This is a multi-part message in MIME format. ---------------1103251122648 Content-Type: text/plain; name="11-50.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="11-50.keywords" Linear Cauchy problems, Banach spaces of entire functions, formal power series, ordinary differential equations ---------------1103251122648 Content-Type: application/postscript; name="smtanhPDE.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="smtanhPDE.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.96.1 Copyright 2007 Radical Eye Software %%Title: smtanhPDE.dvi %%CreationDate: Fri Mar 25 15:21:31 2011 %%Pages: 27 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMR17 CMR12 CMSY8 CMBX10 CMR10 MSBM10 CMR7 CMBX12 %%+ CMTI10 CMMI10 CMMI8 CMR8 CMEX10 CMSY10 CMSY6 CMR9 CMBX8 CMR6 MSBM7 %%+ CMMI6 CMBX6 CMR5 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o smtanhPDE.ps smtanhPDE.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2011.03.25:1521 %%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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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 readonly def /FontBBox{0 -504 2615 1004}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 12 /fi put dup 40 /parenleft put dup 41 /parenright 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 58 /colon put dup 65 /A put dup 68 /D put dup 76 /L put dup 80 /P put dup 82 /R put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 107 /k 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 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 (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 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: 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 11 /ff put dup 12 /fi put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 65 /A put dup 66 /B put dup 67 /C put dup 70 /F put dup 72 /H put dup 73 /I put dup 76 /L put dup 82 /R put dup 84 /T put dup 87 /W 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 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 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-53 -251 1139 750}readonly def currentdict end currentfile eexec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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 readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX8 %!PS-AdobeFont-1.1: CMBX8 1.0 %%CreationDate: 1991 Aug 20 16:36:07 % 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 (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 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 107 /k put dup 112 /p put dup 113 /q put readonly def /FontBBox{-59 -250 1235 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % 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 (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 19 /acute put dup 39 /quoteright 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 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 57 /nine put dup 64 /at put dup 65 /A put dup 67 /C put dup 70 /F put dup 73 /I put dup 76 /L put dup 77 /M put dup 82 /R put dup 83 /S 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 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 120 /x put readonly def /FontBBox{-39 -250 1036 750}readonly def currentdict end currentfile eexec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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 3 /asteriskmath put dup 20 /lessequal put dup 50 /element put dup 80 /P put dup 106 /bar put dup 121 /dagger put readonly def /FontBBox{-4 -948 1329 786}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % 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 (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 2 /multiply put dup 14 /openbullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 56 /universal put dup 65 /A put dup 68 /D put dup 71 /G put dup 74 /J put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec 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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 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 8 /braceleftbig put dup 9 /bracerightbig 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 27 /bracerightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 80 /summationtext put dup 81 /producttext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CAC6A7BEB5D02276E511FFAF2AE11910 DE076F24311D94D07CACC323F360887F1EA11BDDA7927FF3325986FDB0ABDFC8 8E4B40E7988921D551EC0867EBCA44C05657F0DC913E7B3004A5F3E1337B6987 FEBC45F989C8DC6DC0AD577E903F05D0D54208A0AE7F28C734F130C133B48422 BED48639A2B74E4C08F2E710E24A99F347E0F4394CE64EACB549576E89044E52 EABE595BC964156D9D8C2BAB0F49664E951D7C1A3D1789C47F03C7051A63D5E8 DF04FAAC47351E82CAE0794AA9692C6452688A74A7A6A7AD09B8A9783C235EC1 EA2156261B8FB331827145DE315B6EC1B3D8B67B3323F761EAF4C223BB214C4C 6B062D1B281F5041D068319F4911058376D8EFBA59884BA3318C5BC95684F281 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % 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 (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 1 /Delta put dup 33 /exclam 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 readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 12 /beta put dup 13 /gamma put dup 14 /delta put dup 18 /theta put dup 20 /kappa put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 27 /sigma put dup 38 /sigma1 put dup 58 /period put dup 59 /comma put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 68 /D put dup 74 /J put dup 76 /L put dup 83 /S put dup 84 /T put dup 88 /X put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 100 /d 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 112 /p put dup 113 /q put dup 115 /s put dup 116 /t put dup 120 /x put dup 122 /z put readonly def /FontBBox{-24 -250 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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 12 /beta put dup 13 /gamma put dup 14 /delta put dup 16 /zeta put dup 18 /theta put dup 20 /kappa put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 38 /sigma1 put dup 39 /phi1 put dup 44 /arrowhookleft 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 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 72 /H put dup 73 /I 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 88 /X put dup 96 /lscript 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 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n 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 readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D456A09920001A3D2199583EB9B84B4DEE08E3D12939E321 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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 18 /grave put dup 40 /parenleft put dup 41 /parenright 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 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M 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 87 /W 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 readonly def /FontBBox{-163 -250 1146 969}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 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 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 16 /dotlessi put dup 18 /grave put dup 19 /acute put dup 21 /breve put dup 33 /exclam put dup 38 /ampersand 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 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 81 /Q 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 93 /bracketright put dup 94 /circumflex 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 readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 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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 50 /two put readonly def /FontBBox{-27 -250 1122 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 3 /asteriskmath put dup 20 /lessequal put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 68 /D put dup 74 /J put dup 80 /P put dup 81 /Q put dup 106 /bar put dup 121 /dagger put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec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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 48 /zero put dup 49 /one put dup 50 /two put dup 53 /five put dup 67 /C put dup 77 /M put dup 83 /S put dup 97 /a put dup 99 /c put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 110 /n put dup 112 /p put dup 114 /r put dup 116 /t put readonly def /FontBBox{-34 -251 988 750}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. 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Fn(X)1798 5764 y Fp(\014)s Fy(\025)p Fo(0)1947 5566 y Fn(X)1946 5762 y Fp(n)p Fy(\025)p Fo(0)2094 5653 y Fq(v)2138 5668 y Fp(n;\014)2248 5653 y Fs(\()p Fq(T)13 b Fs(\))2394 5591 y Fq(t)2427 5558 y Fp(n)p 2394 5632 80 4 v 2394 5715 a Fq(n)p Fs(!)2494 5591 y Fq(z)2540 5558 y Fp(\014)p 2494 5632 94 4 v 2500 5715 a Fq(\014)5 b Fs(!)2597 5653 y Fq(:)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 3809 100 a Fs(4)75 399 y(As)30 b Fq(V)21 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))26 b Fm(2)f(G)744 413 y Fp(q)797 399 y Fs(\()q Fq(\016)873 413 y Fo(1)913 399 y Fq(;)15 b(\016)993 413 y Fo(2)1033 399 y Fs(;)g Fq(\033)s Fs(\))q(,)30 b(there)h(exists)g(a)g(p)s(ositiv)m (e)g(constan)m(t)h Fq(c)2521 413 y Fo(1)2591 399 y Fs(suc)m(h)e(that) 1438 677 y Fm(jj)q Fq(v)1533 692 y Fp(n;\014)1642 677 y Fs(\()p Fq(T)13 b Fs(\))p Fm(j)q(j)1829 708 y Fp(\014)s Fo(;)p Fp(\033)2032 615 y Fq(\016)2075 582 y Fp(n)2072 639 y Fo(1)2123 615 y Fq(\016)2166 571 y Fp(\014)2163 641 y Fo(2)p 1964 656 318 4 v 1964 739 a Fs(\()p 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Fm(\025)25 b Fs(0.)75 1630 y Fe(Lemma)35 b(1)45 b Fr(We)33 b(have)g Fs(\()p Fq(n)20 b Fs(+)g Fq(\014)5 b Fs(\)!)p Fq(=)p Fs(\()p Fq(n)p Fs(!)p Fq(\014)g Fs(!