ft(h)35 b(=)-51 b Fs(2)p Fw(])p Fs(\000)736 1785 y Fw(~)732 1806 y Ft(\016)s(;)813 1785 y Fw(~)809 1806 y Ft(\016)s Fw([)30 b(for)f(a)g(certain)1385 1785 y(~)1381 1806 y Ft(\016)h(>)c Fw(0.)42 b(The)30 b(previous)f(theorem)g(th)n(us)g(applies,)h(whic)n(h)g(com-)515 1906 y(pletes)d(the)h(pro)r(of.)2208 b Fi(\003)639 2105 y Fw(Let)30 b(us)g(no)n(w)f(consider)g(a)g(phase,)h Ft(')p Fw(\()p Ft(x;)14 b(h;)g(t)p Fw(\),)31 b(smo)r(oth)e(in)h(\()p Ft(x;)14 b(h;)g(t)p Fw(\))28 b Fs(2)f Fn(R)2926 2075 y Fp(d)2984 2105 y Fs(\002)20 b Fn(R)f Fs(\002)h Fn(R)3293 2075 y Fp(m)3356 2105 y Fw(,)515 2205 y(and)1267 2326 y Ft(I)1303 2338 y Fp(f)1346 2326 y Fw(\()p Ft(r)n(;)14 b(h;)g(t)p Fw(\))24 b(=)1709 2213 y Fr(Z)1755 2402 y Fq(R)1797 2385 y FC(d)1849 2326 y Fw(e)1886 2292 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(x;h;t)p Fu(\))2178 2326 y Ft(f)9 b Fw(\()p Ft(x;)14 b(h;)g(t)p Fw(\))19 b(d)p Ft(x:)515 2558 y Fl(Theorem)31 b(8)41 b Fv(Supp)l(ose)31 b Fs(r)1395 2570 y Fp(x)1437 2558 y Ft(')p Fw(\()p Ft(x)1570 2570 y Fu(0)1608 2558 y Ft(;)14 b Fw(0)p Ft(;)g(t)1754 2570 y Fu(0)1790 2558 y Fw(\))24 b(=)e(0)p Fv(,)30 b Fw(det)14 b(D)2222 2528 y Fu(2)2222 2578 y Fp(x)2264 2558 y Ft(')p Fw(\()p Ft(x)2397 2570 y Fu(0)2436 2558 y Ft(;)g Fw(0)p Ft(;)g(t)2582 2570 y Fu(0)2618 2558 y Fw(\))24 b Fs(6)p Fw(=)e(0)p Fv(,)30 b(and)1623 2737 y Ft(@)1667 2749 y Fp(h)1710 2737 y Ft(')p Fw(\()p Ft(x)1843 2749 y Fu(0)1881 2737 y Ft(;)14 b Fw(0)p Ft(;)g(t)2027 2749 y Fu(0)2064 2737 y Fw(\))23 b Fs(6)p Fw(=)g(0)p Ft(:)515 2917 y Fv(Then,)42 b(ther)l(e)e(exist)f(a)g Ft(\016)k(>)d Fw(0)f 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3315 31 3 v 2285 3348 a Fk(2)2369 3258 y Fo(1)2342 3283 y Fr(X)2345 3460 y Fp(j)s Fu(=0)2476 3362 y Ft(a)2520 3374 y Fp(j)2555 3362 y Fw(\()p Ft(t)p Fw(\))p Ft(r)2688 3328 y Fo(\000)2752 3301 y FC(j)p 2751 3315 29 3 v 2751 3348 a Fk(2)515 3641 y Fv(uniformly)30 b(in)g Ft(t)23 b Fs(2)p 1127 3575 68 4 v 24 w Ft(B)t Fv(,)30 b(wher)l(e)g Ft(\022)r Fw(\()p Ft(t)p Fw(\))h Fv(and)f Ft(a)1854 3653 y Fp(j)1889 3641 y Fw(\()p Ft(t)p Fw(\))g Fv(ar)l(e)g(smo)l(oth)h(in)f (the)f(neighb)l(orho)l(o)l(d)k(of)p 3267 3575 V 30 w Ft(B)t Fv(.)515 3821 y Fl(Pro)s(of:)53 b Fw(In)n(terpreting)19 b(\()p Ft(x;)14 b(t)p Fw(\))24 b Fs(2)g Fn(R)1622 3791 y Fp(d)1664 3821 y Fs(\002)t Fn(R)1793 3791 y Fp(m)1876 3821 y Fw(as)c(a)g(parameter,)g(the)h(previous)e(lemma)h(holds,)515 3921 y(since)25 b Ft(@)760 3933 y Fp(h)803 3921 y Ft(')p Fw(\()p Ft(x)936 3933 y Fu(0)974 3921 y Ft(;)14 b Fw(0)p Ft(;)g(t)1120 3933 y Fu(0)1157 3921 y Fw(\))23 b Fs(6)p Fw(=)g(0.)35 b(Moreo)n(v)n(er,)23 b(in)n(terpreting)i(\()p Ft(h;)14 b(t)p Fw(\))23 b Fs(2)h Fn(R)14 b Fs(\002)g Fn(R)2733 3890 y Fp(m)2820 3921 y Fw(as)25 b(a)f(parameter,)515 4020 y(the)41 b(theorem)f(5)g(also)g(holds,)j(since)d Fs(r)1810 4032 y Fp(x)1852 4020 y Ft(')p Fw(\()p Ft(x)1985 4032 y Fu(0)2024 4020 y Ft(;)14 b Fw(0)p Ft(;)g(t)2170 4032 y Fu(0)2206 4020 y Fw(\))45 b(=)g(0)40 b(and)g(det)14 b(D)2841 3990 y Fu(2)2841 4041 y Fp(x)2884 4020 y Ft(')p Fw(\()p Ft(x)3017 4032 y Fu(0)3055 4020 y Ft(;)g Fw(0)p Ft(;)g(t)3201 4032 y Fu(0)3237 4020 y Fw(\))45 b Fs(6)p Fw(=)515 4120 y(0.)63 b(Consequen)n(tly)-7 b(,)38 b(there)f(exist)f(a)h Ft(\016)k(>)c Fw(0)g(and)f(an)g(arbitrarily)f(small)h(cub)r(e,)k Ft(U)33 b Fs(\002)24 b Ft(B)t Fw(,)515 4219 y(con)n(taining)i(\()p Ft(x)997 4231 y Fu(0)1035 4219 y Ft(;)14 b(t)1102 4231 y Fu(0)1139 4219 y Fw(\))28 b(suc)n(h)g(that)g(for)f Ft(f)9 b Fw(\()p Ft(x;)14 b(h;)g(t)p Fw(\))27 b(of)h(the)g(stipulated)g (form)802 4447 y(p)p Ft(:)p Fw(v)q Ft(:)953 4334 y Fr(Z)999 4523 y Fq(R)1059 4447 y Fw(e)1096 4413 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(x;h;t)p Fu(\))1398 4391 y Ft(f)9 b Fw(\()p Ft(x;)14 b(h;)g(t)p Fw(\))p 1398 4428 314 4 v 1531 4504 a Ft(h)1740 4447 y Fw(d)p Ft(h)23 b Fw(=)f Ft(\033)s(\031)s Fw(i)p Ft(f)9 b Fw(\()p Ft(x;)14 b Fw(0)p Ft(;)g(t)p Fw(\)e)2411 4413 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(x;)p Fu(0)p Fp(;t)p Fu(\))2717 4447 y Fw(+)k Ft(O)r Fw(\()p Ft(r)2936 4413 y Fo(\0001)3060 4447 y Fw(\))181 b(\(4\))515 4684 y(uniformly)27 b(in)h(\()p Ft(x;)14 b(t)p Fw(\))24 b Fs(2)p 1270 4617 66 4 v 23 w Ft(U)k Fs(\002)p 1438 4617 68 4 v 18 w Ft(B)j Fw(and)1043 4817 y Fr(Z)1089 5006 y Fq(R)1131 4989 y FC(d)1184 4930 y Fw(e)1221 4896 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(x;)p Fu(0)p Fp(;t)p Fu(\))1507 4930 y Ft(f)9 b Fw(\()p Ft(x;)14 b Fw(0)p Ft(;)g(t)p Fw(\))k(d)p Ft(x)24 b Fs(\030)f Fw(e)2074 4896 y Fu(i)p Fp(r)r(\022)r Fu(\()p Fp(t)p Fu(\))2239 4930 y Ft(r)2278 4896 y Fo(\000)2341 4874 y FC(d)p 2341 4883 31 3 v 2342 4916 a Fk(2)2427 4826 y Fo(1)2400 4851 y Fr(X)2402 5028 y Fp(j)s Fu(=0)2534 4930 y Ft(a)2578 4942 y Fp(j)2612 4930 y Fw(\()p Ft(t)p Fw(\))p Ft(r)2745 4896 y Fo(\000)2809 4869 y FC(j)p 2809 4883 29 3 v 2809 4916 a Fk(2)3273 4930 y Fw(\(5\))1905 5255 y(21)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 515 523 a Fw(uniformly)27 b(and)h(smo)r(othly)f (in)h Ft(t)23 b Fs(2)p 1643 456 68 4 v 23 w Ft(B)t Fw(,)28 b(b)r(oth)g(when)g Ft(r)e Fs(!)d(1)p Fw(.)639 623 y(By)28 b(F)-7 b(ubini's)28 b(and)f(the)h(dominated)g(con)n(v)n(ergence)d (theorems)854 848 y(p)p Ft(:)p Fw(v)q Ft(:)1005 735 y Fr(Z)1051 924 y Fq(R)1124 792 y Fw(1)p 1121 829 48 4 v 1121 905 a Ft(h)1178 848 y(I)1214 860 y Fp(f)1258 848 y Fw(\()p Ft(r)n(;)14 b(h;)g(t)p Fw(\))19 b(d)p Ft(h)k Fw(=)1732 735 y Fr(Z)1779 924 y Fq(R)1821 907 y FC(d)1873 848 y Fw(p)p Ft(:)p Fw(v)q Ft(:)2024 735 y Fr(Z)2071 924 y Fq(R)2130 848 y Fw(e)2167 814 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(x;h;t)p Fu(\))2470 792 y Ft(f)9 b Fw(\()p Ft(x;)14 b(h;)g(t)p Fw(\))p 2470 829 314 4 v 2602 905 a Ft(h)2811 848 y Fw(d)p Ft(h)19 b Fw(d)p Ft(x;)515 1077 y Fw(where)24 b(the)i(dominator)e(is)g(giv)n(en)h(b)n(y)f(the)i(lemma)f (of)g(Theorem)f(2.)35 b(By)25 b(the)g(equation)g(\(4\),)515 1177 y(the)j(ab)r(o)n(v)n(e)e(is)h(equal)h(to)1247 1284 y Fr(Z)1293 1473 y Fq(R)1335 1456 y FC(d)1388 1397 y Ft(\033)s(\031)s Fw(i)p Ft(f)9 b Fw(\()p Ft(x;)14 b Fw(0)p Ft(;)g(t)p Fw(\)e)1855 1363 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(x;)p Fu(0)p Fp(;t)p Fu(\))2160 1397 y Fw(d)p Ft(x)19 b Fw(+)f Ft(O)r Fw(\()p Ft(r)2491 1363 y Fo(\0001)2615 1397 y Fw(\))515 1637 y(uniformly)27 b(in)h Ft(t)23 b Fs(2)p 1121 1571 68 4 v 23 w Ft(B)5 b Fw(.)37 b(The)27 b(equation)g(\(5\))h(then)g(yields)f(the)h(result.)630 b Fi(\003)639 1837 y Fw(In)28 b(the)g(same)f(w)n(a)n(y)f(as)h(w)n(e)h (deriv)n(ed)e(from)i(Theorem)e(2)i(its)f(corollary)-7 b(,)515 2036 y Fl(Corollary)60 b Fv(Supp)l(ose)31 b Fs(r)1354 2048 y Fp(x)1395 2036 y Ft(')p Fw(\()p Ft(x)1528 2048 y Fu(0)1567 2036 y Ft(;)14 b(e)1643 2048 y Fu(0)1680 2036 y Ft(;)g(t)1747 2048 y Fu(0)1784 2036 y Fw(\))23 b(=)g(0)p Fv(,)29 b Fw(det)15 b(D)2216 2006 y Fu(2)2216 2056 y Fp(x)2258 2036 y Ft(')p Fw(\()p Ft(x)2391 2048 y Fu(0)2429 2036 y Ft(;)f(e)2505 2048 y Fu(0)2542 2036 y Ft(;)g(t)2609 2048 y Fu(0)2646 2036 y Fw(\))23 b Fs(6)p Fw(=)g(0)p Fv(,)30 b(and)1609 2218 y Ft(@)1653 2230 y Fp(e)1689 2218 y Ft(')p Fw(\()p Ft(x)1822 2230 y Fu(0)1860 2218 y Ft(;)14 b(e)1936 2230 y Fu(0)1973 2218 y Ft(;)g(t)2040 2230 y Fu(0)2077 2218 y Fw(\))23 b Fs(6)p Fw(=)g(0)p Ft(:)515 2401 y Fv(Then,)k(ther)l(e)e(exist)g(a)h Ft(\016)g(>)c Fw(0)j Fv(and)h(an)f(arbitr)l(arily)i(smal)t(l)f(cub)l(e)f Ft(U)18 b Fs(\002)9 b Ft(B)28 b Fv(c)l(ontaining)e Fw(\()p Ft(x)3205 2413 y Fu(0)3243 2401 y Ft(;)14 b(t)3310 2413 y Fu(0)3347 2401 y Fw(\))515 2500 y Fv(such)28 b(that)f(the)h(fol)t (lowing)j(holds:)39 b(if)28 b Ft(f)9 b Fw(\()p Ft(x;)14 b(e;)g(t)p Fw(\))28 b Fv(is)g(smo)l(oth)h(in)f(the)g(neighb)l(orho)l(o) l(d)i(of)f Fn(R)3262 2470 y Fp(d)3314 2500 y Fs(\002)515 2600 y Fn(R)18 b Fs(\002)p 676 2533 V 18 w Ft(B)34 b Fv(and)c(vanishes)h(on)f Fw(\()p Ft(U)e Fs(\002)18 b Fw(])p Ft(e)1646 2612 y Fu(0)1701 2600 y Fs(\000)g Ft(\016)s(;)c(e)1900 2612 y Fu(0)1955 2600 y Fw(+)k Ft(\016)s Fw([\))2134 2558 y Fu(c)2186 2600 y Fs(\002)p 2269 2533 V 18 w Ft(B)t Fv(,)30 b(then)972 2849 y Fw(p)p Ft(:)p Fw(v)q Ft(:)1123 2736 y Fr(Z)1169 2925 y Fq(R)1311 2793 y Fw(1)p 1239 2830 185 4 v 1239 2906 a Ft(\021)22 b Fs(\000)c Ft(e)1434 2849 y(I)1470 2861 y Fp(f)1513 2849 y Fw(\()p Ft(r)n(;)c(\021)s(;)g(t)p Fw(\))19 b(d)p Ft(\021)27 b Fs(\030)22 b Fw(e)2017 2815 y Fu(i)p Fp(r)r(\022)r Fu(\()p Fp(e;t)p Fu(\))2234 2849 y Ft(r)2273 2815 y Fo(\000)2336 2792 y FC(d)p 2336 2801 31 3 v 2337 2835 a Fk(2)2422 2745 y Fo(1)2395 2770 y Fr(X)2397 2947 y Fp(j)s Fu(=0)2528 2849 y Ft(a)2572 2861 y Fp(j)2607 2849 y Fw(\()p Ft(e;)14 b(t)p Fw(\))p Ft(r)2816 2815 y Fo(\000)2880 2788 y FC(j)p 2879 2802 29 3 v 2879 2835 a Fk(2)515 3131 y Fv(uniformly)27 b(in)g Fw(\()p Ft(e;)14 b(t)p Fw(\))23 b Fs(2)h Fw([)p Ft(e)1323 3143 y Fu(0)1378 3131 y Fs(\000)18 b Ft(\016)s(;)c(e)1577 3143 y Fu(0)1632 3131 y Fw(+)k Ft(\016)t Fw(])11 b Fs(\002)p 1867 3064 68 4 v 12 w Ft(B)t Fv(,)28 b(wher)l(e)f Ft(\022)r Fw(\()p Ft(e;)14 b(t)p Fw(\))27 b Fv(and)g Ft(a)2658 3143 y Fp(j)2693 3131 y Fw(\()p Ft(e;)14 b(t)p Fw(\))27 b Fv(ar)l(e)g(smo)l(oth)g(in)515 3231 y(the)j(neighb)l(orho)l(o)l(d)i (of)f Fw([)p Ft(e)1306 3243 y Fu(0)1361 3231 y Fs(\000)18 b Ft(\016)s(;)c(e)1560 3243 y Fu(0)1615 3231 y Fw(+)k Ft(\016)s Fw(])h Fs(\002)p 1863 3164 V 18 w Ft(B)t Fv(.)639 3430 y Fw(More)27 b(generally)-7 b(,)26 b(b)n(y)i(the)f(argumen)n(t)g (used)h(in)f(Theorem)g(6)g(\(whic)n(h)h(w)n(e)f(rep)r(eat!\),)515 3612 y Fl(Theorem)k(9)41 b Fv(Supp)l(ose)31 b Fs(r)1395 3624 y Fp(x)1437 3612 y Ft(')p Fw(\()p Ft(x)1570 3624 y Fu(0)1608 3612 y Ft(;)14 b(e)1684 3624 y Fu(0)1721 3612 y Ft(;)g(t)1788 3624 y Fu(0)1825 3612 y Fw(\))23 b(=)g(0)p Fv(,)29 b Fw(rank)13 b(D)2305 3582 y Fu(2)2305 3633 y Fp(x)2347 3612 y Ft(')p Fw(\()p Ft(x)2480 3624 y Fu(0)2519 3612 y Ft(;)h(e)2595 3624 y Fu(0)2632 3612 y Ft(;)g(t)2699 3624 y Fu(0)2736 3612 y Fw(\))23 b(=)g Ft(\024)p Fv(,)30 b(and)1609 3795 y Ft(@)1653 3807 y Fp(e)1689 3795 y Ft(')p Fw(\()p Ft(x)1822 3807 y Fu(0)1860 3795 y Ft(;)14 b(e)1936 3807 y Fu(0)1973 3795 y Ft(;)g(t)2040 3807 y Fu(0)2077 3795 y Fw(\))23 b Fs(6)p Fw(=)g(0)p Ft(:)515 3977 y Fv(Then,)42 b(ther)l(e)e(exist)f(a)g Ft(\016)k(>)d Fw(0)f Fv(and)g(an)h(arbitr)l(arily)h(smal)t(l)f(cub)l (e,)h Ft(U)35 b Fs(\002)24 b Ft(B)t Fv(,)42 b(c)l(ontaining)515 4077 y Fw(\()p Ft(x)594 4089 y Fu(0)632 4077 y Ft(;)14 b(t)699 4089 y Fu(0)736 4077 y Fw(\))30 b Fv(such)f(that)h(the)f(fol)t (lowing)j(holds:)40 b(if)31 b Ft(f)9 b Fw(\()p Ft(x;)14 b(e;)g(t)p Fw(\))29 b Fv(is)h(smo)l(oth)g(in)g(the)f(neighb)l(orho)l(o) l(d)515 4176 y(of)h Fn(R)672 4146 y Fp(d)729 4176 y Fs(\002)18 b Fn(R)h Fs(\002)p 974 4110 V 18 w Ft(B)34 b Fv(and)c(vanishes)h(on)f Fw(\()p Ft(U)d Fs(\002)18 b Fw(])p Ft(e)1943 4188 y Fu(0)1999 4176 y Fs(\000)g Ft(\016)s(;)c(e)2198 4188 y Fu(0)2253 4176 y Fw(+)k Ft(\016)s Fw([\))2431 4135 y Fu(c)2484 4176 y Fs(\002)p 2567 4110 V 18 w Ft(B)t Fv(,)30 b(then)1303 4401 y Fw(p)p Ft(:)p Fw(v)q Ft(:)1454 4288 y Fr(Z)1500 4477 y Fq(R)1641 4345 y Fw(1)p 1570 4382 185 4 v 1570 4458 a Ft(\021)21 b Fs(\000)d Ft(e)1764 4401 y(I)1800 4413 y Fp(f)1844 4401 y Fw(\()p Ft(r)n(;)c(\021)s(;)g(t)p Fw(\))19 b(d)p Ft(\021)26 b Fw(=)d Ft(O)r Fw(\()p Ft(r)2447 4367 y Fo(\000)2510 4345 y FC(\024)p 2510 4354 35 3 v 2513 4387 a Fk(2)2559 4401 y Fw(\))515 4641 y Fv(uniformly)30 b(in)g Fw(\()p Ft(e;)14 b(t)p Fw(\))23 b Fs(2)h Fw([)p Ft(e)1329 4653 y Fu(0)1384 4641 y Fs(\000)18 b Ft(\016)s(;)c(e)1583 4653 y Fu(0)1639 4641 y Fw(+)k Ft(\016)s Fw(])g Fs(\002)p 1886 4575 68 4 v 18 w Ft(B)t Fv(.)515 4824 y Fl(Pro)s(of:)64 b Fw(There)27 b(exist)g(indices,)h Ft(j)1604 4836 y Fu(1)1641 4824 y Ft(;)14 b Fs(\001)g(\001)g(\001)28 b Ft(;)14 b(j)1874 4836 y Fp(\024)1917 4824 y Fw(,)28 b(suc)n(h)f(that,)h(for)1562 5006 y Ft(\030)f Fw(=)c(\()p Ft(x)1792 4972 y Fu(\()p Fp(j)1845 4980 y Fk(1)1878 4972 y Fu(\))1909 5006 y Ft(;)14 b Fs(\001)g(\001)g(\001)27 b Ft(;)14 b(x)2154 4972 y Fu(\()p Fp(j)2207 4980 y FC(\024)2246 4972 y Fu(\))2276 5006 y Fw(\))p Ft(;)1905 5255 y Fw(22)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 515 523 a Fw(det)14 b(D)707 493 y Fu(2)707 547 y Fp(\030)745 523 y Fw(\()p Ft(x)824 535 y Fu(0)862 523 y Ft(;)g(t)929 535 y Fu(0)966 523 y Fw(\))23 b Fs(6)p Fw(=)g(0.)36 b(After)28 b(p)r(erm)n(uting)g(the)g(v)-5 b(ariables)26 b(w)n(e)h(write)1433 697 y Ft(x)c Fw(=)g(\()p Ft(\030)t(;)14 b(\037)p Fw(\))56 b(and)27 b Ft(x)2048 709 y Fu(0)2109 697 y Fw(=)c(\()p Ft(\030)2265 709 y Fu(0)2303 697 y Ft(;)14 b(\037)2392 709 y Fu(0)2429 697 y Fw(\))515 870 y(with)25 b(the)f(ob)n(vious)f(de\014nitions)i(of)f Ft(\037)p Fw(,)h Ft(\030)1763 882 y Fu(0)1801 870 y Fw(,)g(and)f Ft(\037)2059 882 y Fu(0)2097 870 y Fw(.)35 b(In)n(terpreting)24 b(\()p Ft(\037;)14 b(t)p Fw(\))25 b(as)f(a)f(parameter,)515 970 y(the)j(ab)r(o)n(v)n(e)e(corollary)f(giv)n(es)h(the)i(existence)g (of)f(a)g Ft(\016)h(>)d Fw(0)i(and)g(an)h(arbitrarily)d(small)j(cub)r (e,)515 1070 y Ft(U)581 1039 y Fo(0)622 1070 y Fs(\002)18 b Fw(\()p Ft(U)803 1039 y Fo(00)863 1070 y Fs(\002)g Ft(B)t Fw(\))23 b Fs(3)h Fw(\()p Ft(\030)1215 1082 y Fu(0)1252 1070 y Fw(;)14 b Ft(\037)1341 1082 y Fu(0)1379 1070 y Ft(;)g(t)1446 1082 y Fu(0)1483 1070 y Fw(\),)28 b(suc)n(h)f(that,)g(letting)h Ft(U)k Fw(=)23 b Ft(U)2461 1039 y Fo(0)2502 1070 y Fs(\002)17 b Ft(U)2650 1039 y Fo(00)2716 1070 y Fs(3)23 b Ft(x)2841 1082 y Fu(0)2879 1070 y Fw(,)k(for)g Ft(f)9 b Fw(\()p Ft(x;)14 b(t)p Fw(\))28 b(of)515 1169 y(the)g(stipulated)g(form)949 1381 y(p)p Ft(:)p Fw(v)q Ft(:)1100 1268 y Fr(Z)1146 1456 y Fq(R)1287 1325 y Fw(1)p 1216 1362 185 4 v 1216 1438 a Ft(\021)21 b Fs(\000)d Ft(e)1424 1268 y Fr(Z)1470 1456 y Fq(R)1512 1440 y FC(\024)1568 1381 y Fw(e)1605 1346 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(\030)r Fu(;)p Fp(\037;\021)r(;t)p Fu(\))1950 1381 y Ft(f)9 b Fw(\()p Ft(\030)t Fw(;)14 b Ft(\037;)g(\021)s(;)g(t)p Fw(\))k(d)p Ft(\030)23 b Fw(d)p Ft(\021)j Fw(=)d Ft(O)r Fw(\()p Ft(r)2801 1346 y Fo(\000)2864 1324 y FC(\024)p 2864 1333 35 3 v 2867 1366 a Fk(2)2913 1381 y Fw(\))515 1612 y(uniformly)k(in)h(\()p Ft(\037;)14 b(e;)g(t)p Fw(\))23 b Fs(2)p 1350 1545 109 4 v 23 w Ft(U)1416 1588 y Fo(00)1477 1612 y Fs(\002)18 b Fw([)p Ft(e)1622 1624 y Fu(0)1678 1612 y Fs(\000)g Ft(\016)s(;)c(e)1877 1624 y Fu(0)1932 1612 y Fw(+)k Ft(\016)s Fw(])g Fs(\002)p 2179 1545 68 4 v 18 w Ft(B)5 b Fw(.)37 b(Therefore,)780 1715 y Fr(Z)826 1904 y Fq(R)868 1887 y FC(d)p Fj(\000)p FC(\024)1000 1828 y Fw(p)p Ft(:)p Fw(v)q Ft(:)1151 1715 y Fr(Z)1198 1904 y Fq(R)1339 1772 y Fw(1)p 1267 1809 185 4 v 1267 1885 a Ft(\021)22 b Fs(\000)c Ft(e)1476 1715 y Fr(Z)1522 1904 y Fq(R)1564 1887 y FC(\024)1620 1828 y Fw(e)1657 1794 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(\030)r Fu(;)p Fp(\037;\021)r(;t)p Fu(\))2001 1828 y Ft(f)9 b Fw(\()p Ft(\030)t Fw(;)14 b Ft(\037;)g(\021)s(;)g(t)p Fw(\))19 b(d)p Ft(\030)k Fw(d)p Ft(\021)f Fw(d)p Ft(\037)h Fw(=)f Ft(O)r Fw(\()p Ft(r)2969 1794 y Fo(\000)3033 1772 y FC(\024)p 3033 1781 35 3 v 3036 1814 a Fk(2)3081 1828 y Fw(\))515 2059 y(uniformly)31 b(in)h(\()p Ft(e;)14 b(t)p Fw(\))30 b Fs(2)g Fw([)p Ft(e)1345 2071 y Fu(0)1400 2059 y Fs(\000)18 b Ft(\016)s(;)c(e)1599 2071 y Fu(0)1655 2059 y Fw(+)k Ft(\016)s Fw(])j Fs(\002)p 1907 1993 68 4 v 20 w Ft(B)5 b Fw(.)48 b(By)32 b(F)-7 b(ubini's)32 b(and)f(the)h(dominated)g(con-)515 2159 y(v)n(ergence)26 b(theorems)h(this)g(last)h(in)n(tegral)e(is)i(equal)f (to)1206 2258 y Fr(Z)1252 2446 y Fq(R)1294 2430 y FC(d)1347 2371 y Fw(p)p Ft(:)p Fw(v)q Ft(:)1498 2258 y Fr(Z)1544 2446 y Fq(R)1686 2314 y Fw(1)p 1614 2351 185 4 v 1614 2428 a Ft(\021)22 b Fs(\000)c Ft(e)1809 2371 y Fw(e)1846 2336 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(x)p Fu(;)p Fp(\021)r(;t)p Fu(\))2134 2371 y Ft(f)9 b Fw(\()p Ft(x)p Fw(;)14 b Ft(\021)s(;)g(t)p Fw(\))19 b(d)p Ft(\021)j Fw(d)p Ft(x;)515 2591 y Fw(and)27 b(hence)h(\(for)f(the)h(same)f(reasons\))f(to)1206 2807 y(p)p Ft(:)p Fw(v)q Ft(:)1357 2694 y Fr(Z)1403 2883 y Fq(R)1463 2694 y Fr(Z)1509 2883 y Fq(R)1551 2866 y FC(d)1686 2751 y Fw(1)p 1614 2788 V 1614 2864 a Ft(\021)c Fs(\000)c Ft(e)1809 2807 y Fw(e)1846 2773 y Fu(i)p Fp(r)r(')p Fu(\()p Fp(x)p Fu(;)p Fp(\021)r(;t)p Fu(\))2134 2807 y Ft(f)9 b Fw(\()p Ft(x)p Fw(;)14 b Ft(\021)s(;)g(t)p Fw(\))19 b(d)p Ft(x)g Fw(d)p Ft(\021)s(;)515 3023 y Fw(where)i(b)r(oth)i(times)g (the)f(dominator)g(is)g(giv)n(en)f(b)n(y)h(the)h(lemma)f(of)g(Theorem)g (2.)34 b(The)23 b(pro)r(of)515 3122 y(is)k(th)n(us)h(complete.)2187 b Fi(\003)515 3495 y Fx(4)134 b(F)-11 b(ourier)35 b(T)-11 b(ransforms)35 b(o)l(v)l(er)h(Lev)l(el)g(Surfaces)g(of)f(An-)716 3644 y(alytic)46 b(F)-11 b(unctions)515 3826 y Fw(W)k(e)28 b(no)n(w)g(consider)f(a)h(real)f(v)-5 b(alued)28 b(function,)h(\010\()p Ft(x)p Fw(\),)g(analytic)f(in)g Ft(x)d Fs(2)f Fn(R)2846 3796 y Fp(d)2885 3826 y Fw(.)39 b(Usually)-7 b(,)28 b(the)515 3926 y(lev)n(el)i(surfaces)f(of)h(\010\()p Ft(x)p Fw(\))h(consist)f(of) g(sev)n(eral)f(connected)h(comp)r(onen)n(ts;)h(let)g(us)f(fo)r(cus)g (on)515 4025 y(some)d(of)g(them,)i(sa)n(y)-7 b(,)26 b(the)i(connected)g (comp)r(onen)n(ts)f(whose)g(reunion)g(is)g(giv)n(en)g(b)n(y)1461 4199 y(\000\()p Ft(e)p Fw(\))c(=)f Fs(f)p Ft(x)h Fs(2)h(R)f Fw(;)37 b(\010\()p Ft(x)p Fw(\))24 b(=)f Ft(e)p Fs(g)515 4373 y Fw(for)k(an)g(appropriate)f(domain)h Fs(R)d(\022)e Fn(R)1741 4343 y Fp(d)1780 4373 y Fw(.)639 4472 y(Giv)n(en)31 b(a)f(function,)j Ft(f)9 b Fw(\()p Ft(x)p Fw(\),)32 b(summable)f(on)f (\000\()p Ft(e)p Fw(\),)i(its)f(F)-7 b(ourier)30 b(transform)g(o)n(v)n (er)f(\000\()p Ft(e)p Fw(\))515 4572 y(is)e(de\014ned)h(as)1282 4693 y Fs(F)8 b Fw(\(\000\()p Ft(e)p Fw(\))p Ft(;)14 b(f)9 b Fw(\)\()p Ft(n)p Fw(\))23 b(=)1881 4580 y Fr(Z)1927 4769 y Fu(\000\()p Fp(e)p Fu(\))2069 4693 y Fw(e)2106 4659 y Fu(i)p Fp(n)p Fo(\001)p Fp(x)2228 4693 y Ft(f)9 b Fw(\()p Ft(x)p Fw(\))19 b(dS\()p Ft(x)p Fw(\))515 4907 y(for)25 b Ft(n)e Fs(2)h Fn(Z)847 4876 y Fp(d)886 4907 y Fw(,)i(where)42 b(dS\()p Ft(x)p Fw(\))27 b(denotes)f(the)h(elemen)n (t)f(of)h(surface)e(on)h(\000\()p Ft(e)p Fw(\).)37 b(In)26 b(this)g(section)515 5006 y(w)n(e)34 b(deriv)n(e)f(the)i(deca)n(y)e(of) i Fs(F)8 b Fw(\(\000\()p Ft(e)p Fw(\))p Ft(;)14 b(f)9 b Fw(\)\()p Ft(n)p Fw(\))34 b(when)h Fs(j)p Ft(n)p Fs(j)f(!)h(1)f Fw(and)g(sho)n(w)g(its)g(uniformit)n(y)1905 5255 y(23)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 515 523 a Fw(when)32 b