\))27 b Fm(\024)e Fs(2)1679 1597 y Fp(n)p Fo(+)p Fp(\014)1857 1630 y Fr(for)33 b(al)5 b(l)33 b Fq(n)p Fr(,)f Fq(\014)f Fm(\025)25 b Fs(0)p Fr(.)75 1817 y Fs(The)f(lemma)h(is)g(a)g (consequence)h(of)f(the)f(classical)j(binomial)e(form)m(ula.)39 b(F)-8 b(rom)26 b(the)f(estimates)h(\(3\))g(w)m(e)f(deduce)75 1930 y(that)75 2182 y(\(4\))860 2118 y Fm(j)p Fq(v)929 2133 y Fp(n;\014)1039 2118 y Fs(\()p Fq(T)13 b Fs(\))p Fm(j)p 860 2161 341 4 v 950 2244 a Fq(n)p Fs(!)p Fq(\014)5 b Fs(!)1236 2182 y Fm(\024)25 b Fq(c)1371 2196 y Fo(1)1426 2182 y Fs(\(1)c(+)f Fm(j)p Fq(T)13 b Fm(j)p Fs(\))1769 2140 y Fp(m)1851 2182 y Fs(exp)i(\()q Fq(\033)s(r)2137 2197 y Fp(b)2171 2182 y Fs(\()p Fq(\014)5 b Fs(\))p Fm(j)p Fq(T)13 b Fm(j)2413 2144 y Fp(q)2452 2182 y Fs(\))2503 2053 y Fn(\022)2597 2120 y Fs(2)p 2580 2161 80 4 v 2580 2244 a Fq(\016)2620 2258 y Fo(1)2670 2053 y Fn(\023)2736 2076 y Fp(n)2799 2053 y Fn(\022)2893 2120 y Fs(2)p 2875 2161 V 2875 2244 a Fq(\016)2915 2258 y Fo(2)2965 2053 y Fn(\023)3032 2076 y Fp(\014)75 2451 y Fs(for)35 b(all)h Fq(T)45 b Fm(2)33 b Fl(C)p Fs(,)j(and)e(for)h(all)h Fq(n)p Fs(,)g Fq(\014)i Fm(\025)33 b Fs(0.)55 b(F)-8 b(rom)35 b(\(4\),)j(w)m(e)d(get)h(that)g Fm(8)p Fq(T)45 b Fm(2)32 b Fl(C)p Fs(,)k Fm(8)p Fq(t)d Fm(2)f Fl(C)j Fs(with)f Fm(j)p Fq(t)p Fm(j)g Fq(<)3597 2414 y Fp(\016)3628 2423 y Fh(1)p 3597 2430 66 4 v 3613 2483 a Fo(2)3708 2451 y Fs(and)75 2574 y Fm(8)p Fq(z)29 b Fm(2)c Fl(C)30 b Fs(with)g Fm(j)p Fq(z)t Fm(j)c Fq(<)814 2537 y Fp(\016)845 2546 y Fh(2)p 814 2553 V 829 2606 a Fo(2)1228 2745 y Fm(j)q Fq(V)20 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))p Fm(j)27 b(\024)1771 2659 y Fn(X)1770 2857 y Fp(\014)s Fy(\025)p Fo(0)1919 2659 y Fn(X)1918 2855 y Fp(n)p Fy(\025)p Fo(0)2076 2682 y Fm(j)p Fq(v)2145 2697 y Fp(n;\014)2255 2682 y Fs(\()p Fq(T)13 b Fs(\))p Fm(j)p 2076 2724 341 4 v 2166 2808 a Fq(n)p Fs(!)p Fq(\014)5 b Fs(!)2427 2745 y Fm(j)p Fq(t)p Fm(j)2510 2708 y Fp(n)2557 2745 y Fm(j)p Fq(z)t Fm(j)2653 2708 y Fp(\014)910 3093 y Fm(\024)25 b Fq(c)1045 3107 y Fo(1)1100 3093 y Fs(\(1)c(+)e Fm(j)p Fq(T)13 b Fm(j)p Fs(\))1443 3051 y Fp(m)1650 3031 y Fs(1)p 1535 3072 277 4 v 1535 3177 a(1)20 b Fm(\000)1701 3132 y Fo(2)p Fy(j)p Fp(t)p Fy(j)p 1701 3156 100 4 v 1718 3208 a Fp(\016)1749 3217 y Fh(1)1837 3006 y Fn(X)1836 3204 y Fp(\014)s Fy(\025)p Fo(0)1985 3093 y Fs(exp)15 b(\()p Fq(\033)s(r)2270 3108 y Fp(b)2305 3093 y Fs(\()p Fq(\014)5 b Fs(\))p Fm(j)p Fq(T)13 b Fm(j)2547 3055 y Fp(q)2586 3093 y Fs(\))2636 2964 y Fn(\022)2713 3031 y Fs(2)p Fm(j)p Fq(z)t Fm(j)p 2713 3072 143 4 v 2744 3155 a Fq(\016)2784 3169 y Fo(2)2865 2964 y Fn(\023)2932 2987 y Fp(\014)2995 3093 y Fq(:)75 3356 y Fs(No)m(w,)38 b(as)e Fq(r)472 3371 y Fp(b)506 3356 y Fs(\()p Fq(\014)5 b Fs(\))35 b Fm(\024)e Fq(\020)7 b Fs(\()p Fq(b)p Fs(\))36 b(for)f(all)i Fq(\014)i Fm(2)33 b Fl(N)p Fs(,)k(w)m(e)f(get)h(that)f (there)f(exists)i(a)f(p)s(ositiv)m(e)g(constan)m(t)h Fq(c)3376 3370 y Fo(1)3416 3356 y Fs(,)g(suc)m(h)e(that)75 3469 y Fm(8)p Fq(T)j Fm(2)24 b Fl(C)p Fs(,)31 b Fm(8)p Fq(t)24 b Fm(2)h Fl(C)30 b Fs(with)g Fm(j)p Fq(t)p Fm(j)c Fq(<)1136 3432 y Fp(\016)1167 3441 y Fh(1)p 1136 3448 66 4 v 1151 3501 a Fo(2)1242 3469 y Fs(and)j Fm(8)p Fq(z)g Fm(2)c Fl(C)30 b Fs(with)g Fm(j)p Fq(z)t Fm(j)c Fq(<)2157 3432 y Fp(\016)2188 3441 y Fh(2)p 2157 3448 V 2173 3501 a Fo(2)2263 3469 y Fs(w)m(e)31 b(ha)m(v)m(e)810 3720 y Fm(j)p Fq(V)21 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))p Fm(j)26 b(\024)f Fq(c)1390 3734 y Fo(1)1445 3720 y Fs(\()q(1)20 b(+)g Fm(j)p Fq(T)13 b Fm(j)p Fs(\))1789 3678 y Fp(m)1996 3659 y Fs(1)p 1881 3699 277 4 v 1881 3805 a(1)20 b Fm(\000)2047 3760 y Fo(2)p Fy(j)p Fp(t)p Fy(j)p 2047 3784 100 4 v 2064 3836 a Fp(\016)2095 3845 y Fh(1)2182 3720 y Fs(exp)15 b(\()p Fq(\033)s(\020)7 b Fs(\()p Fq(b)p Fs(\))p Fm(j)p Fq(T)13 b Fm(j)2698 3683 y Fp(q)2737 3720 y Fs(\))2918 3659 y(1)p 2798 3699 287 4 v 2798 3805 a(1)20 b Fm(\000)2964 3760 y Fo(2)p Fy(j)p Fp(z)s Fy(j)p 2964 3784 111 4 v 2986 3836 a Fp(\016)3017 3845 y Fh(2)3094 3720 y Fq(:)75 4008 y Fs(This)30 b(pro)m(v)m(es)h(the)f(prop)s(osition.)75 4301 y Fg(2.2)112 b(Linear)39 b(op)s(erators)f(de\014ned)g(on)g(the)f (w)m(eigh)m(ted)h(Banac)m(h)g(spaces.)75 4473 y Fs(W)-8 b(e)32 b(de\014ne)d(the)i(in)m(tegration)h(op)s(erator)f Fq(@)1533 4440 y Fy(\000)p Fo(1)1528 4496 y Fp(z)1653 4473 y Fs(:)25 b Fm(A)p Fs(\()p Fl(C)p Fs(\))p Fm(f)p Fq(t;)15 b(z)t Fm(g)27 b(!)e(A)p Fs(\()p Fl(C)p Fs(\))p Fm(f)p Fq(t;)15 b(z)t Fm(g)32 b Fs(as)1281 4648 y Fn(\000)1323 4721 y Fq(@)1376 4684 y Fy(\000)p Fo(1)1371 4744 y Fp(z)1471 4721 y Fq(V)1544 4648 y Fn(\001)1601 4721 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))26 b(:=)2044 4598 y Fn(Z)2135 4624 y Fp(z)2094 4804 y Fo(0)2190 4721 y Fq(V)20 b Fs(\()p Fq(t;)15 b(\030)t(;)g(T)e Fs(\))p Fq(d\030)75 4967 y Fs(and)30 b(recall)h(the)g(classical)i(estimates)75 5196 y(\(5\))1685 5134 y(\000\()p Fq(x)20 b Fs(+)g 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Fm(\000)c Fs(1\)\))p 1482 619 603 4 v 1761 703 a Fq(q)2114 640 y Fs(+)g Fq(\027)31 b(<)25 b(S:)75 919 y Fr(Then)35 b(the)f(op)-5 b(er)g(ator)38 b Fq(T)892 886 y Fp(s)928 919 y Fq(@)981 886 y Fp(\027)976 942 y(t)1025 919 y Fq(@)1078 886 y Fp(\024)1073 946 y(T)1128 919 y Fq(@)1181 886 y Fy(\000)p Fp(S)1176 942 y(z)1321 919 y Fr(is)c(a)h(b)-5 b(ounde)g(d)36 b(line)-5 b(ar)35 b(op)-5 b(er)g(ator)37 b(fr)-5 b(om)2683 818 y Fn(\020)2737 919 y Fm(G)2791 933 y Fp(q)2845 919 y Fs(\()p Fq(\016)2920 933 y Fo(1)2960 919 y Fq(;)15 b(\016)3040 933 y Fo(2)3080 919 y Fs(;)g Fq(\033)s Fs(\))h Fq(;)f Fm(j)q(j)p Fq(:)p Fm(jj)3393 946 y Fp(\016)3424 955 y Fh(1)3459 946 y Fp(;\016)3510 955 y Fh(2)3544 946 y Fo(;)p Fp(\033)3610 818 y Fn(\021)3699 919 y Fr(into)75 1050 y(itself.)42 b(Mor)-5 b(e)g(over,)34 b(ther)-5 b(e)33 b(exists)g(a)g(c)-5 b(onstant)34 b Fq(C)1742 1064 y Fo(5)1807 1050 y Fq(>)25 b Fs(0)33 b Fr(such)g(that)748 1171 y Fn(\014)748 1226 y(\014)778 1171 y(\014)778 1226 y(\014)808 1175 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1474 y Fo(2)1174 1460 y Fs(;)g Fq(\033)s Fs(\))q Fr(.)75 1691 y Fe(Pro)s(of)56 b Fs(F)-8 b(or)1007 1840 y Fq(V)20 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))27 b(=)1499 1754 y Fn(X)1498 1952 y Fp(\014)s Fy(\025)p Fo(0)1647 1754 y Fn(X)1646 1950 y Fp(n)p Fy(\025)p Fo(0)1794 1840 y Fq(v)1838 1855 y Fp(n;\014)1948 1840 y Fs(\()p Fq(T)13 b Fs(\))2094 1779 y Fq(t)2127 1746 y Fp(n)p 2094 1819 80 4 v 2094 1903 a Fq(n)p Fs(!)2194 1779 y Fq(z)2240 1746 y Fp(\014)p 2194 1819 94 4 v 2200 1903 a Fq(\014)5 b Fs(!)2323 1840 y Fm(2)25 b(G)2463 1854 y Fp(q)2516 1840 y Fs(\()p Fq(\016)2591 1854 y Fo(1)2631 1840 y Fq(;)15 b(\016)2711 1854 y Fo(2)2751 1840 y Fs(;)g Fq(\033)s Fs(\))h Fq(;)75 2096 y Fs(w)m(e)31 b(ha)m(v)m(e)780 2172 y Fn(\000)821 2245 y Fq(T)887 2208 y Fp(s)924 2245 y Fq(@)977 2208 y Fp(\027)972 2268 y(t)1020 2245 y Fq(@)1073 2208 y Fp(\024)1068 2268 y(T)1124 2245 y Fq(@)1177 2208 y Fy(\000)p Fp(S)1172 2268 y(z)1283 2245 y Fq(V)1356 2172 y Fn(\001)1413 2245 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))26 b(=)1837 2159 y Fn(X)1830 2357 y Fp(\014)s Fy(\025)p Fp(S)1991 2159 y Fn(X)1990 2355 y Fp(n)p Fy(\025)p Fo(0)2138 2245 y Fs(\()p Fq(T)2239 2208 y Fp(s)2276 2245 y Fq(@)2329 2208 y Fp(\024)2324 2268 y(T)2379 2245 y Fq(v)2423 2260 y Fp(n)p Fo(+)p Fp(\027;\014)s Fy(\000)p Fp(S)2725 2245 y Fs(\))15 b(\()p Fq(T)e Fs(\))2921 2184 y Fq(t)2954 2151 y Fp(n)p 2921 2224 80 4 v 2921 2308 a Fq(n)p Fs(!)3021 2184 y Fq(z)3067 2151 y Fp(\014)p 3021 2224 94 4 v 3027 2308 a Fq(\014)5 b Fs(!)3125 2245 y Fq(:)75 2501 y Fs(Therefore,)30 b(b)m(y)h(de\014nition,)f(w)m(e)h(obtain)g(that)448 2688 y Fn(\014)448 2743 y(\014)478 2688 y(\014)478 2743 y(\014)508 2692 y(\000)550 2765 y Fq(T)616 2728 y Fp(s)653 2765 y Fq(@)706 2728 y Fp(\027)701 2788 y(t)749 2765 y Fq(@)802 2728 y Fp(\024)797 2788 y(T)852 2765 y Fq(@)905 2728 y Fy(\000)p Fp(S)900 2788 y(z)1011 2765 y Fq(V)1085 2692 y Fn(\001)1141 2765 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))1436 2688 y Fn(\014)1436 2743 y(\014)1468 2688 y(\014)1468 2743 y(\014)1498 2801 y Fp(\016)1529 2810 y Fh(1)1564 2801 y Fp(;\016)1615 2810 y Fh(2)1650 2801 y Fo(;)p Fp(\033)1741 2765 y Fs(=)1844 2679 y Fn(X)1837 2877 y Fp(\014)s Fy(\025)p Fp(S)1998 2679 y Fn(X)1997 2875 y Fp(n)p Fy(\025)p Fo(0)2145 2765 y Fm(jj)q Fs(\()p Fq(T)2297 2728 y Fp(s)2333 2765 y Fq(@)2386 2728 y Fp(\024)2381 2788 y(T)2437 2765 y Fq(v)2481 2780 y Fp(n)p Fo(+)p Fp(\027;\014)s Fy(\000)p Fp(S)2782 2765 y Fs(\))i(\()p Fq(T)e Fs(\))p Fm(j)q(j)3019 2797 y Fp(\014)s Fo(;)p Fp(\033)3222 2704 y Fq(\016)3265 2671 y Fp(n)3262 2728 y Fo(1)3313 2704 y Fq(\016)3356 2660 y Fp(\014)3353 2730 y Fo(2)p 3154 2744 318 4 v 3154 2828 a Fs(\()p Fq(n)20 b Fs(+)g Fq(\014)5 b Fs(\)!)75 3056 y(where)298 3255 y Fm(j)q(j)p Fs(\()p Fq(T)450 3218 y Fp(s)487 3255 y Fq(@)540 3218 y Fp(\024)535 3278 y(T)590 3255 y Fq(v)634 3270 y Fp(n)p Fo(+)p Fp(\027;\014)s Fy(\000)p Fp(S)935 3255 y Fs(\))16 b(\()p Fq(T)d Fs(\))p Fm(jj)1173 3287 y Fp(\014)s Fo(;)p Fp(\033)1307 3255 y Fs(=)30 b(sup)1403 3334 y Fp(T)10 b Fy(2)p Fd(C)1564 3255 y Fm(j)q Fs(\()p Fq(T)1691 3218 y Fp(s)1728 3255 y Fq(@)1781 3218 y Fp(\024)1776 3278 y(T)1831 3255 y Fq(v)1875 3270 y Fp(n)p Fo(+)p Fp(\027;\014)s Fy(\000)p Fp(S)2176 3255 y Fs(\))16 b(\()p Fq(T)d Fs(\))p Fm(j)i Fs(\()q(1)20 b(+)g Fm(j)p Fq(T)13 b Fm(j)p Fs(\))2747 3213 y Fy(\000)p Fp(m)2884 3255 y Fs(exp)i(\()p Fm(\000)p Fq(\033)s(r)3240 3270 y Fp(b)3274 3255 y Fs(\()p Fq(\014)5 b Fs(\))p Fm(j)p Fq(T)13 b Fm(j)3516 3218 y Fp(q)3555 3255 y Fs(\))j Fq(:)75 3504 y Fs(F)-8 b(rom)31 b(Cauc)m(h)m(y's)g (theorem,)g(for)f Fq(T)38 b Fm(2)25 b Fl(C)p Fs(,)30 b(w)m(e)h(ha)m(v)m(e)927 3748 y(\()p Fq(T)1028 3710 y Fp(s)1065 3748 y Fq(@)1118 3710 y Fp(\024)1113 3770 y(T)1168 3748 y Fq(v)1212 3763 y Fp(n)p Fo(+)p Fp(\027;\014)s Fy(\000)p Fp(S)1513 3748 y Fs(\))16 b(\()p Fq(T)d Fs(\))25 b(=)1858 3686 y Fq(\024)p Fs(!)p 1831 3727 132 4 v 1831 3810 a(2)p Fq(i\031)1988 3624 y Fn(Z)2039 3830 y Fy(j)o Fp(\030)s Fy(\000)p Fp(T)10 b Fy(j)o Fo(=)p Fp(a)2339 3685 y Fq(T)2405 3652 y Fp(s)2442 3685 y Fq(v)2486 3700 y Fp(n)p Fo(+)p Fp(\027;\014)s Fy(\000)p Fp(S)2787 3685 y Fs(\()p Fq(\030)t Fs(\))p 2339 3727 563 4 v 2407 3823 a(\()p Fq(\030)24 b Fm(\000)c Fq(T)13 b Fs(\))2699 3781 y Fp(\024)p Fo(+1)2911 3748 y Fq(d\030)75 4009 y Fs(where)36 b Fq(a)g(>)f Fs(0)i(can)g(b)s(e)f(c)m(hosen)h(arbitrarily)-8 b(.)60 b(W)-8 b(e)38 b(replace)g Fq(\030)i Fs(b)m(y)d Fq(T)g Fs(+)24 b Fq(a)15 b Fs(exp\()p Fq(i\022)s Fs(\),)38 b(where)f Fq(\022)g Fm(2)f Fs([0)p Fq(;)15 b Fs(2)p Fq(\031)s Fs([)38 b(and)75 4122 y(obtain)326 4346 y Fm(j)p Fs(\()q Fq(T)453 4309 y Fp(s)489 4346 y Fq(@)542 4309 y Fp(\024)537 4369 y(T)593 4346 y Fq(v)637 4361 y Fp(n)p Fo(+)p Fp(\027;\014)s Fy(\000)p Fp(S)938 4346 y Fs(\))15 b(\()p Fq(T)e Fs(\))p Fm(j)26 b(\024)1281 4285 y Fq(\024)p Fs(!)p Fm(j)p Fq(T)13 b Fm(j)1474 4252 y Fp(s)p 1281 4326 231 4 v 1300 4409 a Fs(2)p Fq(\031)s(a)1448 4383 y Fp(\024)1537 4346 y Fm(jj)p Fq(v)1631 4361 y Fp(n)p Fo(+)p Fp(\027;\014)s Fy(\000)p Fp(S)1933 4346 y Fs(\()p Fq(T)g 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Fp(b)2392 5724 y Fs(\()p Fq(\014)25 b Fm(\000)20 b Fq(S)5 b Fs(\)\()p Fm(j)p Fq(T)13 b Fm(j)21 b Fs(+)f Fq(a)p Fs(\))3036 5686 y Fp(q)3075 5724 y Fs(\))15 b Fq(:)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 3809 100 a Fs(6)75 399 y(No)m(w,)31 b(our)f(aim)h(is)f(to)h(estimate)i Fm(jj)p Fs(\()p Fq(T)1364 366 y Fp(s)1401 399 y Fq(@)1454 366 y Fp(\024)1449 425 y(T)1504 399 y Fq(v)1548 414 y Fp(n)p Fo(+)p Fp(\027;\014)s Fy(\000)p Fp(S)1849 399 y Fs(\))16 b(\()p Fq(T)d Fs(\))p Fm(j)q(j)2087 430 y Fp(\014)s Fo(;)p Fp(\033)349 675 y Fm(jj)p Fs(\()q Fq(T)501 637 y Fp(s)537 675 y Fq(@)590 637 y Fp(\024)585 697 y(T)641 675 y Fq(v)685 690 y Fp(n)p Fo(+)p Fp(\027;\014)s Fy(\000)p Fp(S)986 675 y Fs(\))i(\()p Fq(T)e Fs(\))p Fm(j)q(j)1223 707 y Fp(\014)s Fo(;)p Fp(\033)1358 675 y Fm(\024)1454 547 y Fn(\022)1525 675 y Fs(sup)1521 753 y Fp(T)d Fy(2)p Fd(C)1682 675 y Fq(I)1722 690 y Fp(a;\014)1826 675 y Fs(\()p Fq(T)j Fs(\)\(1)22 b(+)e Fm(j)p Fq(T)13 b Fm(j)p Fs(\))2306 637 y Fy(\000)p Fp(m)2443 675 y Fs(exp)i(\()p Fm(\000)p Fq(\033)s(r)2799 690 y Fp(b)2833 675 y Fs(\()p Fq(\014)5 b Fs(\))p Fm(j)p Fq(T)13 b Fm(j)3075 637 y Fp(q)3115 675 y Fs(\))3150 547 y Fn(\023)2050 934 y Fm(\002)i(jj)q Fq(v)2231 949 y Fp(n)p Fo(+)p Fp(\027;\014)s Fy(\000)p Fp(S)2532 934 y Fs(\()p Fq(T)e Fs(\))p Fm(j)q(j)2719 966 y Fp(\014)s Fy(\000)p Fp(S)t Fo(;)p Fp(\033)2945 934 y Fq(:)75 1101 y Fs(So)30 b(w)m(e)h(ha)m(v)m(e)h(to)f(pro)m(v)m(e)g (the)g(follo)m(wing)g(lemma.)75 1289 y Fe(Lemma)k(2)45 b Fr(Ther)-5 b(e)34 b(exists)f(a)g(c)-5 b(ontant)34 b Fq(C)1523 1303 y Fo(6)1588 1289 y Fq(>)25 b Fs(0)33 b Fr(such)f(that)830 1519 y Fs(sup)825 1597 y Fp(T)10 b Fy(2)p Fd(C)986 1519 y Fq(I)1026 1534 y Fp(a;\014)1131 1519 y Fs(\()p Fq(T)j Fs(\)\(1)21 b(+)f Fm(j)p Fq(T)13 b Fm(j)p Fs(\))1610 1481 y Fy(\000)p Fp(m)1747 1519 y Fs(exp)i(\()q Fm(\000)p Fq(\033)s(r)2104 1534 y Fp(b)2138 1519 y Fs(\()p Fq(\014)5 b Fs(\))p Fm(j)p Fq(T)13 b Fm(j)2380 1481 y Fp(q)2419 1519 y Fs(\))26 b Fm(\024)f Fq(C)2641 1533 y Fo(6)2680 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3252 y Fs(1)21 b(+)2128 3191 y Fq(a)p 2016 3231 273 4 v 2016 3315 a Fs(1)g(+)f Fm(j)p Fq(T)13 b Fm(j)2299 3124 y Fn(\023)2366 3147 y Fp(m)2457 3252 y Fm(\024)25 b Fs(\(1)c(+)f Fq(a)p Fs(\))2828 3215 y Fp(m)2895 3252 y Fq(;)75 3501 y Fs(w)m(e)31 b(ha)m(v)m(e)335 3731 y Fq(J)385 3746 y Fp(a;\014)515 3731 y Fm(\024)25 b Fq(\024)p Fs(!)20 b(sup)703 3809 y Fp(T)10 b Fy(2)p Fd(C)874 3669 y Fm(j)p Fq(T)j Fm(j)990 3636 y Fp(s)p 874 3710 154 4 v 904 3793 a Fq(a)952 3767 y Fp(\024)1037 3731 y Fs(\(1)21 b(+)f Fq(a)p Fs(\))1312 3693 y Fp(m)1394 3731 y Fs(exp)15 b(\()q Fq(\033)j Fm(f)q Fq(r)1741 3746 y Fp(b)1775 3731 y Fs(\()p Fq(\014)26 b Fm(\000)20 b Fq(S)5 b Fs(\))20 b Fm(\000)g Fq(r)2226 3746 y Fp(b)2260 3731 y Fs(\()p Fq(\014)5 b Fs(\))p Fm(g)17 b Fs(\()p Fm(j)p Fq(T)c Fm(j)21 b Fs(+)f Fq(a)p Fs(\))2794 3693 y Fp(q)2832 3731 y Fs(\))1283 3990 y Fm(\002)15 b Fs(exp)g(\()p Fq(\033)s(r)1654 4005 y Fp(b)1689 3990 y Fs(\()p Fq(\014)5 b Fs(\))15 b Fm(f)q Fs(\()p Fm(j)p Fq(T)e Fm(j)21 b Fs(+)f Fq(a)p 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4540 317 4 v 2143 4623 a(\()p Fq(n)20 b Fs(+)g(1\))2424 4597 y Fp(b)2494 4561 y Fm(\024)2729 4500 y Fq(S)p 2600 4540 318 4 v 2600 4623 a Fs(\()p Fq(\014)26 b Fs(+)20 b(1\))2883 4597 y Fp(b)2928 4561 y Fq(:)75 4857 y Fs(No)m(w,)34 b(remem)m(b)s(er)e(that)h(the)g(radius)f Fq(a)h Fs(in)f(the)h(Cauc)m(h)m(y's)g(theorem)g(is)g(still)g (arbitrarily)-8 b(.)49 b(As)32 b(suggested)i(b)m(y)75 4970 y(Y.)d(Dubinski)-10 b(\025)-35 b(\020)30 b(in)g([7],)h(let)g(us)f (c)m(ho)s(ose)i Fq(a)25 b(>)g Fs(0)31 b(suc)m(h)f(that)h(\()p Fm(j)p Fq(T)13 b Fm(j)20 b Fs(+)g Fq(a)p Fs(\))2397 4928 y Fp(q)2460 4970 y Fs(=)25 b Fm(j)p Fq(T)13 b Fm(j)2672 4937 y Fp(q)2730 4970 y Fs(+)20 b(1,)31 b(th)m(us)1509 5184 y Fq(a)25 b Fs(=)g(\()p Fm(j)p Fq(T)13 b Fm(j)1829 5147 y Fp(q)1888 5184 y Fs(+)20 b(1\))2060 5142 y Fo(1)p Fp(=q)2188 5184 y Fm(\000)g(j)p Fq(T)13 b Fm(j)p Fq(:)75 5388 y Fs(Then,)30 b(w)m(e)g(obtain)h(that)274 5637 y Fq(J)324 5652 y Fp(a;\014)454 5637 y Fm(\024)25 b Fq(\024)p Fs(!)15 b(exp)h(\()q Fq(\033)s(r)929 5652 y Fp(b)963 5637 y Fs(\()p Fq(\014)5 b Fs(\)\))17 b(exp)1295 5509 y Fn(\022)1361 5637 y Fm(\000)1543 5576 y Fq(\033)s(S)p 1442 5617 V 1442 5700 a Fs(\()p Fq(\014)26 b Fs(+)20 b(1\))1725 5674 y Fp(b)1770 5509 y Fn(\023)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 3809 100 a Fs(7)959 509 y Fm(\002)19 b Fs(sup)1045 587 y Fp(T)10 b Fy(2)p Fd(C)1206 509 y Fm(j)p Fq(T)j Fm(j)1322 471 y Fp(s)1369 313 y Fn(\020)1423 414 y Fs(1)21 b(+)f(\()p Fm(j)p Fq(T)13 b Fm(j)1731 381 y Fp(q)1790 414 y Fs(+)19 b(1\))1961 372 y Fo(1)p Fp(=q)2090 414 y Fm(\000)h(j)p Fq(T)13 b Fm(j)2297 313 y Fn(\021)2352 336 y Fp(m)p 1369 488 1050 4 v 1458 505 a Fn(\020)1512 605 y Fs(\()q Fm(j)p Fq(T)g Fm(j)1664 579 y Fp(q)1722 605 y Fs(+)20 b(1\))1894 564 y Fo(1)p Fp(=q)2023 605 y Fm(\000)g(j)p Fq(T)13 b Fm(j)2230 505 y Fn(\021)2284 527 y Fp(\024)2443 509 y Fs(exp)2597 380 y Fn(\022)2664 509 y Fm(\000)2846 447 y Fq(\033)s(S)p 2745 488 318 4 v 2745 571 a Fs(\()p Fq(\014)26 b Fs(+)20 b(1\))3028 545 y Fp(b)3073 509 y Fm(j)p Fq(T)13 b Fm(j)3189 471 y Fp(q)3227 380 y Fn(\023)3309 509 y Fq(:)75 784 y Fs(T)-8 b(o)32 b(estimate)h(the)f(suprem)m(um)e(on)h(the)h(righ)m(t-hand)f (side,)i(w)m(e)f(distinguish)e(t)m(w)m(o)j(cases.)45 b(F)-8 b(or)33 b(the)e(\014rst)g(case,)75 897 y(using)f(T)-8 b(a)m(ylor's)32 b(form)m(ula)e(at)h(the)g(origin)1095 1103 y(\(1)21 b(+)f Fq(x)p Fs(\))1374 1065 y Fo(1)p Fp(=q)1508 1103 y Fs(=)25 b(1)c(+)1771 1041 y Fq(x)p 1771 1082 52 4 v 1775 1165 a(q)1853 1103 y Fs(+)f Fq(x")p Fs(\()p Fq(x)p Fs(\))p Fq(;)380 b Fm(j)p Fq(x)p Fm(j)26 b Fq(<)f Fs(1)75 1339 y(while)36 b Fq(")p Fs(\()p Fq(x)p Fs(\))h(in)f(the)g (remainder)f(term)h(satis\014es)g(lim)1918 1353 y Fp(x)p Fy(!)p Fo(0)2083 1339 y Fq(")p Fs(\()p Fq(x)p Fs(\))f(=)f(0,)k(there)f (exists)f(a)h(constan)m(t)g(0)e Fm(\024)f Fq(&)42 b(<)34 b Fs(1)75 1452 y(suc)m(h)c(that)1742 1580 y Fm(j)p Fq(")p Fs(\()p Fq(x)p Fs(\))p Fm(j)d Fq(<)2110 1519 y Fs(1)p 2089 1559 90 4 v 2089 1643 a(2)p Fq(q)75 1799 y Fs(for)35 b Fm(j)p Fq(x)p Fm(j)f Fq(<)g(&)7 b Fs(.)56 b(Therefore,)37 b(for)e Fm(j)p Fq(T)13 b Fm(j)34 b Fq(>)f(&)1463 1766 y Fy(\000)p Fo(1)p Fp(=q)1627 1799 y Fs(,)k(there)e(exists)i(a)e (constan)m(t)i Fq(c)2674 1813 y Fo(1)2748 1799 y Fq(>)c Fs(0,)k(indep)s(enden)m(t)e(of)g Fq(\014)41 b Fs(suc)m(h)75 1912 y(that)301 2006 y Fn(\020)355 2107 y Fs(1)20 b(+)g(\()q Fm(j)p Fq(T)13 b Fm(j)663 2074 y Fp(q)721 2107 y Fs(+)20 b(1\))893 2065 y Fo(1)p Fp(=q)1022 2107 y Fm(\000)g(j)p Fq(T)13 b Fm(j)1229 2006 y Fn(\021)1283 2029 y Fp(m)p 301 2181 