Ft(e)f Fw(v)-5 b(aries)31 b(on)g(an)h(appropriate)e(in)n(terv)-5 b(al.)48 b(W)-7 b(e)32 b(then)h(deriv)n(e)e(an)g(analogous)e(result)515 623 y(for)e(the)h(Cauc)n(h)n(y)e(principal)h(v)-5 b(alue)28 b(of)f(suc)n(h)h(a)f(F)-7 b(ourier)26 b(transform.)639 722 y(Let)i Ft(a)832 692 y Fo(0)878 722 y Ft(<)23 b(b)1002 692 y Fo(0)1052 722 y Fw(and)28 b Fs(S)1270 692 y Fo(0)1317 722 y Fw(=)1404 660 y Fr(S)1474 747 y Fp(e)p Fo(2)p Fu(])p Fp(a)1605 731 y Fj(0)1627 747 y Fp(;b)1676 731 y Fj(0)1698 747 y Fu([)1735 722 y Fw(\000\()p Ft(e)p Fw(\))g(b)r(e)g(giv)n(en.)36 b(W)-7 b(e)28 b(assume:)515 922 y Fl(Assumption)i(A)639 1079 y Fs(\017)41 b Fw(\010\()p Ft(x)p Fw(\))31 b Fv(is)f(r)l(e)l(al)g (value)l(d)h(and)f(analytic)h(in)f Fn(R)2061 1049 y Fp(d)2099 1079 y Fv(;)639 1241 y Fs(\017)41 b(r)p Fw(\010\()p Ft(x)p Fw(\))25 b Fs(6)p Fw(=)d(0)29 b Fv(for)i(al)t(l)g Ft(x)23 b Fs(2)h(S)1602 1211 y Fo(0)1625 1241 y Fv(;)639 1403 y Fs(\017)41 b Fw(\000\()p Ft(e)p Fw(\))30 b Fv(is)g(c)l(omp)l(act)h (for)f(al)t(l)h Ft(e)23 b Fs(2)g Fw(])p Ft(a)1775 1373 y Fo(0)1798 1403 y Ft(;)14 b(b)1871 1373 y Fo(0)1894 1403 y Fw([)p Fv(.)515 1660 y Fw(The)38 b(second)g(statemen)n(t)g(in)h (Assumption)f(A)h(and)f(the)h(Implicit)g(F)-7 b(unction)39 b(Theorem)515 1759 y(ensure)31 b(that)h(\000\()p Ft(e)p Fw(\))f(is)h(a)f Fv(r)l(e)l(gular)j(smo)l(oth)g(surfac)l(e)k Fw(for)31 b(an)n(y)g Ft(e)e Fs(2)h Fw(])p Ft(a)2678 1729 y Fo(0)2702 1759 y Ft(;)14 b(b)2775 1729 y Fo(0)2797 1759 y Fw([)q(.)49 b(In)31 b(particular,)515 1869 y(\000\()p Ft(e)p Fw(\))21 b(ma)n(y)f(b)r(e)h(co)n(v)n(ered)e(b)n(y)h (real-analytic)f(lo)r(cal)h(parameterizations,)g Ft(U)2856 1822 y Fp(\033)2809 1869 y Fs(\000)-14 b(!)23 b Fw(\000\()p Ft(e)p Fw(\),)g(where)515 1969 y(eac)n(h)j Ft(U)31 b Fs(\032)23 b Fn(R)937 1938 y Fp(d)p Fo(\000)p Fu(1)1087 1969 y Fw(is)k(op)r(en,)g(eac)n(h)f Ft(\033)s Fw(\()p Ft(u)p Fw(\))h(is)f(a)g(smo)r(oth)h(homeomorphism)e(b)r(et)n(w)n(een)i Ft(U)35 b Fw(and)515 2068 y Ft(\033)s Fw(\()p Ft(U)9 b Fw(\))28 b(in)g(the)g(top)r(ology)e(of)i(\000\()p Ft(e)p Fw(\),)g(and)1659 2178 y Fr([)1627 2360 y Fu(\()p Fp(U)o(;\033)r Fu(\))1797 2257 y Ft(\033)s Fw(\()p Ft(U)9 b Fw(\))24 b(=)f(\000\()p Ft(e)p Fw(\))p Ft(:)515 2526 y Fw(Moreo)n(v)n(er,)f(if)k (t)n(w)n(o)d(of)i(the)g(previous)f(lo)r(cal)g(parameterizations,)f(\()p Ft(U;)14 b(\033)s Fw(\))25 b(and)g(\()p Ft(V)5 b(;)14 b(\034)9 b Fw(\),)26 b(ha)n(v)n(e)515 2625 y(a)h(non)h(empt)n(y)g(o)n (v)n(erlap,)d Fs(D)h Fw(=)d Ft(\033)s Fw(\()p Ft(U)9 b Fw(\))19 b Fs(\\)g Ft(\034)9 b Fw(\()p Ft(V)20 b Fw(\))k Fs(\022)f Fw(\000\()p Ft(e)p Fw(\),)28 b(then)g(the)g(c)n(hange)f(of)h (parameteri-)515 2725 y(zations)f Ft(\034)842 2695 y Fo(\000)p Fu(1)950 2725 y Fs(\016)18 b Ft(\033)s Fw(\()p Ft(u)p Fw(\))28 b(is)f(a)h(real-analytic)d(di\013eomorphism)i(from)h Ft(\033)2661 2695 y Fo(\000)p Fu(1)2750 2725 y Fw(\()p Fs(D)r Fw(\))h(to)e Ft(\034)3055 2695 y Fo(\000)p Fu(1)3145 2725 y Fw(\()p Fs(D)r Fw(\).)639 2825 y(Giv)n(en)k(a)f(lo)r(cal)g (parameterization,)f(\()p Ft(U;)14 b(\033)s Fw(\),)32 b(the)f(restriction)f(of)h(the)g(F)-7 b(ourier)29 b(trans-)515 2924 y(form)e(to)g Ft(\033)s Fw(\()p Ft(U)9 b Fw(\))29 b(giv)n(es)1113 3139 y Fs(F)8 b Fw(\()p Ft(\033)s Fw(\()p Ft(U)h Fw(\))q Ft(;)14 b(f)9 b Fw(\)\()p Ft(n)p Fw(\))83 b(=)1858 3026 y Fr(Z)1904 3214 y Fp(\033)r Fu(\()p Fp(U)6 b Fu(\))2066 3139 y Fw(e)2103 3104 y Fu(i)p Fp(n)p Fo(\001)p Fp(x)2224 3139 y Ft(f)j Fw(\()p Ft(x)p Fw(\))19 b(dS\()p Ft(x)p Fw(\))1710 3375 y(=)1858 3262 y Fr(Z)1904 3451 y Fp(U)1973 3375 y Fw(e)2010 3341 y Fu(i)p Fp(n)p Fo(\001)p Fp(\033)r Fu(\()p Fp(u)p Fu(\))2226 3375 y Ft(f)9 b Fw(\()p Ft(\033)s Fw(\()p Ft(u)p Fw(\)\))p Ft(J)f Fw(\()p Ft(u)p Fw(\))19 b(d)p Ft(u)515 3594 y Fw(for)32 b(an)n(y)f Ft(f)9 b Fw(\()p Ft(x)p Fw(\))33 b(summable)g(on)f Ft(\033)s Fw(\()p Ft(U)9 b Fw(\),)34 b(where)e Ft(J)8 b Fw(\()p Ft(u)p Fw(\))22 b(d)p Ft(u)32 b Fw(is)g(the)h(elemen)n(t)f(of)h (surface.)50 b(Of)515 3693 y(course)28 b(the)i(deca)n(y)e(of)h(suc)n(h) g(an)g(in)n(tegral)f(when)i Fs(j)p Ft(n)p Fs(j)c(!)g(1)j Fw(is)h(studied)f(b)n(y)g(means)g(of)h(the)515 3793 y(stationary)c (phase)h(metho)r(d.)639 3892 y(Let)d Ft(n)f Fw(=)g Ft(r)r(!)28 b Fw(b)r(e)c(the)h(p)r(olar)e(form)g(of)i Ft(n)d Fs(2)i Fn(Z)2014 3862 y Fp(d)2053 3892 y Fw(,)h(so)e Ft(r)j Fw(=)d Fs(j)p Ft(n)p Fs(j)h Fw(and)g Ft(!)h Fs(2)f Fw(S)2830 3862 y Fp(d)p Fo(\000)p Fu(1)2954 3892 y Fw(.)35 b(The)25 b Fv(phase)515 3992 y Fw(in)c(the)h(previous)d(in)n(tegral)h(then)i(b)r (ecomes)f Ft(')p Fw(\()p Ft(u;)14 b(!)s Fw(\))23 b(=)f Ft(!)8 b Fs(\001)d Ft(\033)s Fw(\()p Ft(u)p Fw(\))q(.)35 b(Let)21 b(us)g(consider)f(a)h(p)r(oin)n(t,)515 4092 y Ft(x)562 4104 y Fu(0)622 4092 y Fw(=)i Ft(\033)s Fw(\()p Ft(u)840 4104 y Fu(0)878 4092 y Fw(\),)g(in)e Ft(\033)s Fw(\()p Ft(U)9 b Fw(\).)36 b(Remark)-5 b(ably)e(,)22 b(the)f(fact)h(that)f Ft(u)2274 4104 y Fu(0)2333 4092 y Fw(is)g(or)f(is)i(not)f(a)g(stationary)f(phase)515 4191 y(p)r(oin)n(t)29 b(dep)r(ends)g(on)f Fv(intrinsic)j(pr)l(op)l (erties)h(of)f Fw(\000\()p Ft(e)p Fw(\))g Fv(only)7 b Fw(;)30 b(if)g Ft(u)2509 4203 y Fu(0)2574 4191 y Fw(is)f(stationary)-7 b(,)27 b(the)i(rank)515 4291 y(of)e(D)672 4261 y Fu(2)672 4311 y Fp(u)716 4291 y Ft(')p Fw(\()p Ft(u)850 4303 y Fu(0)887 4291 y Ft(;)14 b(!)s Fw(\))28 b(is)f(also)g(in)n(trinsic.)37 b(Indeed,)515 4463 y Fl(Theorem)31 b(10)41 b Fv(In)34 b(the)g(ab)l(ove)h(cir)l(cumstanc)l(es)e Ft(u)2124 4475 y Fu(0)2195 4463 y Fv(is)h(stationary)h(if,)h(and)e(only)g(if)h Ft(!)i Fv(is)515 4563 y(p)l(erp)l(endicular)c(to)f Fw(\000\()p Ft(e)p Fw(\))g Fv(at)g Ft(x)1468 4575 y Fu(0)1532 4563 y Fw(=)27 b Ft(\033)s Fw(\()p Ft(u)1754 4575 y Fu(0)1791 4563 y Fw(\))p Fv(.)45 b(Then,)34 b(the)e(r)l(ank)g(of)g(the)g(Hessian) g(of)h Ft(')p Fw(\()p Ft(u;)14 b(!)s Fw(\))515 4662 y Fv(at)26 b Ft(u)659 4674 y Fu(0)722 4662 y Fv(is)g(e)l(qual)g(to)g(the) h(numb)l(er)e(of)i(non)f(vanishing)h(princip)l(al)h(curvatur)l(es)e(of) h Fw(\000\()p Ft(e)p Fw(\))f Fv(at)g Ft(x)3316 4674 y Fu(0)3354 4662 y Fv(.)515 4834 y Fl(Pro)s(of:)64 b Fw(Notice)27 b(that)h Ft(u)1326 4846 y Fu(0)1391 4834 y Fw(is)f(stationary)f(i\013) 947 5006 y Fs(r)1016 5018 y Fp(u)1059 5006 y Ft(!)21 b Fs(\001)e Ft(\033)s Fw(\()p Ft(u)1304 5018 y Fu(0)1341 5006 y Fw(\))24 b(=)e(\()p Ft(!)g Fs(\001)c Fw(d)1677 5025 y Fp(u)1716 5009 y Fk(\(1\))1798 5006 y Ft(\033)s Fw(\()p Ft(u)1928 5018 y Fu(0)1966 5006 y Fw(\))p Ft(;)c(:)g(:)g(:)g(;) g(!)21 b Fs(\001)d Fw(d)2343 5025 y Fp(u)2382 5009 y Fk(\()p FC(d)p Fj(\000)p Fk(1\))2540 5006 y Ft(\033)s Fw(\()p Ft(u)2670 5018 y Fu(0)2707 5006 y Fw(\)\))24 b(=)e(0)p Ft(:)1905 5255 y Fw(24)p eop end %%Page: 25 25 TeXDict begin 25 24 bop 515 523 a Fw(Since)28 b(the)f(tangen)n(t)h (plane)f(of)h(\000\()p Ft(e)p Fw(\))f(at)h Ft(x)1824 535 y Fu(0)1889 523 y Fw(is)g(generated)e(b)n(y)1389 704 y Fs(f)p Fw(d)1477 723 y Fp(u)1516 706 y Fk(\()p FC(j)r Fk(\))1596 704 y Ft(\033)s Fw(\()p Ft(u)1726 716 y Fu(0)1763 704 y Fw(\))e(;)37 b Ft(j)28 b Fw(=)22 b(1)p Ft(;)14 b(:)g(:)g(:)f(;)h(d)19 b Fs(\000)f Fw(1)o Fs(g)p Ft(;)515 884 y Fw(the)28 b(\014rst)f(statemen)n(t)h(follo)n(ws.)639 984 y(Supp)r(ose)f Ft(u)1012 996 y Fu(0)1076 984 y Fw(is)g(a)g (stationary)e(phase)h(p)r(oin)n(t)i(and)e(consider)g(an)n(y)h(other)f (lo)r(cal)g(param-)515 1083 y(eterization,)h(\()p Ft(V)5 b(;)14 b(\034)9 b Fw(\),)29 b(of)f(a)g(neigh)n(b)r(orho)r(o)r(d)f(of)h Ft(x)2029 1095 y Fu(0)2090 1083 y Fw(=)23 b Ft(\033)s Fw(\()p Ft(u)2308 1095 y Fu(0)2346 1083 y Fw(\).)38 b(Let)28 b Fs(D)e Fw(=)e Ft(\033)s Fw(\()p Ft(U)9 b Fw(\))19 b Fs(\\)g Ft(\034)9 b Fw(\()p Ft(V)20 b Fw(\))28 b(and)515 1183 y Ft(F)12 b Fw(\()p Ft(v)s Fw(\))36 b(=)e Ft(\033)872 1153 y Fo(\000)p Fu(1)985 1183 y Fs(\016)23 b Ft(\034)9 b Fw(\()p Ft(v)s Fw(\).)60 b(Then,)37 b Ft(F)12 b Fw(\()p Ft(v)s Fw(\))36 b(is)f(a)f(smo)r(oth)h(di\013eomorphism)f(from)h Ft(\034)3050 1153 y Fo(\000)p Fu(1)3139 1183 y Fw(\()p Fs(D)r Fw(\))h(to)515 1283 y Ft(\033)565 1253 y Fo(\000)p Fu(1)654 1283 y Fw(\()p Fs(D)r Fw(\))29 b(satisfying)1545 1382 y Ft(!)21 b Fs(\001)d Ft(\034)9 b Fw(\()p Ft(v)s Fw(\))25 b(=)e Ft(!)e Fs(\001)d Ft(\033)s Fw(\()p Ft(F)12 b Fw(\()p Ft(v)s Fw(\)\))r Ft(:)924 b Fw(\(6\))515 1531 y(Let)29 b Ft(v)705 1543 y Fu(0)767 1531 y Fw(=)c Ft(\034)902 1500 y Fo(\000)p Fu(1)991 1531 y Fw(\()p Ft(x)1070 1543 y Fu(0)1108 1531 y Fw(\),)30 b(so)e(\()p Fs(r)1397 1543 y Fp(u)1441 1531 y Fw(\()p Ft(!)22 b Fs(\001)d Ft(\033)s Fw(\)\)\()p Ft(F)12 b Fw(\()p Ft(v)1872 1543 y Fu(0)1911 1531 y Fw(\)\))25 b(=)g Fs(r)2159 1543 y Fp(u)2203 1531 y Fw(\()p Ft(!)d Fs(\001)d Ft(\033)s Fw(\)\()p Ft(u)2513 1543 y Fu(0)2551 1531 y Fw(\))25 b(=)f(0.)40 b(Then,)29 b(the)h(c)n(hain)515 1630 y(rule)d(applied)h(to)f(\(6\))h(giv)n(es)1196 1811 y(D)1259 1776 y Fu(2)1259 1831 y Fp(v)1299 1811 y Fw(\()p Ft(!)22 b Fs(\001)c Ft(\034)9 b Fw(\)\()p Ft(v)1595 1823 y Fu(0)1634 1811 y Fw(\))23 b(=)g(D)1840 1776 y Fu(2)1840 1831 y Fp(u)1884 1811 y Fw(\()p Ft(!)e Fs(\001)e Ft(\033)s Fw(\)\()p Ft(u)2193 1823 y Fu(0)2230 1811 y Fw(\)\(D)2357 1823 y Fp(v)2398 1811 y Ft(F)12 b Fw(\()p Ft(v)2535 1823 y Fu(0)2572 1811 y Fw(\)\))2636 1776 y Fu(2)2674 1811 y Ft(:)515 1991 y Fw(Since)28 b Ft(F)12 b Fw(\()p Ft(v)s Fw(\))28 b(is)f(a)h(di\013eomorphism,)f(D)1755 2003 y Fp(v)1795 1991 y Ft(F)12 b Fw(\()p Ft(v)1932 2003 y Fu(0)1969 1991 y Fw(\))28 b(is)g(in)n(v)n(ertible)f(and)g(hence)1224 2172 y(rank)13 b(D)1465 2138 y Fu(2)1465 2192 y Fp(v)1505 2172 y Fw(\()p Ft(!)21 b Fs(\001)e Ft(\034)9 b Fw(\)\()p Ft(v)1801 2184 y Fu(0)1840 2172 y Fw(\))23 b(=)g(rank)13 b(D)2224 2138 y Fu(2)2224 2192 y Fp(u)2267 2172 y Fw(\()p Ft(!)22 b Fs(\001)c Ft(\033)s Fw(\)\()p Ft(u)2576 2184 y Fu(0)2614 2172 y Fw(\))p Ft(;)515 2353 y Fw(whic)n(h)27 b(sho)n(ws)g(that)h(this)g(last)f(rank)f(is)i(in)n(trinsic.)639 2452 y(Finally)-7 b(,)33 b(since)f Ft(u)1199 2464 y Fu(0)1267 2452 y Fw(is)g(a)f(stationary)g(phase)g(p)r(oin)n(t,)i Ft(!)h Fw(is)e(p)r(erp)r(endicular)g(to)f(\000\()p Ft(e)p Fw(\))h(at)515 2552 y Ft(x)562 2564 y Fu(0)599 2552 y Fw(,)c(so)1641 2695 y Ft(!)e Fw(=)1826 2638 y Fs(\006r)1960 2650 y Fp(x)2001 2638 y Fw(\010\()p Ft(x)2140 2650 y Fu(0)2178 2638 y Fw(\))p 1816 2676 404 4 v 1816 2752 a Fs(kr)1927 2764 y Fp(x)1969 2752 y Fw(\010\()p Ft(x)2108 2764 y Fu(0)2146 2752 y Fw(\))p Fs(k)2230 2695 y Ft(:)515 2903 y Fw(Moreo)n(v)n(er,)33 b(there)h(exists)g(a)f Ft(j)39 b Fs(2)c(f)p Fw(1)p Ft(;)14 b(:)g(:)g(:)e(;)i(d)p Fs(g)34 b Fw(suc)n(h)g(that)g Ft(!)2425 2873 y Fu(\()p Fp(j)s Fu(\))2546 2903 y Fs(6)p Fw(=)f(0.)56 b(Supp)r(ose)35 b(without)515 3003 y(loss)24 b(of)h(generalit)n(y)f Ft(!)1198 2972 y Fu(\()p Fp(d)p Fu(\))1311 3003 y Fs(6)p Fw(=)f(0,)i(and)g(hence) g Ft(@)1920 3022 y Fp(x)1958 3005 y Fk(\()p FC(d)p Fk(\))2042 3003 y Fw(\010\()p Ft(x)2181 3015 y Fu(0)2219 3003 y Fw(\))e Fs(6)p Fw(=)g(0)p Ft(:)i Fw(Let)g Ft(w)h Fw(=)c(\()p Ft(x)2849 2972 y Fu(\(1\))2939 3003 y Ft(;)14 b(:)g(:)g(:)g(;)g(x)3171 2972 y Fu(\()p Fp(d)p Fo(\000)p Fu(1\))3347 3003 y Fw(\))515 3118 y(and)31 b Ft(w)739 3130 y Fu(0)807 3118 y Fw(=)f(\()p Ft(x)981 3075 y Fu(\(1\))981 3140 y(0)1071 3118 y Ft(;)14 b(:)g(:)g(:)g(;)g(x)1303 3075 y Fu(\()p Fp(d)p Fo(\000)p Fu(1\))1303 3140 y(0)1478 3118 y Fw(\).)51 b(By)31 b(the)h(Implicit)h (F)-7 b(unction)32 b(Theorem)f(there)h(exists)f(a)515 3218 y(function,)22 b Ft(h)p Fw(\()p Ft(w)r Fw(\),)g(smo)r(oth)e(in)g (the)g(neigh)n(b)r(orho)r(o)r(d)f(of)h Ft(w)2245 3230 y Fu(0)2283 3218 y Fw(,)h(suc)n(h)f(that)g(\010\()p Ft(w)r(;)14 b(h)p Fw(\()p Ft(w)r Fw(\)\))25 b(=)e Ft(e)c Fw(and)515 3334 y Ft(h)p Fw(\()p Ft(w)654 3346 y Fu(0)692 3334 y Fw(\))28 b(=)g Ft(x)892 3291 y Fu(\()p Fp(d)p Fu(\))892 3356 y(0)983 3334 y Fw(.)47 b(Hence,)32 b Ft(\015)5 b Fw(\()p Ft(w)r Fw(\))29 b(=)f(\()p Ft(w)r(;)14 b(h)p Fw(\()p Ft(w)r Fw(\)\))33 b(giv)n(es)d(a)g(smo)r(oth)h(lo)r(cal)f (parameterization)515 3433 y(of)c(a)g(neigh)n(b)r(orho)r(o)r(d)f(of)h Ft(x)1336 3445 y Fu(0)1400 3433 y Fw(as)g(a)g(graph)f(of)h(a)g(smo)r (oth)g(function.)37 b(Di\013eren)n(tial)26 b(geometry)515 3533 y(then)f(sho)n(ws)f(that)h(rank)13 b(D)1355 3503 y Fu(2)1355 3554 y Fp(w)1409 3533 y Ft(h)p Fw(\()p Ft(w)r Fw(\))27 b(is)e(equal)f(to)h(the)g(n)n(um)n(b)r(er)g(of)g(non)g(v)-5 b(anishing)25 b(principal)515 3633 y(curv)-5 b(atures)26 b(at)i Ft(x)1068 3645 y Fu(0)1128 3633 y Fs(2)c Fw(\000\()p Ft(e)p Fw(\).)37 b(Since)1427 3813 y(D)1490 3779 y Fu(2)1490 3834 y Fp(w)1544 3813 y Fw(\()p Ft(!)21 b Fs(\001)e Ft(\015)5 b Fw(\)\()p Ft(w)r Fw(\))24 b(=)f Ft(!)2063 3779 y Fu(\()p Fp(d)p Fu(\))2153 3813 y Fw(D)2216 3779 y Fu(2)2216 3834 y Fp(w)2270 3813 y Ft(h)p Fw(\()p Ft(w)r Fw(\))p Ft(;)515 3994 y Fw(the)28 b(pro)r(of)f(is)g(complete.)2010 b Fi(\003)515 4309 y Fl(Join)m(t)46 b(system)f(of)i(parameterizations)83 b Fw(Giv)n(en)40 b(a)f(\014xed)h Ft(e)2643 4321 y Fu(0)2724 4309 y Fs(2)k Fw(])q Ft(a)2891 4279 y Fo(0)2914 4309 y Ft(;)14 b(b)2987 4279 y Fo(0)3010 4309 y Fw([,)43 b(w)n(e)d(no)n(w) 515 4408 y(construct)24 b(a)h(system)g(of)g(parameterizations)d(for)j (\000\()p Ft(e)2228 4420 y Fu(0)2265 4408 y Fw(\))g(compatible)g(with)h (all)f(\000\()p Ft(e)p Fw(\)'s)g(for)515 4508 y Ft(e)36 b Fw(v)-5 b(arying)35 b(in)h(a)g(small)g(neigh)n(b)r(orho)r(o)r(d)f(of) h Ft(e)1975 4520 y Fu(0)2048 4508 y Fw(\(so)g(the)h(deriv)n(ed)e (estimate)h(for)g(F)-7 b(ourier)515 4608 y(transforms)26 b(will)i(b)r(e)g(uniform)f(in)h Ft(e)p Fw(\).)37 b(W)-7 b(e)28 b(mak)n(e)f(the)h(follo)n(wing)e(h)n(yp)r(othesis:)515 4807 y Fl(Assumption)35 b(B)105 b Fv(F)-6 b(or)34 b(every)h Ft(e)30 b Fs(2)h Fw(])p Ft(a)1827 4777 y Fo(0)1850 4807 y Ft(;)14 b(b)1923 4777 y Fo(0)1946 4807 y Fw([)p Fv(,)35 b Fw(\000\()p Ft(e)p Fw(\))f Fv(admits)h(at)e(le)l(ast)h Ft(\024)g Fv(non)g(vanishing)515 4907 y(princip)l(al)e(curvatur)l(es)d (at)g(any)h(p)l(oint,)h(wher)l(e)f Ft(\024)23 b Fi(>)g Fw(1)29 b Fv(is)h(a)g(\014xe)l(d)g(inte)l(ger.)1905 5255 y Fw(25)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 639 523 a Fw(The)26 b(plan)g(is)f(the)i(follo)n (wing:)34 b(starting)25 b(from)h(an)f(arbitrary)f(system)h(of)h (real-analytic)515 623 y(parameterizations)18 b(for)i(\000\()p Ft(e)1425 635 y Fu(0)1462 623 y Fw(\),)j(w)n(e)d(will)h(parametrize)e (\000\()p Ft(e)p Fw(\))h(using)h(the)g(lo)r(cal)e(co)r(ordinates)515 722 y(of)27 b(\000\()p Ft(e)732 734 y Fu(0)769 722 y Fw(\),)h(b)n(y)g(lifting)g(them)g(orthogonally)d(to)j(\000\()p Ft(e)2129 734 y Fu(0)2166 722 y Fw(\))g(\(for)f Ft(e)g Fw(v)n(ery)g(close)g(to)g Ft(e)2973 734 y Fu(0)3010 722 y Fw(\).)639 822 y(Let)36 b Fs(f)p Fw(\()p Fs(U)922 834 y Fp(\014)967 822 y Ft(;)14 b(\015)1047 834 y Fp(\014)1091 822 y Fw(\))p Fs(g)1165 792 y Fp(M)1165 845 y(\014)s Fu(=1)1329 822 y Fw(b)r(e)36 b(a)g(system)f(of)g(real-analytic)f (parameterizations)f(co)n(v)n(ering)515 922 y(\000\()p Ft(e)638 934 y Fu(0)675 922 y Fw(\).)70 b(Since)39 b(\000\()p Ft(e)1151 934 y Fu(0)1188 922 y Fw(\))h(is)e(compact,)j(w)n(e)e(assume) f Ft(M)47 b Fw(to)39 b(b)r(e)g(\014nite.)71 b(Without)39 b(loss)f(of)515 1021 y(generalit)n(y)-7 b(,)37 b(w)n(e)e(also)h(assume) f Ft(\015)1569 1033 y Fp(\014)1614 1021 y Fw(\()p Ft(u)p Fw(\))h(is)g(analytic)g(in)g(a)g(neigh)n(b)r(orho)r(o)r(d)f(of)p 2995 954 97 4 v 36 w Fs(U)3047 1033 y Fp(\014)3092 1021 y Fw(,)j(so)e(the)515 1121 y(expression)26 b Ft(\015)958 1133 y Fp(\014)1003 1121 y Fw(\()p Ft(u)p Fw(\))h(for)g Ft(u)c Fs(2)g Ft(@)5 b Fs(U)1519 1133 y Fp(\014)1592 1121 y Fw(mak)n(es)26 b(sense.)639 1220 y(Let)d Ft(x)830 1232 y Fu(0)890 1220 y Fs(2)h Fw(\000\()p Ft(e)1092 1232 y Fu(0)1129 1220 y Fw(\),)g(sa)n(y)-7 b(,)22 b Ft(x)1410 1232 y Fu(0)1471 1220 y Fw(=)h Ft(\015)1602 1232 y Fp(\014)1646 1220 y Fw(\()p Ft(u)1726 1232 y Fu(0)1763 1220 y Fw(\))g(for)f(a)g(giv) n(en)f Ft(u)2263 1232 y Fu(0)2323 1220 y Fs(2)i(U)2453 1232 y Fp(\014)2521 1220 y Fw(and)f(a)g(giv)n(en)f(1)i Fi(6)f Ft(\014)28 b Fi(6)22 b Ft(M)9 b Fw(.)515 1320 y(Since)28 b Fs(r)801 1332 y Fp(x)843 1320 y Fw(\010\()p Ft(x)p Fw(\))g(is)g(p)r(erp)r(endicular)f(to)g(\000\()p Ft(e)1876 1332 y Fu(0)1913 1320 y Fw(\))h(at)g(an)n(y)f Ft(x)c Fs(2)h Fw(\000\()p Ft(e)2504 1332 y Fu(0)2541 1320 y Fw(\),)k(w)n(e)f(need)h(to)f(solv)n(e)1332 1503 y(\010\()14 b Ft(\015)1481 1515 y Fp(\014)1526 1503 y Fw(\()p Ft(u)p Fw(\))k(+)g Ft(\025)p Fs(r)1856 1515 y Fp(x)1899 1503 y Fw(\010\()p Ft(\015)2034 1515 y Fp(\014)2079 1503 y Fw(\()p Ft(u)p Fw(\)\))c(\))19 b Fs(\000)f Ft(e)23 b Fw(=)f(0)711 b(\(7\))515 1685 y(in)31 b(the)h(neigh)n(b)r(orho)r(o)r (d)e(of)h(\()p Ft(u;)14 b(e;)g(\025)p Fw(\))29 b(=)g(\()p Ft(u)1861 1697 y Fu(0)1898 1685 y Ft(;)14 b(e)1974 1697 y Fu(0)2011 1685 y Ft(;)g Fw(0\).)48 b(The)31 b(deriv)-5 b(ativ)n(e)31 b(at)g(0)g(with)g(resp)r(ect)515 1785 y(to)k Ft(\025)h Fw(of)f(the)g(left-hand)h(side)f(in)h(\(7\))f(is)g Fs(kr)1943 1797 y Fp(x)1985 1785 y Fw(\010\()p Ft(\015)2120 1797 y Fp(\014)2165 1785 y Fw(\()p Ft(u)p Fw(\)\))p Fs(k)2351 1755 y Fu(2)2388 1785 y Fw(,)i(whic)n(h)e(is)g(strictly)g(p)r(ositiv)n (e.)515 1885 y(Hence,)26 b(there)g(exists)f(an)g(analytic)h(function,)g Ft(\025)p Fw(\()p Ft(u;)14 b(e)p Fw(\),)27 b(de\014ned)f(on)f(an)h (arbitrarily)d(small)515 1984 y(cub)r(e,)28 b Fs(U)794 1954 y Fo(0)836 1984 y Fs(\002)18 b Fw(])p Ft(e)981 1996 y Fu(0)1036 1984 y Fs(\000)g Fw(\001)1188 1954 y Fo(0)1212 1984 y Ft(;)c(e)1288 1996 y Fu(0)1343 1984 y Fw(+)k(\001)1495 1954 y Fo(0)1518 1984 y Fw([)24 b Fs(3)f Fw(\()p Ft(u)1723 1996 y Fu(0)1760 1984 y Ft(;)14 b(e)1836 1996 y Fu(0)1873 1984 y Fw(\),)28 b(satisfying)1050 2167 y Ft(\015)1093 2179 y Fp(\014)1138 2167 y Fw(\()p Ft(u)p Fw(\))18 b(+)g Ft(\025)p Fw(\()p Ft(u;)c(e)p Fw(\))p Fs(r)1656 2179 y Fp(x)1698 2167 y Fw(\010\()p Ft(\015)1833 2179 y Fp(\014)1878 2167 y Fw(\()p Ft(u)p Fw(\)\))24 b Fs(2)f Fw(\000\()p Ft(e)p Fw(\),)56 b Ft(\025)p Fw(\()p Ft(u)2486 2179 y Fu(0)2523 2167 y Ft(;)14 b(e)2599 2179 y Fu(0)2636 2167 y Fw(\))23 b(=)g(0)p Ft(:)429 b Fw(\(8\))639 2350 y(Remark)-5 b(ably)e(,)27 b Ft(\025)p Fw(\()p Ft(u;)14 b(e)p Fw(\))28 b(dep)r(ends)g(on)f Ft(\015)1861 2362 y Fp(\014)1906 2350 y Fw(\()p Ft(u)p Fw(\))h(only)-7 b(,)27 b(not)h(on)f(its)h(lo)r (cal)f(co)r(ordinates)f Ft(u)p Fw(:)515 2549 y Fl(Lemma)59 b Fv(Supp)l(ose)30 b Ft(\015)1240 2561 y Fp(\014)1281 2545 y Fj(0)1307 2549 y Fw(\()p Ft(u)1387 2519 y Fo(0)1387 2569 y Fu(0)1424 2549 y Fw(\))24 b(=)e Ft(x)1614 2561 y Fu(0)1682 2549 y Fv(and)30 b(let)g Ft(u)2008 2519 y Fo(0)2053 2549 y Fw(=)23 b