1050 4 v 390 2197 a Fn(\020)444 2298 y Fs(\()p Fm(j)p Fq(T)g Fm(j)595 2272 y Fp(q)654 2298 y Fs(+)20 b(1\))826 2256 y Fo(1)p Fp(=q)954 2298 y Fm(\000)g(j)p Fq(T)13 b Fm(j)1161 2197 y Fn(\021)1216 2220 y Fp(\024)1385 2201 y Fs(=)25 b Fm(j)p Fq(T)13 b Fm(j)1597 2164 y Fp(\024)p Fo(\()p Fp(q)r Fy(\000)p Fo(1\))1831 2006 y Fn(\020)1885 2107 y Fs(1)21 b(+)2152 2071 y Fo(1)p 2052 2086 237 4 v 2052 2141 a Fp(q)r Fy(j)p Fp(T)10 b Fy(j)2177 2122 y Fc(q)r Fk(\000)p Fh(1)2319 2107 y Fs(+)2503 2071 y Fo(1)p 2420 2086 203 4 v 2420 2141 a Fy(j)p Fp(T)g Fy(j)2511 2122 y Fc(q)r Fk(\000)p Fh(1)2632 2107 y Fq(")2689 2006 y Fn(\020)2799 2071 y Fo(1)p 2754 2086 125 4 v 2754 2139 a Fy(j)p Fp(T)g Fy(j)2845 2121 y Fc(q)2889 2006 y Fn(\021\021)2997 2029 y Fp(m)p 1831 2181 1233 4 v 2132 2197 a Fn(\020)2196 2262 y Fo(1)p 2196 2277 36 4 v 2197 2330 a Fp(q)2262 2298 y Fs(+)20 b Fq(")2410 2197 y Fn(\020)2519 2262 y Fo(1)p 2475 2277 125 4 v 2475 2331 a Fy(j)p Fp(T)10 b Fy(j)2566 2312 y Fc(q)2609 2197 y Fn(\021\021)2718 2220 y Fp(\024)3099 2201 y Fm(\024)25 b Fq(c)3234 2215 y Fo(1)3274 2201 y Fm(j)p Fq(T)13 b Fm(j)3390 2164 y Fp(\024)p Fo(\()p Fp(q)r Fy(\000)p Fo(1\))3614 2201 y Fq(:)75 2524 y Fs(Let)33 b Fq(A)p Fs(,)g Fq(B)j Fs(p)s(ositiv)m(e)d(n)m(um)m(b)s(ers,)f(a)g (study)g(of)g(the)g(function)g Fq(f)10 b Fs(\()p Fq(x)p Fs(\))28 b(=)g Fq(x)2511 2491 y Fp(A)2583 2524 y Fs(exp\()p Fm(\000)p Fq(B)5 b(x)p Fs(\))32 b(sho)m(ws)g(that)h(its)f(supre-)75 2636 y(m)m(um)e(yields)g(at)h Fq(x)26 b Fs(=)f Fq(A)p Fm(n)p Fq(B)5 b Fs(,)30 b(th)m(us)515 2879 y(sup)354 2963 y Fp(T)10 b Fy(2)p Fd(C)p Fp(;)p Fy(j)p Fp(T)g Fy(j)611 2944 y Fc(q)644 2963 y Fp(>&)730 2944 y Fk(\000)p Fh(1)828 2879 y Fm(j)p Fq(T)j Fm(j)944 2841 y Fp(s)p Fo(+)p Fp(\024)p Fo(\()p Fp(q)r Fy(\000)p Fo(1\))1270 2879 y Fs(exp)1424 2751 y Fn(\022)1491 2879 y Fm(\000)1673 2817 y Fq(\033)s(S)p 1572 2858 318 4 v 1572 2941 a Fs(\()p Fq(\014)26 b Fs(+)20 b(1\))1855 2915 y Fp(b)1900 2879 y Fm(j)p Fq(T)13 b Fm(j)2016 2841 y Fp(q)2054 2751 y Fn(\023)1204 3252 y Fm(\024)1300 3124 y Fn(\022)1377 3191 y Fq(s)20 b Fs(+)g Fq(\024)p Fs(\()p Fq(q)j Fm(\000)d Fs(1\))p 1377 3232 478 4 v 1536 3315 a Fq(q)s(\033)s(S)1864 3252 y Fs(\()p Fq(\014)26 b Fs(+)20 b(1\))2147 3215 y Fp(b)2182 3124 y Fn(\023)2259 3113 y Fc(s)p Fh(+)p Fc(\024)p Fh(\()p Fc(q)r Fk(\000)p Fh(1\))p 2259 3132 269 3 v 2378 3173 a Fc(q)2557 3252 y Fs(exp)2711 3124 y Fn(\022)2778 3252 y Fm(\000)2859 3191 y Fq(s)f Fs(+)h Fq(\024)p Fs(\()p Fq(q)k Fm(\000)c Fs(1\))p 2859 3232 478 4 v 3075 3315 a Fq(q)3345 3124 y Fn(\023)3427 3252 y Fq(:)75 3458 y Fs(It)30 b(turns)g(out)g(that)h (there)g(exists)g(a)g(constan)m(t)g Fq(c)1735 3472 y Fo(2)1800 3458 y Fq(>)25 b Fs(0,)31 b(indep)s(enden)m(t)e(of)i Fq(\014)5 b Fs(,)31 b(suc)m(h)f(that)520 3765 y(sup)359 3849 y Fp(T)10 b Fy(2)p Fd(C)p Fp(;)p Fy(j)p Fp(T)g Fy(j)616 3830 y Fc(q)649 3849 y Fp(>&)735 3830 y Fk(\000)p Fh(1)833 3765 y Fm(j)p Fq(T)j Fm(j)949 3728 y Fp(s)996 3570 y Fn(\020)1050 3671 y Fs(1)21 b(+)f(\()p Fm(j)p Fq(T)13 b Fm(j)1358 3638 y Fp(q)1417 3671 y Fs(+)20 b(1\))1588 3629 y Fo(1)p Fp(=q)1717 3671 y Fm(\000)g(j)p Fq(T)13 b Fm(j)1924 3570 y Fn(\021)1979 3592 y Fp(m)p 996 3744 1050 4 v 1085 3761 a Fn(\020)1139 3862 y Fs(\()q Fm(j)p Fq(T)g Fm(j)1291 3836 y Fp(q)1349 3862 y Fs(+)20 b(1\))1521 3820 y Fo(1)p Fp(=q)1650 3862 y Fm(\000)g(j)p Fq(T)13 b Fm(j)1857 3761 y Fn(\021)1911 3784 y Fp(\024)2070 3765 y Fs(exp)2224 3637 y Fn(\022)2291 3765 y Fm(\000)2473 3704 y Fq(\033)s(S)p 2372 3744 318 4 v 2372 3827 a Fs(\()p Fq(\014)26 b Fs(+)20 b(1\))2655 3801 y Fp(b)2700 3765 y Fm(j)p Fq(T)13 b Fm(j)2816 3728 y Fp(q)2854 3637 y Fn(\023)2946 3765 y Fm(\024)25 b Fq(c)3081 3779 y Fo(2)3121 3765 y Fq(\014)3187 3686 y Fc(b)p Fh(\()p Fc(s)p Fh(+)p Fc(\024)p Fh(\()p Fc(q)r Fk(\000)p Fh(1\)\))p 3187 3704 344 3 v 3344 3746 a Fc(q)3545 3765 y Fq(:)75 4092 y Fs(F)-8 b(or)27 b(the)e(second)h(case)h Fm(j)p Fq(T)13 b Fm(j)26 b(\024)f Fq(&)1140 4059 y Fy(\000)p Fo(1)p Fp(=q)1303 4092 y Fs(,)i(it)f(is)g(easy)h(to)f(see)g(that)h(there)f(exists)g(a)g (constan)m(t)h Fq(c)3111 4106 y Fo(3)3176 4092 y Fq(>)e Fs(0,)j(indep)s(enden)m(t)75 4204 y(of)j Fq(\014)5 b Fs(,)30 b(suc)m(h)g(that)745 4511 y(sup)584 4595 y Fp(T)10 b Fy(2)p Fd(C)p Fp(;)p Fy(j)p Fp(T)g Fy(j)841 4576 y Fc(q)874 4595 y Fy(\024)p Fp(&)960 4576 y Fk(\000)p Fh(1)1058 4511 y Fm(j)p Fq(T)j Fm(j)1174 4474 y Fp(s)1221 4316 y Fn(\020)1275 4417 y Fs(1)20 b(+)g(\()q Fm(j)p Fq(T)13 b Fm(j)1583 4384 y Fp(q)1641 4417 y Fs(+)20 b(1\))1813 4375 y Fo(1)p Fp(=q)1942 4417 y Fm(\000)g(j)p Fq(T)13 b Fm(j)2149 4316 y Fn(\021)2203 4338 y Fp(m)p 1221 4490 1050 4 v 1310 4507 a Fn(\020)1364 4608 y Fs(\()p Fm(j)p Fq(T)g Fm(j)1515 4582 y Fp(q)1574 4608 y Fs(+)20 b(1\))1746 4566 y Fo(1)p Fp(=q)1874 4608 y Fm(\000)g(j)p Fq(T)13 b Fm(j)2081 4507 y Fn(\021)2136 4530 y Fp(\024)2295 4511 y Fs(exp)2449 4383 y Fn(\022)2516 4511 y Fm(\000)2697 4450 y Fq(\033)s(S)p 2597 4490 318 4 v 2597 4574 a Fs(\()p Fq(\014)25 b Fs(+)20 b(1\))2879 4547 y Fp(b)2924 4511 y Fm(j)p Fq(T)13 b Fm(j)3040 4474 y Fp(q)3079 4383 y Fn(\023)3171 4511 y Fm(\024)25 b Fq(c)3306 4525 y Fo(3)75 4818 y Fs(for)30 b(all)h Fq(\014)g Fm(\025)25 b Fq(S)5 b Fs(.)40 b(Hence,)32 b(w)m(e)f(ha)m(v)m(e)g(t)m(w)m(o)h(constan)m(ts)g Fq(c)1897 4832 y Fo(4)1936 4818 y Fs(,)f Fq(c)2031 4832 y Fo(5)2096 4818 y Fq(>)25 b Fs(0,)31 b(indep)s(enden)m(t)e(of)i Fq(\014)5 b Fs(,)30 b(suc)m(h)g(that)587 5055 y Fq(J)637 5070 y Fp(a;\014)767 5055 y Fm(\024)25 b Fq(c)902 5069 y Fo(4)941 5055 y Fq(\024)p Fs(!)15 b(exp)h(\()q Fq(\033)s(\020)7 b Fs(\()p Fq(b)p Fs(\)\))16 b(exp)1640 4927 y Fn(\022)1706 5055 y Fm(\000)1888 4994 y Fq(\033)s(S)p 1787 5034 V 1787 5118 a Fs(\()p Fq(\014)26 b Fs(+)20 b(1\))2070 5091 y Fp(b)2115 4927 y Fn(\023)2197 5055 y Fq(\014)2253 5018 y Fp(b)p Fo(\()p Fp(s)p Fo(+)p Fp(\024)p Fo(\()p Fp(q)r Fy(\000)p Fo(1\)\))p Fp(=q)2744 5055 y Fm(\024)25 b Fq(c)2879 5069 y Fo(5)2919 5055 y Fq(\014)2985 4976 y Fc(b)p Fh(\()p Fc(s)p Fh(+)p Fc(\024)p Fh(\()p Fc(q)r Fk(\000)p Fh(1\)\))p 2985 4995 344 3 v 3142 5036 a Fc(q)75 5293 y Fs(for)30 b(all)h Fq(\014)g Fm(\025)25 b Fq(S)5 b Fs(.)40 b(This)30 b(pro)m(v)m(es)h(the)g(lemma)g(2.)75 5406 y(No)m(w,)39 b(w)m(e)d(return)f(to)i(the)g(pro)s(of)e(of)h(prop)s(osition)g(2.)59 b(Using)36 b(Lemma)h(2,)h(w)m(e)e(obtain)h(that)g(there)f(exists)h(a)75 5518 y(constan)m(t)32 b Fq(c)481 5532 y Fo(1)546 5518 y Fq(>)25 b Fs(0,)31 b(indep)s(enden)m(t)e(of)h Fq(\014)5 b Fs(,)31 b(suc)m(h)f(that)756 5740 y Fm(j)q(j)p Fs(\()p Fq(T)908 5703 y Fp(s)945 5740 y Fq(@)998 5703 y Fp(\024)993 5763 y(T)1048 5740 y Fq(v)1092 5755 y Fp(n)p Fo(+)p Fp(\027;\014)s Fy(\000)p Fp(S)1393 5740 y Fs(\))16 b(\()p Fq(T)d Fs(\))p Fm(j)q(j)1631 5772 y Fp(\014)s Fo(;)p Fp(\033)1766 5740 y Fm(\024)25 b Fq(c)1901 5754 y Fo(1)1940 5740 y Fq(\014)2006 5661 y Fc(b)p Fh(\()q Fc(s)p Fh(+)p Fc(\024)p Fh(\()p Fc(q)r Fk(\000)p Fh(1\)\))p 2006 5680 V 2163 5721 a Fc(q)2379 5740 y Fm(j)q(j)p Fq(v)2474 5755 y Fp(n)p Fo(+)p Fp(\027;\014)s Fy(\000)p Fp(S)2775 5740 y Fs(\()p Fq(T)13 b Fs(\))p Fm(j)q(j)2962 5772 y Fp(\014)s Fy(\000)p Fp(S)t Fo(;)p Fp(\033)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 3809 100 a Fs(8)75 399 y(for)30 b(all)h Fq(n)25 b Fm(\025)g Fs(0)31 b(and)f(for)g(all)h Fq(\014)g Fm(\025)25 b Fq(S)5 b Fs(,)30 b(and)g(hence)240 518 y Fn(\014)240 573 y(\014)270 518 y(\014)270 573 y(\014)300 522 y(\000)342 596 y Fq(T)408 558 y Fp(s)444 596 y Fq(@)497 558 y Fp(\027)492 618 y(t)541 596 y Fq(@)594 558 y Fp(\024)589 618 y(T)644 596 y Fq(@)697 558 y Fy(\000)p Fp(S)692 618 y(z)803 596 y Fq(V)877 522 y Fn(\001)933 596 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))1228 518 y Fn(\014)1228 573 y(\014)1260 518 y(\014)1260 573 y(\014)1290 632 y Fp(\016)1321 641 y Fh(1)1356 632 y Fp(;\016)1407 641 y Fh(2)1442 632 y Fo(;)p Fp(\033)289 889 y Fm(\024)25 b Fq(c)424 903 y Fo(1)464 889 y Fq(\016)507 850 y Fy(\000)p Fp(\027)504 915 y Fo(1)606 889 y Fq(\016)649 851 y Fp(S)646 911 y Fo(2)722 802 y Fn(X)716 1000 y Fp(\014)s Fy(\025)p Fp(S)876 802 y Fn(X)875 998 y Fp(n)p Fy(\025)p Fo(0)1024 889 y Fq(\014)1090 810 y Fc(b)p Fh(\()p Fc(s)p Fh(+)p Fc(\024)p Fh(\()p Fc(q)r Fk(\000)p Fh(1\)\))p 1090 828 344 3 v 1247 869 a Fc(q)1458 827 y Fs(\()p Fq(n)20 b Fs(+)g Fq(\027)25 b Fs(+)20 b Fq(\014)26 b Fm(\000)20 b Fq(S)5 b Fs(\)!)p 1458 868 652 4 v 1625 951 a(\()p Fq(n)20 b Fs(+)g Fq(\014)5 b Fs(\)!)2134 889 y Fm(j)q(j)p Fq(v)2229 904 y Fp(n)p Fo(+)p Fp(\027;\014)s Fy(\000)p Fp(S)2530 889 y Fs(\()p Fq(T)13 b Fs(\))p Fm(j)q(j)2717 921 y Fp(\014)s Fy(\000)p Fp(S)t Fo(;)p Fp(\033)3090 827 y Fq(\016)3133 789 y Fp(n)p Fo(+)p Fp(\027)3130 853 y Fo(1)3275 827 y Fq(\016)3318 783 y Fp(\014)s Fy(\000)p Fp(S)3315 853 y Fo(2)p 2953 868 V 2953 951 a Fs(\()p Fq(n)20 b Fs(+)g Fq(\027)26 b Fs(+)20 b Fq(\014)25 b Fm(\000)20 b Fq(S)5 b Fs(\)!)3615 889 y Fq(:)75 1175 y Fs(Using)37 b(the)h(estimates)g(\(5\))h(and)d(our)h(h)m(yp)s(otheses) g(that)2097 1130 y Fp(b)p Fo(\()p Fp(s)p Fo(+)p Fp(\024)p Fo(\()p Fp(q)r Fy(\000)p Fo(1\)\))p 2097 1154 393 4 v 2276 1207 a Fp(q)2523 1175 y Fs(+)25 b Fq(\027)42 b(<)36 b(S)5 b Fs(,)39 b(there)f(exists)g(a)f(constan)m(t)75 1288 y Fq(c)114 1302 y Fo(2)179 1288 y Fq(>)25 b Fs(0,)31 b(indep)s(enden)m(t)e(of)i Fq(\014)5 