Ft(\015)2189 2513 y Fo(\000)p Fu(1)2184 2574 y Fp(\014)2225 2557 y Fj(0)2296 2549 y Fs(\016)18 b Ft(\015)2399 2561 y Fp(\014)2444 2549 y Fw(\()p Ft(u)p Fw(\))p Fv(,)30 b(wher)l(e)h Ft(u)e Fv(varies)i(in)1517 2749 y Ft(\015)1565 2714 y Fo(\000)p Fu(1)1560 2774 y Fp(\014)1654 2749 y Fw(\()p Ft(\015)1729 2761 y Fp(\014)1774 2749 y Fw(\()p Fs(U)1858 2761 y Fp(\014)1903 2749 y Fw(\))19 b Fs(\\)f Ft(\015)2070 2761 y Fp(\014)2111 2745 y Fj(0)2138 2749 y Fw(\()p Fs(U)2222 2761 y Fp(\014)2263 2745 y Fj(0)2289 2749 y Fw(\)\))p Ft(:)515 2932 y Fv(De\014ne)30 b Ft(\025)826 2902 y Fo(0)849 2932 y Fw(\()p Ft(u)929 2902 y Fo(0)952 2932 y Ft(;)14 b(e)p Fw(\))31 b Fv(as)f(ab)l(ove)i(with)f(r)l(esp)l(e)l (ct)f(to)h Ft(\015)2019 2944 y Fp(\014)2060 2928 y Fj(0)2086 2932 y Fw(\()p Ft(u)2166 2902 y Fo(0)2189 2932 y Fw(\))p Fv(|while)h Ft(\025)p Fw(\()p Ft(u;)14 b(e)p Fw(\))31 b Fv(was)g(de\014ne)l(d)f(with)515 3032 y(r)l(esp)l(e)l(ct)f(to)h Ft(\015)928 3044 y Fp(\014)973 3032 y Fw(\()p Ft(u)p Fw(\))p Fv(.)38 b(Then,)31 b Ft(\025)p Fw(\()p Ft(u;)14 b(e)p Fw(\))23 b(=)g Ft(\025)1785 3002 y Fo(0)1809 3032 y Fw(\()p Ft(u)1889 3002 y Fo(0)1912 3032 y Ft(;)14 b(e)p Fw(\))30 b Fv(when)g Ft(e)f Fv(is)h(close)h(enough)f(to)g Ft(e)3044 3044 y Fu(0)3081 3032 y Fv(.)639 3231 y Fw(The)40 b(pro)r(of,)i(based)d(on)g(the)g(uniqueness)h(part)f(of)g(the)h (Implicit)g(F)-7 b(unction)40 b(Theo-)515 3331 y(rem,)690 3300 y Fu(7)769 3331 y Fw(is)f(left)g(to)g(the)g(reader.)69 b(Since)39 b(the)g(considered)f(surfaces)g(are)f(compact,)42 b Fv(this)515 3430 y(last)d(lemma)g(ensur)l(es)f(the)i(existenc)l(e)e (of)i(an)f(analytic)h(function,)i Fw(\003\()p Ft(x;)14 b(e)p Fw(\))p Fv(,)42 b(de\014ne)l(d)d(on)515 3530 y Fw(\000\()p Ft(e)638 3542 y Fu(0)675 3530 y Fw(\))22 b Fs(\002)g Fw(])p Ft(e)878 3542 y Fu(0)933 3530 y Fs(\000)c Fw(\001)1085 3500 y Fo(0)1109 3530 y Ft(;)c(e)1185 3542 y Fu(0)1240 3530 y Fw(+)k(\001)1392 3500 y Fo(0)1416 3530 y Fw([)34 b Fv(\(wher)l(e)i Fw(\000\()p Ft(e)1870 3542 y Fu(0)1907 3530 y Fw(\))f Fv(is)g(endowe)l(d)h(with)f(its)g (surfac)l(e)g(structur)l(e\),)515 3629 y(such)30 b(that)1107 3729 y Ft(x)19 b Fw(+)f(\003\()p Ft(x;)c(e)p Fw(\))p Fs(r)1570 3741 y Fp(x)1612 3729 y Fw(\010\()p Ft(x)p Fw(\))25 b Fs(2)e Fw(\000\()p Ft(e)p Fw(\))59 b Fv(and)h Fw(\003\()p Ft(x)2428 3741 y Fu(0)2466 3729 y Ft(;)14 b(e)2542 3741 y Fu(0)2579 3729 y Fw(\))23 b(=)g(0)p Ft(:)486 b Fw(\(9\))639 3978 y(Inciden)n(tally)-7 b(,)28 b(the)g(function)g Ft(\025)p Fw(\()p Ft(u;)14 b(e)p Fw(\))28 b(has)f(the)h(follo)n(wing,)e (in)n(teresting)h(prop)r(erties:)515 4177 y Fl(Lemma)69 b Fv(F)-6 b(or)35 b(every)g Ft(u)d Fs(2)g(U)1502 4147 y Fo(0)1526 4177 y Fv(,)k Ft(\025)p Fw(\()p Ft(u;)14 b(e)1791 4189 y Fu(0)1828 4177 y Fw(\))33 b(=)e(0)p Fv(.)54 b(In)34 b(p)l(articular,)j Ft(@)2678 4197 y Fp(u)2717 4180 y Fk(\()p FC(j)r Fk(\))2798 4177 y Ft(\025)p Fw(\()p Ft(u;)14 b(e)3002 4189 y Fu(0)3039 4177 y Fw(\))32 b(=)g(0)i Fv(for)515 4277 y(any)c Ft(j)e Fw(=)22 b(1)p Ft(;)14 b(:)g(:)g(:)g(;)g(d)k Fs(\000)g Fw(1)p Fv(.)639 4476 y Fw(The)28 b(pro)r(of)f(is)g(left)i(to)e(the)h(reader.)515 4676 y Fl(Lemma)59 b Ft(@)925 4688 y Fp(e)961 4676 y Ft(\025)p Fw(\()p Ft(u)1089 4688 y Fu(0)1126 4676 y Ft(;)14 b(e)1202 4688 y Fu(0)1239 4676 y Fw(\))23 b Ft(>)g Fw(0)p Ft(:)p 515 4828 1146 4 v 607 4882 a FB(7)642 4905 y Fm(Of)g(course,)g (this)h(uniqueness)h(holds)e Fh(lo)l(c)l(al)t(ly)p Fm(.)1905 5255 y Fw(26)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 515 523 a Fl(Pro)s(of:)64 b Fw(In)27 b(fact,)h(since)g(\010\()p Ft(x)1471 535 y Fu(0)1527 523 y Fw(+)18 b Ft(\025)p Fw(\()p Ft(u)1738 535 y Fu(0)1776 523 y Ft(;)c(e)p Fw(\))p Fs(r)1953 535 y Fp(x)1994 523 y Fw(\010\()p Ft(x)2133 535 y Fu(0)2171 523 y Fw(\)\))24 b(=)f Ft(e)p Fw(,)k(the)h(c)n(hain)f(rule)g(giv)n(es)1455 730 y Ft(@)1499 742 y Fp(e)1534 730 y Ft(\025)p Fw(\()p Ft(u)1662 742 y Fu(0)1700 730 y Ft(;)14 b(e)1776 742 y Fu(0)1813 730 y Fw(\))23 b(=)2165 674 y(1)p 1966 711 441 4 v 1966 787 a Fs(kr)2077 799 y Fp(x)2118 787 y Fw(\010\()p Ft(x)2257 799 y Fu(0)2295 787 y Fw(\))p Fs(k)2369 763 y Fu(2)2416 730 y Ft(:)3314 940 y Fi(\003)639 1139 y Fw(The)33 b(equation)g(\(8\))g(p)r(ermits)g(to)g(de\014ne)g(the)h (follo)n(wing)e(lo)r(cal)g(parameterization)f(of)515 1239 y(\000\()p Ft(e)p Fw(\):)1275 1339 y Ft(\033)s Fw(\()p Ft(u;)14 b(e)p Fw(\))23 b(=)f Ft(\015)1666 1351 y Fp(\014)1711 1339 y Fw(\()p Ft(u)p Fw(\))d(+)f Ft(\025)p Fw(\()p Ft(u;)c(e)p Fw(\))p Fs(r)2230 1351 y Fp(x)2272 1339 y Fw(\010\()p Ft(\015)2407 1351 y Fp(\014)2452 1339 y Fw(\()p Ft(u)p Fw(\)\))p Ft(;)515 1479 y Fw(where)27 b Ft(u)22 b Fs(2)i(U)964 1449 y Fo(0)1015 1479 y Fw(and)j Fs(j)p Ft(e)19 b Fs(\000)f Ft(e)1379 1491 y Fu(0)1415 1479 y Fs(j)24 b Ft(<)e Fw(\001)1618 1449 y Fo(0)1642 1479 y Fw(.)37 b(Notice)27 b(that)h Ft(\033)s Fw(\()p Ft(u;)14 b(e)p Fw(\))28 b(is)f(real-analytic.)35 b(Indeed,)515 1678 y Fl(Lemma)45 b Ft(\033)s Fw(\()p Ft(u;)14 b(e)p Fw(\))24 b Fv(is)f(a)g(r)l(e)l(al-analytic)i(di\013e)l (omorphism)g(in)f(a)f(neighb)l(orho)l(o)l(d)j(of)d Fw(\()p Ft(u)3135 1690 y Fu(0)3172 1678 y Ft(;)14 b(e)3248 1690 y Fu(0)3285 1678 y Fw(\))p Fv(.)515 1878 y Fl(Pro)s(of:)60 b Fw(Notice)25 b(that)g(the)g(am)n(bien)n(t)g(space,)g Fn(R)2028 1847 y Fp(d)2067 1878 y Fw(,)g(is)g(generated)f(b)n(y)h(the)g (tangen)n(t)g(v)n(ectors)515 1977 y(of)38 b(\000\()p Ft(e)743 1989 y Fu(0)780 1977 y Fw(\))g(at)g Ft(x)1009 1989 y Fu(0)1084 1977 y Fw(\(namely)-7 b(,)41 b(d)1480 1996 y Fp(u)1519 1980 y Fk(\()p FC(j)r Fk(\))1600 1977 y Ft(\015)1643 1989 y Fp(\014)1687 1977 y Fw(\()p Ft(u)1767 1989 y Fu(0)1804 1977 y Fw(\))e(for)e(1)j Fi(6)g Ft(j)45 b Fi(6)40 b Ft(d)25 b Fs(\000)g Fw(1\))38 b(and)g Fs(r)2894 1989 y Fp(x)2936 1977 y Fw(\010\()p Ft(x)3075 1989 y Fu(0)3113 1977 y Fw(\).)68 b(The)515 2077 y(p)r(en)n(ultimate)39 b(lemma)g(and)g(a)f(direct)h(computation)f(sho)n(w)g(that)h(the)g (columns)g(of)g(the)515 2176 y(matrix)27 b(D)848 2191 y Fu(\()p Fp(u;e)p Fu(\))995 2176 y Ft(\033)s Fw(\()p Ft(u)1125 2188 y Fu(0)1162 2176 y Ft(;)14 b(e)1238 2188 y Fu(0)1275 2176 y Fw(\))28 b(in)g(canonical)e(basis)h(are)1495 2344 y(d)1541 2363 y Fp(u)1580 2347 y Fk(\()p FC(j)r Fk(\))1661 2344 y Ft(\015)1704 2356 y Fp(\014)1748 2344 y Fw(\()p Ft(u)1828 2356 y Fu(0)1865 2344 y Fw(\))111 b(for)f(1)23 b Fi(6)g Ft(j)28 b Fi(6)22 b Ft(d)d Fs(\000)f Fw(1)1187 2469 y Ft(@)1231 2481 y Fp(e)1267 2469 y Ft(\025)p Fw(\()p Ft(u)1395 2481 y Fu(0)1432 2469 y Ft(;)c(e)1508 2481 y Fu(0)1545 2469 y Fw(\))p Fs(r)1646 2481 y Fp(x)1688 2469 y Fw(\010\()p Ft(x)1827 2481 y Fu(0)1865 2469 y Fw(\))111 b(for)f Ft(j)29 b Fw(=)22 b Ft(d;)515 2636 y Fw(whic)n(h)k(are)g(linearly)g(indep)r(enden)n(t)i(b)n(y)e(the)h (last)g(lemma.)36 b(The)27 b(In)n(v)n(erse)e(Mapping)i(Theo-)515 2736 y(rem)g(then)h(completes)f(the)h(pro)r(of.)1705 b Fi(\003)639 2935 y Fw(W)-7 b(e)38 b(th)n(us)f(select)g(an)f (arbitrarily)f(small)i(cub)r(e,)j Fs(U)33 b(\002)24 b Fw(])p Ft(e)2447 2947 y Fu(0)2502 2935 y Fs(\000)18 b Fw(\001)q Ft(;)c(e)2731 2947 y Fu(0)2786 2935 y Fw(+)k(\001[)39 b Fs(3)g Fw(\()p Ft(u)3174 2947 y Fu(0)3211 2935 y Ft(;)14 b(e)3287 2947 y Fu(0)3324 2935 y Fw(\),)515 3035 y(con)n(tained)31 b(in)h Fs(U)1056 3005 y Fo(0)1101 3035 y Fs(\002)21 b Fw(])p Ft(e)1249 3047 y Fu(0)1304 3035 y Fs(\000)d Fw(\001)1456 3005 y Fo(0)1480 3035 y Ft(;)c(e)1556 3047 y Fu(0)1611 3035 y Fw(+)k(\001)1763 3005 y Fo(0)1787 3035 y Fw([,)33 b(suc)n(h)f(that)g Ft(\033)s Fw(\()p Ft(u;)14 b(e)p Fw(\))32 b(is)g(a)f(real-analytic)f(di\013eo-)515 3135 y(morphism)d(in)h(the)g (neigh)n(b)r(orho)r(o)r(d)e(of)p 1759 3068 61 4 v 28 w Fs(U)g(\002)18 b Fw([)p Ft(e)1982 3147 y Fu(0)2038 3135 y Fs(\000)g Fw(\001)p Ft(;)c(e)2266 3147 y Fu(0)2321 3135 y Fw(+)k(\001].)639 3234 y(Let)26 b Ft(')p Fw(\()p Ft(u)p Fw(;)14 b Ft(e;)g(!)s Fw(\))23 b(=)g Ft(!)18 b Fs(\001)d Ft(\033)s Fw(\()p Ft(u;)f(e)p Fw(\))26 b(b)r(e)g(the)h(an)n (ticipated)f(phase,)f(where)h Ft(u)c Fs(2)i(U)8 b Fw(,)27 b Fs(j)p Ft(e)15 b Fs(\000)f Ft(e)3231 3246 y Fu(0)3268 3234 y Fs(j)23 b Ft(<)515 3334 y Fw(\001,)43 b(and)d Ft(!)46 b Fs(2)e Fw(S)1067 3304 y Fp(d)p Fo(\000)p Fu(1)1191 3334 y Fw(.)74 b(By)40 b(Theorem)f(10,)j(the)f(asso)r(ciated)d (stationary)h(phase)g(p)r(oin)n(ts)515 3433 y(and)34 b(rank)12 b(D)923 3403 y Fu(2)923 3454 y Fp(u)967 3433 y Ft(')34 b Fw(at)g(these)g(p)r(oin)n(ts)g(are)f(in)n(trinsic)h(prop)r (erties)f(of)g(\000\()p Ft(e)p Fw(\).)57 b(Let)34 b Ft(!)3050 3445 y Fu(0)3120 3433 y Fs(2)g Fw(S)3255 3403 y Fp(d)p Fo(\000)p Fu(1)515 3533 y Fw(b)r(e)39 b(arbitrarily)e(\014xed.)71 b(It)39 b(app)r(ears)f(that)h(if)h Ft(x)2063 3545 y Fu(0)2139 3533 y Fw(is)f(a)g(stationary)e(phase)i(p)r(oin)n(t,)j(then)515 3633 y Ft(@)559 3645 y Fp(e)594 3633 y Ft(')p Fw(\()p Ft(u)728 3645 y Fu(0)766 3633 y Fw(;)14 b Ft(e)842 3645 y Fu(0)879 3633 y Ft(;)g(!)968 3645 y Fu(0)1004 3633 y Fw(\))28 b(is)g(also)e(\\in)n(trinsic")h(in)h(the)g(follo)n(wing)e (sense:)515 3800 y Fl(Theorem)31 b(11)41 b Fv(Supp)l(ose)30 b Ft(x)1420 3812 y Fu(0)1481 3800 y Fw(=)23 b Ft(\033)s Fw(\()p Ft(u)1699 3812 y Fu(0)1736 3800 y Ft(;)14 b(e)1812 3812 y Fu(0)1849 3800 y Fw(\))30 b Fv(is)g(a)g(stationary)h(phase)g(p)l (oint.)39 b(Then,)1407 4007 y Ft(@)1451 4019 y Fp(e)1487 4007 y Ft(')p Fw(\()p Ft(u)1621 4019 y Fu(0)1659 4007 y Fw(;)14 b Ft(e)1735 4019 y Fu(0)1771 4007 y Ft(;)g(!)1860 4019 y Fu(0)1897 4007 y Fw(\))23 b(=)2199 3951 y Fs(\006)p Fw(1)p 2050 3988 404 4 v 2050 4064 a Fs(kr)2161 4076 y Fp(x)2203 4064 y Fw(\010\()p Ft(x)2342 4076 y Fu(0)2380 4064 y Fw(\))p Fs(k)2463 4007 y Ft(:)515 4245 y Fl(Pro)s(of:)57 b Fw(By)23 b(Theorem)g(10,)g(since)g Ft(x)1680 4257 y Fu(0)1741 4245 y Fw(is)h(stationary)-7 b(,)22 b Ft(!)2280 4257 y Fu(0)2340 4245 y Fw(=)h Fs(\006)2537 4205 y Fo(r)2592 4213 y FC(x)2629 4205 y Fu(\010\()p Fp(x)2740 4213 y Fk(0)2772 4205 y Fu(\))p 2502 4226 330 4 v 2502 4274 a Fo(kr)2591 4282 y FC(x)2629 4274 y Fu(\010\()p Fp(x)2740 4282 y Fk(0)2772 4274 y Fu(\))p Fo(k)2842 4245 y Fw(.)36 b(In)23 b(particular,)961 4427 y Ft(@)1005 4439 y Fp(e)1041 4427 y Ft(')p Fw(\()p Ft(u)1175 4439 y Fu(0)1212 4427 y Fw(;)14 b Ft(e)1288 4439 y Fu(0)1325 4427 y Ft(;)g(!)1414 4439 y Fu(0)1451 4427 y Fw(\))83 b(=)g Ft(!)1766 4439 y Fu(0)1821 4427 y Fs(\001)19 b Ft(@)1907 4439 y Fp(e)1942 4427 y Ft(\033)s Fw(\()p Ft(u)2072 4439 y Fu(0)2110 4427 y 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Fw(Consequen)n(tly)-7 b(,)31 b Ft(@)1218 535 y Fp(e)1254 523 y Ft(')p Fw(\()p Ft(u)1388 535 y Fu(0)1425 523 y Fw(;)14 b Ft(e)1501 535 y Fu(0)1538 523 y Ft(;)g(!)1627 535 y Fu(0)1664 523 y Fw(\))29 b Fs(6)p Fw(=)f(0)j(if)g Ft(x)2017 535 y Fu(0)2086 523 y Fw(is)g(a)f(stationary)f(phase)i(p)r(oin)n(t.)47 b(Hence,)515 623 y(whether)27 b Fs(r)902 635 y Fp(u)946 623 y Ft(')p Fw(\()p Ft(u)1080 635 y Fu(0)1117 623 y Fw(;)14 b Ft(e)1193 635 y Fu(0)1230 623 y Ft(;)g(!)1319 635 y Fu(0)1356 623 y Fw(\))23 b Fs(6)p Fw(=)g(0)k(or)1598 794 y Fs(r)1667 806 y Fp(u)1710 794 y Ft(')p Fw(\()p Ft(u)1844 806 y Fu(0)1882 794 y Fw(;)14 b Ft(e)1958 806 y Fu(0)1995 794 y Ft(;)g(!)2084 806 y Fu(0)2120 794 y Fw(\))84 b(=)e(0)p Ft(;)1426 918 y Fw(rank)13 b(D)1667 884 y Fu(2)1667 939 y Fp(u)1710 918 y Ft(')p Fw(\()p Ft(u)1844 930 y Fu(0)1882 918 y Fw(;)h Ft(e)1958 930 y Fu(0)1995 918 y Ft(;)g(!)2084 930 y Fu(0)2120 918 y Fw(\))84 b Fi(>)e Ft(\024;)1631 1043 y(@)1675 1055 y Fp(e)1710 1043 y Ft(')p Fw(\()p Ft(u)1844 1055 y Fu(0)1882 1043 y Fw(;)14 b Ft(e)1958 1055 y Fu(0)1995 1043 y Ft(;)g(!)2084 1055 y Fu(0)2120 1043 y Fw(\))84 b Fs(6)p Fw(=)e(0)p Ft(:)515 1214 y Fw(By)29 b(Theorems)g(1,)h(6,)f(the)h(corollary)e(of)h (Theorem)g(2,)h(and)f(Theorem)g(9)h(there)f(exists)g(an)515 1314 y(arbitrarily)e(small)h(cub)r(e,)i Ft(U)e Fs(\002)18 b Ft(B)29 b Fs(\032)c(U)i(\002)19 b Fn(R)1928 1284 y Fp(d)1967 1314 y Fw(,)29 b(con)n(taining)f(\()p Ft(u)2504 1326 y Fu(0)2541 1314 y Ft(;)14 b(!)2630 1326 y Fu(0)2667 1314 y Fw(\))29 b(and)g(an)f(arbitrarily)515 1413 y(small)f(0)c Ft(<)f(\016)k(<)d Fw(\001)28 b(suc)n(h)f(that)h(the)g(follo)n(wing)e (holds:)722 1570 y(If)37 b Ft(f)9 b Fw(\()p Ft(u)p Fw(;)14 b Ft(e;)g(!)s Fw(\))35 b(is)h(smo)r(oth)h(in)f(the)h(neigh)n(b)r(orho)r (o)r(d)e(of)h Fn(R)2524 1540 y Fp(d)p Fo(\000)p Fu(1)2672 1570 y Fs(\002)24 b Fn(R)g Fs(\002)p 2934 1503 68 4 v 24 w Ft(B)41 b Fw(and)722 1670 y(v)-5 b(anishes)27 b(on)h(\()p Ft(U)f Fs(\002)18 b Fw(])p Ft(e)1425 1682 y Fu(0)1481 1670 y Fs(\000)g Ft(\016)s(;)c(e)1680 1682 y Fu(0)1735 1670 y Fw(+)k Ft(\016)s Fw([\))1913 1628 y Fu(c)1965 1670 y Fs(\002)p 2048 1603 V 18 w Ft(B)5 b Fw(,)27 b(then)1481 1841 y Ft(I)1517 1853 y Fp(f)1561 1841 y Fw(\()p Ft(r)r Fw(;)14 b Ft(e;)g(!)s Fw(\))23 b(=)g Ft(O)r Fw(\()p Ft(r)2079 1807 y Fo(\000)2143 1784 y FC(\024)p 2143 1793 35 3 v 2146 1827 a Fk(2)2191 1841 y Fw(\))56 b(and)1276 2057 y(p)p Ft(:)p Fw(v)q Ft(:)1428 1944 y Fr(Z)1474 2133 y Fq(R)1615 2001 y Fw(1)p 1544 2038 185 4 v 1544 2114 a Ft(\021)21 b Fs(\000)d Ft(e)1738 2057 y(I)1774 2069 y Fp(f)1817 2057 y Fw(\()p Ft(r)r Fw(;)c Ft(\021)s(;)g(!)s Fw(\))20 b(d)p Ft(\021)26 b Fw(=)d Ft(O)r Fw(\()p Ft(r)2450 2023 y Fo(\000)2513 2001 y FC(\024)p 2513 2010 35 3 v 2516 2043 a Fk(2)2562 2057 y Fw(\))p Ft(;)722 2258 y Fw(where)k(b)r(oth)h(estimates)g(are)e(uniform)i(in)g(\()p Ft(e;)14 b(!)s Fw(\))23 b Fs(2)g Fw([)p Ft(e)2429 2270 y Fu(0)2484 2258 y Fs(\000)18 b Ft(\016)s(;)c(e)2683 2270 y Fu(0)2739 2258 y Fw(+)k Ft(\016)s Fw(])g Fs(\002)p 2986 2191 68 4 v 18 w Ft(B)t Fw(.)639 2415 y(Of)37 b(course,)g(the)g (previous)f(cub)r(e)g(and)h(asso)r(ciated)e(constructions)g(dep)r(end)i (on)g(the)515 2514 y(\014xed)j Ft(e)767 2526 y Fu(0)804 2514 y Fw(,)k Ft(!)923 2526 y Fu(0)960 2514 y Fw(,)g(and)c Ft(x)1248 2526 y Fu(0)1331 2514 y Fs(2)k Fw(\000\()p Ft(e)1553 2526 y Fu(0)1591 2514 y Fw(\).)76 b(Rep)r(eating)40 b(this)h(pro)r(cedure)e(for)h(all)h Ft(x)k Fs(2)g Fw(\000\()p Ft(e)3287 2526 y Fu(0)3324 2514 y Fw(\),)515 2614 y(where)26 b Ft(e)793 2626 y Fu(0)857 2614 y Fw(and)h Ft(!)1070 2626 y Fu(0)1134 2614 y Fw(are)f(still)h(\014xed,)g(one)g(obtains)f(a)h (system)f(of)h(real-analytic)e(di\013eomor-)515 2713 y(phisms,)k Fs(f)p Fw(\()p Ft(U)948 2725 y Fp(x)1008 2713 y Fs(\002)18 b Fw(])p Ft(e)1153 2725 y Fu(0)1208 2713 y Fs(\000)g Ft(\016)1328 2725 y Fp(x)1370 2713 y Ft(;)c(e)1446 2725 y Fu(0)1501 2713 y Fw(+)k Ft(\016)1621 2725 y Fp(x)1663 2713 y Fw([)c Ft(;)g(\033)1784 2725 y Fp(x)1826 2713 y Fw(\))p Fs(g)1900 2740 y Fp(x)p Fo(2)p Fu(\000\()p Fp(e)2081 2748 y Fk(0)2113 2740 y Fu(\))2143 2713 y Fw(,)29 b(and)f(a)h(family)f(of)h(cub)r(es,)g Fs(f)p Ft(B)3136 2725 y Fp(\013)3183 2713 y Fs(g)3225 2683 y Fp(N)3225 2734 y(\013)p Fu(=1)3356 2713 y Fw(,)515 2813 y(satisfying)39 b(the)h(ab)r(o)n(v)n(e)e(prop)r(erties,)j(from)f (whic)n(h)f(one)g(extracts)g(a)g(\014nite)h(subsystem,)515 2913 y Fs(f)p Fw(\()p Ft(U)646 2925 y Fp(\013)711 2913 y Fs(\002)18 b Fw(])p Ft(e)856 2925 y Fu(0)911 2913 y Fs(\000)g Ft(\016)1031 2925 y Fp(\013)1079 2913 y Ft(;)c(e)1155 2925 y Fu(0)1210 2913 y Fw(+)k Ft(\016)1330 2925 y Fp(\013)1377 2913 y Fw([)c Ft(;)g(\033)1498 2925 y Fp(\013)1546 2913 y Fw(\))p Fs(g)1620 2883 y Fp(N)1620 2933 y(\013)p Fu(=1)1751 2913 y Ft(;)28 b Fw(whic)n(h)f(co)n(v)n(ers)f(\000\()p Ft(e)2410 2925 y Fu(0)2447 2913 y Fw(\):)1525 3165 y(\000\()p Ft(e)1648 3177 y Fu(0)1685 3165 y Fw(\))e(=)1862 3061 y Fp(N)1846 3086 y Fr([)1828 3262 y Fp(\013)p Fu(=1)1969 3165 y Ft(\033)2016 3177 y Fp(\013)2064 3165 y Fw(\()p 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4491 y(\000\()p Ft(e)638 4503 y Fu(0)675 4491 y Fw(\),)25 b(\006\()p Ft(x;)14 b(e)p Fw(\))25 b(describ)r(es)e(a)h(whole)g(\\closed")e(surface)i (included)g(in)h(\000\()p Ft(e)p Fw(\))f(\(whic)n(h)g(should)515 4590 y(therefore)h(corresp)r(onds)g(to)i(\000\()p Ft(e)p Fw(\),)g(as)f(long)g(as)g(\000\()p Ft(e)2127 4602 y Fu(0)2164 4590 y Fw(\))h(and)f(\000\()p Ft(e)p Fw(\))h(ha)n(v)n(e)e(the)i(same)f (n)n(um)n(b)r(er)515 4690 y(of)h(connected)h(comp)r(onen)n(ts\).)36 b(In)28 b(other)f(w)n(ords,)f(one)i(exp)r(ects)1562 4942 y(\000\()p Ft(e)p Fw(\))c(=)1862 4839 y Fp(N)1846 4864 y Fr([)1828 5039 y Fp(\013)p Fu(=1)1969 4942 y Ft(\033)2016 4954 y Fp(\013)2064 4942 y Fw(\()p Ft(U)2153 4954 y Fp(\013)2200 4942 y Ft(;)14 b(e)p Fw(\))p Ft(:)900 b Fw(\(11\))1905 5255 y(28)p eop end %%Page: 29 29 TeXDict begin 29 28 bop 515 523 a Fw(Ho)n(w)n(ev)n(er,)25 b(it)j(is)g(easier)e(to)i(shorten)f Ft(\016)1708 493 y Fo(0)1759 523 y Fw(in)g(order)g(to)g Fv(pr)l(ove)35 b Fw(the)28 b(previous)f(relation.)639 623 y(This)e(ma)n(y)e(b)r(e)i (done)f(in)h(the)g(follo)n(wing)e(w)n(a)n(y:)34 b(supp)r(ose)24 b(b)n(y)g(con)n(tradiction)f(there)h(do)r(es)515 722 y(not)k(exist)g(a)g Ft(\016)k Fw(suc)n(h)c(that)g(the)h(relation)f (\(11\))g(holds)g(for)f(all)h Ft(e)c Fs(2)h Fw(])p Ft(e)2659 734 y Fu(0)2714 722 y Fs(\000)19 b Ft(\016)s(;)14 b(e)2914 734 y Fu(0)2969 722 y Fw(+)k Ft(\016)s Fw([,)29 b(where)515 822 y(0)i Ft(<)h(\016)j Fi(6)d Ft(\016)894 792 y Fo(0)917 822 y Fw(.)53 b(Then)34 b(there)f(exist)f(a)h(sequence,)h Fs(f)p Ft(e)2165 834 y Fp(n)2209 822 y Fs(g)p Fw(,)g(con)n(v)n(erging)d (to)i Ft(e)2873 834 y Fu(0)2910 822 y Fw(,)h(and)f(p)r(oin)n(ts,)515 932 y Ft(x)562 944 y Fp(n)607 932 y Fw(,)28 b(on)f(\000\()p Ft(e)896 944 y Fp(n)941 932 y Fw(\),)g(suc)n(h)g(that)g Ft(x)1436 944 y Fp(n)1514 932 y Ft(=)-51 b Fs(2)1583 870 y Fr(S)1653 891 y Fp(N)1653 957 y(\013)p Fu(=1)1798 932 y Ft(\033)1845 944 y Fp(\013)1893 932 y Fw(\()p Ft(U)1982 944 y Fp(\013)2029 932 y Ft(;)14 b(e)2105 944 y Fp(n)2150 932 y Fw(\).)37 b(Letting)27 b Fs(S)j Fw(=)2700 870 y Fr(S)2769 957 y Fo(j)p Fp(e)p Fo(\000)p Fp(e)2903 965 y Fk(0)2936 957 y Fo(j)p Fp(<\016)3058 932 y Fw(\000\()p Ft(e)p Fw(\),)e(the)515 1054 y Ft(x)562 1066 y Fp(n)607 1054 y Fw('s)35 b(lie)g(in)g(the)h(compact)e(set)p 1547 987 57 4 v 35 w Fs(S)7 b Fw(,)36 b(so)f(they)g(accum)n(ulate)f(to)n(w)n (ards)f(a)i(certain)f Ft(x)3135 1023 y Fo(\003)3209 1054 y Fs(2)p 3299 987 V 35 w(S)7 b Fw(.)515 1153 y(Going)27 b(to)g(a)g(subsequence,)h(again)e(denoted)i(b)n(y)f Ft(x)2120 1165 y Fp(n)2189 1153 y Fs(!)c Ft(x)2342 1123 y Fo(\003)2380 1153 y Fw(,)28 b(one)f(\014nds)1050 1336 y(\010\()p Ft(x)1189 1301 y Fo(\003)1227 1336 y Fw(\))d(=)e(\010\()30 b(lim)1462 1386 y Fp(n)p Fo(!1)1650 1336 y Ft(x)1697 1348 y Fp(n)1743 1336 y Fw(\))23 b(=)52 b(lim)1886 1386 y Fp(n)p Fo(!1)2073 1336 y Fw(\010\()p Ft(x)2212 1348 y Fp(n)2258 