b Fs(,)30 b(suc)m(h)h(that)995 1576 y Fq(\014)1061 1497 y Fc(b)p Fh(\()p Fc(s)p Fh(+)p Fc(\024)p Fh(\()p Fc(q)r Fk(\000)p Fh(1\)\))p 1061 1516 344 3 v 1218 1557 a Fc(q)1429 1515 y Fs(\()p Fq(n)20 b Fs(+)g Fq(\027)26 b Fs(+)20 b Fq(\014)25 b Fm(\000)20 b Fq(S)5 b Fs(\)!)p 1429 1555 652 4 v 1596 1639 a(\()p Fq(n)20 b Fs(+)g Fq(\014)5 b Fs(\)!)2116 1576 y Fm(\024)25 b Fq(c)2251 1590 y Fo(2)2301 1515 y Fq(\014)2367 1436 y Fc(b)p Fh(\()p Fc(s)p Fh(+)p Fc(\024)p Fh(\()p Fc(q)r Fk(\000)p Fh(1\)\))p 2367 1454 344 3 v 2524 1495 a Fc(q)p 2301 1555 424 4 v 2412 1639 a Fq(\014)2468 1612 y Fp(S)t Fy(\000)p Fp(\027)2760 1576 y Fm(\024)g Fq(c)2895 1590 y Fo(2)75 1817 y Fs(for)30 b(all)h Fq(\014)g Fm(\025)25 b Fq(S)5 b Fs(.)40 b(T)-8 b(o)31 b(conclude)g(the)g(pro)s(of)e(of)i (Prop)s(osition)f(2,)h(there)g(exists)g(a)g(constan)m(t)g Fq(c)3205 1831 y Fo(3)3270 1817 y Fq(>)25 b Fs(0)31 b(suc)m(h)f(that) 761 1937 y Fn(\014)761 1991 y(\014)791 1937 y(\014)791 1991 y(\014)821 1940 y(\000)863 2014 y Fq(T)929 1976 y Fp(s)965 2014 y Fq(@)1018 1976 y Fp(\027)1013 2036 y(t)1062 2014 y Fq(@)1115 1976 y Fp(\024)1110 2036 y(T)1165 2014 y Fq(@)1218 1976 y Fy(\000)p Fp(S)1213 2036 y(z)1324 2014 y Fq(V)1397 1940 y Fn(\001)1454 2014 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))1749 1937 y Fn(\014)1749 1991 y(\014)1781 1937 y(\014)1781 1991 y(\014)1811 2050 y Fp(\016)1842 2059 y Fh(1)1877 2050 y Fp(;\016)1928 2059 y Fh(2)1962 2050 y Fo(;)p Fp(\033)2054 2014 y Fm(\024)25 b Fq(c)2189 2028 y Fo(3)2229 2014 y Fq(\016)2272 1976 y Fy(\000)p Fp(\027)2269 2040 y Fo(1)2371 2014 y Fq(\016)2414 1976 y Fp(S)2411 2036 y Fo(2)2481 2014 y Fm(jj)p Fq(V)20 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))p Fm(j)r(j)2951 2041 y Fp(\016)2982 2050 y Fh(1)3017 2041 y Fp(;\016)3068 2050 y Fh(2)3102 2041 y Fo(;)p Fp(\033)75 2223 y Fs(for)30 b(all)h Fq(V)21 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))26 b Fm(2)f(G)875 2237 y Fp(q)928 2223 y Fs(\()q Fq(\016)1004 2237 y Fo(1)1044 2223 y Fq(;)15 b(\016)1124 2237 y Fo(2)1164 2223 y Fs(;)g Fq(\033)s Fs(\))q(.)216 2384 y(In)44 b(the)h(next)g(prop) s(osition,)j(w)m(e)d(study)f(the)h(con)m(tin)m(uit)m(y)i(of)e(the)g(op) s(erator)g(of)g(m)m(ultiplication)h(b)m(y)f(a)75 2497 y(holomorphic)30 b(function.)75 2732 y Fe(Prop)s(osition)36 b(3)46 b Fr(L)-5 b(et)42 b Fq(a)p Fs(\()p Fq(t;)15 b(z)t Fs(\))26 b(=)1264 2646 y Fn(X)1263 2843 y Fp(\014)s Fy(\025)p Fo(0)1412 2646 y Fn(X)1412 2842 y Fp(n)p Fy(\025)p Fo(0)1560 2732 y Fq(a)1608 2747 y Fp(n;\014)1727 2670 y Fq(t)1760 2637 y Fp(n)p 1727 2711 80 4 v 1727 2794 a Fq(n)p Fs(!)1827 2670 y Fq(z)1873 2637 y Fp(\014)p 1827 2711 94 4 v 1833 2794 a Fq(\014)5 b Fs(!)1972 2732 y Fr(a)42 b(holomorphic)i(function)e (ne)-5 b(ar)42 b(the)g(origin)g(in)g Fl(C)3788 2699 y Fo(2)3827 2732 y Fr(.)75 2937 y(We)32 b(de\014ne)1381 3103 y Fm(j)p Fq(a)p Fm(j)p Fs(\()p Fq(t;)15 b(z)t Fs(\))27 b(=)1792 3017 y Fn(X)1791 3215 y Fp(\014)s Fy(\025)p Fo(0)1940 3017 y Fn(X)1939 3213 y Fp(n)p Fy(\025)p Fo(0)2087 3103 y Fm(j)p Fq(a)2160 3118 y Fp(n;\014)2270 3103 y Fm(j)2320 3042 y Fq(t)2353 3009 y Fp(n)p 2320 3082 80 4 v 2320 3166 a Fq(n)p Fs(!)2420 3042 y Fq(z)2466 3009 y Fp(\014)p 2420 3082 94 4 v 2426 3166 a Fq(\014)5 b Fs(!)2524 3103 y Fq(:)75 3358 y Fr(Then)33 b(ther)-5 b(e)33 b(exists)g(a)g(c)-5 b(onstant)35 b Fq(C)1294 3372 y Fo(7)1358 3358 y Fq(>)25 b Fs(0)p Fr(,)33 b(such)g(that)852 3555 y Fm(j)q(j)p Fq(a)p Fs(\()p Fq(t;)15 b(z)t Fs(\))p Fq(V)21 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))p Fm(j)r(j)1561 3582 y Fp(\016)1592 3591 y Fh(1)1627 3582 y Fp(;\016)1678 3591 y Fh(2)1712 3582 y Fo(;)p Fp(\033)1804 3555 y Fm(\024)25 b Fq(C)1965 3569 y Fo(7)2004 3555 y Fm(j)p Fq(a)p Fm(j)p Fs(\()p Fq(\016)2177 3569 y Fo(1)2218 3555 y Fq(;)15 b(\016)2298 3569 y Fo(2)2338 3555 y Fs(\))g Fm(j)q(j)p Fq(V)21 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))p Fm(j)q(j)2860 3582 y Fp(\016)2891 3591 y Fh(1)2925 3582 y Fp(;\016)2976 3591 y Fh(2)3011 3582 y Fo(;)p Fp(\033)75 3752 y Fr(for)33 b(al)5 b(l)33 b Fq(V)21 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))26 b Fm(2)f(G)885 3766 y Fp(q)938 3752 y Fs(\()q Fq(\016)1014 3766 y Fo(1)1053 3752 y Fq(;)15 b(\016)1133 3766 y Fo(2)1174 3752 y Fs(;)g Fq(\033)s Fs(\))q Fr(.)75 3982 y Fe(Pro)s(of)56 b Fs(F)-8 b(or)1007 4130 y Fq(V)20 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))27 b(=)1499 4044 y Fn(X)1498 4242 y Fp(\014)s Fy(\025)p Fo(0)1647 4044 y Fn(X)1646 4240 y Fp(n)p Fy(\025)p Fo(0)1794 4130 y Fq(v)1838 4145 y Fp(n;\014)1948 4130 y Fs(\()p Fq(T)13 b Fs(\))2094 4069 y Fq(t)2127 4036 y Fp(n)p 2094 4110 80 4 v 2094 4193 a Fq(n)p Fs(!)2194 4069 y Fq(z)2240 4036 y Fp(\014)p 2194 4110 94 4 v 2200 4193 a Fq(\014)5 b Fs(!)2323 4130 y Fm(2)25 b(G)2463 4144 y Fp(q)2516 4130 y Fs(\()p Fq(\016)2591 4144 y Fo(1)2631 4130 y Fq(;)15 b(\016)2711 4144 y Fo(2)2751 4130 y Fs(;)g Fq(\033)s Fs(\))h Fq(;)75 4386 y Fs(w)m(e)31 b(ha)m(v)m(e)524 4557 y Fq(t)557 4524 y Fp(n)600 4533 y Fh(0)p 521 4598 120 4 v 521 4681 a Fq(n)576 4695 y Fo(0)615 4681 y Fs(!)660 4557 y Fq(z)706 4524 y Fp(\014)746 4533 y Fh(0)p 660 4598 125 4 v 664 4681 a Fq(\014)715 4695 y Fo(0)756 4681 y Fs(!)795 4619 y Fq(V)20 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))27 b(=)1306 4532 y Fn(X)1286 4730 y Fp(\014)s Fy(\025)p Fp(\014)1424 4739 y Fh(0)1495 4532 y Fn(X)1473 4728 y Fp(n)p Fy(\025)p Fp(n)1614 4737 y Fh(0)2065 4557 y Fq(n)p Fs(!)p Fq(\014)5 b Fs(!)p 1673 4598 946 4 v 1673 4681 a Fq(n)1728 4695 y Fo(0)1767 4681 y Fs(!\()p Fq(n)21 b Fm(\000)f Fq(n)2049 4695 y Fo(0)2088 4681 y Fs(\)!)p Fq(\014)2199 4695 y Fo(0)2239 4681 y Fs(!\()p Fq(\014)26 b Fm(\000)20 b Fq(\014)2518 4695 y Fo(0)2558 4681 y Fs(\)!)2628 4619 y Fq(v)2672 4634 y Fp(n)p Fy(\000)p Fp(n)2813 4643 y Fh(0)2848 4634 y Fp(;\014)s Fy(\000)p Fp(\014)3006 4643 y Fh(0)3043 4619 y Fs(\()p Fq(T)13 b Fs(\))3189 4557 y Fq(t)3222 4524 y Fp(n)p 3189 4598 80 4 v 3189 4681 a Fq(n)p Fs(!)3290 4557 y Fq(z)3336 4524 y Fp(\014)p 3290 4598 94 4 v 3296 4681 a Fq(\014)5 b Fs(!)3393 4619 y Fq(:)75 4907 y Fs(Therefore,)30 b(b)m(y)h(de\014nition,)f(w)m(e)h (obtain)g(that)157 5038 y Fn(\014)157 5092 y(\014)157 5147 y(\014)157 5201 y(\014)187 5038 y(\014)187 5092 y(\014)187 5147 y(\014)187 5201 y(\014)230 5108 y Fq(t)263 5075 y Fp(n)306 5084 y Fh(0)p 227 5148 120 4 v 227 5232 a Fq(n)282 5246 y Fo(0)321 5232 y Fs(!)366 5108 y Fq(z)412 5075 y Fp(\014)452 5084 y Fh(0)p 366 5148 125 4 v 370 5232 a Fq(\014)421 5246 y Fo(0)461 5232 y Fs(!)501 5169 y Fq(V)20 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))869 5038 y Fn(\014)869 5092 y(\014)869 5147 y(\014)869 5201 y(\014)901 5038 y(\014)901 5092 y(\014)901 5147 y(\014)901 5201 y(\014)931 5260 y Fp(\016)962 5269 y Fh(1)997 5260 y Fp(;\016)1048 5269 y Fh(2)1082 5260 y Fo(;)p Fp(\033)1174 5169 y Fs(=)1290 5083 y Fn(X)1270 5281 y Fp(\014)s Fy(\025)p Fp(\014)1408 5290 y Fh(0)1479 5083 y Fn(X)1457 5279 y Fp(n)p Fy(\025)p Fp(n)1598 5288 y Fh(0)1648 5038 y Fn(\014)1648 5092 y(\014)1648 5147 y(\014)1648 5201 y(\014)1678 5038 y(\014)1678 5092 y(\014)1678 5147 y(\014)1678 5201 y(\014)2110 5108 y Fq(n)p Fs(!)p Fq(\014)5 b Fs(!)p 1718 5148 946 4 v 1718 5232 a Fq(n)1773 5246 y Fo(0)1812 5232 y Fs(!\()p Fq(n)20 b Fm(\000)g Fq(n)2093 5246 y Fo(0)2132 5232 y Fs(\)!)p Fq(\014)2243 5246 y Fo(0)2284 5232 y Fs(!\()p Fq(\014)26 b Fm(\000)20 b Fq(\014)2563 5246 y Fo(0)2603 5232 y Fs(\)!)2673 5169 y Fq(v)2717 5184 y Fp(n)p Fy(\000)p Fp(n)2858 5193 y Fh(0)2892 5184 y Fp(;\014)s Fy(\000)p Fp(\014)3050 5193 y Fh(0)3088 5169 y Fs(\()p Fq(T)13 b Fs(\))3224 5038 y Fn(\014)3224 5092 y(\014)3224 5147 y(\014)3224 5201 y(\014)3255 5038 y(\014)3255 5092 y(\014)3255 5147 y(\014)3255 5201 y(\014)3285 5260 y Fp(\014)s Fo(;)p Fp(\033)3488 5108 y Fq(\016)3531 5075 y Fp(n)3528 5132 y Fo(1)3579 5108 y Fq(\016)3622 5064 y Fp(\014)3619 5134 y Fo(2)p 3420 5148 318 4 v 3420 5232 a Fs(\()p Fq(n)20 b Fs(+)g Fq(\014)5 b Fs(\)!)3748 5169 y Fq(;)75 5458 y Fs(where)159 5551 y Fn(\014)159 5605 y(\014)159 5660 y(\014)159 5714 y(\014)190 5551 y(\014)190 5605 y(\014)190 5660 y(\014)190 5714 y(\014)622 5621 y Fq(n)p Fs(!)p Fq(\014)g Fs(!)p 230 5662 946 4 v 230 5745 a Fq(n)285 5759 y Fo(0)324 5745 y Fs(!\()p Fq(n)20 b Fm(\000)g Fq(n)605 5759 y Fo(0)644 5745 y Fs(\)!)p Fq(\014)755 5759 y Fo(0)796 5745 y Fs(!\()p Fq(\014)26 b Fm(\000)20 b Fq(\014)1075 5759 y Fo(0)1115 5745 y Fs(\)!)1185 5682 y Fq(v)1229 5697 y Fp(n)p Fy(\000)p Fp(n)1370 5706 y Fh(0)1404 5697 y Fp(;\014)s Fy(\000)p Fp(\014)1562 5706 y Fh(0)1600 5682 y Fs(\()p Fq(T)13 b Fs(\))1736 5551 y Fn(\014)1736 5605 y(\014)1736 5660 y(\014)1736 5714 y(\014)1767 5551 y(\014)1767 5605 y(\014)1767 5660 y(\014)1767 5714 y(\014)1797 5773 y Fp(\014)s Fo(;)p Fp(\033)1932 5682 y Fs(=)2430 5621 y Fq(n)p Fs(!)p Fq(\014)5 b Fs(!)p 2038 5662 V 2038 5745 a Fq(n)2093 5759 y Fo(0)2132 5745 y Fs(!\()p Fq(n)20 b Fm(\000)g Fq(n)2413 5759 y Fo(0)2452 5745 y Fs(\)!)p Fq(\014)2563 5759 y Fo(0)2604 5745 y Fs(!