1336 y Fw(\))23 b(=)52 b(lim)2401 1386 y Fp(n)p Fo(!1)2588 1336 y Ft(e)2627 1348 y Fp(n)2695 1336 y Fw(=)22 b Ft(e:)515 1567 y Fw(Therefore,)40 b Ft(x)975 1537 y Fo(\003)1053 1567 y Fw(lies)f(in)1316 1504 y Fr(S)1385 1525 y Fp(N)1385 1592 y(\013)p Fu(=1)1530 1567 y Ft(\033)1577 1579 y Fp(\013)1625 1567 y Fw(\()p Ft(U)1714 1579 y Fp(\013)1787 1567 y Fs(\002)26 b Fw(])p Ft(e)1940 1579 y Fu(0)1995 1567 y Fs(\000)18 b Ft(\016)2118 1537 y Fo(0)2142 1567 y Ft(;)c(e)2218 1579 y Fu(0)2273 1567 y Fw(+)k Ft(\016)2396 1537 y Fo(0)2419 1567 y Fw([\),)43 b(while)c(the)g Ft(x)2969 1579 y Fp(n)3015 1567 y Fw('s)f(do)h(not,)515 1666 y(con)n(tradicting)26 b(the)i(fact)g(that)g(this)g(last)f(region)f(is)i(op)r(en)f(in)h Fn(R)2516 1636 y Fp(d)2555 1666 y Fw(.)639 1766 y(Indeed,)515 1949 y Fl(Theorem)j(12)41 b Fv(Ther)l(e)c(exists)e(a)h Ft(\016)h Fs(2)d Fw(]0)p Ft(;)14 b(\016)1918 1918 y Fo(0)1941 1949 y Fw([)36 b Fv(such)g(that)f(for)i(any)f Ft(e)d Fs(2)i Fw(])p Ft(e)2898 1961 y Fu(0)2953 1949 y Fs(\000)18 b Ft(\016)s(;)c(e)3152 1961 y Fu(0)3207 1949 y Fw(+)k Ft(\016)s Fw([)q Fv(,)515 2048 y Fw(\006\()p Ft(x;)c(e)p Fw(\))28 b Fv(is)h(a)f(r)l(e)l(al-analytic)i(di\013e)l(omorphism)h(b)l (etwe)l(en)d Fw(\000\()p Ft(e)2425 2060 y Fu(0)2462 2048 y Fw(\))g Fv(and)h Fw(\000\()p Ft(e)p Fw(\))f Fv(\(endowe)l(d)h(with) 515 2148 y(their)h(surfac)l(e)g(structur)l(es\).)515 2330 y Fl(Pro)s(of:)94 b Fw(The)37 b(previous)g(paragraph)e(sho)n(ws)h (that)i(\006\()p Ft(x;)14 b(e)p Fw(\))38 b(is)g(surjectiv)n(e)f(for)g Ft(e)g Fw(close)515 2430 y(enough)g(to)g Ft(e)961 2442 y Fu(0)998 2430 y Fw(.)66 b(By)37 b(a)g(similar)g(argumen)n(t)f(supp)r (ose)h(there)g(do)r(es)g(not)h(exist)f(a)g Ft(\016)43 b(>)38 b Fw(0)515 2530 y(suc)n(h)30 b(that)h(\006\()p Ft(x;)14 b(e)p Fw(\))32 b(is)f(injectiv)n(e)g(for)f(ev)n(ery)g Ft(e)e Fs(2)h Fw(])p Ft(e)2153 2542 y Fu(0)2208 2530 y Fs(\000)18 b Ft(\016)s(;)c(e)2407 2542 y Fu(0)2462 2530 y Fw(+)k Ft(\016)s Fw([.)47 b(Then,)32 b(there)f(exist)f(a)515 2629 y(sequence,)d Fs(f)p Ft(e)963 2641 y Fp(n)1007 2629 y Fs(g)p Fw(,)g(con)n(v)n(erging)f(to)h Ft(e)1653 2641 y Fu(0)1690 2629 y Fw(,)h(and)f(p)r(oin)n(ts,)h Ft(x)2222 2641 y Fp(n)2267 2629 y Fw(,)g(on)f(\000\()p Ft(e)2556 2641 y Fp(n)2601 2629 y Fw(\),)h(suc)n(h)g(that)1460 2812 y Ft(x)1507 2824 y Fp(n)1575 2812 y Fw(=)23 b(\006\()p Ft(y)1796 2824 y Fp(n)1841 2812 y Ft(;)14 b(e)1917 2824 y Fp(n)1962 2812 y Fw(\))23 b(=)g(\006\()p Ft(z)2236 2824 y Fp(n)2281 2812 y Ft(;)14 b(e)2357 2824 y Fp(n)2402 2812 y Fw(\))515 2995 y(for)34 b(distinct)i(p)r(oin)n(ts)f Ft(y)1255 3007 y Fp(n)1335 2995 y Fw(and)f Ft(z)1542 3007 y Fp(n)1622 2995 y Fw(on)h(\000\()p Ft(e)1868 3007 y Fu(0)1905 2995 y Fw(\).)59 b(Since)35 b(the)h Ft(x)2441 3007 y Fp(n)2486 2995 y Fw('s)f(accum)n(ulate)f(to)n(w)n(ards)f(a)515 3094 y(certain)27 b Ft(x)839 3064 y Fo(\003)901 3094 y Fs(2)d Fw(\000\()p Ft(e)1103 3106 y Fu(0)1140 3094 y Fw(\),)29 b Ft(y)1265 3106 y Fp(n)1338 3094 y Fw(and)e Ft(z)1538 3106 y Fp(n)1611 3094 y Fw(are)g(ev)n(en)n(tually)g(in)h(the) h(same)e(co)r(ordinates)f(neigh)n(b)r(or-)515 3194 y(ho)r(o)r(d,)h(con) n(tradicting)g(the)h(fact)f(that)h(eac)n(h)f Ft(\033)1971 3206 y Fp(\013)2046 3194 y Fw(is)h(a)f(di\013eomorphism.)535 b Fi(\003)639 3393 y Fw(W)-7 b(e)31 b(limit)g(our)f(considerations)e (to)i(the)h(system)f Fs(f)p Fw(\()p Ft(U)2339 3405 y Fp(\013)2404 3393 y Fs(\002)18 b Fw(])p Ft(e)2549 3405 y Fu(0)2605 3393 y Fs(\000)g Ft(\016)s(;)c(e)2804 3405 y Fu(0)2859 3393 y Fw(+)k Ft(\016)s Fw([)c Ft(;)g(\033)3103 3405 y Fp(\013)3151 3393 y Fw(\))p Fs(g)3225 3363 y Fp(N)3225 3414 y(\013)p Fu(=1)3356 3393 y Fw(,)515 3493 y(where)27 b Ft(\016)k Fw(is)c(sp)r(eci\014ed)h(b)n(y)f(the)h(previous)f(theorem.) 36 b(W)-7 b(e)28 b(ha)n(v)n(e)e(pro)n(v)n(en:)515 3675 y Fl(Theorem)31 b(13)41 b Fv(L)l(et)29 b Ft(!)1252 3687 y Fu(0)1312 3675 y Fs(2)23 b Fw(S)1436 3645 y Fp(d)p Fo(\000)p Fu(1)1560 3675 y Fv(,)30 b Ft(e)1654 3687 y Fu(0)1713 3675 y Fs(2)24 b Fw(])p Ft(a)1859 3645 y Fo(0)1882 3675 y Ft(;)14 b(b)1955 3645 y Fo(0)1978 3675 y Fw([)p Fv(,)30 b(and)f Ft(")23 b(>)g Fw(0)28 b Fv(b)l(e)h(arbitr)l(arily)i (\014xe)l(d.)38 b(Under)515 3775 y(the)29 b(assumptions)h(A)f(and)h(B,) g(ther)l(e)f(exists)g(a)h(\014nite)f(family)i(of)f(cub)l(es)f(of)h (diameters)h(less)515 3875 y(than)f Ft(")p Fv(,)1382 3974 y Fs(f)p Ft(U)1481 3986 y Fp(\013)1546 3974 y Fs(\002)18 b Fw(])p Ft(e)1691 3986 y Fu(0)1747 3974 y Fs(\000)g Ft(\016)s(;)c(e)1946 3986 y Fu(0)2001 3974 y Fw(+)k Ft(\016)s Fw([)g Fs(\002)g Ft(B)5 b Fs(g)2358 3940 y Fp(N)2358 3995 y(\013)p Fu(=1)2488 3974 y Ft(;)515 4124 y Fv(wher)l(e)30 b Fw([)p Ft(e)811 4136 y Fu(0)867 4124 y Fs(\000)18 b Ft(\016)s(;)c(e)1066 4136 y Fu(0)1121 4124 y Fw(+)k Ft(\016)s Fw(])23 b Fs(\032)g Fw(])p Ft(a)1445 4094 y Fo(0)1468 4124 y Ft(;)14 b(b)1541 4094 y Fo(0)1564 4124 y Fw([)30 b Fv(and)g Ft(!)1830 4136 y Fu(0)1890 4124 y Fs(2)23 b Ft(B)k Fs(\032)c Fn(R)2206 4094 y Fp(d)p Fo(\000)p Fu(1)2330 4124 y Fv(,)30 b(and)g(functions,)1219 4323 y Ft(\033)1266 4335 y Fp(\013)1323 4323 y Fw(:)37 b Ft(U)1440 4335 y Fp(\013)1505 4323 y Fs(\002)18 b Fw(])p Ft(e)1650 4335 y Fu(0)1706 4323 y Fs(\000)g Ft(\016)s(;)c(e)1905 4335 y Fu(0)1960 4323 y Fw(+)k Ft(\016)s Fw([)23 b Fs(!)2324 4244 y Fr([)2235 4426 y Fo(j)p Fp(e)p Fo(\000)p Fp(e)2369 4434 y Fk(0)2402 4426 y Fo(j)p Fp(<\016)2520 4323 y Fw(\000\()p Ft(e)p Fw(\))515 4598 y Fv(such)30 b(that)f(the)h(fol)t(lowing)i (holds:)613 4764 y(1.)42 b(F)-6 b(or)33 b(every)h Ft(\013)p Fv(,)48 b Ft(\033)1276 4776 y Fp(\013)1324 4764 y Fw(\()p Ft(u;)14 b(e)p Fw(\))33 b Fv(is)g(a)g(r)l(e)l(al-analytic)h(di\013e)l (omorphism)h(fr)l(om)f(a)f(neighb)l(or-)722 4864 y(ho)l(o)l(d)e(of)p 1011 4797 104 4 v 31 w Ft(U)1068 4876 y Fp(\013)1133 4864 y Fs(\002)18 b Fw([)p Ft(e)1278 4876 y Fu(0)1334 4864 y Fs(\000)g Ft(\016)s(;)c(e)1533 4876 y Fu(0)1588 4864 y Fw(+)k Ft(\016)s Fw(])30 b Fv(to)g(its)f(image.)1905 5255 y Fw(29)p eop end %%Page: 30 30 TeXDict begin 30 29 bop 613 523 a Fv(2.)42 b(F)-6 b(or)30 b(al)t(l)h Ft(e)23 b Fs(2)g Fw(])p Ft(e)1198 535 y Fu(0)1254 523 y Fs(\000)18 b Ft(\016)s(;)c(e)1453 535 y Fu(0)1508 523 y Fw(+)k Ft(\016)s Fw([)p Fv(,)30 b Fw(\000\()p Ft(e)p Fw(\))23 b(=)1975 461 y Fr(S)2044 481 y Fp(N)2044 548 y(\013)p Fu(=1)2189 523 y Ft(\033)2236 535 y Fp(\013)2284 523 y Fw(\()p Ft(U)2373 535 y Fp(\013)2420 523 y Ft(;)14 b(e)p Fw(\))p Fv(.)2573 493 y Fu(8)613 709 y Fv(3.)42 b(L)l(et)24 b(us)f(denote)i(by)f Ft(I)1367 666 y Fu(\()p Fp(\013)p Fu(\))1360 734 y Fp(f)1466 709 y Fw(\()p Ft(r)r Fw(;)14 b Ft(e;)g(!)s Fw(\))25 b Fv(the)f(oscil)t(latory)i(inte)l(gr)l (al)e(of)h(amplitude)g Ft(f)9 b Fw(\()p Ft(u)p Fw(;)14 b Ft(e;)g(!)s Fw(\))722 809 y Fv(with)23 b(r)l(esp)l(e)l(ct)f(to)g(the) h(phase)h Ft(')1657 821 y Fp(\013)1704 809 y Fw(\()p Ft(u)p Fw(;)14 b Ft(e;)g(!)s Fw(\))22 b(=)h Ft(!)5 b Fs(\001)r Ft(\033)2223 821 y Fp(\013)2271 809 y Fw(\()p Ft(u;)14 b(e)p Fw(\))p Fv(.)36 b(If)22 b Ft(f)9 b Fw(\()p Ft(u)p Fw(;)14 b Ft(e;)g(!)s Fw(\))22 b Fv(is)g(smo)l(oth)h(in)722 909 y(the)g(neighb)l(orho)l(o)l(d)i(of)e Fn(R)1489 879 y Fp(d)p Fo(\000)p Fu(1)1615 909 y Fs(\002)r Fn(R)r Fs(\002)p 1811 842 68 4 v 2 w Ft(B)k Fv(and)c(vanishes)h(on)e Fw(\()p Ft(U)2579 921 y Fp(\013)2645 909 y Fs(\002)c Fw(])p Ft(e)2790 921 y Fu(0)2846 909 y Fs(\000)g Ft(\016)s(;)c(e)3045 921 y Fu(0)3100 909 y Fw(+)k Ft(\016)s Fw([\))3278 867 y Fu(c)3314 909 y Fs(\002)p 722 942 V 722 1008 a Ft(B)t Fv(,)31 b(then)1553 1119 y Ft(I)1596 1076 y Fu(\()p Fp(\013)p Fu(\))1589 1144 y Fp(f)1695 1119 y Fw(\()p Ft(r)r Fw(;)14 b Ft(e;)g(!)s Fw(\))24 b(=)e Ft(O)r Fw(\()p Ft(r)2213 1085 y Fo(\000)2277 1063 y FC(\024)p 2277 1072 35 3 v 2280 1105 a Fk(2)2325 1119 y Fw(\))60 b Fv(and)1380 1327 y Fw(p)p Ft(:)p Fw(v)q Ft(:)1531 1214 y Fr(Z)1577 1403 y Fq(R)1719 1271 y Fw(1)p 1647 1308 185 4 v 1647 1384 a Ft(\021)22 b Fs(\000)c Ft(e)1842 1327 y(I)1878 1339 y Fp(f)1921 1327 y Fw(\()p Ft(r)r Fw(;)c Ft(\021)s(;)g(!)s Fw(\))20 b(d)p Ft(\021)26 b Fw(=)d Ft(O)r Fw(\()p Ft(r)2554 1293 y Fo(\000)2617 1270 y FC(\024)p 2617 1279 35 3 v 2620 1313 a Fk(2)2666 1327 y Fw(\))p Ft(;)722 1534 y Fv(wher)l(e)31 b(b)l(oth)f(estimates)g(ar)l(e)g(uniform)g(in)g Fw(\()p Ft(e;)14 b(!)s Fw(\))23 b Fs(2)g Fw([)p Ft(e)2416 1546 y Fu(0)2472 1534 y Fs(\000)18 b Ft(\016)s(;)c(e)2671 1546 y Fu(0)2726 1534 y Fw(+)k Ft(\016)s Fw(])g Fs(\002)p 2973 1467 68 4 v 18 w Ft(B)5 b Fv(.)639 1717 y Fw(W)-7 b(e)30 b(no)n(w)f(construct)g(a)g(smo)r(oth,)h(join)n(t)f(\\partition)g (of)g(unit)n(y")g(sub)r(ordinated)g(to)h(the)515 1816 y(system)d(of)h(neigh)n(b)r(orho)r(o)r(ds)e Fs(f)p Ft(\033)1528 1828 y Fp(\013)1575 1816 y Fw(\()p Ft(U)1664 1828 y Fp(\013)1711 1816 y Ft(;)14 b(e)p Fw(\))p Fs(g)1861 1786 y Fp(N)1861 1837 y(\013)p Fu(=1)1992 1816 y Fw(.)515 1999 y Fl(Theorem)31 b(14)41 b Fv(L)l(et)31 b Fw(0)c Ft(<)1366 1977 y Fw(~)1362 1999 y Ft(\016)j(<)c(\016)35 b Fv(and)d Fs(S)h Fw(=)1930 1937 y Fr(S)1999 2025 y Fo(j)p Fp(e)p Fo(\000)p Fp(e)2133 2033 y Fk(0)2166 2025 y Fo(j)p Fp(<)2241 2009 y Fu(~)2238 2025 y Fp(\016)2288 1999 y Fw(\000\()p Ft(e)p Fw(\))p Fv(.)44 b(Ther)l(e)33 b(exists)e(a)i(family)g(of)515 2118 y(functions,)d Fs(f)p Ft(\037)995 2130 y Fp(\013)1042 2118 y Fw(\()p Ft(x)p Fw(\))q Fs(g)1196 2088 y Fp(N)1196 2139 y(\013)p Fu(=1)1327 2118 y Fv(,)g(smo)l(oth)g(in)g Ft(x)23 b Fs(2)h Fn(R)1975 2088 y Fp(d)2013 2118 y Fv(,)30 b(satisfying:)639 2284 y Fs(\017)41 b Fw(0)23 b Fi(6)f Ft(\037)926 2296 y Fp(\013)997 2284 y Fi(6)g Fw(1)p Fv(,)639 2450 y Fs(\017)41 b Fw(supp)30 b Ft(\037)975 2462 y Fp(\013)1045 2450 y Fs(\032)23 b Ft(\033)1180 2462 y Fp(\013)1228 2450 y Fw(\()p Ft(U)1317 2462 y Fp(\013)1383 2450 y Fs(\002)18 b Fw(])p Ft(e)1528 2462 y Fu(0)1583 2450 y Fs(\000)g Ft(\016)s(;)c(e)1782 2462 y Fu(0)1837 2450 y Fw(+)k Ft(\016)s Fw([\))p Fv(,)639 2627 y Fs(\017)722 2565 y Fr(P)810 2585 y Fp(N)810 2652 y(\013)p Fu(=1)955 2627 y Ft(\037)1007 2639 y Fp(\013)1054 2627 y Fw(\()p Ft(x)p Fw(\))24 b(=)f(1)29 b Fv(for)i(al)t(l)g Ft(x)23 b Fs(2)p 1749 2560 57 4 v 24 w(S)6 b Fv(.)515 2825 y Fl(Pro)s(of:)64 b Fw(Notice)27 b(that)p 1278 2758 V 28 w Fs(S)j(\032)1445 2762 y Fr(S)1515 2783 y Fp(N)1515 2850 y(\013)p Fu(=1)1660 2825 y Ft(\033)1707 2837 y Fp(\013)1755 2825 y Fw(\()p Ft(U)1844 2837 y Fp(\013)1909 2825 y Fs(\002)18 b Fw(])p Ft(e)2054 2837 y Fu(0)2110 2825 y Fs(\000)g Ft(\016)s(;)c(e)2309 2837 y Fu(0)2364 2825 y Fw(+)k Ft(\016)s Fw([\),)28 b(so)f(in)h(particular)790 3007 y Fs(f)p Ft(\033)879 3019 y Fu(1)916 3007 y Fw(\()p Ft(U)1005 3019 y Fu(1)1060 3007 y Fs(\002)19 b Fw(])p Ft(e)1206 3019 y Fu(0)1261 3007 y Fs(\000)f Ft(\016)s(;)c(e)1460 3019 y Fu(0)1515 3007 y Fw(+)k Ft(\016)s Fw([\))p Ft(;)c(:)g(:)g(:)g(;) g(\033)1925 3019 y Fp(N)1989 3007 y Fw(\()p Ft(U)2078 3019 y Fp(N)2159 3007 y Fs(\002)k Fw(])p Ft(e)2304 3019 y Fu(0)2359 3007 y Fs(\000)g Ft(\016)s(;)c(e)2558 3019 y Fu(0)2614 3007 y Fw(+)k Ft(\016)s Fw([\))p Ft(;)c Fn(R)2889 2973 y Fp(d)2946 3007 y Fs(n)p 3006 2941 V 18 w(S)7 b(g)515 3190 y Fw(is)22 b(a)h(\014nite)g(op)r(en)g(co)n(v)n(ering)d(of)j Fn(R)1531 3160 y Fp(d)1570 3190 y Fw(.)35 b(After)23 b(discarding)f Ft(\037)2285 3202 y Fp(N)6 b Fu(+1)2432 3190 y Fw(\()p Ft(x)p Fw(\),)25 b(an)n(y)d(partition)g(of)h(unit)n(y)-7 b(,)515 3290 y Fs(f)p Ft(\037)609 3302 y Fp(\013)656 3290 y Fw(\()p Ft(x)p Fw(\))q Fs(g)810 3254 y Fp(N)6 b Fu(+1)810 3312 y Fp(\013)p Fu(=1)956 3290 y Fw(,)29 b(sub)r(ordinated)f(to)g(the)g(previous)g(co)n(v)n(ering)e(satis\014es) h(the)i(stated)f(prop)r(er-)515 3389 y(ties.)2651 b Fi(\003)515 3705 y Fl(F)-8 b(ourier)32 b(T)-8 b(ransforms)83 b Fw(W)-7 b(e)29 b(are)e(ready)g(to)h(compute)g(\(uniform!\))38 b(deca)n(ys)27 b(of)h(F)-7 b(ourier)515 3804 y(transforms,)30 b Fs(F)8 b Fw(\(\000\()p Ft(e)p Fw(\))q Ft(;)14 b(f)9 b Fw(\),)31 b(for)g(suitable)g(amplitudes)g(and)g(deriv)n(e)f(an)h (analogous)d(result)515 3904 y(for)f(Cauc)n(h)n(y)f(principal)h(v)-5 b(alues.)515 4087 y Fl(Theorem)31 b(15)41 b Fv(L)l(et)34 b Ft(n)e Fw(=)f Ft(r)r(!)38 b Fv(b)l(e)d(the)f(p)l(olar)i(form)f(of)h Ft(n)31 b Fs(2)h Fn(Z)2506 4056 y Fp(d)2545 4087 y Fv(,)k(wher)l(e)g Ft(n)31 b Fs(6)p Fw(=)g(0)p Fv(,)36 b(and)f(let)515 4186 y(the)27 b(amplitude,)j Ft(f)9 b Fw(\()p Ft(x)p Fw(\))p Fv(,)29 b(b)l(e)f(smo)l(oth)g(in)f Ft(x)d Fs(2)f Fn(R)1959 4156 y Fp(d)1998 4186 y Fv(.)38 b(Consider)29 b(any)f(interval)g Fw([)p Ft(a;)14 b(b)p Fw(])23 b Fs(\032)f Fw(])p Ft(a)3211 4156 y Fo(0)3235 4186 y Ft(;)14 b(b)3308 4156 y Fo(0)3330 4186 y Fw([)q Fv(.)515 4286 y(Under)29 b(Assumptions)h(A)f(and)h(B,) 1507 4469 y Fs(F)8 b Fw(\(\000\()p Ft(e)p Fw(\))p Ft(;)14 b(f)9 b Fw(\)\()p Ft(n)p Fw(\))24 b(=)e Ft(O)r Fw(\()p Ft(r)2242 4434 y Fo(\000)2306 4412 y FC(\024)p 2306 4421 35 3 v 2309 4454 a Fk(2)2354 4469 y Fw(\))515 4651 y Fv(uniformly)30 b(in)g Fw(\()p Ft(e;)14 b(!)s Fw(\))23 b Fs(2)h Fw([)p Ft(a;)14 b(b)p Fw(])k Fs(\002)g Fw(S)1602 4621 y Fp(d)p Fo(\000)p Fu(1)1725 4651 y Fv(.)p 515 4725 1146 4 v 607 4781 a FB(8)642 4804 y Fm(In)23 b(other)h(w)n(ords,)e FJ(f)p Fm(\()p Ff(U)1244 4812 y FA(\013)1289 4804 y Ff(;)11 b(\033)1360 4812 y FA(\013)1404 4804 y Fm(\()p FJ(\001)p Ff(;)h(e)p Fm(\)\))q FJ(g)1606 4781 y FA(N)1606 4825 y(\013)p FB(=1)1750 4804 y Fm(is)22 b(a)h(system)g(of)g(real-analytic)g (parameterizations)h(of)e(\000\()p Ff(e)p Fm(\))515 4883 y(for)h Fh(any)29 b Ff(e)19 b FJ(2)h Fm(])p Ff(e)935 4892 y FB(0)985 4883 y FJ(\000)15 b Ff(\016)s(;)c(e)1153 4892 y FB(0)1203 4883 y Fm(+)16 b Ff(\016)s Fm([)o(.)1905 5255 y Fw(30)p eop end %%Page: 31 31 TeXDict begin 31 30 bop 515 523 a Fl(Pro)s(of:)97 b Fw(Let)38 b Ft(e)1069 535 y Fu(0)1148 523 y Fs(2)j Fw([)p Ft(a;)14 b(b)p Fw(])38 b(and)h Ft(!)1670 535 y Fu(0)1748 523 y Fs(2)j Fw(S)1891 493 y Fp(d)p Fo(\000)p Fu(1)2053 523 y Fw(b)r(e)d(\014xed.)70 b(By)38 b(Theorems)g(13)g(and)g(14,)515 623 y(there)i(exist)h(a)g Ft(\016)48 b(>)d Fw(0,)e(a)e(cub)r(e,)j Ft(B)50 b Fs(3)45 b Ft(!)1919 635 y Fu(0)1956 623 y Fw(,)f(a)d(join)n (t)g(system)f(of)h(parameterizations,)515 722 y Fs(f)p Fw(\()p Ft(U)646 734 y Fp(\013)711 722 y Fs(\002)18 b Fw(])p Ft(e)856 734 y Fu(0)911 722 y Fs(\000)g Fw(2)p Ft(\016)s(;)c(e)1152 734 y Fu(0)1207 722 y Fw(+)k(2)p Ft(\016)s Fw([)c Ft(;)g(\033)1493 734 y Fp(\013)1540 722 y Fw(\))q Fs(g)1615 692 y Fp(N)1615 743 y(\013)p Fu(=1)1746 722 y Fw(,)21 b(and)f(a)g(join)n(t)g(partition)g(of)g(unit)n (y)-7 b(,)22 b Fs(f)p Ft(\037)2939 734 y Fp(\013)2986 722 y Fs(g)3028 692 y Fp(N)3028 743 y(\013)p Fu(=1)3159 722 y Fw(,)f(sum-)515 822 y(ming)27 b(at)h(1)f(on)1008 760 y Fr(S)1077 847 y Fo(j)p Fp(e)p Fo(\000)p Fp(e)1211 855 y Fk(0)1244 847 y Fo(j)p Fp(<\016)1366 822 y Fw(\000\()p Ft(e)p Fw(\),)h(suc)n(h)f(that)h(the)g(follo)n(wing)f(holds:)36 b(Let)790 1066 y Ft(I)833 1031 y Fu(\()p Fp(\013)p Fu(\))932 1066 y Fw(\()p Ft(r)r Fw(;)14 b Ft(e;)g(!)s Fw(\))24 b(=)1314 953 y Fr(Z)1360 1141 y Fp(U)1405 1149 y FC(\013)1466 1066 y Fw(e)1503 1031 y Fu(i)p Fp(r)r(!)r Fo(\001)p Fp(\033)1657 1039 y FC(\013)1698 1031 y Fu(\()p Fp(u;e)p Fu(\))1845 1066 y Ft(f)9 b Fw(\()p Ft(\033)1974 1078 y Fp(\013)2021 1066 y Fw(\()p Ft(u;)14 b(e)p Fw(\)\))p Ft(\037)2293 1078 y Fp(\013)2341 1066 y Fw(\()p Ft(\033)2420 1078 y Fp(\013)2468 1066 y Fw(\()p Ft(u;)g(e)p Fw(\)\))p Ft(J)2734 1078 y Fp(\013)2781 1066 y Fw(\()p Ft(u;)g(e)p Fw(\))k(d)p Ft(u;)515 1303 y Fw(where)27 b Ft(J)801 1315 y Fp(\013)848 1303 y Fw(\()p Ft(u;)14 b(e)p Fw(\))k(d)p Ft(u)28 b Fw(is)f(the)h (elemen)n(t)g(of)f(surface)g(of)h Ft(\033)2225 1315 y Fp(\013)2272 1303 y Fw(\()p Ft(U)2361 1315 y Fp(\013)2409 1303 y Ft(;)14 b(e)p Fw(\).)36 b(Then,)1348 1567 y Fs(F)8 b Fw(\(\000\()p Ft(e)p Fw(\))p Ft(;)14 b(f)9 b Fw(\)\()p Ft(r)r(!)s Fw(\))24 b(=)2026 1463 y Fp(N)1995 1488 y Fr(X)1991 1664 y Fp(\013)p Fu(=1)2132 1567 y Ft(I)2175 1533 y Fu(\()p Fp(\013)p Fu(\))2275 1567 y Fw(\()p Ft(r)r Fw(;)14 b Ft(e;)g(!)s Fw(\))515 1837 y(for)24 b(ev)n(ery)g(\()p Ft(e;)14 b(!)s Fw(\))23 b Fs(2)h Fw([)p Ft(e)1215 1849 y Fu(0)1270 1837 y Fs(\000)18 b Ft(\016)s(;)c(e)1469 1849 y Fu(0)1524 1837 y Fw(+)k Ft(\016)s Fw(])c Fs(\002)p 1763 1770 68 4 v 14 w Ft(B)t Fw(.)36 b(Since)25 b Ft(\037)2155 1849 y Fp(\013)2216 1837 y Fs(\016)14 b Ft(\033)2319 1849 y Fp(\013)2366 1837 y Fw(,)26 b(and)f(hence)g(the)h(in)n(tegrand)e (of)515 1937 y Ft(I)558 1907 y Fu(\()p Fp(\013)p Fu(\))657 1937 y Fw(\()p Ft(r)r Fw(;)14 b Ft(e;)g(!)s Fw(\))21 b(v)-5 b(anish)20 b(outside)g Ft(U)1534 1949 y Fp(\013)1585 1937 y Fs(\002)t Fw(])o Ft(e)1715 1949 y Fu(0)1770 1937 y Fs(\000)e Fw(2)p Ft(\016)s(;)c(e)2011 1949 y Fu(0)2066 1937 y Fw(+)k(2)p Ft(\016)s Fw([,)k(b)n(y)d(Theorem)h(13,)h Ft(I)2921 1907 y Fu(\()p Fp(\013)p Fu(\))3020 1937 y Fw(\()p Ft(r)r Fw(;)14 b Ft(e;)g(!)s Fw(\))23 b(=)515 2036 y Ft(O)r Fw(\()p Ft(r)651 2006 y Fo(\000)714 1984 y FC(\024)p 714 1993 35 3 v 717 2026 a Fk(2)763 2036 y Fw(\))28 b(uniformly)f(in)h(\()p Ft(e;)14 b(!)s Fw(\))23 b Fs(2)g Fw([)p Ft(e)1656 2048 y Fu(0)1712 2036 y Fs(\000)18 b Ft(\016)s(;)c(e)1911 2048 y Fu(0)1966 2036 y Fw(+)k Ft(\016)s Fw(])h Fs(\002)p 2214 1970 68 4 v 18 w Ft(B)t Fw(.)37 b(Th)n(us,)1485 2219 y Fs(F)8 b Fw(\(\000\()p Ft(e)p Fw(\))p Ft(;)14 b(f)9 b Fw(\)\()p Ft(r)r(!)s Fw(\))24 b(=)f Ft(O)r Fw(\()p Ft(r)2265 2185 y Fo(\000)2328 2162 y FC(\024)p 2328 2171 35 3 v 2331 2205 a Fk(2)2377 2219 y Fw(\))515 2402 y(uniformly)29 b(in)g([)p Ft(e)1055 2414 y Fu(0)1111 2402 y Fs(\000)18 b Ft(\016)s(;)c(e)1310 2414 y Fu(0)1365 2402 y Fw(+)k Ft(\016)s Fw(])i Fs(\002)p 1615 2335 68 4 v 19 w Ft(B)t Fw(.)41 b(Since)30 b Ft(e)2004 2414 y Fu(0)2066 2402 y Fs(2)c Fw([)p Ft(a;)14 b(b)p Fw(])29 b(and)g Ft(!)2554 2414 y Fu(0)2617 2402 y Fs(2)d Fw(S)2744 2371 y Fp(d)p Fo(\000)p Fu(1)2897 2402 y Fw(are)i(arbitrary) -7 b(,)515 2501 y(the)28 b(result)f(follo)n(ws)g(from)g(the)h (compactness)e(of)i([)p Ft(a;)14 b(b)p Fw(])k Fs(\002)g Fw(S)2386 2471 y Fp(d)p Fo(\000)p Fu(1)2510 2501 y Fw(.)781 b Fi(\003)515 2817 y Fl(Cauc)m(h)m(y)41 b(Principal)e(V)-8 b(alue)83 b Fw(An)35 b(analogous)c(result)j(for)f(principal)h(v)-5 b(alues)33 b(ma)n(y)h(b)r(e)515 2916 y(deriv)n(ed)27 b(when)g(a\013ecting)h(the)g(amplitude)g(b)n(y)f(a)g(cuto\013)h (function.)639 3016 y(W)-7 b(e)25 b(establish)e(\014rst)h(that)g(the)g (elemen)n(t)g(of)g(surface)f(on)g(an)n(y)h(lo)r(cal)f(part)g(of)h (\000\()p Ft(e)p Fw(\))g(v)-5 b(aries)515 3115 y(smo)r(othly)38 b(in)h(\()p Ft(u;)14 b(e)p Fw(\).)69 b(Ha)n(ving)37 b(in)i(hand)g(a)f (common)g(system)g(of)g(parameterizations,)515 3215 y Ft(\033)562 3227 y Fp(\013)609 3215 y Fw(\()p Ft(u;)14 b(e)p Fw(\),)35 b(for)e Ft(u)g Fs(2)g Ft(U)1214 3227 y Fp(\013)1295 3215 y Fw(and)h Fs(j)p Ft(e)22 b Fs(\000)g Ft(e)1673 3227 y Fu(0)1710 3215 y Fs(j)33 b Ft(<)f(\016)37 b Fw(\(where)c Ft(e)2254 3227 y Fu(0)2324 3215 y Fs(2)h Fw(])p Ft(a)2480 3185 y Fo(0)2503 3215 y Ft(;)14 b(b)2576 3185 y Fo(0)2599 3215 y Fw([)34 b(is)f(arbitrarily)f(\014xed\),)515 3315 y(let)i(us)g(compute)g(this)h(elemen)n(t)f(of)g(surface.)55 b(T)-7 b(o)34 b(this)g(end,)i(one)d(considers)g(the)i(deter-)515 3426 y(minan)n(ts,)i Ft(M)954 3383 y Fu(\()p Fp(l)p Fu(\))945 3436 y Fp(\013)1031 3426 y Fw(,)h(of)d(the)h(submatrices)f(of)g(format) g(\()p Ft(d)24 b Fs(\000)g Fw(1\))f Fs(\002)g Fw(\()p Ft(d)i Fs(\000)e Fw(1\))35 b(obtained)g(from)515 3542 y([)p Ft(@)582 3561 y Fp(u)621 3544 y Fk(\()p FC(k)q Fk(\))707 3542 y Ft(\033)757 3499 y Fu(\()p Fp(l)p Fu(\))754 3551 y Fp(\013)834 3542 y Fw(\()p Ft(u;)14 b(e)p Fw(\)])1045 3565 y Fp(k)q(;l)1155 3542 y Fw(b)n(y)27 b(remo)n(ving)f(its)i Ft(l)1773 3512 y Fp(th)1868 3542 y Fw(column,)g(where)f Ft(\033)2469 3554 y Fp(\013)2539 3542 y Fw(=)c(\()p Ft(\033)2709 3499 y Fu(\(1\))2706 3551 y Fp(\013)2799 3542 y Ft(;)14 b(:)g(:)g(:)f(;)h(\033)3033 3499 y Fu(\()p Fp(d)p Fu(\))3030 3551 y Fp(\013)3124 3542 y Fw(\).)38 b(Let)860 3746 y Ft(J)906 3758 y Fp(\013)953 3746 y Fw(\()p Ft(u;)14 b(e)p Fw(\))23 b(=)g(\()p Ft(M)1374 3711 y Fu(\(1\))1365 3766 y Fp(\013)1463 3746 y Fw(\()p Ft(u;)14 b(e)p Fw(\))p Ft(;)g Fs(\000)p Ft(M)1843 3711 y Fu(\(2\))1834 3766 y Fp(\013)1930 3746 y Fw(\()p Ft(u;)g(e)p Fw(\))p Ft(;)g Fs(\001)g(\001)g(\001)27 b Ft(;)14 b Fw(\()p Fs(\000)p Fw(1\))2487 3711 y Fp(d)p Fo(\000)p Fu(1)2611 3746 y Ft(M)2701 3711 y Fu(\()p Fp(d)p Fu(\))2692 3766 y Fp(\013)2791 3746 y Fw(\()p Ft(u;)g(e)p Fw(\)\))p Ft(:)515 3928 y Fw(By)27 b(de\014nition)h(the)g(elemen)n(t)g(of)f(surface)g(is)h Fs(k)p Ft(J)2010 3940 y Fp(\013)2056 3928 y Fw(\()p Ft(u;)14 b(e)p Fw(\))p Fs(k)k Fw(d)p Ft(u)p Fw(.)515 4111 y Fl(Theorem)31 b(16)41 b Fs(k)p Ft(J)1145 4123 y Fp(\013)1192 4111 y Fw(\()p Ft(u;)14 b(e)p Fw(\))p Fs(k)22 b Fw(=)h Fs(kr)1643 4123 y Fp(x)1684 4111 y Fw(\010\()p Ft(\033)1823 4123 y Fp(\013)1871 4111 y Fw(\()p Ft(u;)14 b(e)p Fw(\)\))p Fs(k)k(j)c Fw(det)g(D)2380 4126 y Fu(\()p Fp(u;e)p Fu(\))2527 4111 y Ft(\033)2574 4123 y Fp(\013)2622 4111 y Fw(\()p Ft(u;)g(e)p Fw(\))p Fs(j)p Ft(:)515 4294 y Fl(Pro)s(of:)64 b Fw(The)27 b(c)n(hain)g(rule)h(applied)f(to)h(\010\()p Ft(\033)1925 4306 y Fp(\013)1973 4294 y Fw(\()p Ft(u;)14 b(e)p Fw(\)\))23 b(=)f Ft(e)28 b Fw(giv)n(es)1200 4476 y Fs(r)1269 4488 y Fp(x)1311 4476 y Fw(\010\()p Ft(\033)1450 4488 y Fp(\013)1498 4476 y Fw(\()p Ft(u;)14 b(e)p Fw(\)\)D)1781 4491 y Fu(\()p Fp(u;e)p Fu(\))1927 4476 y Ft(\033)1974 4488 y Fp(\013)2022 4476 y Fw(\()p Ft(u;)g(e)p Fw(\))23 b(=)g([0)31 b Fs(\001)14 b(\001)g(\001)33 b Fw(0)18 b(1])p Ft(;)515 4659 y Fw(whic)n(h)26 b(w)n(e)g(abbreviate)g Fs(r)t Fw(\010D)t Ft(\033)1527 4671 y Fp(\013)1598 4659 y Fw(=)d Ft(e)1725 4671 y Fp(d)1763 4629 y Fu(t)1793 4659 y Fw(.)37 b(Since)27 b Ft(\033)2116 4671 y Fp(\013)2163 4659 y Fw(\()p Ft(u;)14 b(e)p Fw(\))27 b(is)f(a)g(di\013eomorphism,)g (D)t Ft(\033)3248 4671 y Fp(\013)3323 4659 y Fw(is)515 4759 y(in)n(v)n(ertible.)36 b(Th)n(us,)1167 4976 y Fs(r)t Fw(\010)23 b(=)g Ft(e)1450 4988 y Fp(d)1488 4941 y Fu(t)1518 4976 y Fw(\(D)t Ft(\033)1664 4988 y Fp(\013)1713 4976 y Fw(\))1745 4941 y Fo(\000)p Fu(1)1857 4976 y Fw(=)2080 4920 y(1)p 1955 4957 292 4 v 1955 5033 a(det)14 b(D)t Ft(\033)2198 5045 y Fp(\013)2256 4976 y Ft(e)2295 4988 y Fp(d)2334 4941 y Fu(t)2364 4976 y Fw(\(adjD)t Ft(\033)2623 4988 y Fp(\013)2672 4976 y Fw(\))p Ft(;)1905 5255 y Fw(31)p eop end %%Page: 32 32 TeXDict begin 32 31 bop 515 523 a Fw(where)27 b(adj)g(stands)h(for)f (the)h(classical)e(adjoin)n(t.)37 b(The)27 b(result)h(follo)n(ws)e (from)1369 706 y Ft(e)1408 718 y Fp(d)1446 671 y Fu(t)1476 706 y Fw(\(adjD)t Ft(\033)1735 718 y Fp(\013)1784 706 y Fw(\))d(=)g(\()p Fs(\000)p Fw(1\))2098 671 y Fp(d)p Fo(\000)p Fu(1)2221 706 y Ft(J)2267 718 y Fp(\013)2314 706 y Fw(\()p Ft(u;)14 b(e)p Fw(\))p Ft(:)3314 888 y Fi(\003)515 1187 y Fl(Corollary)60 b Fs(k)p Ft(J)1056 1199 y Fp(\013)1103 1187 y Fw(\()p Ft(u;)14 b(e)p Fw(\))p Fs(k)29 b Fv(is)h(smo)l(oth)g(in)g Fw(\()p Ft(u;)14 b(e)p Fw(\))23 b Fs(2)g Ft(U)2181 1199 y Fp(\013)2247 1187 y Fs(\002)18 b Fw(])p Ft(e)2392 1199 y Fu(0)2447 1187 y Fs(\000)g Ft(\016)s(;)c(e)2646 1199 y Fu(0)2702 1187 y Fw(+)k Ft(\016)s Fw([)p Fv(.)639 1386 y Fw(F)-7 b(or)30 b(\014xed)f Ft(!)1045 1398 y Fu(0)1109 1386 y Fs(2)f Fw(S)1238 1356 y Fp(d)p Fo(\000)p Fu(1)1391 1386 y Fw(and)i Ft(e)1594 1398 y Fu(0)1658 1386 y Fs(2)d Fw(])p Ft(a)1807 1356 y Fo(0)1830 1386 y Ft(;)14 b(b)1903 1356 y Fo(0)1926 1386 y Fw([,)31 b(let)f Fs(f)p Fw(\()p Ft(U)2256 1398 y Fp(\013)2321 1386 y Fs(\002)18 b Fw(])p Ft(e)2466 1398 y Fu(0)2522 1386 y Fs(\000)g Fw(3)p Ft(\016)r(;)c(e)2762 1398 y Fu(0)2818 1386 y Fw(+)k(3)p Ft(\016)r Fw([)c Ft(;)g(\033)3103 1398 y Fp(\013)3151 1386 y Fw(\))p Fs(g)3225 1356 y Fp(N)3225 1407 y(\013)p Fu(=1)3356 1386 y Fw(,)515 1486 y Ft(B)27 b Fs(3)c Ft(!)735 1498 y Fu(0)772 1486 y Fw(,)k(and)g Fs(f)p Ft(\037)1077 1498 y Fp(\013)1124 1486 y Fs(g)1166 1456 y Fp(N)1166 1507 y(\013)p Fu(=1)1296 1486 y Fw(,)g(summing)g(at)g (1)f(on)1984 1424 y Fr(S)2054 1511 y Fo(j)p Fp(e)p Fo(\000)p Fp(e)2188 1519 y Fk(0)2221 1511 y Fo(j)p Fp(<)p Fu(2)p Fp(\016)2376 1486 y Fw(\000\()p Ft(e)p Fw(\),)h(b)r(e)g(giv)n(en)f(b)n (y)g(Theorems)515 1604 y(13)j(and)g(14.)43 b(Let)30 b(us)g(de\014ne)g (a)g(cuto\013)g(function,)h(0)26 b Fi(6)h Ft(\037)2313 1616 y Fp(e)2344 1624 y Fk(0)2381 1604 y Fw(\()p Ft(x)p Fw(\))h Fi(6)e Fw(1,)k(smo)r(oth)g(in)g Ft(x)d Fs(2)g Fn(R)3317 1574 y Fp(d)3356 1604 y Fw(,)515 1704 y(suc)n(h)g(that)868 1674 y Fu(9)1211 1833 y Ft(\037)1263 1845 y Fp(e)1294 1853 y Fk(0)1331 1833 y Fw(\()p Ft(x)p Fw(\))e(=)1554 1716 y Fr(\032)1658 1782 y Fw(1)138 b(if)h Fs(j)p Fw(\010\()p Ft(x)p Fw(\))19 b Fs(\000)f Ft(e)2360 1794 y Fu(0)2397 1782 y Fs(j)23 b Ft(<)g(\016)o(;)1658 1882 y Fw(0)138 b(if)h Fs(j)p Fw(\010\()p Ft(x)p Fw(\))19 b Fs(\000)f Ft(e)2360 1894 y Fu(0)2397 1882 y Fs(j)23 b Ft(>)g Fw(2)p Ft(\016)o(:)515 2032 y Fw(Notice)i(that)g Ft(\037)1002 2044 y Fp(e)1033 2052 y Fk(0)1070 2032 y Fw(\()p Ft(x)p Fw(\))h(is)e(analytic)g(in)h(the)h(neigh)n(b)r(orho)r(o)r(d)d(of)i (\000\()p Ft(e)2568 2044 y Fu(0)2605 2032 y Fw(\)|whic)n(h)g(will)g(b)r (ecome)515 2132 y(imp)r(ortan)n(t)i(later.)36 b(Under)28 b(Assumptions)f(A)h(and)g(B,)515 2314 y Fl(Theorem)j(17)41 b Fv(F)-6 b(or)30 b Ft(f)9 b Fw(\()p Ft(x)p Fw(\))30 b Fv(smo)l(oth)h(in)e Ft(x)24 b Fs(2)f Fn(R)1996 2284 y Fp(d)2065 2314 y Fv(and)30 b Ft(r)c(>)c Fw(0)1137 2539 y(p)p Ft(:)p Fw(v)q Ft(:)1288 2426 y Fr(Z)1334 2615 y Fq(R)1475 2483 y Fw(1)p 1404 2520 185 4 v 1404 2596 a Ft(\021)g Fs(\000)c Ft(e)1598 2539 y Fs(F)8 b Fw(\(\000\()p Ft(\021)s Fw(\))q Ft(;)14 b(\037)1948 2551 y Fp(e)1979 2559 y Fk(0)2016 2539 y Ft(f)9 b Fw(\)\()p Ft(r)r(!)s Fw(\))19 b(d)p Ft(\021)27 b Fw(=)22 b Ft(O)r Fw(\()p Ft(r)2612 2505 y Fo(\000)2676 2483 y FC(\024)p 2676 2492 35 3 v 2679 2525 a Fk(2)2724 2539 y Fw(\))515 2777 y Fv(uniformly)30 b(in)g Fw(\()p Ft(e;)14 b(!)s Fw(\))23 b Fs(2)h Fw([)p Ft(e)1354 2789 y Fu(0)1409 2777 y Fs(\000)18 b Ft(\016)s(;)c(e)1608 2789 y Fu(0)1663 2777 y Fw(+)k Ft(\016)s Fw(])h Fs(\002)f Fw(S)1957 2747 y Fp(d)p Fo(\000)p Fu(1)2081 2777 y Fv(.)515 2959 y Fl(Pro)s(of:)64 b Fw(The)27 b(considered)g(principal)g(v)-5 b(alue)28 b(is)f(equal)g(to)1363 3219 y(p)p Ft(:)p Fw(v)q Ft(:)1514 3106 y Fr(Z)1560 3294 y Fq(R)1702 3162 y Fw(1)p 1630 3200 185 4 v 1630 3276 a Ft(\021)22 b Fs(\000)c Ft(e)1873 3115 y Fp(N)1842 3140 y Fr(X)1838 3316 y Fp(\013)p Fu(=1)1979 3219 y Ft(I)2022 3175 y Fu(\()p Fp(\013)p Fu(\))2015 3243 y Fp(F)2057 3251 y FC(\013)2122 3219 y Fw(\()p Ft(r)r Fw(;)c Ft(\021)s(;)g(!)s Fw(\))19 b(d)p Ft(\021)s(;)515 3477 y Fw(where)27 b Ft(F)808 3489 y Fp(\013)856 3477 y Fw(\()p Ft(u;)14 b(\021)s Fw(\))23 b(=)f(\()p Ft(\037)1243 3489 y Fp(e)1274 3497 y Fk(0)1312 3477 y Ft(f)9 b(\037)1414 3489 y Fp(\013)1461 3477 y Fw(\)\()p Ft(\033)1572 3489 y Fp(\013)1620 3477 y Fw(\()p Ft(u;)14 b(\021)s Fw(\)\))p Fs(k)p Ft(J)1933 3489 y Fp(\013)1980 3477 y Fw(\()p Ft(u;)g(\021)s Fw(\))p Fs(k)p Ft(:)28 b Fw(Notice)f(that)1155 3710 y Ft(\037)1207 3722 y Fp(e)1238 3730 y Fk(0)1275 3710 y Fw(\()p Ft(\033)1354 3722 y Fp(\013)1402 3710 y Fw(\()p Ft(u;)14 b(\021)s Fw(\)\))24 b(=)1738 3593 y Fr(\032)1842 3659 y Fw(1)138 b(if)h Fs(j)p Ft(\021)21 b Fs(\000)e Ft(e)2417 3671 y Fu(0)2453 3659 y Fs(j)24 b Ft(<)e(\016)o(;)1842 3759 y Fw(0)138 b(if)h Fs(j)p Ft(\021)21 b Fs(\000)e Ft(e)2417 3771 y Fu(0)2453 3759 y Fs(j)24 b Ft(>)e Fw(2)p Ft(\016)o(:)515 3951 y Fw(Hence,)28 b Ft(F)838 3963 y Fp(\013)885 3951 y Fw(\()p Ft(u;)14 b(\021)s Fw(\))28 b(is)g(smo)r(oth)f(on)g Fn(R)1658 3920 y Fp(d)p Fo(\000)p Fu(1)1800 3951 y Fs(\002)18 b Fn(R)28 b Fw(and)f(v)-5 b(anishes)28 b(when)1378 4133 y(\()p Ft(u;)14 b(\021)s Fw(\))32 b Ft(=)-51 b Fs(2)24 b Ft(U)1730 4145 y Fp(\013)1795 4133 y Fs(\002)18 b Fw(])p Ft(e)1940 4145 y Fu(0)1995 4133 y Fs(\000)g Fw(2)p Ft(\016)s(;)c(e)2236 4145 y Fu(0)2291 4133 y Fw(+)k(2)p Ft(\016)s Fw([)c Ft(:)515 4316 y Fw(By)37 b(Theorem)f(13)h(the)h(principal)f(v)-5 b(alue)37 b(under)g(consideration)f(is)i(th)n(us)f Ft(O)r Fw(\()p Ft(r)3054 4286 y Fo(\000)3118 4263 y FC(\024)p 3118 4272 35 3 v 3121 4306 a Fk(2)3166 4316 y Fw(\))h(uni-)515 4416 y(formly)29 b(in)h(\()p Ft(e;)14 b(!)s Fw(\))26 b Fs(2)g Fw([)p Ft(e)1243 4428 y Fu(0)1299 4416 y Fs(\000)18 b Ft(\016)s(;)c(e)1498 4428 y Fu(0)1553 4416 y Fw(+)k Ft(\016)s Fw(])i Fs(\002)p 1803 4349 68 4 v 19 w Ft(B)t Fw(.)43 b(Since)30 b Ft(!)2207 4428 y Fu(0)2270 4416 y Fs(2)c Fw(S)2397 4385 y Fp(d)p Fo(\000)p Fu(1)2551 4416 y Fw(is)j(arbitrarily)f(\014xed,)i(the)515 4515 y(result)d(follo)n(ws)g(from)g(compactness)f(of)i(S)1836 4485 y Fp(d)p Fo(\000)p Fu(1)1960 4515 y Fw(.)1331 b Fi(\003)p 515 4668 1146 4 v 607 4722 a FB(9)642 4745 y Fm(Since)24 b(the)g(amplitude)f(w)n(e)h(will)d(use)j(do)r(es)g(not)f (dep)r(end)i(on)f Ff(!)r Fm(,)e(the)j(follo)n(wing)d(cuto\013)i (function)g(do)r(es)515 4824 y(not)g(dep)r(end)h(on)f Ff(!)1027 4833 y FB(0)1062 4824 y Fm(;)f(in)g(other)h(circumstances)h (if)e(it)g(do)r(es,)h(the)h(follo)n(wing)e(argumen)n(t)h(still)f (holds.)1905 5255 y Fw(32)p eop end %%Page: 33 33 TeXDict begin 33 32 bop 515 523 a Fl(Analyticit)m(y)38 b(of)f(F)-8 b(ourier)38 b(T)-8 b(ransform)83 b Fw(W)-7 b(e)33 b(close)f(this)g(section)g(b)n(y)h(sho)n(wing)e(that)515 623 y(for)d(an)n(y)f(\014xed)h Ft(n)d Fs(2)f Fn(Z)1210 593 y Fp(d)1278 623 y Fw(the)k(follo)n(wing)g(F)-7 b(ourier)27 b(transform,)g Fs(F)8 b Fw(\(\000\()p Ft(e)p Fw(\))q Ft(;)14 b(f)9 b Fw(\)\()p Ft(n)p Fw(\),)29 b(is)f(analytic)515 722 y(at)g Ft(e)656 734 y Fu(0)722 722 y Fw(when)g(the)h(amplitude,)h Ft(f)9 b Fw(\()p Ft(x)p Fw(\),)29 b(is)g(analytic)e(in)i(the)g(neigh)n (b)r(orho)r(o)r(d)f(of)g(\000\()p Ft(e)3097 734 y Fu(0)3134 722 y Fw(\))d Fs(\032)f Fn(R)3340 692 y Fp(d)515 822 y Fw(\(where)g Ft(e)823 834 y Fu(0)883 822 y Fs(2)f Fw(])p Ft(a)1028 792 y Fo(0)1051 822 y Ft(;)14 b(b)1124 792 y Fo(0)1147 822 y Fw([)24 b(is)h(arbitrarily)d(\014xed\).)36 b(T)-7 b(o)24 b(this)g(end)h(the)f(di\013eomorphism)g(b)r(et)n(w)n(een) 515 922 y(\000\()p Ft(e)638 934 y Fu(0)675 922 y Fw(\))k(and)g(\000\()p Ft(e)p Fw(\),)g(\006\()p Ft(x;)14 b(e)p Fw(\),)28 b(de\014ned)h(in)f (the)g(relation)f(\(10\))g(is)h(helpful,)h(b)r(ecause,)e(giv)n(en)g(a) 515 1021 y(system)h(of)g Fv(disjoint)37 b Fw(op)r(en)29 b(neigh)n(b)r(orho)r(o)r(ds)d(on)i(\000\()p Ft(e)2186 1033 y Fu(0)2223 1021 y Fw(\))h(co)n(v)n(ering)d(all)i(\000\()p Ft(e)2849 1033 y Fu(0)2886 1021 y Fw(\))g(except)g(a)g(set)515 1121 y(of)j(area)f(zero,)i(its)g(lifting)g(to)f(\000\()p Ft(e)p Fw(\))h(via)f(\006\()p Ft(x;)14 b(e)p Fw(\))32 b(also)f(co)n(v)n(ers)e(the)j(whole)f(\000\()p Ft(e)p Fw(\))h(except)f(a)515 1220 y(set)c(of)h(area)e(zero.)639 1320 y(In)i(details,)f(let)h Fs(f)p Fw(\()p Ft(U)1280 1332 y Fp(\013)1346 1320 y Fs(\002)18 b Fw(])p Ft(e)1491 1332 y Fu(0)1546 1320 y Fs(\000)g Ft(\016)s(;)c(e)1745 1332 y Fu(0)1800 1320 y Fw(+)k Ft(\016)s Fw([)c Ft(;)g(\033)2044 1332 y Fp(\013)2092 1320 y Fw(\))p Fs(g)2166 1290 y Fp(N)2166 1341 y(\013)p Fu(=1)2325 1320 y Fw(b)r(e)28 b(giv)n(en)f(b)n(y)g (Theorem)g(13.)515 1468 y Fl(Theorem)k(18)41 b Fv(Ther)l(e)31 b(exists)e(a)h(\014nite,)g(joint)g(system)g(of)h(lo)l(c)l(al)f(p)l(ar)l (ameterizations,)1369 1616 y Fs(f)p Fw(\()p Ft(V)1491 1628 y Fp(\014)1555 1616 y Fs(\002)18 b Fw(])p Ft(e)1700 1628 y Fu(0)1755 1616 y Fs(\000)g Ft(\016)s(;)c(e)1954 1628 y Fu(0)2010 1616 y Fw(+)k Ft(\016)s Fw([)c Ft(;)g(\033)2254 1628 y Fp(\014)2299 1616 y Fw(\))p Fs(g)2373 1582 y Fp(M)2373 1637 y(\014)s Fu(=1)2501 1616 y Ft(;)515 1764 y Fv(such)30 b(that:)639 1903 y Fs(\017)41 b Fv(F)-6 b(or)22 b(al)t(l)h Ft(\014)28 b Fs(2)23 b(f)p Fw(1)p Ft(;)14 b(:)g(:)g(:)f(;)h(M)8 b Fs(g)22 b Fv(ther)l(e)g(exists)f(an)h Ft(\013)2136 1915 y Fp(\014)2204 1903 y Fs(2)h(f)p Fw(1)p Ft(;)14 b(:)g(:)g(:)f(;)h(N)9 b Fs(g)21 b Fv(such)h(that)g Ft(V)3080 1915 y Fp(\014)3148 1903 y Fs(\022)h Ft(U)3293 1915 y Fp(\013)3336 1924 y FC(\014)722 2002 y Fv(and)31 b Ft(\033)931 2014 y Fp(\014)999 2002 y Fw(=)22 b Ft(\033)1133 2014 y Fp(\013)1176 2023 y FC(\014)1230 2002 y Fi(\026)9 b Ft(V)1322 2014 y Fp(\014)1367 2002 y Fv(;)639 2155 y Fs(\017)41 b Fv(F)-6 b(or)37 b(al)t(l)g Ft(e)d Fs(2)h Fw(])p Ft(e)1234 2167 y Fu(0)1289 2155 y Fs(\000)18 b Ft(\016)s(;)c(e)1488 2167 y Fu(0)1544 2155 y Fw(+)k Ft(\016)s Fw([)36 b Fv(the)g(c)l(o)l(or)l(dinates)h(neighb)l(orho)l(o)l(ds)i Fs(f)p Ft(\033)2931 2167 y Fp(\014)2975 2155 y Fw(\()p Ft(V)3055 2167 y Fp(\014)3101 2155 y Ft(;)14 b(e)p Fw(\))p Fs(g)3251 2125 y Fp(M)3251 2178 y(\014)s Fu(=1)722 2254 y Fv(ar)l(e)30 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y(case)e(where)g Ft(e)g Fw(is)h(in)g(the)g(range)e(of)i(\010\()p Ft(x)p Fw(\);)i(otherwise,)e (since)g(b)n(y)f(assumption)g(\010\()p Ft(x)p Fw(\))i(admits)515 2661 y(a)28 b(holomorphic)f(extension,)h(one)g(ma)n(y)f(sligh)n(tly)h (c)n(hange)f(the)i(domain)f(of)g(in)n(tegration)f(in)515 2761 y(\(12\))g(and)g(deduce)h(that)g Ft(G)p Fw(\()p Ft(n;)14 b(e)p Fw(\))28 b(deca)n(ys)e(exp)r(onen)n(tially)-7 b(.)639 2860 y(The)31 b(pro)r(of)f(of)g(the)h(follo)n(wing)f(elemen)n (tary)g(lemma,)h(based)f(on)g(Cauc)n(h)n(y's)g(theorem,)515 2960 y(is)d(left)i(to)e(the)h(reader.)515 3159 y Fl(Lemma)73 b Fv(Consider)38 b(a)f(function,)i Ft(f)9 b Fw(\()p Ft(\021)s Fw(\))p Fv(,)39 b(c)l(ontinuous)d(in)g Ft(\021)j Fs(2)d Fw([)p Ft(a;)14 b(b)p Fw(])36 b Fv(and)h(analytic)h(at)515 3259 y Ft(e)22 b Fs(2)i Fw(])p Ft(a;)14 b(b)p Fw([)o Fv(.)39 b(Then,)1131 3504 y Fw(lim)1123 3551 y Fp(z)r Fo(!)p Fp(e)1105 3603 y(z)r Fo(2)p Fq(C)1226 3611 y Fk(+)1286 3391 y Fr(Z)1369 3411 y Fp(b)1332 3579 y(a)1441 3448 y Ft(f)9 b Fw(\()p Ft(\021)s Fw(\))p 1427 3485 188 4 v 1427 3561 a Ft(\021)21 b Fs(\000)d Ft(z)1643 3504 y Fw(d)p Ft(\021)26 b Fw(=)d Ft(\031)s Fw(i)p Ft(f)9 b Fw(\()p Ft(e)p Fw(\))18 b(+)h(p)p Ft(:)p Fw(v)q Ft(:)2323 3391 y Fr(Z)2406 3411 y Fp(b)2369 3579 y(a)2476 3448 y Ft(f)9 b Fw(\()p Ft(\021)s Fw(\))p 2463 3485 185 4 v 2463 3561 a Ft(\021)21 b Fs(\000)d Ft(e)2675 3504 y Fw(d)p Ft(\021)s(:)639 3917 y Fw(F)-7 b(or)30 b(\014xed)f Ft(!)1045 3929 y Fu(0)1109 3917 y Fs(2)f Fw(S)1238 3887 y Fp(d)p Fo(\000)p Fu(1)1391 3917 y Fw(and)i Ft(e)1594 3929 y Fu(0)1658 3917 y Fs(2)d Fw(])p Ft(a)1807 3887 y Fo(0)1830 3917 y Ft(;)14 b(b)1903 3887 y Fo(0)1926 3917 y Fw([,)31 b(let)f Fs(f)p Fw(\()p Ft(U)2256 3929 y Fp(\013)2321 3917 y Fs(\002)18 b Fw(])p Ft(e)2466 3929 y Fu(0)2522 3917 y Fs(\000)g Fw(3)p Ft(\016)r(;)c(e)2762 3929 y Fu(0)2818 3917 y Fw(+)k(3)p Ft(\016)r Fw([)c Ft(;)g(\033)3103 3929 y Fp(\013)3151 3917 y Fw(\))p Fs(g)3225 3887 y Fp(N)3225 3937 y(\013)p Fu(=1)3356 3917 y Fw(,)515 4016 y Ft(B)27 b 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Ft(e)1725 4624 y Fs(F)8 b Fw(\(\000\()p Ft(\021)s Fw(\))q Ft(;)14 b(\037)2075 4636 y Fp(e)2106 4644 y Fk(0)2143 4624 y Ft(P)e Fw(\)\()p Ft(n)p Fw(\))k(d)p Ft(\021)j Fw(+)2556 4511 y Fr(Z)2602 4700 y Fq(T)2641 4683 y FC(d)2694 4624 y Fw(e)2731 4590 y Fu(i)p Fp(n)p Fo(\001)p Fp(x)2862 4568 y Fw(1)f Fs(\000)g Ft(\037)3057 4580 y Fp(e)3088 4588 y Fk(0)3125 4568 y Fw(\()p Ft(x)p Fw(\))p 2862 4605 376 4 v 2893 4681 a(\010\()p Ft(x)p Fw(\))i Fs(\000)e Ft(e)3262 4624 y Fw(d)p Ft(x;)515 4868 y Fv(wher)l(e)30 b Ft(P)12 b Fw(\()p Ft(x)p Fw(\))24 b(=)1179 4836 y Fu(1)p 1047 4850 298 4 v 1047 4897 a Fo(kr)1136 4905 y FC(x)1173 4897 y Fu(\010\()p Fp(x)p Fu(\))p Fo(k)1354 4868 y Fv(.)1905 5255 y Fw(34)p eop end %%Page: 35 35 TeXDict begin 35 34 bop 515 523 a Fl(Pro)s(of:)64 b Fw(Notice)27 b(that)1002 651 y Fr(Z)1048 839 y Fq(T)1087 823 y FC(d)1140 764 y Fw(e)1177 729 y Fu(i)p Fp(n)p Fo(\001)p Fp(x)1350 707 y Ft(\037)1402 719 y Fp(e)1433 727 y Fk(0)1470 707 y Fw(\()p Ft(x)p Fw(\))p 1308 744 316 4 v 1308 820 a(\010\()p Ft(x)p Fw(\))19 b Fs(\000)f Ft(z)1652 764 y Fw(d)p Ft(x)24 b Fw(=)1891 660 y Fp(N)1860 685 y Fr(X)1856 860 y Fp(\013)p Fu(=1)1997 651 y Fr(Z)2044 839 y Fq(T)2083 823 y FC(d)2135 764 y Fw(e)2172 729 y Fu(i)p Fp(n)p Fo(\001)p Fp(x)2303 707 y Ft(\037)2355 719 y Fp(e)2386 727 y Fk(0)2423 707 y Fw(\()p Ft(x)p Fw(\))p Ft(\037)2586 719 y Fp(\013)2635 707 y Fw(\()p Ft(x)p Fw(\))p 2303 744 444 4 v 2367 820 a(\010\()p Ft(x)p Fw(\))c Fs(\000)e Ft(z)2775 764 y Fw(d)p Ft(x:)515 1020 y Fw(The)27 b(c)n(hange)g(of)g(v)-5 b(ariables)27 b Ft(x)c Fw(=)g Ft(\033)1604 1032 y Fp(\013)1652 1020 y Fw(\()p Ft(u;)14 b(\021)s Fw(\))27 b(applied)h(to)g(the)f(ab)r(o)n(v) n(e)g(giv)n(es)636 1177 y Fp(N)606 1202 y Fr(X)602 1378 y Fp(\013)p Fu(=1)743 1168 y Fr(Z)789 1357 y Fq(R)849 1168 y Fr(Z)895 1357 y Fq(R)937 1340 y FC(d)p Fj(\000)p Fk(1)1063 1281 y Fw(e)1100 1247 y Fu(i)p Fp(n)p Fo(\001)p Fp(\033)1218 1255 y FC(\013)1260 1247 y Fu(\()p Fp(u;\021)r Fu(\))1422 1225 y Ft(\037)1474 1237 y Fp(e)1505 1245 y Fk(0)1542 1225 y Fw(\()p Ft(\033)1621 1237 y Fp(\013)1669 1225 y Fw(\()p Ft(u;)14 b(\021)s Fw(\)\))p Ft(\037)1946 1237 y Fp(\013)1993 1225 y Fw(\()p Ft(\033)2072 1237 y Fp(\013)2120 1225 y Fw(\()p Ft(u;)g(\021)s Fw(\)\))p 1422 1262 925 4 v 1790 1338 a Ft(\021)21 b Fs(\000)e Ft(z)2356 1281 y Fs(j)14 b Fw(det)g(D)2585 