\()p Fq(\014)26 b Fm(\000)20 b Fq(\014)2883 5759 y Fo(0)2922 5745 y Fs(\)!)3008 5682 y Fm(jj)q Fq(v)3103 5697 y Fp(n)p Fy(\000)p Fp(n)3244 5706 y Fh(0)3278 5697 y Fp(;\014)s Fy(\000)p Fp(\014)3436 5706 y Fh(0)3474 5682 y Fs(\()p Fq(T)13 b Fs(\))p Fm(jj)3661 5714 y Fp(\014)s Fo(;)p Fp(\033)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 3809 100 a Fs(9)75 399 y(and)540 511 y Fm(j)q(j)p Fq(v)635 526 y Fp(n)p Fy(\000)p Fp(n)776 535 y Fh(0)810 526 y Fp(;\014)s Fy(\000)p Fp(\014)968 535 y Fh(0)1006 511 y Fs(\()p Fq(T)13 b Fs(\))p Fm(j)q(j)1193 543 y Fp(\014)s Fo(;)p Fp(\033)1328 511 y Fs(=)29 b(sup)1424 590 y Fp(T)10 b Fy(2)p Fd(C)1585 511 y Fm(j)p Fq(v)1654 526 y Fp(n)p Fy(\000)p Fp(n)1795 535 y Fh(0)1829 526 y Fp(;\014)s Fy(\000)p Fp(\014)1987 535 y Fh(0)2025 511 y Fs(\()p Fq(T)j Fs(\))p Fm(j)j Fs(\(1)21 b(+)f Fm(j)p Fq(T)13 b Fm(j)p Fs(\))2545 470 y Fy(\000)p Fp(m)2682 511 y Fs(exp)i(\()p Fm(\000)p Fq(\033)s(r)3038 526 y Fp(b)3073 511 y Fs(\()p Fq(\014)5 b Fs(\))p Fm(j)p Fq(T)13 b Fm(j)3315 474 y Fp(q)3354 511 y Fs(\))75 712 y(for)30 b(all)h Fq(n)25 b Fm(\025)g Fq(n)571 726 y Fo(0)640 712 y Fs(and)30 b(for)g(all)i Fq(\014)e Fm(\025)25 b Fq(\014)1311 726 y Fo(0)1351 712 y Fs(.)75 825 y(Observ)m(e)30 b(that)439 996 y Fm(j)q(j)p Fq(v)534 1011 y Fp(n)p Fy(\000)p Fp(n)675 1020 y Fh(0)709 1011 y Fp(;\014)s Fy(\000)p Fp(\014)867 1020 y Fh(0)905 996 y Fs(\()p Fq(T)13 b Fs(\))p Fm(j)q(j)1092 1028 y Fp(\014)s Fo(;)p Fp(\033)1227 996 y Fm(\024)29 b Fs(sup)1323 1075 y Fp(T)10 b Fy(2)p Fd(C)1484 996 y Fm(j)p Fq(v)1553 1011 y Fp(n)p Fy(\000)p Fp(n)1694 1020 y Fh(0)1728 1011 y Fp(;\014)s Fy(\000)p Fp(\014)1886 1020 y Fh(0)1924 996 y Fs(\()p Fq(T)j Fs(\))p Fm(j)j Fs(\(1)21 b(+)f Fm(j)p Fq(T)13 b Fm(j)p Fs(\))2444 954 y Fy(\000)p Fp(m)2581 996 y Fs(exp)i(\()p Fm(\000)p Fq(\033)s(r)2937 1011 y Fp(b)2972 996 y Fs(\()p Fq(\014)25 b Fm(\000)20 b Fq(\014)3225 1010 y Fo(0)3265 996 y Fs(\))p Fm(j)p Fq(T)13 b Fm(j)3416 959 y Fp(q)3455 996 y Fs(\))75 1223 y(since)31 b Fq(r)339 1238 y Fp(b)373 1223 y Fs(\()p Fq(\014)5 b Fs(\))26 b Fm(\025)f Fq(r)662 1238 y Fp(b)697 1223 y Fs(\()p Fq(\014)g Fm(\000)20 b Fq(\014)950 1237 y Fo(0)990 1223 y Fs(\))31 b(for)f(all)h Fq(\014)g Fm(\025)25 b Fq(\014)1550 1237 y Fo(0)1589 1223 y Fs(.)75 1336 y(Therefore,)30 b(w)m(e)h(ha)m(v)m(e)1078 1449 y Fm(jj)p Fq(v)1172 1464 y Fp(n)p Fy(\000)p Fp(n)1313 1473 y Fh(0)1347 1464 y Fp(;\014)s Fy(\000)p Fp(\014)1505 1473 y Fh(0)1543 1449 y Fs(\()p Fq(T)13 b Fs(\))p Fm(j)q(j)1730 1481 y Fp(\014)s Fo(;)p Fp(\033)1865 1449 y Fm(\024)25 b(jj)p Fq(v)2055 1464 y Fp(n)p Fy(\000)p Fp(n)2196 1473 y Fh(0)2231 1464 y Fp(;\014)s Fy(\000)p Fp(\014)2389 1473 y Fh(0)2426 1449 y Fs(\()p Fq(T)13 b Fs(\))p Fm(j)q(j)2613 1481 y Fp(\014)s Fy(\000)p Fp(\014)2751 1490 y Fh(0)2785 1481 y Fo(;)p Fp(\033)75 1599 y Fs(for)30 b(all)h Fq(\014)g Fm(\025)25 b Fq(\014)569 1613 y Fo(0)609 1599 y Fs(.)75 1712 y(Using)31 b(the)f(upp)s(er)e(men)m(tioned)j(inequalities,)h(w)m (e)f(obtain)331 1816 y Fn(\014)331 1871 y(\014)331 1925 y(\014)331 1980 y(\014)361 1816 y(\014)361 1871 y(\014)361 1925 y(\014)361 1980 y(\014)404 1886 y Fq(t)437 1853 y Fp(n)480 1862 y Fh(0)p 401 1927 120 4 v 401 2010 a Fq(n)456 2024 y Fo(0)495 2010 y Fs(!)540 1886 y Fq(z)586 1853 y Fp(\014)626 1862 y Fh(0)p 540 1927 125 4 v 544 2010 a Fq(\014)595 2024 y Fo(0)636 2010 y Fs(!)675 1948 y Fq(V)20 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))1043 1816 y Fn(\014)1043 1871 y(\014)1043 1925 y(\014)1043 1980 y(\014)1075 1816 y(\014)1075 1871 y(\014)1075 1925 y(\014)1075 1980 y(\014)1105 2038 y Fp(\016)1136 2047 y Fh(1)1171 2038 y Fp(;\016)1222 2047 y Fh(2)1256 2038 y Fo(;)p Fp(\033)1348 1948 y Fm(\024)1454 1886 y Fq(\016)1497 1846 y Fp(n)1540 1855 y Fh(0)1494 1913 y Fo(1)p 1454 1927 126 4 v 1457 2010 a Fq(n)1512 2024 y Fo(0)1551 2010 y Fs(!)1599 1886 y Fq(\016)1642 1842 y Fp(\014)1682 1851 y Fh(0)1639 1913 y Fo(2)p 1599 1927 123 4 v 1602 2010 a Fq(\014)1653 2024 y Fo(0)1693 2010 y Fs(!)1767 1862 y Fn(X)1747 2059 y Fp(\014)s Fy(\025)p Fp(\014)1885 2068 y Fh(0)1956 1862 y Fn(X)1934 2058 y Fp(n)p Fy(\025)p Fp(n)2075 2067 y Fh(0)2408 1886 y Fq(n)p Fs(!)p Fq(\014)5 b Fs(!)p 2134 1927 710 4 v 2134 2010 a(\()p Fq(n)20 b Fm(\000)g Fq(n)2390 2024 y Fo(0)2429 2010 y Fs(\)!\()p Fq(\014)27 b Fm(\000)20 b Fq(\014)2744 2024 y Fo(0)2783 2010 y Fs(\)!)2864 1886 y(\()p Fq(n)g Fm(\000)g Fq(n)3120 1900 y Fo(0)3179 1886 y Fs(+)g Fq(\014)25 b Fm(\000)20 b Fq(\014)3488 1900 y Fo(0)3528 1886 y Fs(\)!)p 2864 1927 725 4 v 3067 2010 a(\()p Fq(n)h Fs(+)e Fq(\014)5 b Fs(\)!)1084 2300 y Fm(\002)15 b(jj)p Fq(v)1264 2315 y Fp(n)p Fy(\000)p Fp(n)1405 2324 y Fh(0)1439 2315 y Fp(;\014)s Fy(\000)p Fp(\014)1597 2324 y Fh(0)1635 2300 y Fs(\()p Fq(T)e Fs(\))p Fm(j)q(j)1822 2332 y Fp(\014)s Fy(\000)p Fp(\014)1960 2341 y Fh(0)1994 2332 y Fo(;)p Fp(\033)2227 2238 y Fq(\016)2270 2198 y Fp(n)p Fy(\000)p Fp(n)2411 2207 y Fh(0)2267 2264 y Fo(1)2450 2238 y Fq(\016)2493 2194 y Fp(\014)s Fy(\000)p Fp(\014)2631 2203 y Fh(0)2490 2264 y Fo(2)p 2086 2279 V 2086 2362 a Fs(\()p Fq(n)20 b Fm(\000)g Fq(n)2342 2376 y Fo(0)2401 2362 y Fs(+)g Fq(\014)25 b Fm(\000)20 b Fq(\014)2710 2376 y Fo(0)2750 2362 y Fs(\)!)2821 2300 y Fq(:)75 2499 y Fs(On)34 b(the)h(other)g (hand,)g(using)f(the)h(estimates)h(\(5\),)h(there)e(exists)h(a)f (constan)m(t)h Fq(c)2859 2513 y Fo(1)2931 2499 y Fq(>)c Fs(0)j(\(indep)s(enden)m(t)f(on)h Fq(n)75 2612 y Fs(and)30 b Fq(\014)5 b Fs(\),)31 b(suc)m(h)f(that)1412 2711 y Fq(n)p Fs(!)p Fq(\014)5 b Fs(!)p 1137 2751 710 4 v 1137 2835 a(\()p Fq(n)21 b Fm(\000)e Fq(n)1393 2849 y Fo(0)1432 2835 y Fs(\)!\()p Fq(\014)27 b Fm(\000)20 b Fq(\014)1747 2849 y Fo(0)1787 2835 y Fs(\)!)1867 2711 y(\()p Fq(n)g Fm(\000)g Fq(n)2123 2725 y Fo(0)2182 2711 y Fs(+)g Fq(\014)26 b Fm(\000)20 b Fq(\014)2492 2725 y Fo(0)2531 2711 y Fs(\)!)p 1867 2751 725 4 v 2071 2835 a(\()p Fq(n)g Fs(+)g Fq(\014)5 b Fs(\)!)2627 2772 y Fm(\024)25 b Fq(c)2762 2786 y Fo(1)75 2966 y Fs(for)30 b(all)h Fq(n)25 b Fm(\025)g Fq(n)571 2980 y Fo(0)640 2966 y Fs(and)30 b(for)g(all)i Fq(\014)e Fm(\025)25 b Fq(\014)1311 2980 y Fo(0)1351 2966 y Fs(,)31 b(therefore)853 3071 y Fn(\014)853 3125 y(\014)853 3180 y(\014)853 3234 y(\014)883 3071 y(\014)883 3125 y(\014)883 3180 y(\014)883 3234 y(\014)926 3141 y Fq(t)959 3108 y Fp(n)1002 3117 y Fh(0)p 924 3181 120 4 v 924 3265 a Fq(n)979 3279 y Fo(0)1018 3265 y Fs(!)1063 3141 y Fq(z)1109 3108 y Fp(\014)1149 3117 y Fh(0)p 1063 3181 125 4 v 1067 3265 a Fq(\014)1118 3279 y Fo(0)1158 3265 y Fs(!)1197 3202 y Fq(V)21 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))1566 3071 y Fn(\014)1566 3125 y(\014)1566 3180 y(\014)1566 3234 y(\014)1597 3071 y(\014)1597 3125 y(\014)1597 3180 y(\014)1597 3234 y(\014)1628 3293 y Fp(\016)1659 3302 y Fh(1)1693 3293 y Fp(;\016)1744 3302 y Fh(2)1779 3293 y Fo(;)p Fp(\033)1870 3202 y Fm(\024)25 b Fq(c)2005 3216 y Fo(1)2055 3141 y Fq(\016)2098 3100 y Fp(n)2141 3109 y Fh(0)2095 3167 y Fo(1)p 2055 3181 126 4 v 2058 3265 a Fq(n)2113 3279 y Fo(0)2152 3265 y Fs(!)2201 3141 y Fq(\016)2244 3097 y Fp(\014)2284 3106 y Fh(0)2241 3167 y Fo(2)p 2201 3181 123 4 v 2204 3265 a Fq(\014)2255 3279 y Fo(0)2294 3265 y Fs(!)2348 3202 y Fm(jj)p Fq(V)c Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))p Fm(j)q(j)2818 3229 y Fp(\016)2849 3238 y Fh(1)2884 3229 y Fp(;\016)2935 3238 y Fh(2)2970 3229 y Fo(;)p Fp(\033)3051 3202 y Fq(:)75 3435 y Fs(So)30 b(that,)h(there)g(exists)g(a)g(constan)m(t)g Fq(c)1388 3449 y Fo(2)1454 3435 y Fq(>)24 b Fs(0)31 b(suc)m(h)f(that)662 3660 y Fm(jj)p Fq(a)p Fs(\()p Fq(t;)15 b(z)t Fs(\))p Fq(V)22 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))p Fm(j)q(j)1371 3687 y Fp(\016)1402 3696 y Fh(1)1436 3687 y Fp(;\016)1487 3696 y Fh(2)1522 3687 y Fo(;)p Fp(\033)1614 3660 y Fm(\024)1726 3574 y Fn(X)1710 3772 y Fp(\014)1750 3781 y Fh(0)1784 3772 y Fy(\025)p Fo(0)1907 3574 y Fn(X)1889 3770 y Fp(n)1932 3779 y Fh(0)1966 3770 y Fy(\025)p Fo(0)2072 3528 y Fn(\014)2072 3583 y(\014)2072 3638 y(\014)2072 3692 y(\014)2102 3528 y(\014)2102 3583 y(\014)2102 3638 y(\014)2102 3692 y(\014)2132 3660 y Fq(a)2180 3675 y Fp(n)2223 3684 y Fh(0)2258 3675 y Fp(;\014)2318 3684 y Fh(0)2368 3599 y Fq(t)2401 3566 y Fp(n)2444 3575 y Fh(0)p 2366 3639 120 4 v 2366 3723 a Fq(n)2421 3737 y Fo(0)2460 3723 y Fs(!)2505 3599 y Fq(z)2551 3566 y Fp(\014)2591 3575 y Fh(0)p 2505 3639 125 4 v 2509 3723 a Fq(\014)2560 3737 y Fo(0)2600 3723 y Fs(!)2639 3660 y Fq(V)21 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))3008 3528 y Fn(\014)3008 3583 y(\014)3008 3638 y(\014)3008 3692 y(\014)3039 3528 y(\014)3039 3583 y(\014)3039 3638 y(\014)3039 3692 y(\014)3070 3751 y Fp(\016)3101 3760 y Fh(1)3136 3751 y Fp(\016)3167 3760 y Fh(2)3201 3751 y Fo(;)p Fp(\033)423 4001 y Fm(\024)536 3914 y Fn(X)519 4112 y Fp(\014)559 4121 y Fh(0)593 4112 y Fy(\025)p Fo(0)717 3914 y Fn(X)699 4110 y Fp(n)742 4119 y Fh(0)776 4110 y Fy(\025)p Fo(0)881 4001 y Fm(j)q Fq(a)955 4016 y Fp(n)998 4025 y Fh(0)1032 4016 y Fp(;\014)1092 4025 y Fh(0)1130 4001 y Fm(j)1171 3869 y Fn(\014)1171 3923 y(\014)1171 3978 y(\014)1171 4033 y(\014)1201 3869 y(\014)1201 3923 y(\014)1201 3978 y(\014)1201 4033 y(\014)1244 3939 y Fq(t)1277 3906 y Fp(n)1320 3915 y Fh(0)p 1241 3980 120 4 v 1241 4063 a Fq(n)1296 4077 y Fo(0)1335 4063 y Fs(!)1380 3939 y Fq(z)1426 3906 y Fp(\014)1466 3915 y Fh(0)p 1380 3980 125 4 v 1384 4063 a Fq(\014)1435 4077 y Fo(0)1475 4063 y Fs(!)1515 4001 y Fq(V)20 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))1883 3869 y Fn(\014)1883 