1296 y Fu(\()p Fp(u;\021)r Fu(\))2737 1281 y Ft(\033)2784 1293 y Fp(\013)2831 1281 y Fw(\()p Ft(u;)g(\021)s Fw(\))p Fs(j)19 b Fw(d)p Ft(u)f Fw(d)p Ft(\021)s(;)515 1533 y Fw(whic)n(h,)27 b(b)n(y)h(Theorem)e(16,)h(is)h(equal)f(to)1083 1685 y Fp(N)1053 1710 y Fr(X)1049 1886 y Fp(\013)p Fu(=1)1190 1676 y Fr(Z)1236 1865 y Fq(R)1380 1733 y Fw(1)p 1306 1770 188 4 v 1306 1846 a Ft(\021)22 b Fs(\000)c Ft(z)1518 1676 y Fr(Z)1564 1865 y Fu(\000\()p Fp(\021)r Fu(\))1711 1789 y Fw(e)1748 1755 y Fu(i)p Fp(n)p Fo(\001)p Fp(x)1870 1789 y Ft(\037)1922 1801 y Fp(e)1953 1809 y Fk(0)1990 1789 y Fw(\()p Ft(x)p Fw(\))p Ft(\037)2153 1801 y Fp(\013)2201 1789 y Fw(\()p Ft(x)p Fw(\))p Ft(P)12 b Fw(\()p Ft(x)p Fw(\))20 b(dS)q(\()p Ft(x)p Fw(\))f(d)p Ft(\021)s(;)515 2045 y Fw(where)45 b(dS)q(\()p Ft(x)p Fw(\))28 b(denotes)f(the)h (elemen)n(t)g(of)g(surface.)36 b(Hence,)1039 2160 y Fr(Z)1086 2349 y Fq(T)1125 2332 y FC(d)1177 2273 y Fw(e)1214 2239 y Fu(i)p Fp(n)p Fo(\001)p Fp(x)1387 2217 y Ft(\037)1439 2229 y Fp(e)1470 2237 y Fk(0)1507 2217 y Fw(\()p Ft(x)p Fw(\))p 1345 2254 316 4 v 1345 2330 a(\010\()p Ft(x)p Fw(\))20 b Fs(\000)e Ft(z)1689 2273 y Fw(d)p Ft(x)24 b Fw(=)1894 2160 y Fr(Z)1940 2349 y Fq(R)2083 2217 y Fw(1)p 2010 2254 188 4 v 2010 2330 a Ft(\021)d Fs(\000)d Ft(z)2208 2273 y Fs(F)8 b Fw(\(\000\()p Ft(\021)s Fw(\))q Ft(;)14 b(\037)2558 2285 y Fp(e)2589 2293 y Fk(0)2625 2273 y Ft(P)e Fw(\))19 b(d)p Ft(\021)s(:)515 2502 y Fw(Notice)35 b(that)h Ft(\037)1023 2514 y Fp(e)1054 2522 y Fk(0)1091 2502 y Ft(P)47 b Fw(is)35 b(analytic)g(in)g(the)h(neigh)n(b)r(orho)r(o) r(d)e(of)h(\000\()p Ft(e)p Fw(\))h(when)f Fs(j)p Ft(e)23 b Fs(\000)h Ft(e)3120 2514 y Fu(0)3156 2502 y Fs(j)36 b Ft(<)g(\016)s Fw(.)515 2601 y(Hence,)31 b(b)n(y)g(Theorem)e(19,)i Fs(F)8 b Fw(\(\000\()p Ft(\021)s Fw(\))q Ft(;)14 b(\037)1748 2613 y Fp(e)1779 2621 y Fk(0)1816 2601 y Ft(P)e Fw(\))31 b(is)f(analytic)g(at)g Ft(\021)i Fw(=)27 b Ft(e)p Fw(.)46 b(The)31 b(result)f(follo)n(ws)515 2701 y(from)d(the)h(lemma)f(and)h (the)g(dominated)f(con)n(v)n(ergence)f(theorem.)699 b Fi(\003)639 2900 y Fw(The)28 b(desired)f(deca)n(y)g(follo)n(ws:)515 3080 y Fl(Theorem)k(21)41 b Fv(L)l(et)f Ft(n)i Fw(=)f Ft(r)r(!)j Fv(b)l(e)c(the)h(p)l(olar)g(form)g(of)g Ft(n)h Fs(2)g Fn(Z)2587 3050 y Fp(d)2626 3080 y Fv(,)h(wher)l(e)e Ft(n)h Fs(6)p Fw(=)g(0)p Fv(,)g(and)515 3180 y(c)l(onsider)31 b(any)f(interval)g Fw([)p Ft(a;)14 b(b)p Fw(])23 b Fs(\032)f Fw(])p Ft(a)1646 3150 y Fo(0)1670 3180 y Ft(;)14 b(b)1743 3150 y Fo(0)1765 3180 y Fw([)p Fv(.)39 b(Under)30 b(Assumptions)f(A)h (and)g(B)1624 3360 y Ft(G)p Fw(\()p Ft(n;)14 b(e)p Fw(\))23 b(=)g Ft(O)r Fw(\()p Ft(r)2126 3326 y Fo(\000)2189 3303 y FC(\024)p 2189 3312 35 3 v 2192 3346 a Fk(2)2238 3360 y Fw(\))515 3540 y Fv(uniformly)30 b(in)g Fw(\()p Ft(e;)14 b(!)s Fw(\))23 b Fs(2)h Fw([)p Ft(a;)14 b(b)p Fw(])k Fs(\002)g Fw(S)1602 3510 y Fp(d)p Fo(\000)p Fu(1)1725 3540 y Fv(.)515 3720 y Fl(Pro)s(of:)64 b Fw(F)-7 b(or)27 b Ft(!)1039 3732 y Fu(0)1076 3720 y Fw(,)g Ft(e)1165 3732 y Fu(0)1202 3720 y Fw(,)h(etc.)37 b(as)27 b(ab)r(o)n(v)n(e,)f(b)n (y)i(Theorems)e(15)h(and)g(17)1368 3900 y Ft(\031)s Fw(i)p Fs(F)8 b Fw(\(\000\()p Ft(e)p Fw(\))q Ft(;)14 b(P)e Fw(\)\()p Ft(n)p Fw(\))23 b(=)g Ft(O)r Fw(\()p Ft(r)2192 3866 y Fo(\000)2256 3843 y FC(\024)p 2256 3852 V 2259 3886 a Fk(2)2304 3900 y Fw(\))56 b(and)1114 4127 y(p)p Ft(:)p Fw(v)q Ft(:)1265 4014 y Fr(Z)1348 4034 y Fo(1)1311 4203 y(\0001)1528 4071 y Fw(1)p 1457 4108 185 4 v 1457 4184 a Ft(\021)21 b Fs(\000)d Ft(e)1651 4127 y Fs(F)8 b Fw(\(\000\()p Ft(\021)s Fw(\))q Ft(;)14 b(\037)2001 4139 y Fp(e)2032 4147 y Fk(0)2069 4127 y Ft(P)e Fw(\)\()p Ft(n)p Fw(\))19 b(d)p Ft(\021)26 b Fw(=)d Ft(O)r Fw(\()p Ft(r)2636 4093 y Fo(\000)2699 4070 y FC(\024)p 2699 4079 35 3 v 2702 4113 a Fk(2)2748 4127 y Fw(\))515 4339 y(uniformly)k(on)g([)q Ft(e)1071 4351 y Fu(0)1126 4339 y Fs(\000)18 b Ft(\016)s(;)c(e)1325 4351 y Fu(0)1380 4339 y Fw(+)k Ft(\016)s Fw(])h Fs(\002)p 1628 4273 68 4 v 18 w Ft(B)t Fw(.)37 b(Moreo)n(v)n(er,)25 b(b)n(y)i(the)h(sc)n(holium)f(of)h(Theorem)e(1)1344 4454 y Fr(Z)1390 4643 y Fq(T)1429 4626 y FC(d)1482 4567 y Fw(e)1519 4533 y Fu(i)p Fp(n)p Fo(\001)p Fp(x)1650 4511 y Fw(1)18 b Fs(\000)g Ft(\037)1845 4523 y Fp(e)1876 4531 y Fk(0)1913 4511 y Fw(\()p Ft(x)p Fw(\))p 1650 4548 376 4 v 1681 4624 a(\010\()p Ft(x)p Fw(\))i Fs(\000)e Ft(e)2053 4567 y Fw(d)p Ft(x)24 b Fw(=)f Ft(O)r Fw(\()p Ft(r)2394 4533 y Fo(\0001)2517 4567 y Fw(\))515 4807 y(uniformly)37 b(on)g([)p Ft(e)1090 4819 y Fu(0)1145 4807 y Fs(\000)18 b Ft(\016)s(;)c(e)1344 4819 y Fu(0)1400 4807 y Fw(+)k Ft(\016)s Fw(])25 b Fs(\002)p 1660 4740 68 4 v 24 w Ft(B)t Fw(.)66 b(Since)38 b(\()p Ft(e)2114 4819 y Fu(0)2151 4807 y Ft(;)14 b(!)2240 4819 y Fu(0)2277 4807 y Fw(\))37 b(is)g(arbitrary)f(in)h([)p Ft(a;)14 b(b)p Fw(])25 b Fs(\002)f Fw(S)3232 4777 y Fp(d)p Fo(\000)p Fu(1)3356 4807 y Fw(,)515 4907 y(whic)n(h)j(is)h(compact,)f(the)h(result)f(follo) n(ws.)1480 b Fi(\003)1905 5255 y Fw(35)p eop end %%Page: 36 36 TeXDict begin 36 35 bop 515 523 a Fx(6)134 b(Application)46 b(to)f(Generalized)h(Laplacians)515 705 y Fw(In)34 b(this)g(section)g (w)n(e)g(apply)g(the)g(previous)f(results)h(to)g(Green's)f(functions)i (of)f(concrete)515 805 y(Laplacians)29 b(on)i Fn(Z)1104 774 y Fp(d)1143 805 y Fw(.)47 b(W)-7 b(e)31 b(fo)r(cus)g(on)f(t)n(w)n (o)g(sp)r(eci\014c)h(examples:)43 b(the)31 b Fv(standar)l(d)j(L)l (aplacian)515 904 y Fw(and)27 b(the)h Fv(Molchanov{V)-6 b(ainb)l(er)l(g)33 b(L)l(aplacian)p Fw(.)639 1004 y(A)n(t)d(a)g(higher) f(lev)n(el)h(of)f(generalit)n(y)g(let)h(us)g(consider)f(a)g(simple)h (graph)f(without)h(lo)r(op,)515 1103 y(whose)f(set)h(of)h(v)n(ertices)e (is)h(denoted)g(b)n(y)g Ft(X)36 b Fw(\(where)30 b Ft(X)37 b Fw(is)30 b(allo)n(w)n(ed)e(to)j(b)r(e)f(in\014nite\).)46 b(F)-7 b(or)515 1203 y Ft(m;)14 b(n)33 b Fs(2)h Ft(X)7 b Fw(,)35 b(d\()p Ft(m;)14 b(n)p Fw(\))34 b(denotes)f(the)i Fv(gr)l(aph)h(distanc)l(e)41 b Fw(b)r(et)n(w)n(een)34 b Ft(m)g Fw(and)g Ft(n)p Fw(,)h(that)f(is,)i(the)515 1303 y(length)d(of)h(the)g(minimal)f(c)n(hain)g(joining)h Ft(m)f Fw(and)h Ft(n)f Fw(in)h(the)g(graph)e(\()p Fs(1)i Fw(if)g Ft(m)f Fw(and)h Ft(n)f Fw(lie)515 1402 y(in)28 b(t)n(w)n(o)g(di\013eren)n(t)g(connected)g(comp)r(onen)n(ts\).)39 b(Of)28 b(course,)f(d)i(is)f(a)f(metric)i(distance)f(with)515 1502 y(v)-5 b(alues)33 b(in)h Fn(N)23 b Fs([)g(f1g)p Fw(.)55 b(Notice)34 b(also)f(that)h(\()p Ft(X)r(;)14 b Fw(d\))34 b(determines)g(the)g(graph)f(completely)-7 b(,)515 1602 y(since)27 b Fs(f)p Ft(m;)14 b(n)o Fs(g)27 b Fw(is)h(an)f(edge)g(if)h(and)g(only)f(if)h(d\()p Ft(m;)14 b(n)p Fw(\))24 b(=)e(1.)639 1701 y(W)-7 b(e)23 b(are)f(in)n(terested)g (in)h(the)g(Hilb)r(ert)g(space,)g 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Fu(d\()p Fp(m;n)p Fu(\)=1)2199 2441 y Ft( )s Fw(\()p Ft(m)p Fw(\))p Ft(;)515 2689 y Fw(where)j Ft( )f Fs(2)d Ft(l)940 2659 y Fu(2)977 2689 y Fw(\()p Ft(X)7 b Fw(\).)37 b(In)28 b(particular,)e(\(\001)p Ft(\016)1825 2701 y Fp(n)1871 2689 y Fw(\)\()p Ft(m)p Fw(\))d(=)g Fl(1)2199 2701 y Fo(V)2241 2709 y FC(n)2285 2689 y Fw(\()p Ft(m)p Fw(\),)29 b(where)1413 2865 y Fs(V)1464 2877 y Fp(n)1532 2865 y Fw(=)23 b Fs(f)p Ft(m)f Fs(2)h Ft(X)30 b Fw(;)37 b(d\()p Ft(m;)14 b(n)p Fw(\))23 b(=)g(1)o Fs(g)p Ft(:)639 3041 y Fw(In)31 b(the)h(sequel)e(w)n(e)h(supp)r(ose)f(that)h (the)h(degrees)d(of)i(the)g(v)n(ertices)f(of)h(the)g(considered)515 3141 y(graph)36 b(are)h(b)r(ounded,)j(in)e(other)f(w)n(ords,)i(sup)2016 3161 y Fp(n)2075 3141 y Fw(#)p Fs(V)2195 3153 y Fp(n)2280 3141 y Fi(6)g Ft(B)j Fw(for)37 b(a)g(certain)g Ft(B)43 b(<)d Fs(1)p Fw(.)3309 3111 y Fu(12)515 3241 y Fw(Then,)32 b(\001)g Fv(is)h(a)g(b)l(ounde)l(d,)i(selfadjoint)g(op)l(er)l(ator)e (on)g Ft(l)2255 3210 y Fu(2)2292 3241 y Fw(\()p Ft(X)7 b Fw(\),)31 b(as)f(the)i(reader)d(ma)n(y)h(easily)515 3340 y(v)n(erify)-7 b(.)639 3539 y(W)g(e)34 b(are)e(no)n(w)g(in)n (terested)g(in)i(the)f(case)f(where)g Ft(X)39 b Fw(=)31 b Fn(Z)2432 3509 y Fp(d)2504 3539 y Fw(and)i(the)g(graph)f(distance,) 515 3639 y(d\()p Ft(m;)14 b(n)p Fw(\),)21 b(is)e(translational)f(in)n (v)-5 b(arian)n(t.)32 b(In)20 b(these)f(circumstances)f(w)n(e)h(call)f (\001)i(a)f Fv(gener)l(alize)l(d)515 3739 y(L)l(aplacian)p Fw(.)53 b(Then,)34 b(d\()p Ft(m;)14 b(n)p Fw(\))33 b(is)f(a)g(function) h(of)f Ft(m)22 b Fs(\000)f Ft(n)32 b Fw(only)-7 b(.)51 b(Hence,)34 b(letting)f Fs(V)k Fw(=)31 b Fs(V)3342 3751 y Fu(0)515 3838 y Fw(\(where)g(0)e Fs(2)h Fn(Z)1002 3808 y Fp(d)1073 3838 y Fw(denotes)h(the)h(origin\),)f(the)h(considered)f (graph)f(is)i(clearly)e(determined)515 3938 y(b)n(y)d(\()p Fn(Z)717 3908 y Fp(d)756 3938 y Ft(;)14 b Fs(V)7 b Fw(\).)37 b(In)28 b(particular,)1531 4131 y(\001)p Ft( )s Fw(\()p Ft(n)p Fw(\))c(=)1886 4052 y Fr(X)1883 4230 y Fp(v)r Fo(2V)2023 4131 y Ft( )s Fw(\()p Ft(n)19 b Fw(+)f Ft(v)s Fw(\))p Ft(;)515 4385 y Fw(where)27 b(#)p Fs(V)j Fi(6)22 b Ft(B)32 b Fw(is)c(still)g(assumed)f(to)g(b)r(e)h(\014nite.)639 4485 y(Recall)35 b(that)g(the)g(F)-7 b(ourier)34 b(transform)g(of)h Ft( )j Fs(2)d Ft(l)2229 4455 y Fu(2)2266 4485 y Fw(\()p Fn(Z)2353 4455 y Fp(d)2392 4485 y Fw(\))h(is)e(the)i(follo)n(wing)d (function,)515 4585 y(de\014ned)28 b(for)f Ft(x)c Fs(2)h Fn(T)1132 4554 y Fp(d)1171 4585 y Fw(:)1214 4739 y Fr(b)1199 4761 y Ft( )s Fw(\()p Ft(x)p Fw(\))h(=)d(\()p Fs(F)8 b Ft( )s Fw(\)\()p Ft(x)p Fw(\))25 b(=)d(\(2)p Ft(\031)s Fw(\))2047 4727 y Fo(\000)2110 4704 y FC(d)p 2110 4713 31 3 v 2111 4747 a Fk(2)2188 4682 y Fr(X)2168 4866 y Fp(n)p Fo(2)p Fq(Z)2293 4849 y FC(d)2342 4761 y Fw(e)2379 4727 y Fu(i)p Fp(n)p Fo(\001)p Fp(x)2500 4761 y Ft( )s Fw(\()p Ft(n)p Fw(\))p Ft(:)p 515 4929 1146 4 v 577 4983 a FB(12)642 5006 y Fm(The)i(cardinalit)n(y)g(of)f(a)h(giv)n(en)g(set,)g Ff(A)p Fm(,)f(is)g(denoted)i(b)n(y)f(#)p Ff(A)p Fm(.)1905 5255 y Fw(36)p eop end %%Page: 37 37 TeXDict begin 37 36 bop 515 523 a Fw(The)27 b Fv(symb)l(ol)37 b Fw(of)28 b(\001,)g(denoted)g(b)n(y)1616 502 y Fr(b)1605 523 y Fw(\001,)g(is)f(the)h(lifting)g(of)g(\001)g(via)f(the)h(F)-7 b(ourier)26 b(transform:)1710 674 y Fr(b)1698 695 y Fw(\001)e(=)e Fs(F)8 b Fw(\001)p Fs(F)2083 661 y Fo(\000)p Fu(1)2172 695 y Ft(:)515 868 y Fl(Theorem)31 b(22)41 b Fv(L)l(et)33 b Fw(\001)g Fv(b)l(e)g(the)g(gener)l(alize)l(d)h(L)l(aplacian)h(asso)l (ciate)l(d)g(with)e(a)g(given)h Fs(V)h(\032)515 967 y Fn(Z)570 937 y Fp(d)609 967 y Fv(.)j(Then,)31 b(its)f(symb)l(ol)g(is)h (the)e(multiplic)l(ation)j(by)1352 1152 y Fw(\010\()p Ft(x)p Fw(\))24 b(=)1638 1073 y Fr(X)1634 1251 y Fp(v)r Fo(2V)1775 1152 y Fw(e)1812 1117 y Fu(i)p Fp(v)r Fo(\001)p Fp(x)1951 1152 y Fw(=)2042 1073 y Fr(X)2038 1251 y Fp(v)r Fo(2V)2179 1152 y Fw(cos)13 b(\()p Ft(v)22 b Fs(\001)c Ft(x)p Fw(\))q Ft(;)515 1415 y Fv(wher)l(e)30 b Ft(x)24 b Fs(2)f Fn(T)953 1385 y Fp(d)992 1415 y Fv(.)515 1587 y Fl(Pro)s(of:)849 1566 y Fr(b)838 1587 y Fw(\001)28 b(maps)1167 1565 y Fr(b)1152 1587 y Ft( )s Fw(\()p Ft(x)p Fw(\))h(to)e(the)h(follo)n(wing)f(function:)762 1776 y(\(2)p Ft(\031)s Fw(\))918 1742 y Fo(\000)981 1720 y FC(d)p 981 1729 31 3 v 982 1762 a Fk(2)1059 1697 y Fr(X)1039 1881 y Fp(n)p Fo(2)p Fq(Z)1164 1865 y FC(d)1212 1776 y Fw(e)1249 1742 y Fu(i)p Fp(n)p Fo(\001)p Fp(x)1388 1697 y Fr(X)1385 1875 y Fp(v)r Fo(2V)1525 1776 y Ft( )s Fw(\()p Ft(n)19 b Fw(+)f Ft(v)s Fw(\))83 b(=)g(\(2)p Ft(\031)s Fw(\))2228 1742 y Fo(\000)2290 1720 y FC(d)p 2290 1729 V 2291 1762 a Fk(2)2352 1697 y Fr(X)2349 1875 y Fp(v)r Fo(2V)2509 1697 y Fr(X)2489 1881 y Fp(n)p Fo(2)p Fq(Z)2614 1865 y FC(d)2663 1776 y Fw(e)2700 1742 y Fu(i\()p Fp(n)p Fo(\000)p Fp(v)r Fu(\))o Fo(\001)p Fp(x)2960 1776 y Ft( )s Fw(\()p Ft(n)p Fw(\))1924 2014 y(=)2075 1935 y Fr(X)2072 2113 y Fp(v)r Fo(2V)2212 2014 y Fw(e)2249 1980 y Fo(\000)p Fu(i)p Fp(v)r Fo(\001)p Fp(x)2432 1992 y Fr(b)2417 2014 y Ft( )s Fw(\()p Ft(x)p Fw(\))p Ft(:)515 2265 y Fw(Notice)27 b(that)917 2437 y Ft(v)f Fs(2)e(V)53 b(\()-14 b(\))46 b Fw(d\()p Ft(v)s(;)14 b Fw(0\))23 b(=)g(1)46 b Fs(\()-14 b(\))46 b Fw(d\(0)p Ft(;)14 b Fs(\000)p Ft(v)s Fw(\))23 b(=)g(1)45 b Fs(\()-14 b(\))47 b(\000)p Ft(v)25 b Fs(2)f(V)7 b Ft(:)515 2609 y Fw(Hence,)41 b(denoting)e(b)n(y)f Fs(V)1335 2579 y Fu(+)1428 2609 y Fw(the)h(set)g(of)g Ft(v)44 b Fs(2)e(V)j Fw(whose)38 b(\014rst)h(non)f(zero)g(co)r (ordinate)f(is)515 2709 y(p)r(ositiv)n(e,)961 2815 y Fr(X)958 2993 y Fp(v)r Fo(2V)1098 2893 y Fw(e)1135 2859 y Fo(\000)p Fu(i)p Fp(v)r Fo(\001)p Fp(x)1326 2893 y Fw(=)1417 2815 y Fr(X)1413 2993 y Fp(v)r Fo(2V)1554 2893 y Fw(e)1591 2859 y Fu(i)p Fp(v)r Fo(\001)p Fp(x)1729 2893 y Fw(=)1844 2815 y Fr(X)1817 2997 y Fp(v)r Fo(2V)1944 2981 y Fk(+)2004 2893 y Fw(2)14 b(cos)o(\()p Ft(v)22 b Fs(\001)c Ft(x)p Fw(\))24 b(=)2500 2815 y Fr(X)2497 2993 y Fp(v)r Fo(2V)2637 2893 y Fw(cos)13 b Ft(v)22 b Fs(\001)d Ft(x;)515 3149 y Fw(whic)n(h)27 b(completes)h(the)g(pro)r (of.)1822 b Fi(\003)639 3348 y Fw(Since)28 b(the)g(range)e(of)i(\010\() p Ft(x)p Fw(\))g(is)g(equal)f(to)h(the)g(sp)r(ectrum)f(of)h(\001,)g (one)f(obtains:)515 3547 y Fl(Corollary)60 b Fw(sp)r(ec\(\001\))24 b(=)f([min)14 b(\010)p Ft(;)g Fw(#)p Fs(V)7 b Fw(])14 b Ft(:)639 3746 y Fw(Notice)28 b(that)g(\010\()p Ft(x)p Fw(\))g(is)g(analytic)f(at)g(eac)n(h)g Ft(x)d Fs(2)f Fn(T)2171 3716 y Fp(d)2210 3746 y Fw(.)37 b(Hence,)1436 3919 y Ft(E)29 b Fw(=)22 b Fs(f)p Ft(x)h Fs(2)h Fn(T)1859 3884 y Fp(d)1921 3919 y Fw(;)37 b Fs(r)2050 3931 y Fp(x)2092 3919 y Fw(\010\()p Ft(x)p Fw(\))24 b(=)e(0)p Fs(g)515 4091 y Fw(has)28 b(Leb)r(esgue)g(measure)g(zero.)39 b(In)29 b(particular,)e(sp)r(ec\(\001\))j(is)e(purely)h(absolutely)f(con)n (tin-)515 4191 y(uous.)718 4160 y Fu(13)639 4290 y Fw(The)g Fv(Gr)l(e)l(en)-8 b('s)29 b(function)34 b Fw(of)28 b(\001)g(is)f (de\014ned)h(as)1375 4462 y Ft(G)p Fw(\()p Ft(m;)14 b(n)p Fw(;)g Ft(z)t Fw(\))23 b(=)g Fs(h)p Ft(\016)1924 4474 y Fp(m)2001 4462 y Fs(j)14 b Fw(\(\001)k Fs(\000)g Ft(z)t Fw(\))2315 4428 y Fo(\000)p Fu(1)2404 4462 y Ft(\016)2441 4474 y Fp(n)2486 4462 y Fs(i)p 515 4532 1146 4 v 577 4588 a FB(13)642 4612 y Fm(This)26 b(ma)n(y)h(b)r(e)f(seen)i(in)e(the)h (follo)n(wing)f(w)n(a)n(y:)37 b Ff(L)1951 4588 y FB(2)1986 4612 y Fm(\()p Fg(T)2057 4588 y FA(d)2094 4612 y Fm(\))25 b(=)f Ff(L)2273 4588 y FB(2)2307 4612 y Fm(\()p Fg(T)2378 4588 y FA(d)2421 4612 y FJ(n)6 b Ff(E)t Fm(\).)40 b(Moreo)n(v)n(er,)27 b(b)n(y)g(the)g(In)n(v)n(erse)515 4691 y(F)-6 b(unction)33 b(Theorem)f(\010\()p Ff(x)p Fm(\))g(is)f(in)n(v)n(ertible)g(on)h Fg(T)1849 4667 y FA(d)1895 4691 y FJ(n)9 b Ff(E)t Fm(.)55 b(Let)32 b(\011\()p Ff(y)r Fm(\))h(b)r(e)f(its)f(in)n(v)n(erse,)j(so)d (the)i(c)n(hange)g(of)515 4770 y(v)l(ariables)28 b Ff(y)h Fm(=)e(\010\()p Ff(x)p Fm(\))i(giv)n(es)g(a)f(unitary)g(equiv)l(alence) i(b)r(et)n(w)n(een)h Ff(L)2329 4746 y FB(2)2363 4770 y Fm(\()p Fg(T)2434 4746 y FA(d)2471 4770 y Fm(\))d(with)h(the)g(Leb)r (esgue)h(measure)515 4848 y(and)24 b Ff(L)700 4825 y FB(2)734 4848 y Fm(\(\010\()p Fg(T)883 4825 y FA(d)924 4848 y FJ(n)s Ff(E)t Fm(\)\))g(with)f(the)h(measure)g FJ(kr)1746 4856 y FA(y)1783 4848 y Fm(\011\()p Ff(y)r Fm(\))q FJ(k)16 b Fm(d)p Ff(y)r Fm(.)31 b(Via)23 b(this)g(unitary)h (equiv)l(alence)h(the)f(op)r(erator)515 4927 y(of)32 b(m)n(ultiplication)g(b)n(y)h(\010\()p Ff(x)p Fm(\))g(is)e(lifted)h(to) h(the)g(op)r(erator)g(of)f(m)n(ultiplication)h(b)n(y)g Ff(y)r Fm(,)h(so)e(b)n(y)h(de\014nition)515 5006 y FJ(kr)609 5014 y FA(y)646 5006 y Fm(\011\()p Ff(y)r Fm(\))q FJ(k)16 b Fm(d)p Ff(y)10 b Fd(\026)e Fm(\010\()p Fg(T)1087 4983 y FA(d)1129 5006 y FJ(n)t Ff(E)t Fm(\))23 b(is)g(a)h(sp)r(ectral)g (measure)g(for)f(\001|and)g(is)g(clearly)h(absolutely)h(con)n(tin)n (uous.)1905 5255 y Fw(37)p eop end %%Page: 38 38 TeXDict begin 38 37 bop 515 523 a Fw(for)27 b Ft(m;)14 b(n)23 b Fs(2)h Fn(Z)959 493 y Fp(d)1026 523 y Fw(and)j Ft(z)g Fs(2)d Fn(C)1392 535 y Fu(+)1447 523 y Fw(.)37 b(Since)28 b(\001)g(is)g(translational)e(in)n(v)-5 b(arian)n(t,)27 b(this)h(last)f(function)515 623 y(dep)r(ends)f(on)f Ft(m)14 b Fs(\000)g Ft(n)26 b Fw(and)f Ft(z)k Fw(only)-7 b(.)36 b(In)26 b(fact,)g Ft(G)p Fw(\()p Ft(m;)14 b(n)p Fw(;)g Ft(z)t Fw(\))23 b(=)g Ft(G)p Fw(\(0)p Ft(;)14 b(n)g Fs(\000)g Ft(m)p Fw(;)g Ft(z)t Fw(\))25 b(so)g(w)n(e)g(denote)515 722 y(this)33 b(latter)g(simply)g(b)n(y)g Ft(G)p Fw(\()p Ft(n)23 b Fs(\000)e Ft(m;)14 b(z)t Fw(\).)53 b(Since)34 b(the)f(F)-7 b(ourier)32 b(transform)g(is)h(unitary)-7 b(,)35 b(for)515 822 y(an)n(y)27 b Ft(n)22 b Fs(2)i Fn(Z)878 792 y Fp(d)945 822 y Fw(and)j Ft(z)f Fs(2)e Fn(C)1310 834 y Fu(+)1252 1014 y Ft(G)p Fw(\()p Ft(n;)14 b(z)t Fw(\))83 b(=)g Fs(h)1788 992 y Fr(b)1774 1014 y Ft(\016)1811 1026 y Fu(0)1848 1014 y Fw(\()p Ft(x)p Fw(\))15 b Fs(j)f Fw(\()2055 993 y Fr(b)2043 1014 y Fw(\001)19 b Fs(\000)f Ft(z)t Fw(\))2289 980 y Fo(\000)p Fu(1)2396 992 y Fr(b)2378 1014 y Ft(\016)2415 1026 y Fp(n)2460 1014 y Fw(\()p Ft(x)p Fw(\))q Fs(i)2604 1026 y Fu(2)1594 1193 y Fw(=)83 b(\(2)p Ft(\031)s Fw(\))1898 1159 y Fo(\000)p Fp(d)2003 1080 y Fr(Z)2049 1269 y Fq(T)2088 1252 y FC(d)2229 1137 y Fw(e)2266 1107 y Fu(i)p Fp(n)p Fo(\001)p Fp(x)p 2150 1174 316 4 v 2150 1250 a Fw(\010\()p Ft(x)p Fw(\))20 b Fs(\000)e Ft(z)2494 1193 y Fw(d)p Ft(x:)515 1419 y Fw(By)27 b(Theorem)g(20)f(the)i(limit)1544 1602 y Ft(G)p Fw(\()p Ft(n;)14 b(e)p Fw(\))23 b(=)49 b(lim)1928 1649 y Fp(z)r Fo(!)p Fp(e)1910 1700 y(z)r Fo(2)p Fq(C)2031 1708 y Fk(+)2091 1602 y Ft(G)p Fw(\()p Ft(n;)14 b(z)t Fw(\))515 1877 y(exists)34 b(for)g(an)n(y)f Ft(e)44 b(=)-52 b Fs(2)35 b Ft(E)5 b Fw(.)58 b(W)-7 b(e)35 b(will)f(compute)h(its)f (deca)n(y)g(when)g Fs(j)p Ft(n)p Fs(j)h(!)f(1)h Fw(in)g(t)n(w)n(o)e (par-)515 1977 y(ticular)26 b(examples:)36 b(the)28 b Fv(standar)l(d)h(discr)l(ete)h(L)l(aplacian)36 b Fw(and)27 b(the)g Fv(Molchanov{V)-6 b(ainb)l(er)l(g)515 2076 y(L)l(aplacian)p Fw(,)27 b(whic)n(h)e(are)f(describ)r(ed)g(in)h(the)g(rest)g(of)f(the)i (section.)35 b(Both)25 b(are)e(imp)r(ortan)n(t)i(in)515 2176 y(mathematical)c(ph)n(ysics,)h(since)f(b)r(oth)h(are)e (discretizations)g(of)h(the)h(con)n(tin)n(uous)e(Laplacian)515 2276 y(on)27 b Fn(R)690 2245 y Fp(d)756 2276 y Fw(\(up)i(to)e(an)g (additiv)n(e)h(constan)n(t\).)515 2491 y Fl(Standard)42 b(Laplacian)85 b Fw(The)36 b Fv(standar)l(d)i(L)l(aplacian)44 b Fw(is)36 b(the)g(adjacency)f(op)r(erator)f(of)515 2591 y(the)28 b(usual)f(grid)g(on)g Fn(Z)1216 2561 y Fp(d)1255 2591 y Fw(;)h(it)g(is)f(sp)r(eci\014ed)h(b)n(y)g(the)g(graph)e (distance)1247 2850 y(d\()p Ft(m;)14 b(n)p Fw(\))23 b(=)g Fs(j)p Ft(m)18 b Fs(\000)g Ft(n)p Fs(j)1898 2862 y Fu(1)1958 2850 y Fw(=)2089 2746 y Fp(d)2046 2771 y Fr(X)2049 2948 y Fp(j)s Fu(=1)2180 2850 y Fs(j)p Ft(m)2276 2816 y Fu(\()p Fp(j)s Fu(\))2381 2850 y Fs(\000)g Ft(n)2514 2816 y Fu(\()p Fp(j)s Fu(\))2601 2850 y Fs(j)p Ft(;)515 3116 y Fw(so)27 b(the)h(set)f(of)h(immediate)g(neigh)n(b)r(ors)e(of)i(the)g(origin)e (is)986 3298 y Fs(V)k Fw(=)23 b Fs(f)p Fw(\()p Fs(\006)p Fw(1)p Ft(;)14 b Fw(0)p Ft(;)g(:)g(:)g(:)e(;)i Fw(0\))p Ft(;)g Fw(\(0)p Ft(;)g Fs(\006)p Fw(1)p Ft(;)g(:)g(:)g(:)e(;)i Fw(0\))p Ft(;)g(:)g(:)g(:)f(;)h Fw(\(0)p Ft(;)g Fw(0)p Ft(;)g(:)g(:)g(:)f(;)h Fs(\006)p Fw(1\))o Fs(g)p Ft(:)515 3481 y Fw(Hence,)27 b(b)n(y)h(Theorem)e(22)h(the)h(sym)n(b)r(ol)f(of)g (the)h(standard)f(Laplacian)f(is)h(the)h(op)r(erator)e(of)515 3580 y(m)n(ultiplication)h(b)n(y)1570 3757 y(\010\()p Ft(x)p Fw(\))d(=)e(2)1950 3653 y Fp(d)1908 3678 y Fr(X)1910 3855 y Fp(j)s Fu(=1)2042 3757 y Fw(cos)13 b Ft(x)2214 3723 y Fu(\()p Fp(j)s Fu(\))2301 3757 y Ft(:)515 3995 y Fw(Th)n(us,)25 b(the)f(sp)r(ectrum)h(of)f(\001)h(is)f(purely)g (absolutely)f(con)n(tin)n(uous)h(and)g(equal)g(to)g([)p Fs(\000)p Fw(2)p Ft(d)o(;)14 b Fw(2)p Ft(d)p Fw(].)515 4094 y(Notice)27 b(that)1241 4194 y Fs(r)1310 4206 y Fp(x)1352 4194 y Fw(\010\()p Ft(x)p Fw(\))d(=)f(\()p Fs(\000)p Fw(2)14 b(sin)f Ft(x)1950 4160 y Fu(\(1\))2039 4194 y Ft(;)h(:)g(:)g(:)g(;)g Fs(\000)p Fw(2)g(sin)e Ft(x)2506 4160 y Fu(\()p Fp(d)p Fu(\))2597 4194 y Fw(\))p Ft(;)515 4343 y Fw(whic)n(h)31 b(v)-5 b(anishes)30 b(only)h(if)g(cos)13 b Ft(x)1522 4313 y Fu(\()p Fp(j)s Fu(\))1638 4343 y Fw(=)28 b Fs(\006)p Fw(1)j(for)f(all)h Ft(j)i Fw(=)c(1)p Ft(;)14 b(:)g(:)g(:)f(;)h(d)p Fw(,)32 b(in)f(whic)n(h)g(case)f(\010\()p Ft(x)p Fw(\))g Fs(2)515 4443 y(f\000)p Fw(2)p Ft(d;)14 b Fs(\000)p Fw(2)p Ft(d)i Fw(+)i(4)p Ft(;)c(:)g(:)g(:)f(;)h Fw(2)p Ft(d)k Fs(\000)g Fw(4)p Ft(;)c Fw(2)p Ft(d)p Fs(g)p Fw(.)36 b(In)28 b(particular,)e(the)i(lev)n(el)f(surfaces)g(of)g (\010\()p Ft(x)p Fw(\),)1437 4625 y(\000\()p Ft(e)p Fw(\))c(=)g Fs(f)p Ft(x)g Fs(2)g Fn(T)1948 4591 y Fp(d)2011 4625 y Fw(;)37 b(\010\()p Ft(x)p Fw(\))24 b(=)e Ft(e)p Fs(g)p Ft(;)515 4807 y Fw(are)k(regular)g(for)h(all)h Ft(e)j(=)-51 b Fs(2)24 b(f\000)p Fw(2)p Ft(d;)14 b Fs(\000)p Fw(2)p Ft(d)i Fw(+)i(4)p Ft(;)c(:)g(:)g(:)f(;)h Fw(2)p Ft(d)k Fs(\000)g Fw(4)p Ft(;)c Fw(2)p Ft(d)o Fs(g)p Fw(.)639 4907 y(Let)39 b(us)g(sho)n(w)f(that)i(for)e(suc)n(h)h Ft(e)p Fw('s)f(the)i(lev)n(el)e(surfaces)g(are)g(exempt)h(of)g (planarit)n(y)-7 b(,)515 5006 y(except)27 b(for)g Ft(e)c Fw(=)g(0:)1905 5255 y(38)p eop end %%Page: 39 39 TeXDict begin 39 38 bop 515 523 a Fl(Theorem)31 b(23)41 b Fv(L)l(et)c Ft(E)k Fw(=)35 b Fs(f\000)p Fw(2)p Ft(d;)14 b Fs(\000)p Fw(2)p Ft(d)j Fw(+)h(4)p Ft(;)c(:)g(:)g(:)f(;)h Fw(2)p Ft(d)k Fs(\000)g Fw(4)p Ft(;)c Fw(2)p Ft(d)o Fs(g)23 b([)h(f)p Fw(0)p Fs(g)36 b Fv(and)h(supp)l(ose)h Ft(e)e Fs(2)515 623 y Fw([)p Fs(\000)p Fw(2)p Ft(d)o(;)14 b Fw(2)p Ft(d)p Fw(])h Fs(n)f Ft(E)5 b Fv(.)38 b(Then,)30 b Fw(\000\()p Ft(e)p Fw(\))e Fv(admits)h(at)f(le)l(ast)g(one)h(non)f (vanishing)h(princip)l(al)i(curvatur)l(e)515 722 y(at)e(any)i(p)l (oint.)515 899 y Fl(Pro)s(of:)68 b Fw(Let)29 b Ft(x)1039 911 y Fu(0)1102 899 y Fs(2)d Fw(\000\()p Ft(e)p Fw(\))j(b)r(e)g (\014xed.)41 b(By)29 b(c)n(hoice)f(of)h Ft(e)p Fw(,)g(\000\()p Ft(e)p Fw(\))g(is)g(regular,)e(so)i(there)f(exists)515 1015 y(a)h Ft(j)35 b Fw(suc)n(h)30 b(that)g(sin)14 b Ft(x)1190 972 y Fu(\()p Fp(j)s Fu(\))1190 1037 y(0)1304 1015 y Fs(6)p Fw(=)27 b(0.)43 b(After)31 b(renaming)e(the)h(v)-5 b(ariables,)29 b(the)i(equation)e(de\014ning)515 1114 y(\000\()p Ft(e)p Fw(\))f(in)f(a)h(neigh)n(b)r(orho)r(o)r(d)e(of)i Ft(x)1527 1126 y Fu(0)1592 1114 y Fw(th)n(us)g(b)r(ecomes)1186 1291 y(2)14 b(cos)e Ft(u)1414 1257 y Fu(\(1\))1521 1291 y Fw(+)18 b Fs(\001)c(\001)g(\001)k Fw(+)h(2)14 b(cos)e Ft(u)2031 1257 y Fu(\()p Fp(d)p Fo(\000)p Fu(1\))2224 1291 y Fw(+)18 b(2)c(cos)f Ft(h)23 b Fw(=)f Ft(e;)523 b Fw(\(13\))515 1468 y(where)37 b(sin)14 b Ft(h)39 b Fs(6)p Fw(=)h(0)d(on)g(a)h(certain)f(in)n(terv)-5 b(al.)66 b(In)38 b(particular,)h(the)f(deriv)-5 b(ativ)n(e)37 b(with)h(re-)515 1567 y(sp)r(ect)g(to)g Ft(h)f Fw(of)h(the)g(left-hand) h(side)e(in)h(\(13\))g(is)g(not)g(zero.)66 b(Therefore,)39 b(writing)f Ft(u)h Fw(=)515 1667 y(\()p Ft(u)595 1637 y Fu(\(1\))684 1667 y Ft(;)14 b(:)g(:)g(:)f(;)h(u)916 1637 y Fu(\()p Fp(d)p Fo(\000)p Fu(1\))1091 1667 y Fw(\),)28 b(there)g(exists)f(an)g(implicit)i(function,)f(h\()p Ft(u)p Fw(\),)g(suc)n(h)f(that)1142 1844 y(2)14 b(cos)e Ft(u)1370 1809 y Fu(\(1\))1477 1844 y Fw(+)19 b Fs(\001)14 b(\001)g(\001)k Fw(+)g(2)c(cos)e Ft(u)1987 1809 y Fu(\()p Fp(d)p Fo(\000)p Fu(1\))2181 1844 y Fw(+)18 b(2)c(cos)e(h\()p Ft(u)p Fw(\))23 b(=)g Ft(e)515 2020 y Fw(for)32 b(all)h Ft(u)g Fw(in)h(a)f(neigh)n(b)r(orho)r(o)r(d)f(of)h Ft(u)1702 2032 y Fu(0)1739 2020 y Fw(,)h(where)f(\()p Ft(u)2122 2032 y Fu(0)2159 2020 y Ft(;)14 b Fw(h\()p Ft(u)2322 2032 y Fu(0)2359 2020 y Fw(\)\))34 b(is)f(the)h(p)r(erm)n(uted)f Ft(x)3121 2032 y Fu(0)3159 2020 y Fw(.)54 b(The)515 2120 y(n)n(um)n(b)r(er)25 b(of)g(non)g(v)-5 b(anishing)25 b(principal)g(curv)-5 b(atures)24 b(at)h Ft(x)2323 2132 y Fu(0)2386 2120 y Fw(is)h(then)g(giv)n(en)e(b)n(y)h(the)h(rank)e(of) 515 2220 y(D)578 2190 y Fu(2)578 2240 y Fp(u)622 2220 y Fw(h\()p Ft(u)748 2232 y Fu(0)785 2220 y Fw(\).)37 b(Indeed,)27 b(for)g(an)n(y)f Ft(j)i Fw(=)22 b(1)p Ft(;)14 b(:)g(:)g(:)f(;)h(d)k Fs(\000)e Fw(1,)27 b(di\013eren)n(tiating)g(the)g (previous)f(equation)515 2319 y(with)i(resp)r(ect)f(to)h Ft(u)1138 2289 y Fu(\()p Fp(j)s Fu(\))1252 2319 y Fw(giv)n(es)1450 2496 y Fs(\000)14 b Fw(sin)f Ft(u)1692 2462 y Fu(\()p Fp(j)s Fu(\))1797 2496 y Fs(\000)18 b Fw(sin)c Ft(h)g(@)2102 2515 y Fp(u)2141 2499 y Fk(\()p FC(j)r Fk(\))2221 2496 y Ft(h)23 b Fw(=)f(0)p Ft(;)787 b Fw(\(14\))515 2673 y(where)27 b Ft(h)c Fw(=)f(h\()p Ft(u)p Fw(\).)38 b(Consequen)n(tly)-7 b(,)536 2849 y Fs(\000)14 b Fw(sin)f Ft(h)h Fw(D)855 2815 y Fu(2)855 2870 y Fp(u)899 2849 y Fw(h\()p Ft(u)p Fw(\))23 b(=)g(diag\(cos)13 b Ft(u)1526 2815 y Fu(\(1\))1615 2849 y Ft(;)h(:)g(:)g(:)g(;)g Fw(cos)e Ft(u)1972 2815 y Fu(\()p Fp(d)p Fo(\000)p Fu(1\))2148 2849 y Fw(\))18 b(+)g(cos)13 b Ft(h)h Fw([)p Ft(@)2535 2869 y Fp(u)2574 2852 y Fk(\()p FC(j)r Fk(\))2654 2849 y Fw(h\()p Ft(u)p Fw(\))p Ft(@)2856 2869 y Fp(u)2895 2852 y Fk(\()p FC(k)q Fk(\))2981 2849 y Fw(h\()p Ft(u)p Fw(\))q(])3163 2814 y Fp(d)p Fo(\000)p Fu(1)3163 2875 y Fp(j;k)q Fu(=1)3335 2849 y Ft(:)515 3026 y Fw(Since)30 b(sin)13 b Ft(h)27 b Fs(6)p Fw(=)f(0,)k(it)h(su\016ces)e(to)h(sho)n(w)f(that)h(the)g(righ) n(t)f(side)h(in)g(the)g(previous)f(equation)515 3126 y(do)r(es)i(not)h(v)-5 b(anish)32 b(in)h(a)e(neigh)n(b)r(orho)r(o)r(d)g (of)h Ft(u)1967 3138 y Fu(0)2004 3126 y Fw(.)50 b(By)32 b(the)g(equation)g(\(14\))f(this)i(matrix)e(is)515 3225 y(equal)c(to)899 3441 y(diag\(cos)14 b Ft(u)1258 3407 y Fu(\(1\))1346 3441 y Ft(;)g(:)g(:)g(:)g(;)g Fw(cos)f Ft(u)1704 3407 y Fu(\()p Fp(d)p Fo(\000)p Fu(1\))1879 3441 y Fw(\))19 b(+)2037 3385 y(cos)13 b Ft(h)p 2023 3422 201 4 v 2023 3504 a Fw(sin)2125 3469 y Fu(2)2176 3504 y Ft(h)2234 3441 y Fw([sin)g Ft(u)2420 3407 y Fu(\()p Fp(j)s Fu(\))2521 3441 y Fw(sin)g Ft(u)2684 3407 y Fu(\()p Fp(k)q Fu(\))2777 3441 y Fw(])2800 3405 y Fp(d)p Fo(\000)p Fu(1)2800 3466 y Fp(j;k)q Fu(=1)2972 3441 y Ft(:)236 b Fw(\(15\))639 3647 y(Supp)r(ose)30 b(b)n(y)f(con)n(tradiction)g (there)g(exists)h(an)f Ft(u)g 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4596 y Fu(\()p Fp(j)s Fu(\))1547 4630 y Fw(=)22 b Fs(\000)14 b Fw(cos)f Ft(h)55 b Fw(for)27 b Ft(j)h Fw(=)22 b(1)p Ft(;)14 b(:)g(:)g(:)g(;)g(d)k Fs(\000)g Fw(1)515 4807 y(at)g(suc)n(h)h(an)f Ft(u)p Fw(.)34 b(Hence,)21 b(b)n(y)d(the)h(equation)f(\(13\),)j(2\(2)q Fs(\000)q Ft(d)p Fw(\))14 b(cos)d Ft(h)23 b Fw(=)g Ft(e)p Fw(.)34 b(If)19 b Ft(d)k Fw(=)g(2,)d(then)f Ft(e)k Fw(=)f(0,)515 4907 y(con)n(trary)17 b(to)j(our)f(assumption.)34 b(Th)n(us,)21 b(the)f(previous)f(situation)g(w)n(ould)g(o)r(ccur)h(only)f(when)515 5006 y Ft(d)31 b(>)h Fw(2,)h(in)g(whic)n(h)g(case)e(cos)13 b Ft(h)32 b Fw(=)f Fs(\000)1745 4974 y Fp(e)p 1684 4988 153 4 v 1684 5035 a Fu(2)p Fp(d)p Fo(\000)p Fu(4)1880 5006 y Fw(and)h(cos)13 b Ft(u)2219 4976 y Fu(\()p Fp(j)s Fu(\))2337 5006 y Fw(=)2504 4974 y Fp(e)p 2443 4988 V 2443 5035 a Fu(2)p Fp(d)p Fo(\000)p Fu(4)2638 5006 y Fw(for)32 b Ft(j)37 b Fw(=)31 b(1)p Ft(;)14 b(:)g(:)g(:)f(;)h(d)22 b Fs(\000)f Fw(1.)1905 5255 y(39)p eop end %%Page: 40 40 TeXDict begin 40 39 bop 515 523 a Fw(Then,)30 b(cos)868 493 y Fu(2)919 523 y Ft(h)c Fw(=)g(cos)1195 493 y Fu(2)1246 523 y Ft(u)1294 493 y Fu(\()p Fp(j)s Fu(\))1410 523 y Fw(for)j(an)n(y)g Ft(j)5 b Fw(,)30 b(so)f(sin)14 b Ft(u)2058 493 y Fu(\()p Fp(j)s Fu(\))2158 523 y Fw(sin)g Ft(u)2322 493 y Fu(\()p Fp(k)q Fu(\))2440 523 y Fw(=)26 b Fs(\006)14 b Fw(sin)2711 488 y Fu(2)2762 523 y Ft(h)p Fw(.)43 b(Consequen)n(tly)-7 b(,)515 623 y(the)24 b(\()p Ft(j;)14 b(k)s Fw(\)-th)25 b(elemen)n(t)g(of)f(the)h(considered)e(matrix,)h(\(15\),)h(is)f(0)g (when)g Ft(j)k Fw(=)23 b Ft(k)s Fw(,)i(but)g Fs(\006)3277 590 y Fp(e)p 3216 604 153 4 v 3216 651 a Fu(2)p Fp(d)p Fo(\000)p Fu(4)515 722 y Fw(otherwise.)41 b(This)30 b(last)f(quan)n (tit)n(y)g(di\013ers)h(from)f(zero)f(\(since)i Ft(e)35 b(=)-51 b Fs(2)26 b Ft(E)5 b Fw(\),)31 b(whic)n(h)e(pro)n(vides)f(a)515 822 y(con)n(tradiction.)639 922 y(In)35 b(conclusion,)h(at)e(an)n(y)g (p)r(oin)n(t)h(in)g(a)f(neigh)n(b)r(orho)r(o)r(d)g(of)g Ft(x)2537 934 y Fu(0)2575 922 y Fw(,)i(\000\()p 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Ft(G)p Fw(\()p Ft(n;)14 b(z)t Fw(\))23 b(=)f Ft(O)r Fw(\()p Fs(j)p Ft(n)p Fs(j)2433 1880 y Fo(\000)2496 1857 y Fk(1)p 2496 1866 29 3 v 2496 1900 a(2)2538 1914 y Fw(\))515 2198 y Fv(when)36 b Fs(j)p Ft(n)p Fs(j)f(!)g(1)p Fv(,)k(uniformly)e(in)f Ft(e)g Fv(on)g(e)l(ach)i(c)l(omp)l(act)f(and)f(uniformly)i(in)e Ft(!)h Fs(2)f Fw(S)3230 2168 y Fp(d)p Fo(\000)p Fu(1)3354 2198 y Fv(,)515 2298 y(wher)l(e)30 b Ft(n)23 b Fw(=)g Fs(j)p Ft(n)p Fs(j)p Ft(!)32 b Fv(is)e(the)g(p)l(olar)h(form)g(of)f Ft(n)23 b Fs(6)p Fw(=)g(0)p Fv(.)515 2613 y Fl(Molc)m(hano)m(v{V)-8 b(ain)m(b)s(erg)21 b(Laplacian)85 b Fw(In)18 b(order)f(to)h(a)n(v)n (oid)f(con)n(v)n(exit)n(y)g(problems,)j(Mol-)515 2713 y(c)n(hano)n(v)26 b(and)i(V)-7 b(ain)n(b)r(erg)28 b(ha)n(v)n(e)f (suggested)g(to)h(c)n(hange)f(the)h(discretization)g(of)g(the)g(Lapla-) 515 2812 y(cian.)50 b(They)32 b(ha)n(v)n(e)g(based)f(their)h (construction)g(on)g(the)h(2)2381 2782 y Fp(d)2451 2812 y Fw(full-diagonal)e(neigh)n(b)r(ors)g(of)515 2912 y(elemen)n(ts)f(in)g Fn(Z)1011 2882 y Fp(d)1050 2912 y Fw(,)i(instead)e(of)g(their)g(2)p Ft(d)g Fw(immediate)h(neigh)n(b)r(ors.)43 b(The)31 b(constan)n(t)e (energy)515 3012 y(surfaces)22 b(of)h(the)g(resulting)g(op)r(erator)e (are)h(strictly)h(con)n(v)n(ex)f(in)h(an)n(y)f(dimension,)i(as)f(sho)n (wn)515 3111 y(b)r(elo)n(w.)639 3211 y(Explicitly)-7 b(,)24 b(the)g Fv(Molchanov{V)-6 b(ainb)l(er)l(g)29 b(L)l(aplacian)j Fw(\(or)22 b Fv(diagonal)28 b(L)l(aplacian)6 b Fw(\))26 b(is)e(the)515 3311 y(adjacency)19 b(op)r(erator)f(of)i(the)h (translational)d(in)n(v)-5 b(arian)n(t)19 b(graph)g(sp)r(eci\014ed)h(b) n(y)g(the)g(follo)n(wing)515 3410 y(set)27 b(of)h(p)r(oin)n(ts)f (adjacen)n(t)h(to)f(the)h(origin:)960 3593 y Fs(V)i Fw(=)23 b Fs(f)p Fw(\()p Ft(v)1246 3559 y Fu(\(1\))1335 3593 y Ft(;)14 b(:)g(:)g(:)f(;)h(v)1562 3559 y Fu(\()p Fp(d)p Fu(\))1653 3593 y Fw(\))24 b(;)36 b Ft(v)1811 3559 y Fu(\()p Fp(j)s Fu(\))1922 3593 y Fs(2)23 b(f)p Fw(1)p Ft(;)14 b Fs(\000)p Fw(1)n Fs(g)55 b Fw(for)27 b Ft(j)h Fw(=)23 b(1)p Ft(;)14 b(:)g(:)g(:)f(;)h(d)p Fs(g)p Ft(:)515 3775 y Fw(By)30 b(an)h(elemen)n(tary)e(com)n(binatorial)g(argumen)n(t,) i Ft(n)d Fs(2)g Fn(Z)2336 3745 y Fp(d)2406 3775 y Fw(is)j(in)g(the)g (comp)r(onen)n(t)f(of)h(the)515 3875 y(origin)h(if)i(and)f(only)g(if)h (the)f Ft(n)1473 3845 y Fu(\()p Fp(j)s Fu(\))1560 3875 y Fw('s)g(are)g(all)g(ev)n(en)g(or)f(all)h(o)r(dd.)54 b(Indeed,)35 b(the)f(considered)515 3975 y(graph)26 b(consists)h(of)h (2)1193 3945 y Fp(d)p Fo(\000)p Fu(1)1343 3975 y Fw(connected)g(comp)r (onen)n(ts)f(with)h(set)g(of)f(represen)n(tativ)n(es)1028 4157 y Fs(f)p Fw(\(0)p Ft(;)14 b(n)1231 4123 y Fu(\(2\))1319 4157 y Ft(;)g(:)g(:)g(:)f(;)h(n)1553 4123 y Fu(\()p Fp(d)p Fu(\))1644 4157 y Fw(\))23 b(;)37 b Ft(n)1809 4123 y Fu(\()p Fp(j)s Fu(\))1919 4157 y Fs(2)23 b(f)p Fw(0)p Ft(;)14 b Fw(1)o Fs(g)55 b Fw(for)27 b Ft(j)h Fw(=)22 b(2)p Ft(;)14 b(:)g(:)g(:)f(;)h(d)p Fs(g)p Ft(:)515 4340 y Fw(The)27 b(graph)g(is)g(also)g(sp)r(eci\014ed)h(b)n(y)f(the)h(follo) n(wing)f(metric:)554 4568 y(d\()p Ft(m;)14 b(n)p Fw(\))23 b(=)935 4451 y Fr(\032)1039 4517 y Fs(j)p Ft(m)18 b Fs(\000)g Ft(n)p Fs(j)1309 4529 y Fo(1)1490 4517 y Fw(if)28 b(the)g(comp)r(onen)n (ts)f(of)h Ft(m)18 b Fs(\000)g Ft(n)28 b Fw(ha)n(v)n(e)e(the)i(same)f (parit)n(y)-7 b(,)1039 4617 y Fs(1)340 b Fw(otherwise,)515 4812 y(where)27 b Fs(j)p Ft(n)p Fs(j)851 4824 y Fo(1)944 4812 y Fw(=)c(max)1186 4782 y Fp(d)1186 4833 y(j)s Fu(=1)1319 4812 y Fs(j)p Ft(n)1392 4782 y Fu(\()p Fp(j)s Fu(\))1479 4812 y Fs(j)p Fw(.)639 4911 y(Remark)-5 b(ably)e(,)27 b(the)h(sym)n(b)r(ol)f(of)h(the)g(Molc)n(hano)n(v{V)-7 b(ain)n(b)r(erg)24 b(Laplacian)i(factorizes:)1905 5255 y(40)p eop end %%Page: 41 41 TeXDict begin 41 40 bop 515 523 a Fl(Theorem)31 b(24)1069 502 y Fr(b)1057 523 y Fw(\001)22 b Fv(is)f(the)h(op)l(er)l(ator)g(of)g (multiplic)l(ation)h(by)e Fw(\010\()p Ft(x)p Fw(\))j(=)f(2)2705 493 y Fp(d)2757 523 y Fw(cos)13 b Ft(x)2929 493 y Fu(\(1\))3032 523 y Ft(:)h(:)g(:)g Fw(cos)f Ft(x)3315 493 y Fu(\()p Fp(d)p Fu(\))3406 523 y Fv(,)515 623 y(wher)l(e)30 b Ft(x)24 b Fw(=)e(\()p Ft(x)986 593 y Fu(\(1\))1076 623 y Ft(;)14 b(:)g(:)g(:)g(;)g(x)1308 593 y Fu(\()p Fp(d)p Fu(\))1399 623 y Fw(\))23 b Fs(2)g Fn(T)1587 593 y Fp(d)1627 623 y Fv(.)515 799 y Fl(Pro)s(of:)75 b Fw(Let)31 b(us)h(denote)f(b)n(y) g Fs(f)p Ft(e)1583 811 y Fu(1)1619 799 y Ft(;)14 b(:)g(:)g(:)g(;)g(e) 1843 811 y Fp(d)1881 799 y Fs(g)31 b Fw(the)h(standard)e(basis)h(of)g Fn(Z)2812 769 y Fp(d)2851 799 y Fw(.)48 b(By)31 b(Theorem)515 898 y(22)26 b(the)i(sym)n(b)r(ol)g(of)f(\001)h(is)f(the)h(m)n (ultiplication)g(b)n(y)846 1152 y(\010\()p Ft(x)p Fw(\))84 b(=)1252 1073 y Fr(X)1248 1251 y Fp(v)r Fo(2V)1389 1152 y Fw(e)1426 1117 y Fu(i)p Fp(x)p Fo(\001)p Fp(v)1592 1152 y Fw(=)1711 1073 y Fr(X)1707 1251 y Fp(v)r Fo(2V)1848 1152 y Fw(e)1885 1116 y Fu(i)p Fp(x)p Fo(\001)1962 1072 y Fc(P)2031 1091 y FC(d)2031 1135 y(j)r Fk(=1)2144 1116 y Fp(v)2179 1091 y Fk(\()p FC(j)r Fk(\))2255 1116 y Fp(e)2286 1124 y FC(j)2373 1152 y Fw(=)2491 1073 y Fr(X)2488 1251 y Fp(v)r Fo(2V)2669 1048 y Fp(d)2633 1073 y Fr(Y)2628 1250 y Fp(j)s Fu(=1)2757 1152 y Fw(e)2794 1117 y Fu(i)p Fp(x)p Fo(\001)p Fp(v)2906 1092 y Fk(\()p FC(j)r Fk(\))2982 1117 y Fp(e)3013 1125 y FC(j)1101 1455 y Fw(=)1288 1351 y Fp(d)1253 1376 y Fr(Y)1248 1553 y Fp(j)s Fu(=1)1363 1455 y Fw(\(e)1432 1421 y Fu(i)p Fp(x)p Fo(\001)p Fp(e)1540 1429 y FC(j)1593 1455 y Fw(+)18 b(e)1713 1421 y Fo(\000)p Fu(i)p Fp(x)p Fo(\001)p Fp(e)1873 1429 y FC(j)1908 1455 y Fw(\))51 b(=)f(2)2148 1421 y Fp(d)2241 1351 y(d)2205 1376 y Fr(Y)2201 1553 y Fp(j)s Fu(=1)2329 1455 y Fw(cos)13 b Ft(x)2501 1421 y Fu(\()p Fp(j)s Fu(\))2589 1455 y Ft(;)515 1715 y Fw(as)27 b(claimed.)2397 b Fi(\003)639 1914 y Fw(Consequen)n(tly)-7 b(,)28 b(the)h(sp)r(ectrum)g(of)f(\001)h(is)f (purely)g(absolutely)g(con)n(tin)n(uous)f(and)i(equal)515 2014 y(to)616 1946 y Fr(\002)651 2014 y Fs(\000)p Fw(2)758 1984 y Fp(d)796 2014 y Ft(;)14 b Fw(2)875 1984 y Fp(d)913 1946 y Fr(\003)947 2014 y Fw(.)37 b(Moreo)n(v)n(er,)515 2213 y Fl(Lemma)77 b Fv(Supp)l(ose)40 b Fw(0)f Ft(<)h Fs(j)p Ft(e)p Fs(j)g Ft(<)f Fw(2)1682 2183 y Fp(d)1720 2213 y Fv(.)67 b(Then,)42 b(for)e(al)t(l)g Ft(x)h Fs(2)f Fw(\000\()p Ft(e)p Fw(\))p Fv(,)i Fs(r)2809 2225 y Fp(x)2851 2213 y Fw(\010\()p Ft(x)p Fw(\))f Fs(6)p Fw(=)e(0)p Fv(.)66 b(In)515 2313 y(p)l(articular,)31 b Fw(\000\()p Ft(e)p Fw(\))f Fv(de\014nes)g(a)g(r)l(e)l(gular)g(surfac)l(e)g(for)g(such)g (an)g Ft(e)p Fv(.)515 2512 y Fl(Pro)s(of:)112 b Fw(If)44 b Ft(x)50 b Fs(2)g Fw(\000\()p Ft(e)p Fw(\),)e(then)c(cos)13 b Ft(x)1790 2482 y Fu(\()p Fp(j)s Fu(\))1927 2512 y Fs(6)p Fw(=)50 b(0)43 b(for)g Ft(j)55 b Fw(=)49 b(1)p Ft(;)14 b(:)g(:)g(:)f(;)h(d)p Fw(,)48 b(so)43 b Ft(@)2975 2531 y Fp(x)3013 2514 y Fk(\()p FC(j)r Fk(\))3093 2512 y Fw(\010\()p Ft(x)p Fw(\))50 b(=)515 2623 y Fs(\000)p Fw(2)622 2593 y Fp(d)660 2623 y Ft(e)14 b Fw(tan)f Ft(x)893 2593 y Fu(\()p Fp(j)s Fu(\))980 2623 y Fw(.)62 b(Th)n(us,)38 b Fs(kr)1420 2635 y Fp(x)1462 2623 y Fw(\010\()p Ft(x)p Fw(\))q Fs(k)1676 2593 y Fu(2)1749 2623 y Fw(=)f(4)1893 2593 y Fp(d)1931 2623 y Ft(e)1970 2593 y Fu(2)2021 2561 y Fr(P)2109 2581 y Fp(d)2109 2648 y(j)s Fu(=1)2241 2623 y Fw(tan)2361 2592 y Fu(2)2412 2623 y Ft(x)2459 2593 y Fu(\()p Fp(j)s Fu(\))2547 2623 y Fw(,)h(whic)n(h)e(di\013ers)g(from)f (0,)515 2738 y(since)27 b Ft(e)c Fs(6)p Fw(=)f Fs(\006)p Fw(2)974 2708 y Fp(d)1012 2738 y Fw(.)2279 b Fi(\003)639 2937 y Fw(Let)22 b(us)f(in)n(v)n(estigate)f(the)i(constan)n(t)e(energy) g(surfaces)h(asso)r(ciated)f(with)h(\010\()p Ft(x)p Fw(\).)36 b(Firstly)-7 b(,)515 3037 y(let)28 b(us)f(consider)g(the)h(co)n(v)n (ering)1538 3016 y(~)1533 3037 y(\000\()p Ft(e)p Fw(\))23 b(=)g Fs(f)p Ft(x)g Fs(2)g Fn(R)2049 3007 y Fp(d)2111 3037 y Fw(;)37 b(\010\()p Ft(x)p Fw(\))24 b(=)f Ft(e)o Fs(g)p Fw(.)639 3137 y(If)32 b Ft(e)d Fw(=)g(0,)990 3116 y(~)985 3137 y(\000\()p Ft(e)p Fw(\))i(consists)g(of)g(the)h(h)n(yp)r (erplanes)e(of)h(equation)g Ft(x)2677 3106 y Fu(\()p Fp(j)s Fu(\))2794 3137 y Fw(=)d(\(2)p Ft(k)c Fw(+)d(1\))3198 3104 y Fp(\031)p 3197 3118 41 4 v 3201 3165 a Fu(2)3280 3137 y Fw(for)515 3236 y Ft(k)37 b Fs(2)e Fn(Z)g Fw(and)f Ft(j)40 b Fw(=)34 b(1)p Ft(;)14 b(:)g(:)g(:)f(;)h(d)p Fw(.)58 b(These)34 b(h)n(yp)r(erplanes)g(divide)g Fn(R)2493 3206 y Fp(d)2567 3236 y Fw(in)n(to)g(op)r(en)g(h)n(yp)r(ercub)r(es,)515 3336 y(whic)n(h)29 b(w)n(e)f(call)h Fv(c)l(el)t(ls)p Fw(.)42 b(The)29 b(cells)g(admit)g(a)f(go)r(o)r(d)h(bicoloration)e(in)i (the)h(follo)n(wing)e(sense:)515 3435 y(starting)g(from)h(a)g(set)g(of) g(t)n(w)n(o)g(colors,)f(sa)n(y)-7 b(,)29 b(red)g(and)g(blue,)h(it)f(is) h(p)r(ossible)f(to)g(pain)n(t)g(eac)n(h)515 3535 y(cell)e(in)g(suc)n(h) f(a)h(w)n(a)n(y)e(that)i(the)g(2)p Ft(d)g