3923 y(\014)1883 3978 y(\014)1883 4033 y(\014)1915 3869 y(\014)1915 3923 y(\014)1915 3978 y(\014)1915 4033 y(\014)1945 4091 y Fp(\016)1976 4100 y Fh(1)2011 4091 y Fp(;\016)2062 4100 y Fh(2)2096 4091 y Fo(;)p Fp(\033)2188 4001 y Fm(\024)25 b Fq(c)2323 4015 y Fo(2)2378 4001 y Fm(j)p Fq(a)p Fm(j)16 b Fs(\()p Fq(\016)2567 4015 y Fo(1)2607 4001 y Fq(;)f(\016)2687 4015 y Fo(2)2727 4001 y Fs(\))g Fm(j)q(j)p Fq(V)20 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))p Fm(j)r(j)3248 4028 y Fp(\016)3279 4037 y Fh(1)3314 4028 y Fp(;\016)3365 4037 y Fh(2)3399 4028 y Fo(;)p Fp(\033)3481 4001 y Fq(:)75 4243 y Fs(This)30 b(pro)m(v)m(es)h(prop)s(osition)f(3.)216 4395 y(In)i(the)h(next)g(prop)s(osition,)g(w)m(e)g(analyse)g(the)g (action)h(of)f(some)g(di\013eren)m(tial)h(op)s(erator)f(on)f(functions) g(of)75 4507 y Fm(G)129 4521 y Fp(q)167 4507 y Fs(\()p Fq(\016)242 4521 y Fo(1)282 4507 y Fq(;)15 b(\016)362 4521 y Fo(2)403 4507 y Fs(;)g Fq(\033)s Fs(\))31 b(whic)m(h)f(are)h(p)s (olynomial)f(in)g(the)h(v)-5 b(ariable)31 b Fq(z)t Fs(.)75 4668 y Fe(Prop)s(osition)36 b(4)46 b Fr(L)-5 b(et)30 b Fq(s)p Fr(,)f Fq(\027)6 b Fr(,)30 b Fq(\024)g Fr(and)g Fq(\022)i Fr(non)e(ne)-5 b(gative)29 b(inte)-5 b(gers)30 b(and)h(let)e Fq(W)13 b Fs(\()p Fq(t;)i(z)t(;)g(T)e Fs(\))27 b Fm(2)d(G)3149 4682 y Fp(q)3203 4668 y Fs(\()p Fq(\016)3278 4682 y Fo(1)p Fp(;)p Fo(0)3373 4668 y Fq(;)15 b(\016)3453 4682 y Fo(2)3493 4668 y Fs(;)g Fq(\033)3585 4682 y Fo(0)3625 4668 y Fs(\))30 b Fr(with)75 4780 y Fq(W)13 b Fs(\()p Fq(t;)i(z)t(;)g(T)e Fs(\))36 b Fr(p)-5 b(olynomial)38 b(in)d(the)h(variable)f Fq(z)t Fr(.)50 b(Then)35 b(ther)-5 b(e)36 b(exist)g Fq(\016)2422 4794 y Fo(1)2491 4780 y Fq(>)30 b Fs(0)p Fr(,)36 b(smal)5 b(l)36 b(enough)f(\()p Fq(\016)3333 4794 y Fo(1)3403 4780 y Fm(\024)30 b Fq(\016)3544 4794 y Fo(1)p Fp(;)p Fo(0)3638 4780 y Fr(\))35 b(and)75 4893 y Fq(\033)s Fr(,)e(big)f(enough)h(\()p Fq(\033)28 b(>)d(\033)909 4907 y Fo(0)949 4893 y Fr(\))32 b(with)1636 5045 y Fq(\033)1688 5059 y Fo(0)1727 5045 y Fq(r)1768 5060 y Fp(b)1803 5045 y Fs(\()p Fq(d)p Fs(\))21 b(+)2042 4983 y(1)p 2042 5024 46 4 v 2042 5107 a(2)2122 5045 y Fm(\024)k Fq(\033)n(;)75 5232 y Fr(such)44 b(that)h(the)f(series)919 5158 y Fn(\000)961 5232 y Fq(T)1027 5199 y Fp(s)1063 5232 y Fq(@)1116 5199 y Fp(\027)1111 5254 y(t)1160 5232 y Fq(@)1213 5199 y Fp(\024)1208 5259 y(T)1263 5232 y Fq(@)1316 5199 y Fp(\022)1311 5254 y(z)1356 5232 y Fq(W)1455 5158 y Fn(\001)1511 5232 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))45 b Fr(b)-5 b(elongs)44 b(to)g Fm(G)2347 5246 y Fp(q)2400 5232 y Fs(\()q Fq(\016)2476 5246 y Fo(1)2516 5232 y Fq(;)15 b(\016)2596 5246 y Fo(2)2636 5232 y Fs(;)g Fq(\033)s Fs(\))q Fr(.)74 b(Mor)-5 b(e)g(over,)48 b(ther)-5 b(e)44 b(exists)g(a)75 5345 y(c)-5 b(onstant)34 b Fq(C)502 5359 y Fo(8)567 5345 y Fq(>)25 b Fs(0)33 b Fr(such)g(that)681 5425 y Fn(\014)681 5479 y(\014)681 5534 y(\014)711 5425 y(\014)711 5479 y(\014)711 5534 y(\014)741 5428 y(\020)796 5529 y Fq(T)862 5492 y Fp(s)898 5529 y Fq(@)951 5492 y Fp(\027)946 5552 y(t)995 5529 y Fq(@)1048 5492 y Fp(\024)1043 5552 y(T)1098 5529 y Fq(@)1151 5492 y Fp(\022)1146 5552 y(z)1190 5529 y Fq(W)1289 5428 y Fn(\021)1358 5529 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))1653 5425 y Fn(\014)1653 5479 y(\014)1653 5534 y(\014)1685 5425 y(\014)1685 5479 y(\014)1685 5534 y(\014)1715 5593 y Fp(\016)1746 5602 y Fh(1)1781 5593 y Fp(;\016)1832 5602 y Fh(2)1867 5593 y Fo(;)p Fp(\033)1958 5529 y Fm(\024)25 b Fq(C)2119 5543 y Fo(8)2159 5529 y Fq(\016)2202 5491 y Fy(\000)p Fp(\027)2199 5555 y Fo(1)2301 5529 y Fq(\016)2344 5491 y Fy(\000)p Fp(\022)2341 5555 y Fo(2)2454 5529 y Fm(jj)p Fq(W)13 b Fs(\()p Fq(t;)i(z)t(;)g(T)e Fs(\))p Fm(j)q(j)2949 5556 y Fp(\016)2980 5565 y Fh(1)p Fc(;)p Fh(0)3065 5556 y Fp(;\016)3116 5565 y Fh(2)3150 5556 y Fo(;)p Fp(\033)3210 5565 y Fh(0)75 5740 y Fr(for)33 b(al)5 b(l)33 b Fq(W)13 b Fs(\()p Fq(t;)i(z)t(;)g(T)e Fs(\))26 b Fm(2)f(G)910 5754 y Fp(q)964 5740 y Fs(\()p Fq(\016)1039 5754 y Fo(1)p Fp(;)p Fo(0)1134 5740 y Fq(;)15 b(\016)1214 5754 y Fo(2)1254 5740 y Fs(;)g Fq(\033)1346 5754 y Fo(0)1386 5740 y Fs(\))33 b Fr(such)f(that)i Fq(W)13 b Fs(\()p Fq(t;)i(z)t(;)g(T)e Fs(\))34 b Fr(is)e(a)h(p)-5 b(olynomial)36 b(in)c(the)h(variable)h Fq(z)t Fr(.)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 3764 100 a Fs(10)75 399 y Fe(Pro)s(of)26 b Fs(F)-8 b(or)31 b Fq(W)13 b Fs(\()p Fq(t;)i(z)t(;)g(T)e Fs(\))26 b Fm(2)f(G)1077 413 y Fp(q)1131 399 y Fs(\()p Fq(\016)1206 413 y Fo(1)p Fp(;)p Fo(0)1301 399 y Fq(;)15 b(\016)1381 413 y Fo(2)1421 399 y Fs(;)g Fq(\033)1513 413 y Fo(0)1553 399 y Fs(\))30 b(and)1296 683 y Fq(W)13 b Fs(\()p Fq(t;)i(z)t(;)g(T)e Fs(\))26 b(=)1861 570 y Fp(d)1813 597 y Fn(X)1812 794 y Fp(\014)s Fo(=0)1962 597 y Fn(X)1961 793 y Fp(n)p Fy(\025)p Fo(0)2109 683 y Fq(w)2174 698 y Fp(n;\014)2284 683 y Fs(\()p Fq(T)13 b Fs(\))2430 622 y Fq(t)2463 589 y Fp(n)p 2430 662 80 4 v 2430 746 a Fq(n)p Fs(!)2530 622 y Fq(z)2576 589 y Fp(\014)p 2530 662 94 4 v 2536 746 a Fq(\014)5 b Fs(!)75 969 y(p)s(olynomial)31 b(in)f(the)g(v)-5 b(ariable)31 b Fq(z)t Fs(,)g(w)m(e)g(ha)m(v)m(e)789 1144 y Fn(\020)843 1245 y Fq(T)909 1207 y Fp(s)946 1245 y Fq(@)999 1207 y Fp(\027)994 1267 y(t)1042 1245 y Fq(@)1095 1207 y Fp(\024)1090 1267 y(T)1145 1245 y Fq(@)1198 1207 y Fp(\022)1193 1267 y(z)1238 1245 y Fq(W)1337 1144 y Fn(\021)1406 1245 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))26 b(=)1827 1131 y Fp(d)p Fy(\000)p Fp(\022)1824 1158 y Fn(X)1823 1356 y Fp(\014)s Fo(=0)1972 1158 y Fn(X)1972 1354 y Fp(n)p Fy(\025)p Fo(0)2120 1245 y Fs(\()p Fq(T)2221 1207 y Fp(s)2258 1245 y Fq(@)2311 1207 y Fp(\024)2306 1267 y(T)2361 1245 y Fq(w)2426 1260 y Fp(n)p Fo(+)p Fp(\027;\014)s Fo(+)p Fp(\022)2715 1245 y Fs(\))16 b(\()p Fq(T)d Fs(\))2912 1183 y Fq(t)2945 1150 y Fp(n)p 2912 1224 80 4 v 2912 1307 a Fq(n)p Fs(!)3012 1183 y Fq(z)3058 1150 y Fp(\014)p 3012 1224 94 4 v 3018 1307 a Fq(\014)5 b Fs(!)3115 1245 y Fq(:)75 1530 y Fs(Therefore,)30 b(b)m(y)h(de\014nition,)f(w)m(e)h (obtain)g(that)75 1806 y(\(7\))457 1702 y Fn(\014)457 1756 y(\014)457 1811 y(\014)487 1702 y(\014)487 1756 y(\014)487 1811 y(\014)517 1705 y(\020)572 1806 y Fq(T)638 1769 y Fp(s)674 1806 y Fq(@)727 1769 y Fp(\027)722 1829 y(t)771 1806 y Fq(@)824 1769 y Fp(\024)819 1829 y(T)874 1806 y Fq(@)927 1769 y Fp(\022)922 1829 y(z)966 1806 y Fq(W)1065 1705 y Fn(\021)1134 1806 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))1429 1702 y Fn(\014)1429 1756 y(\014)1429 1811 y(\014)1461 1702 y(\014)1461 1756 y(\014)1461 1811 y(\014)1491 1869 y Fp(\016)1522 1878 y Fh(1)1557 1869 y Fp(;\016)1608 1878 y Fh(2)1643 1869 y Fo(;)p Fp(\033)1734 1806 y Fs(=)1834 1693 y Fp(d)p Fy(\000)p Fp(\022)1831 1720 y Fn(X)1830 1917 y Fp(\014)s Fo(=0)1979 1720 y Fn(X)1979 1916 y Fp(n)p Fy(\025)p Fo(0)2127 1806 y Fm(jj)p Fs(\()q Fq(T)2279 1769 y Fp(s)2315 1806 y Fq(@)2368 1769 y Fp(\024)2363 1829 y(T)2418 1806 y Fq(w)2483 1821 y Fp(n)p Fo(+)p Fp(\027;\014)s Fo(+)p Fp(\022)2773 1806 y Fs(\))i(\()p Fq(T)e Fs(\))p Fm(j)q(j)3010 1838 y Fp(\014)s Fo(;)p Fp(\033)3213 1745 y Fq(\016)3256 1712 y Fp(n)3253 1769 y Fo(1)3304 1745 y Fq(\016)3347 1700 y Fp(\014)3344 1771 y Fo(2)p 3145 1785 318 4 v 3145 1869 a Fs(\()p Fq(n)20 b Fs(+)g Fq(\014)5 b Fs(\)!)75 2092 y(where)289 2286 y Fm(jj)q Fs(\()p Fq(T)441 2248 y Fp(s)477 2286 y Fq(@)530 2248 y Fp(\024)525 2308 y(T)581 2286 y Fq(w)646 2301 y Fp(n)p Fo(+)p Fp(\027;\014)s Fo(+)p Fp(\022)935 2286 y Fs(\))16 b(\()p Fq(T)d Fs(\))p Fm(jj)1173 2318 y Fp(\014)s Fo(;)p Fp(\033)1307 2286 y Fs(=)30 b(sup)1403 2364 y Fp(T)10 b Fy(2)p Fd(C)1564 2286 y Fm(j)q Fs(\()p Fq(T)1691 2248 y Fp(s)1728 2286 y Fq(@)1781 2248 y Fp(\024)1776 2308 y(T)1831 2286 y Fq(w)1896 2301 y Fp(n)p Fo(+)p Fp(\027;\014)s Fo(+)p Fp(\022)2185 2286 y Fs(\))16 b(\()p Fq(T)d Fs(\))p Fm(j)j Fs(\(1)21 b(+)f Fm(j)p Fq(T)13 b Fm(j)p Fs(\))2756 2244 y Fy(\000)p Fp(m)2893 2286 y Fs(exp)i(\()p Fm(\000)p Fq(\033)s(r)3249 2301 y Fp(b)3283 2286 y Fs(\()p Fq(\014)5 b Fs(\))p Fm(j)p Fq(T)13 b Fm(j)3525 2248 y Fp(q)3565 2286 y Fs(\))i Fq(:)75 2530 y Fs(As)30 b(in)g(the)h(pro)s(of)f(of)g (Prop)s(osition)h(2,)g(using)e(Cauc)m(h)m(y's)i(theorem,)g(w)m(e)g(ha)m (v)m(e,)h(for)e(all)h Fq(T)38 b Fm(2)25 b Fl(C)307 2767 y Fm(j)p Fs(\()p Fq(T)433 2729 y Fp(s)470 2767 y Fq(@)523 2729 y Fp(\024)518 2789 y(T)573 2767 y Fq(w)638 2782 y Fp(n)p Fo(+)p Fp(\027;\014)s Fo(+)p Fp(\022)928 2767 y Fs(\))15 b(\()p Fq(T)e Fs(\))p Fm(j)26 b(\024)1271 2705 y Fq(\024)p Fs(!)p Fm(j)p Fq(T)13 b Fm(j)1464 2672 y Fp(s)p 1271 2746 231 4 v 1290 2829 a Fs(2)p Fq(\031)s(a)1438 2803 y Fp(\024)1527 2767 y Fm(jj)p Fq(w)1642 2782 y Fp(n)p Fo(+)p Fp(\027;\014)s Fo(+)p Fp(\022)1932 2767 y Fs(\()p Fq(T)g Fs(\))p Fm(jj)2119 2799 y Fp(\014)s Fo(+)p Fp(\022)r Fo(;)p Fp(\033)2312 2808 y Fh(0)861 3059 y Fm(\002)947 2935 y Fn(Z)1038 2961 y Fo(2)p Fp(\031)997 3141 y Fo(0)1135 3059 y Fs(\(1)21 b(+)f Fm(j)p Fq(T)33 b Fs(+)20 b Fq(a)15 b Fs(exp\()p Fq(i\034)10 b Fs(\))p Fm(j)p Fs(\))1944 3017 y Fp(m)2026 3059 y Fs(exp)15 b(\()p Fq(\033)2267 3073 y Fo(0)2306 3059 y Fq(r)2347 3074 y Fp(b)2382 3059 y Fs(\()p Fq(\014)25 b Fs(+)20 b Fq(\022)s Fs(\))p Fm(j)p Fq(T)33 b Fs(+)20 b Fq(a)15 b Fs(exp\()p Fq(i\034)10 b Fs(\))p Fm(j)3245 