Fw(neigh)n(b)r(ors)f(of)g(an) n(y)g(red)h(cell)g(are)f(blue)h(and)f Fv(vic)l(e)k(versa)p Fw(.)515 3635 y(Let)d(us)h(accomplish)f(this,)h(the)g(cell)f(con)n (taining)g(the)h(origin)e(b)r(eing)i(pain)n(ted)g(in)f(red.)639 3734 y(If)f Ft(e)c Fw(=)h(2)911 3704 y Fp(d)949 3734 y Fw(,)j(then)1189 3713 y(~)1184 3734 y(\000\()p Ft(e)p Fw(\))f(is)g(a)f(discrete)h(set)g(consisting)f(of)g(the)i(cen)n(ters)e (of)h(the)g(red)f(cells.)515 3834 y(On)34 b(the)i(other)e(hand,)j(if)e Ft(e)g Fw(=)g Fs(\000)p Fw(2)1639 3804 y Fp(d)1676 3834 y Fw(,)i(then)1938 3813 y(~)1933 3834 y(\000)o(\()p Ft(e)p Fw(\))f(consists)e(of)g(the)i(cen)n(ters)e(of)h(the)g(blue)515 3934 y(cells.)639 4033 y(When)24 b Ft(x)f Fw(v)-5 b(aries)22 b(con)n(tin)n(uously)-7 b(,)23 b(\010\()p Ft(x)p Fw(\))h(c)n(hanges)d (sign)h(eac)n(h)g(time)i(one)e(of)h(the)g(previous)515 4133 y(h)n(yp)r(erplanes)33 b(is)h(crossed.)54 b(It)35 b(follo)n(ws)e(that)h(the)g(connected)g(comp)r(onen)n(ts)g(of)3084 4112 y(~)3079 4133 y(\000\()p Ft(e)p Fw(\))g(are)515 4232 y(enclosed)d(in)g(the)h(red)f(cells)h(when)g Ft(e)d(>)g Fw(0,)j(eac)n(h)f(red)g(cell)h(con)n(taining)e(one)h(comp)r(onen)n(t.) 515 4332 y(Moreo)n(v)n(er,)24 b(these)j(comp)r(onen)n(ts)g(are)f(all)h (congruen)n(t.)35 b(The)27 b(situation)g(is)g(the)g(same)g(when)515 4432 y Ft(e)22 b(<)h Fw(0,)k(but)i(replacing)d(the)i(red)f(cells)g (with)i(the)f(blue)f(ones.)639 4531 y(Finally)-7 b(,)26 b(\000\()p Ft(e)p Fw(\))f(is)g(obtained)g(from)g(the)h(previous)e (surface)g(b)n(y)h(restricting)2989 4510 y(~)2984 4531 y(\000\()p Ft(e)p Fw(\))h(to)f(the)515 4631 y(torus,)32 b(where)g Ft(e)e Fs(2)1156 4564 y Fr(\003)1190 4631 y Fs(\000)p Fw(2)1297 4601 y Fp(d)1335 4631 y Ft(;)14 b Fw(2)1414 4601 y Fp(d)1452 4564 y Fr(\002)1508 4631 y Fs(n)21 b(f)p Fw(0)p Fs(g)31 b Fw(is)h(\014xed.)50 b(It)32 b(follo)n(ws)f(that)h(\000\()p Ft(e)p Fw(\))h(consists)e(of)h(2)3256 4601 y Fp(d)p Fo(\000)p Fu(1)515 4731 y Fw(iden)n(tical)27 b(connected)h(comp)r(onen)n(ts.)639 4830 y(As)g(Molc)n(hano)n(v)e(and)h (V)-7 b(ain)n(b)r(erg)27 b(conjectured,)515 5006 y Fl(Theorem)k(25)41 b Fv(F)-6 b(or)30 b Fw(0)23 b Ft(<)f Fs(j)p Ft(e)p Fs(j)h Ft(<)g Fw(2)1602 4976 y Fp(d)1670 5006 y Fv(any)30 b(c)l(omp)l(onent)g (of)g Fw(\000\()p Ft(e)p Fw(\))g Fv(is)g(strictly)g(c)l(onvex.)1905 5255 y Fw(41)p eop end %%Page: 42 42 TeXDict begin 42 41 bop 515 523 a Fl(Pro)s(of:)69 b Fw(Supp)r(ose)30 b(0)c Ft(<)f(e)h(<)g Fw(2)1527 493 y Fp(d)1565 523 y Fw(,)k(the)g(other)f(case)g(b)r(eing)g(similar.)42 b(Then,)30 b(without)g(loss)515 623 y(of)d(generalit)n(y)f(the)i(considered)f (comp)r(onen)n(t)g(is)1283 869 y Fs(f)p Ft(x)c Fs(2)1473 777 y Fr(i)1512 869 y Fs(\000)1587 813 y Ft(\031)p 1587 850 51 4 v 1591 926 a Fw(2)1647 869 y Ft(;)1694 813 y(\031)p 1694 850 V 1698 926 a Fw(2)1754 777 y Fr(h)1793 794 y Fp(d)1855 869 y Fw(;)37 b(2)1957 835 y Fp(d)2049 765 y(d)2014 790 y Fr(Y)2009 967 y Fp(j)s Fu(=1)2138 869 y Fw(cos)13 b Ft(x)2310 835 y Fu(\()p Fp(j)s Fu(\))2420 869 y Fw(=)23 b Ft(e)p Fs(g)p Ft(:)515 1138 y Fw(Let)33 b Ft(m)f Fw(=)g Ft(d)23 b Fs(\000)f Fw(1)32 b(and)h Ft(h)g Fw(=)f Ft(x)1490 1108 y Fu(\()p Fp(d)p Fu(\))1581 1138 y Fw(.)54 b(The)33 b(equation)f(de\014ning)i(the)f(previous)f(comp)r (onen)n(t)515 1238 y(b)r(ecomes)1384 1337 y(2)1426 1303 y Fp(d)1478 1337 y Fw(cos)13 b Ft(x)1650 1303 y Fu(\(1\))1753 1337 y Ft(:)h(:)g(:)g Fw(cos)f Ft(x)2036 1303 y Fu(\()p Fp(m)p Fu(\))2165 1337 y Fw(cos)g Ft(h)23 b Fw(=)f Ft(e:)515 1479 y Fw(Since)28 b(eac)n(h)e(factor)h(in)h(the)g(ab)r(o)n(v)n(e)e(is) i(p)r(ositiv)n(e,)920 1648 y Ft(d)14 b Fw(ln)g(2)k(+)g(ln)c(cos)f Ft(x)1458 1614 y Fu(\(1\))1566 1648 y Fw(+)18 b Fs(\001)c(\001)g(\001)k Fw(+)h(ln)14 b(cos)f Ft(x)2103 1614 y Fu(\()p Fp(m)p Fu(\))2236 1648 y Fw(+)18 b(ln)c(cos)f Ft(h)19 b Fs(\000)f Fw(ln)c Ft(e)22 b Fw(=)h(0)p Ft(:)257 b Fw(\(16\))639 1817 y(Since)19 b(the)g(considered)f(comp)r(onen)n(t)h(is)f(symmetric)h (with)g(resp)r(ect)f(to)h(the)g(h)n(yp)r(erplanes)515 1917 y Ft(x)562 1887 y Fu(\()p Fp(j)s Fu(\))674 1917 y Fw(=)25 b(0)k(and)g Ft(x)1045 1887 y Fu(\()p Fp(j)s Fu(\))1157 1917 y Fw(=)c Ft(x)1294 1887 y Fu(\()p Fp(l)p Fu(\))1372 1917 y Fw(,)30 b(where)e Ft(j;)g(l)f Fs(2)f(f)p Fw(1)p Ft(;)14 b(:)g(:)g(:)e(;)i(d)p Fs(g)29 b Fw(are)f(distinct,)i(it) f(su\016ces)g(to)g(sho)n(w)515 2016 y(the)f(result)f(on)h(the)g (fundamen)n(tal)g(domain)f Ft(h)d Fi(6)f Ft(x)2130 1986 y Fu(\(1\))2243 2016 y Fi(6)g Fs(\001)14 b(\001)g(\001)23 b Fi(6)g Ft(x)2586 1986 y Fu(\()p Fp(m)p Fu(\))2725 2016 y Fi(6)g Fw(0.)37 b(There,)27 b Ft(h)c Fs(6)p Fw(=)g(0)515 2116 y(since)k Ft(e)c Fs(6)p Fw(=)f(2)909 2086 y Fp(d)948 2116 y Fw(.)639 2216 y(The)h(deriv)-5 b(ativ)n(e)22 b(with)h(resp)r (ect)f(to)g Ft(h)h Fw(of)f(the)h(left)g(side)g(in)g(\(16\))f(is)g Fs(\000)14 b Fw(tan)f Ft(h)p Fw(,)24 b(whic)n(h)e(do)r(es)515 2315 y(not)27 b(v)-5 b(anish.)38 b(Consequen)n(tly)-7 b(,)27 b(an)g(implicit)i(function,)f Ft(h)23 b Fw(=)g(h\()p Ft(x;)14 b(e)p Fw(\),)29 b(satisfying)e(\(16\))g(in)h(a)515 2415 y(neigh)n(b)r(orho)r(o)r(d)e(of)g(an)h(arbitrarily)e(\014xed)i(p)r (oin)n(t)g(in)g(the)h(fundamen)n(tal)f(domain)f(exists,)h(is)515 2514 y(analytic,)g(and)g(induces)h(a)f(lo)r(cal)g(parameterization)f (of)h(the)h(previous)f(comp)r(onen)n(t.)639 2614 y(It)36 b(th)n(us)f(su\016ces)f(to)h(sho)n(w)f(that)h(D)1788 2584 y Fu(2)1788 2635 y Fp(x)1831 2614 y Fw(h\()p Ft(x;)14 b(e)p Fw(\))35 b(is)g(p)r(ositiv)n(e)g(de\014nite.)59 b(Di\013eren)n(tiating)515 2714 y(\(16\))27 b(with)h(resp)r(ect)f(to)h Ft(x)1312 2684 y Fu(\()p Fp(j)s Fu(\))1427 2714 y Fw(giv)n(es)1433 2883 y Fs(\000)14 b Fw(tan)f Ft(x)1692 2848 y Fu(\()p Fp(j)s Fu(\))1798 2883 y Fs(\000)18 b Fw(tan)13 b Ft(h)h(@)2120 2902 y Fp(x)2158 2885 y Fk(\()p FC(j)r Fk(\))2238 2883 y Ft(h)23 b Fw(=)f(0)p Ft(;)515 3052 y Fw(where)27 b Ft(h)c Fw(=)f(h\()p Ft(x;)14 b(e)p Fw(\).)38 b(Di\013eren)n(tiating)28 b(this)f(last)h(equation)f(with)h(resp)r(ect)f(to)h Ft(x)3040 3022 y Fu(\()p Fp(l)p Fu(\))3145 3052 y Fw(giv)n(es)818 3224 y Fs(\000)14 b Fw(sec)1003 3194 y Fu(2)1054 3224 y Ft(x)1101 3194 y Fu(\()p Fp(j)s Fu(\))1207 3224 y Fs(\000)k Fw(sec)1397 3194 y Fu(2)1448 3224 y Ft(h)c Fw(\()p Ft(@)1586 3243 y Fp(x)1624 3227 y Fk(\()p FC(j)r Fk(\))1704 3224 y Ft(h)p Fw(\))1784 3194 y Fu(2)1839 3224 y Fs(\000)k Fw(tan)c Ft(h)g(@)2167 3194 y Fu(2)2162 3255 y Fp(x)2200 3238 y Fk(\()p FC(j)r Fk(\))2280 3224 y Ft(h)83 b Fw(=)f(0)110 b(if)28 b Ft(l)d Fw(=)e Ft(j)5 b(;)956 3325 y Fs(\000)14 b Fw(sec)1141 3295 y Fu(2)1192 3325 y Ft(h)g(@)1298 3344 y Fp(x)1336 3327 y Fk(\()p FC(j)r Fk(\))1415 3325 y Ft(h@)1507 3344 y Fp(x)1545 3327 y Fk(\()p FC(l)p Fk(\))1618 3325 y Ft(h)k Fs(\000)g Fw(tan)c Ft(h)g(@)2007 3344 y Fp(x)2045 3327 y Fk(\()p FC(j)r Fk(\))2125 3325 y Ft(@)2169 3344 y Fp(x)2207 3327 y Fk(\()p FC(l)p Fk(\))2280 3325 y Ft(h)83 b Fw(=)f(0)110 b(otherwise.)515 3495 y(Let)26 b Ft(a)706 3507 y Fp(j)s(l)785 3495 y Fw(=)d Fs(\000)14 b Fw(tan)f Ft(h)h(@)1191 3514 y Fp(x)1229 3497 y Fk(\()p FC(j)r Fk(\))1309 3495 y Ft(@)1353 3514 y Fp(x)1391 3497 y Fk(\()p FC(l)p Fk(\))1464 3495 y Ft(h)p Fw(.)36 b(Since)26 b Fs(\000)14 b Fw(tan)f Ft(h)23 b(>)g Fw(0,)j(it)g(su\016ces)g(to)g(sho)n (w)f([)p Ft(a)2981 3507 y Fp(j)s(l)3037 3495 y Fw(])j Ft(>)22 b Fw(0.)36 b(By)515 3594 y(the)28 b(ab)r(o)n(v)n(e,)874 3764 y([)p Ft(a)941 3776 y Fp(j)s(l)998 3764 y Fw(])f(=)c(diag\(sec) 1428 3729 y Fu(2)1479 3764 y Ft(x)1526 3729 y Fu(\(1\))1615 3764 y Ft(;)14 b(:)g(:)g(:)g(;)g Fw(sec)1906 3729 y Fu(2)1957 3764 y Ft(x)2004 3729 y Fu(\()p Fp(d)p Fo(\000)p Fu(1\))2181 3764 y Fw(\))k(+)g(sec)2421 3729 y Fu(2)2472 3764 y Ft(h)c Fw([)p Ft(@)2601 3783 y Fp(x)2639 3766 y Fk(\()p FC(j)r Fk(\))2719 3764 y Ft(h@)2811 3783 y Fp(x)2849 3766 y Fk(\()p FC(l)p Fk(\))2921 3764 y Ft(h)p Fw(])t Ft(:)515 3933 y Fw(Clearly)-7 b(,)25 b(the)i(\014rst)f(term)h(of)f(the)h(righ)n (t-hand)e(side)h(in)h(the)g(previous)e(equation)h(is)g(strictly)515 4032 y(p)r(ositiv)n(e.)33 b(Moreo)n(v)n(er,)18 b(the)i(second)e(term)h (is)g(non)g(negativ)n(e,)h(since)e(all)h(its)g(principal)g(minors)515 4132 y(are)29 b(zero)h(except)g(the)h(\014rst,)g(whic)n(h)f(is)h(a)f (square.)44 b(Hence,)31 b([)p Ft(a)2483 4144 y Fp(j)s(l)2540 4132 y Fw(])h Ft(>)27 b Fw(0,)k(whic)n(h)f(completes)515 4232 y(the)e(pro)r(of.)2443 b Fi(\003)639 4431 y Fw(Theorem)30 b(21)g(then)i(giv)n(es)e(an)g(optimal)h(deca)n(y)f(for)h(the)g(Green's) f(function,)j Ft(G)p Fw(\()p Ft(n;)14 b(e)p Fw(\),)515 4530 y(asso)r(ciated)26 b(with)i(the)g(Molc)n(hano)n(v{V)-7 b(ain)n(b)r(erg)24 b(Laplacian.)36 b(Explicitly:)515 4730 y Fl(Corollary)60 b Fv(L)l(et)30 b Ft(E)e Fw(=)22 b Fs(f\000)p Fw(2)1437 4700 y Fp(d)1475 4730 y Ft(;)14 b Fw(0)p Ft(;)g Fw(2)1633 4700 y Fp(d)1670 4730 y Fs(g)30 b Fv(and)g(supp)l(ose)g Ft(e)23 b Fs(2)2346 4662 y Fr(\002)2381 4730 y Fs(\000)p Fw(2)2488 4700 y Fp(d)2526 4730 y Ft(;)14 b Fw(2)2605 4700 y Fp(d)2643 4662 y Fr(\003)2696 4730 y Fs(n)k Ft(E)5 b Fv(.)39 b(Then,)1285 4925 y Ft(G)p Fw(\()p Ft(n;)14 b(e)p Fw(\))24 b(=)48 b(lim)1669 4973 y Fp(z)r Fo(!)p Fp(e)1651 5024 y(z)r Fo(2)p Fq(C)1772 5032 y Fk(+)1833 4925 y Ft(G)p Fw(\()p Ft(n;)14 b(z)t Fw(\))23 b(=)f Ft(O)r Fw(\()p Fs(j)p Ft(n)p Fs(j)2395 4891 y Fo(\000)2458 4865 y FC(d)p Fj(\000)p Fk(1)p 2458 4878 104 3 v 2496 4911 a(2)2576 4925 y Fw(\))1905 5255 y(42)p eop end %%Page: 43 43 TeXDict begin 43 42 bop 515 523 a Fv(when)36 b Fs(j)p Ft(n)p Fs(j)f(!)g(1)p Fv(,)k(uniformly)e(in)f Ft(e)g Fv(on)g(e)l(ach)i(c)l(omp)l(act)f(and)f(uniformly)i(in)e Ft(!)h Fs(2)f Fw(S)3230 493 y Fp(d)p Fo(\000)p Fu(1)3354 523 y Fv(,)515 623 y(wher)l(e)30 b Ft(n)23 b Fw(=)g Fs(j)p Ft(n)p Fs(j)p Ft(!)32 b Fv(is)e(the)g(p)l(olar)h(form)g(of)f Ft(n)23 b Fs(6)p Fw(=)g(0)p Fv(.)1857 1063 y Fw(*)k(*)g(*)515 1304 y Fb(App)s(endix:)51 b(Estimate)37 b(Using)i(Bessel)e(F)-9 b(unction)639 1504 y Fw(Let)32 b(\001)g(b)r(e)g(the)g Fv(standar)l(d)42 b Fw(discrete)31 b(Laplacian)f(in)i(dimension)g Ft(d)e Fi(>)g Fw(4.)49 b(Its)32 b(Green's)515 1603 y(function)c(is)f (denoted)h(b)n(y)1331 1755 y Ft(G)p Fw(\()p Ft(n)19 b Fs(\000)f Ft(m;)c(z)t Fw(\))23 b(=)f Fs(h)p Ft(\016)1944 1767 y Fp(m)2021 1755 y Fs(j)14 b Fw(\(\001)19 b Fs(\000)f Ft(z)s Fw(\))2335 1721 y Fo(\000)p Fu(1)2425 1755 y Ft(\016)2462 1767 y Fp(n)2507 1755 y Fs(i)p Ft(;)515 1907 y Fw(where)27 b Ft(m;)14 b(n)22 b Fs(2)i Fn(Z)1071 1877 y Fp(d)1110 1907 y Fw(,)k Ft(z)e Fs(2)d Fn(C)1364 1919 y Fu(+)1420 1907 y Fw(,)k(and)h Ft(\016)j Fw(is)d(the)g(Kronec)n(k)n(er)d(delta.)37 b(By)27 b(Kato's)f(form)n(ula)1303 2117 y Ft(G)p Fw(\()p Ft(n;)14 b(z)t Fw(\))23 b(=)f(i)1709 2004 y Fr(Z)1792 2025 y Fu(0)1755 2193 y Fo(\0001)1891 2117 y Fw(e)1928 2083 y Fo(\000)p Fu(i)p Fp(tz)2062 2117 y Fs(h)p Ft(\016)2131 2129 y Fu(0)2183 2117 y Fs(j)14 b Fw(e)2257 2083 y Fu(i)p Fp(t)p Fu(\001)2359 2117 y Ft(\016)2396 2129 y Fp(n)2441 2117 y Fs(i)19 b Fw(d)p Ft(t:)640 b Fw(\(17\))515 2343 y(Recall)34 b(that)i(the)f(sym)n(b)r(ol)g(of)g(\001)g(is)g(the)h(m)n (ultiplication)f(b)n(y)g(\010\()p Ft(x)p Fw(\))i(=)e(2)2877 2280 y Fr(P)2964 2301 y Fp(d)2964 2367 y(j)s Fu(=1)3097 2343 y Fw(cos)13 b Ft(x)3269 2312 y Fu(\()p Fp(j)s Fu(\))3356 2343 y Fw(,)515 2461 y(where)27 b Ft(x)c Fw(=)g(\()p Ft(x)992 2431 y Fu(\(1\))1082 2461 y Ft(;)14 b(:)g(:)g(:)f(;)h(x)1313 2431 y Fu(\()p Fp(d)p Fu(\))1404 2461 y Fw(\))24 b Fs(2)f Fn(T)1593 2431 y Fp(d)1632 2461 y Fw(.)37 b(Hence,)1058 2695 y Fs(h)p Ft(\016)1127 2707 y Fu(0)1178 2695 y Fs(j)14 b Fw(e)1252 2660 y Fu(i)p Fp(t)p Fu(\001)1354 2695 y Ft(\016)1391 2707 y Fp(n)1436 2695 y Fs(i)84 b Fw(=)e(\(2)p Ft(\031)s Fw(\))1855 2660 y Fo(\000)p Fp(d)2000 2591 y(d)1965 2616 y Fr(Y)1960 2793 y Fp(j)s Fu(=1)2089 2582 y Fr(Z)2172 2602 y Fp(\031)2135 2770 y Fo(\000)p Fp(\031)2246 2695 y Fw(e)2283 2660 y Fu(i)p Fp(n)2343 2635 y Fk(\()p FC(j)r Fk(\))2418 2660 y Fp(k)2459 2695 y Fw(e)2496 2660 y Fu(2i)p Fp(t)12 b Fu(cos)f Fp(k)2744 2695 y Fw(d)p Ft(k)1552 2998 y Fw(=)1739 2894 y Fp(d)1704 2919 y Fr(Y)1699 3096 y Fp(j)s Fu(=1)1828 2998 y Fw(i)1851 2964 y Fu(\()p Fp(n)1918 2939 y Fk(\()p FC(j)r Fk(\))1994 2964 y Fu(\))2024 2998 y Ft(J)2070 3017 y Fp(n)2111 3000 y Fk(\()p FC(j)r Fk(\))2191 2998 y Fw(\(2)p Ft(t)p Fw(\))515 3250 y(at)25 b(an)n(y)f Ft(n)f Fw(=)g(\()p Ft(n)1011 3220 y Fu(\(1\))1100 3250 y Ft(;)14 b(:)g(:)g(:)g(;)g(n)1335 3220 y Fu(\()p Fp(d)p Fu(\))1425 3250 y Fw(\))25 b(and)h Ft(z)g Fs(2)d Fn(C)1845 3262 y Fu(+)1900 3250 y Fw(,)j(where)f Ft(J)2233 3262 y Fp(m)2296 3250 y Fw(\()p Ft(t)p Fw(\))f(=)2532 3217 y Fu(1)p 2511 3231 75 4 v 2511 3279 a(2)p Fp(\031)2609 3183 y Fr(R)2664 3204 y Fp(\031)2648 3279 y Fo(\000)p Fp(\031)2759 3250 y Fw(e)2796 3220 y Fu(i)p Fp(mk)2914 3250 y Fw(e)2951 3220 y Fo(\000)p Fu(i)p Fp(t)11 b Fu(sin)g Fp(k)3206 3250 y Fw(d)p Ft(k)28 b Fw(is)515 3350 y(the)g(Bessel)f (function.)639 3449 y(It)h(is)g(w)n(ell)f(kno)n(wn)g(that)h(there)f (exist)h(a)f(univ)n(ersal)f(constan)n(t,)h Ft(C)6 b Fw(,)28 b(suc)n(h)f(that)1294 3639 y Fs(j)p Ft(J)1363 3651 y Fp(m)1427 3639 y Fw(\()p Ft(t)p Fw(\))p Fs(j)c Fi(6)1718 3583 y Ft(C)p 1665 3620 172 4 v 1665 3708 a Fs(j)p Ft(m)p Fs(j)1794 3656 y Fk(1)p 1794 3665 29 3 v 1794 3699 a(3)1901 3639 y Fw(and)56 b Fs(j)p Ft(J)2160 3651 y Fp(m)2223 3639 y Fw(\()p Ft(t)p Fw(\))p Fs(j)23 b Fi(6)2493 3583 y Ft(C)p 2461 3620 129 4 v 2461 3708 a Fs(j)p Ft(t)p Fs(j)2547 3656 y Fk(1)p 2547 3665 29 3 v 2547 3699 a(3)515 3851 y Fw(for)j(an)n(y)g Ft(m)d Fs(2)g Fn(Z)28 b Fw(and)e Ft(t)d Fs(2)h Fn(R)i Fw(\(see)h([L]\).)37 b(Consequen)n(tly)-7 b(,)26 b(since)h Ft(d)c Fi(>)g Fw(4,)j(for)h(an)f(arbitrarily)515 3951 y(\014xed)h Ft(")c(>)g Fw(0)645 4103 y Fs(jh)p Ft(\016)737 4115 y Fu(0)788 4103 y Fs(j)14 b Fw(e)862 4069 y Fu(i)p Fp(t)p Fu(\001)965 4103 y Ft(\016)1002 4115 y Fp(n)1047 4103 y Fs(ij)23 b Fi(6)g Ft(C)d Fs(j)p Ft(t)p Fs(j)1368 4069 y Fo(\000)1430 4047 y Fk(1)p 1430 4056 V 1430 4089 a(3)1472 4103 y Fs(j)p Ft(t)p Fs(j)1548 4069 y Fo(\000)1610 4047 y Fk(1)p 1610 4056 V 1610 4089 a(3)1652 4103 y Fs(j)p Ft(t)p Fs(j)1728 4069 y Fo(\000)1790 4047 y Fk(1)p 1790 4056 V 1790 4089 a(3)1833 4103 y Fw(\()p Fs(j)p Ft(t)p Fs(j)1941 4069 y Fo(\000)2003 4047 y FC(")p 2003 4056 V 2003 4089 a Fk(3)2045 4103 y Fs(j)p Ft(n)2118 4069 y Fu(\()p Fp(j)2171 4077 y Fk(4)2204 4069 y Fu(\))2234 4103 y Fs(j)2257 4069 y Fo(\000)2319 4043 y Fk(1)p Fj(\000)p FC(")p 2319 4056 102 3 v 2356 4089 a Fk(3)2434 4103 y Fw(\))p Fs(j)p Ft(n)2539 4069 y Fu(\()p Fp(j)2592 4077 y Fk(5)2625 4069 y Fu(\))2655 4103 y Fs(j)2678 4069 y Fo(\000)2740 4047 y Fk(1)p 2740 4056 29 3 v 2740 4089 a(3)2797 4103 y Ft(:)14 b(:)g(:)f Fs(j)p Ft(n)2980 4069 y Fu(\()p Fp(j)3033 4078 y FC(d)3068 4069 y Fu(\))3099 4103 y Fs(j)3122 4069 y Fo(\000)3183 4047 y Fk(1)p 3183 4056 V 3183 4089 a(3)3226 4103 y Ft(;)515 4255 y Fw(where)29 b(\()p Ft(n)839 4225 y Fu(\()p Fp(j)892 4233 y Fk(1)925 4225 y Fu(\))955 4255 y Ft(;)14 b(:)g(:)g(:)g(;)g(n)1190 4225 y Fu(\()p Fp(j)1243 4234 y FC(d)1278 4225 y Fu(\))1308 4255 y Fw(\))31 b(is)f(a)f(p)r(erm)n(utation)h(of)g(\()p Ft(n)2185 4225 y Fu(\(1\))2274 4255 y Ft(;)14 b(:)g(:)g(:)g(;)g(n)2509 4225 y Fu(\()p Fp(d)p Fu(\))2599 4255 y Fw(\).)45 b(This)30 b(last)g(estimate,)515 4355 y(the)e(equation)f(\(17\),)g(and)g(the)h (dominated)g(con)n(v)n(ergence)d(theorem)i(then)h(giv)n(e)956 4524 y Fs(j)p Ft(G)p Fw(\()p Ft(n;)14 b(e)19 b Fw(+)f(i0\))p Fs(j)23 b Fi(6)f Fv(Const)g Fs(j)p Ft(n)1839 4490 y Fu(\()p Fp(j)1892 4498 y Fk(4)1925 4490 y Fu(\))1955 4524 y Fs(j)1978 4490 y Fo(\000)2040 4464 y Fk(1)p Fj(\000)p FC(")p 2040 4477 102 3 v 2076 4510 a Fk(3)2155 4524 y Fs(j)p Ft(n)2228 4490 y Fu(\()p Fp(j)2281 4498 y Fk(5)2314 4490 y Fu(\))2344 4524 y Fs(j)2367 4490 y Fo(\000)2429 4467 y Fk(1)p 2429 4476 29 3 v 2429 4510 a(3)2485 4524 y Ft(:)14 b(:)g(:)g Fs(j)p Ft(n)2669 4490 y Fu(\()p Fp(j)2722 4499 y FC(d)2757 4490 y Fu(\))2787 4524 y Fs(j)2810 4490 y Fo(\000)2872 4467 y Fk(1)p 2872 4476 V 2872 4510 a(3)2914 4524 y Ft(:)515 4676 y Fw(Since)24 b Ft(")f Fw(and)h(the)g(p)r(erm)n(utation)g(of)g(\() p Ft(n)1732 4646 y Fu(\(1\))1821 4676 y Ft(;)14 b(:)g(:)g(:)f(;)h(n) 2055 4646 y Fu(\()p Fp(d)p Fu(\))2146 4676 y Fw(\))24 b(are)f(arbitrary)-7 b(,)22 b(it)j(follo)n(ws)d Fv(a)27 b(fortiori)515 4776 y Fw(that)1516 4875 y Ft(G)p Fw(\()p Ft(n;)14 b(e)k Fw(+)g(i0\))23 b(=)g Ft(O)r Fw(\()p Fs(j)p Ft(n)p Fs(j)2241 4841 y Fo(\000)2303 4819 y Fk(1)p 2303 4828 V 2303 4861 a(3)2346 4875 y Fw(\))515 5006 y(in)28 b(dimension)f(at)h(least)f(4.)1905 5255 y(43)p eop end %%Page: 44 44 TeXDict begin 44 43 bop 515 523 a Fx(References)515 688 y FH([ES])120 b(L.)22 b(Erd\177)-38 b(os,)23 b(M.)f(Salmhofer,)i Fa(De)l(c)l(ay)h(of)e(the)i(F)-6 b(ourier)25 b(T)-6 b(r)l(ansform)25 b(of)e(Surfac)l(es)i(with)f(V)-6 b(an-)772 780 y(ishing)27 b(Curvatur)l(e)p FH(.)h(Pre-prin)n(t.)e(\(2006\))515 904 y([ESY])62 b(L.)47 b(Erd\177)-38 b(os,)53 b(M.)47 b(Salmhofer,)53 b(H.-T.)47 b(Y)-6 b(au,)52 b Fa(Quantum)47 b(Di\013usion)g(of)g(the)g(R)l(andom)772 996 y(Schr\177)-39 b(odinger)29 b(Evolution)f(in)f(the)i(Sc)l(aling)e(Limit)p FH(.)f(Pre-prin)n(t.)f(\(2006\))515 1120 y([JP])124 b(V.)31 b(Jak)l(\024)-34 b(si)n(\023)e(c,)34 b(P)-6 b(.)31 b(P)n(oulin,)j Fa(Sc)l(attering)h(fr)l(om)d(Sp)l(arse)i(Potential)g(on)f(Gr)l(aphs)p FH(.)g(Pre-prin)n(t.)772 1211 y(\(2006\))515 1336 y([L])167 b(L.)43 b(Landau,)48 b Fa(Monotonicity)c(and)g(Bounds)h(on)f(Bessel)g (F)-6 b(unctions)p FH(.)45 b(Mathematical)772 1427 y(Ph)n(ysics)29 b(and)g(Quan)n(tum)f(Field)i(Theory)-6 b(,)30 b(Electronic)h(Journal)f (of)g(Di\013eren)n(tial)g(Equa-)772 1519 y(tions,)c(conf.)h(04.)f (\(2000\))515 1643 y([Li])146 b(W.)32 b(Littman,)j Fa(F)-6 b(ourier)34 b(T)-6 b(r)l(ansforms)35 b(of)e(Surfac)l(e-Carrie)l(d)j(Me) l(asur)l(es)g(and)e(Di\013er)l(en-)772 1734 y(tiability)29 b(of)f(Surfac)l(e)j(A)n(ver)l(ages)p FH(.)e(Bulletin)g(of)f(the)f (American)h(Mathematical)h(So)r(ciet)n(y)-6 b(,)772 1826 y(v)n(ol.)26 b(69.)g(\(1963\))515 1950 y([MV])87 b(S.)33 b(Molc)n(hano)n(v,)k(B.)d(V)-6 b(ain)n(b)r(erg,)35 b Fa(Sc)l(attering)i(on)e(the)h(System)g(of)e(the)i(Sp)l(arse)g(Bumps:) 772 2042 y(Multidimensional)27 b(Case)p FH(.)f(Applicable)g(Analysis,)g (v)n(ol.)g(71\(1-4\).)h(\(1999\))515 2166 y([P])163 b(P)-6 b(.)31 b(P)n(oulin,)i Fa(The)g(Molchanov{V)-6 b(ainb)l(er)l(g)35 b(L)l(aplacian)p FH(.)d(Pro)r(ceedings)g(of)g(the)f(American)772 2257 y(Mathematical)c(So)r(ciet)n(y)-6 b(.)26 b(\(2005\))515 2382 y([St])142 b(E.)28 b(Stein,)h 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