3021 y Fp(q)3285 3059 y Fs(\))15 b Fq(d\034)75 3261 y Fs(where)21 b Fq(a)k(>)g Fs(0)d(can)g(b)s(e)f(c)m (hosen)h(arbitrarily)g(and)f Fq(\033)1769 3275 y Fo(0)1830 3261 y Fs(will)h(b)s(e)f(c)m(hosen)h(later.)39 b(Using)22 b(the)f(triangular)h(inequalit)m(y)-8 b(,)75 3374 y(w)m(e)31 b(get)208 3606 y Fm(j)p Fs(\()q Fq(T)335 3568 y Fp(s)371 3606 y Fq(@)424 3568 y Fp(\024)419 3628 y(T)475 3606 y Fq(w)540 3621 y Fp(n)p Fo(+)p Fp(\027;\014)s Fo(+)p Fp(\022)829 3606 y Fs(\))16 b(\()p Fq(T)d Fs(\))p Fm(j)26 b(\024)1180 3544 y Fq(\024)p Fs(!)p 1172 3585 93 4 v 1172 3668 a Fq(a)1220 3642 y Fp(\024)1275 3606 y Fm(j)p Fq(T)13 b Fm(j)1391 3568 y Fp(s)1443 3606 y Fm(j)q(j)p Fq(w)1559 3621 y Fp(n)p Fo(+)p Fp(\027;\014)s Fo(+)p Fp(\022)1848 3606 y Fs(\()p Fq(T)g Fs(\))p Fm(j)q(j)2035 3638 y Fp(\014)s Fo(+)p Fp(\022)r Fo(;)p Fp(\033)2228 3647 y Fh(0)1444 3828 y Fm(\002)i Fs(\(1)20 b(+)g Fm(j)p Fq(T)13 b Fm(j)21 b Fs(+)f Fq(a)p Fs(\))2032 3786 y Fp(m)2114 3828 y Fs(exp)15 b(\()p Fq(\033)2355 3842 y Fo(0)2395 3828 y Fq(r)2436 3843 y Fp(b)2470 3828 y Fs(\()p Fq(\014)26 b Fs(+)20 b Fq(\022)s Fs(\)\()p Fm(j)p Fq(T)13 b Fm(j)20 b Fs(+)g Fq(a)p Fs(\))3099 3790 y Fp(q)3137 3828 y Fs(\))c Fq(:)75 3990 y Fs(As)30 b(in)g(\(6\),)i(w)m(e)f(de\014ne)722 4226 y Fq(I)762 4241 y Fp(a;\014)s Fo(;)p Fp(\033)922 4250 y Fh(0)960 4226 y Fs(\()p Fq(T)13 b Fs(\))26 b(:=)1261 4165 y Fq(\024)p Fs(!)p 1253 4205 V 1253 4289 a Fq(a)1301 4262 y Fp(\024)1356 4226 y Fm(j)p Fq(T)13 b Fm(j)1472 4189 y Fp(s)1524 4226 y Fs(\()q(1)20 b(+)g Fm(j)p Fq(T)13 b Fm(j)21 b Fs(+)f Fq(a)p Fs(\))2027 4184 y Fp(m)2109 4226 y Fs(exp)15 b(\()p Fq(\033)2350 4240 y Fo(0)2389 4226 y Fq(r)2430 4241 y Fp(b)2465 4226 y Fs(\()p Fq(\014)26 b Fs(+)19 b Fq(\022)s Fs(\)\()p Fm(j)p Fq(T)13 b Fm(j)21 b Fs(+)f Fq(a)p Fs(\))3094 4189 y Fp(q)3132 4226 y Fs(\))c Fq(:)75 4446 y Fs(No)m(w,)31 b(our)f(aim)h(is)f(to)h(estimate)i Fm(jj)p Fs(\()p Fq(T)1364 4413 y Fp(s)1401 4446 y Fq(@)1454 4413 y Fp(\024)1449 4473 y(T)1504 4446 y Fq(w)1569 4461 y Fp(n)p Fo(+)p Fp(\027;\014)s Fo(+)p Fp(\022)1859 4446 y Fs(\))15 b(\()p Fq(T)e Fs(\))p Fm(j)q(j)2096 4478 y Fp(\014)s Fo(;)p Fp(\033)75 4712 y Fs(\(8\))165 b Fm(jj)q Fs(\()p Fq(T)507 4674 y Fp(s)544 4712 y Fq(@)597 4674 y Fp(\024)592 4734 y(T)647 4712 y Fq(w)712 4727 y Fp(n)p Fo(+)p Fp(\027;\014)s Fo(+)p Fp(\022)1001 4712 y Fs(\))16 b(\()p Fq(T)d Fs(\))p Fm(jj)1239 4744 y Fp(\014)s Fo(;)p Fp(\033)1373 4712 y Fm(\024)1469 4584 y Fn(\022)1541 4712 y Fs(sup)1536 4791 y Fp(T)d Fy(2)p Fd(C)1697 4712 y Fq(I)1737 4727 y Fp(a;\014)s Fo(;)p Fp(\033)1897 4736 y Fh(0)1936 4712 y Fs(\()p Fq(T)j Fs(\)\(1)22 b(+)e Fm(j)p Fq(T)13 b Fm(j)p Fs(\))2416 4674 y Fy(\000)p Fp(m)2553 4712 y Fs(exp)i(\()p Fm(\000)p Fq(\033)s(r)2909 4727 y Fp(b)2943 4712 y Fs(\()p Fq(\014)5 b Fs(\))p Fm(j)p Fq(T)13 b Fm(j)3185 4674 y Fp(q)3225 4712 y Fs(\))3260 4584 y Fn(\023)2035 4961 y Fm(\002)i(j)q(j)p Fq(w)2237 4976 y Fp(n)p Fo(+)p Fp(\027;\014)s Fo(+)p Fp(\022)2526 4961 y Fs(\()p Fq(T)e Fs(\))p Fm(j)q(j)2713 4993 y Fp(\014)s Fo(+)p Fp(\022)r Fo(;)p Fp(\033)2906 5002 y Fh(0)2960 4961 y Fq(:)75 5123 y Fs(So)30 b(w)m(e)h(ha)m(v)m(e)h(to)f(pro)m(v)m(e) g(the)g(follo)m(wing)g(lemma.)75 5302 y Fe(Lemma)k(3)45 b Fr(Ther)-5 b(e)34 b(exists)f(a)g(c)-5 b(ontant)34 b Fq(C)1523 5316 y Fo(9)1588 5302 y Fq(>)25 b Fs(0)33 b Fr(such)f(that)995 5496 y Fs(sup)990 5575 y Fp(T)10 b Fy(2)p Fd(C)1151 5496 y Fq(I)1191 5511 y Fp(a;\014)s Fo(;)p Fp(\033)1351 5520 y Fh(0)1390 5496 y Fs(\()p Fq(T)j Fs(\)\(1)21 b(+)f Fm(j)p Fq(T)13 b Fm(j)p Fs(\))1869 5458 y Fy(\000)p Fp(m)2007 5496 y Fs(exp)i(\()p Fm(\000)p Fq(\033)s(r)2363 5511 y Fp(b)2397 5496 y Fs(\()p Fq(\014)5 b Fs(\))p Fm(j)p Fq(T)13 b Fm(j)2639 5458 y Fp(q)2678 5496 y Fs(\))26 b Fm(\024)f Fq(C)2900 5510 y Fo(9)75 5740 y Fr(for)33 b(al)5 b(l)33 b Fs(0)26 b Fm(\024)f Fq(\014)30 b Fm(\024)25 b Fq(d)c Fm(\000)f Fq(\022)s Fr(.)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 3764 100 a Fs(11)75 399 y Fe(Pro)s(of)56 b Fs(As)31 b(in)f(the)g(pro)s(of)g(of)h(Lemma)f(2,)h(w)m(e)g(denote)g (b)m(y)f Fq(J)2151 414 y Fp(a;\014)s Fo(;)p Fp(\033)2311 423 y Fh(0)2346 414 y Fp(;\033)2443 399 y Fs(the)g(suprem)m(um)f(to)i (estimate)854 600 y Fq(J)904 615 y Fp(a;\014)s Fo(;)p Fp(\033)1064 624 y Fh(0)1099 615 y Fp(;\033)1191 600 y Fs(:=)e(sup)1312 678 y Fp(T)10 b Fy(2)p Fd(C)1473 600 y Fq(I)1513 615 y Fp(a;\014)s Fo(;)p Fp(\033)1673 624 y Fh(0)1712 600 y Fs(\()p Fq(T)j Fs(\)\(1)21 b(+)f Fm(j)p Fq(T)13 b Fm(j)p Fs(\))2191 562 y Fy(\000)p Fp(m)2328 600 y Fs(exp)i(\()q Fm(\000)p Fq(\033)s(r)2685 615 y Fp(b)2719 600 y Fs(\()p Fq(\014)5 b Fs(\))p Fm(j)p Fq(T)13 b Fm(j)2961 562 y Fp(q)3000 600 y Fs(\))j Fq(:)75 856 y Fs(By)31 b(de\014nition)f(of)g Fq(I)765 871 y Fp(a;\014)s Fo(;)p Fp(\033)925 880 y Fh(0)964 856 y Fs(\()p Fq(T)13 b Fs(\),)31 b(w)m(e)g(ha)m(v)m(e)383 1115 y Fq(J)433 1130 y Fp(a;\014)s Fo(;)p Fp(\033)593 1139 y Fh(0)628 1130 y Fp(;\033)719 1115 y Fs(=)25 b Fq(\024)p Fs(!)c(sup)907 1194 y Fp(T)10 b Fy(2)p Fd(C)1079 1053 y Fm(j)p Fq(T)j Fm(j)1195 1020 y Fp(s)p 1079 1094 154 4 v 1109 1177 a Fq(a)1157 1151 y Fp(\024)1257 987 y Fn(\022)1334 1053 y Fs(1)21 b(+)f Fm(j)p Fq(T)13 b Fm(j)20 b Fs(+)g Fq(a)p 1334 1094 432 4 v 1414 1177 a Fs(1)g(+)g Fm(j)p Fq(T)13 b Fm(j)1776 987 y Fn(\023)1843 1009 y Fp(m)1925 1115 y Fs(exp)i(\()p Fq(\033)2166 1129 y Fo(0)2206 1115 y Fq(r)2247 1130 y Fp(b)2281 1115 y Fs(\()p Fq(\014)26 b Fs(+)20 b Fq(\022)s Fs(\)\()p Fm(j)p Fq(T)13 b Fm(j)20 b Fs(+)g Fq(a)p Fs(\))2910 1077 y Fp(q)2968 1115 y Fm(\000)g Fq(\033)s(r)3155 1130 y Fp(b)3190 1115 y Fs(\()p Fq(\014)5 b Fs(\))p Fm(j)p Fq(T)13 b Fm(j)3432 1077 y Fp(q)3471 1115 y Fs(\))j Fq(:)75 1372 y Fs(As)30 b(in)g(the)h(pro)s(of)f(of)g (Lemma)h(2,)g(as)f Fm(j)p Fq(T)13 b Fm(j)26 b(2)f Fs([0)p Fq(;)15 b Fs(+)p Fm(1)p Fs([)31 b(and)f Fq(a)p Fs(,)g Fq(m)25 b Fm(\025)g Fs(0,)1009 1497 y Fn(\022)1086 1564 y Fs(1)20 b(+)g Fm(j)p Fq(T)13 b Fm(j)21 b Fs(+)f Fq(a)p 1086 1604 V 1165 1688 a Fs(1)h(+)f Fm(j)p Fq(T)13 b Fm(j)1528 1497 y Fn(\023)1595 1520 y Fp(m)1687 1625 y Fs(=)1782 1497 y Fn(\022)1849 1625 y Fs(1)21 b(+)2128 1564 y Fq(a)p 2016 1604 273 4 v 2016 1688 a Fs(1)g(+)f Fm(j)p Fq(T)13 b Fm(j)2299 1497 y Fn(\023)2366 1520 y Fp(m)2457 1625 y Fm(\024)25 b Fs(\(1)c(+)f Fq(a)p Fs(\))2828 1588 y Fp(m)2895 1625 y Fq(;)75 1871 y Fs(and)30 b(w)m(e)h(ha)m(v)m(e)226 2098 y Fq(J)276 2113 y Fp(a;\014)s Fo(;)p Fp(\033)436 2122 y Fh(0)471 2113 y Fp(;\033)563 2098 y Fm(\024)25 b Fq(\024)p Fs(!)20 b(sup)751 2176 y Fp(T)10 b Fy(2)p Fd(C)923 2036 y Fm(j)p Fq(T)j Fm(j)1039 2003 y Fp(s)p 923 2077 154 4 v 953 2160 a Fq(a)1001 2134 y Fp(\024)1086 2098 y Fs(\(1)21 b(+)e Fq(a)p Fs(\))1360 2060 y Fp(m)1443 2098 y Fs(exp)c(\()p Fm(f)p Fq(\033)1729 2112 y Fo(0)1769 2098 y Fq(r)1810 2113 y Fp(b)1844 2098 y Fs(\()p Fq(\014)26 b Fs(+)20 b Fq(\022)s Fs(\))g Fm(\000)g Fq(\033)s(r)2335 2113 y Fp(b)2369 2098 y Fs(\()p Fq(\014)5 b Fs(\))p Fm(g)17 b Fs(\()p Fm(j)p Fq(T)c Fm(j)20 b Fs(+)g Fq(a)p Fs(\))2902 2060 y Fp(q)2941 2098 y Fs(\))1303 2354 y Fm(\002)15 b Fs(exp)g(\()p Fq(\033)s(r)1674 2369 y Fp(b)1709 2354 y Fs(\()p Fq(\014)5 b Fs(\))15 b Fm(f)q Fs(\()p Fm(j)p Fq(T)e Fm(j)21 b Fs(+)f Fq(a)p Fs(\))2242 2317 y Fp(q)2301 2354 y Fm(\000)g(j)p Fq(T)13 b Fm(j)2508 2317 y Fp(q)2546 2354 y Fm(g)p Fs(\))75 2520 y(using)31 b(the)g(equalit)m(y)i Fq(\015)865 2534 y Fo(1)904 2520 y Fq(\015)951 2534 y Fo(2)1011 2520 y Fm(\000)21 b Fq(\015)1150 2534 y Fo(3)1189 2520 y Fq(\015)1236 2534 y Fo(4)1302 2520 y Fs(=)26 b Fq(\015)1446 2534 y Fo(1)1486 2520 y Fs(\()p Fq(\015)1568 2534 y Fo(2)1628 2520 y Fm(\000)21 b Fq(\015)1767 2534 y Fo(3)1806 2520 y Fs(\))g(+)g Fq(\015)2001 2534 y Fo(3)2040 2520 y Fs(\()p Fq(\015)2122 2534 y Fo(1)2183 2520 y Fm(\000)f Fq(\015)2321 2534 y Fo(4)2361 2520 y Fs(\))31 b(in)g(the)g(exp)s(onen)m (tial.)44 b(No)m(w,)32 b(w)m(e)g(will)f(\014x)75 2633 y Fq(\033)s Fs(.)41 b(It)30 b(is)h(necessary)g(that)g Fq(\033)1038 2647 y Fo(0)1102 2633 y Fm(\024)25 b Fq(\033)s Fs(.)41 b(Therefore,)30 b(w)m(e)h(can)g(sho)m(w)f(that)h(for)f(all)h(0) 26 b Fm(\024)f Fq(\014)31 b Fm(\024)25 b Fq(d)20 b Fm(\000)g Fq(\022)877 2834 y(\033)929 2848 y Fo(0)969 2834 y Fq(r)1010 2849 y Fp(b)1044 2834 y Fs(\()p Fq(\014)26 b Fs(+)20 b Fq(\022)s Fs(\))g Fm(\000)f Fq(\033)s(r)1534 2849 y Fp(b)1569 2834 y Fs(\()p Fq(\014)5 b Fs(\))26 b Fm(\024)f Fq(\033)1869 2848 y Fo(0)1908 2834 y Fq(r)1949 2849 y Fp(b)1984 2834 y Fs(\()p Fq(d)p Fs(\))c Fm(\000)f Fq(\033)s(r)2309 2849 y Fp(b)2344 2834 y Fs(\(0\))26 b(=)f Fq(\033)2633 2848 y Fo(0)2672 2834 y Fq(r)2713 2849 y Fp(b)2748 2834 y Fs(\()p Fq(d)p Fs(\))c Fm(\000)f Fq(\033)n(:)75 3035 y Fs(Let)31 b Fq(\033)i Fs(\()p Fq(\033)c Fm(\025)c Fq(\033)587 3049 y Fo(0)627 3035 y Fs(\))30 b(suc)m(h)g(that)1633 3186 y Fq(\033)f Fm(\025)c Fq(\033)1862 3200 y Fo(0)1901 3186 y Fq(r)1942 3201 y Fp(b)1976 3186 y Fs(\()p Fq(d)p Fs(\))d(+)2216 3125 y(1)p 2216 3165 46 4 v 2216 3249 a(2)2271 3186 y Fq(;)75 3373 y Fs(then)30 b(w)m(e)h(ha)m(v)m(e)h(that) 1427 3501 y Fq(\033)1479 3515 y Fo(0)1518 3501 y Fq(r)1559 3516 y Fp(b)1594 3501 y Fs(\()p Fq(\014)25 b Fs(+)20 b Fq(\022)s Fs(\))g Fm(\000)g Fq(\033)s(r)2084 3516 y Fp(b)2118 3501 y Fs(\()p Fq(\014)5 b Fs(\))27 b Fm(\024)e(\000)2448 3439 y Fs(1)p 2448 3480 V 2448 3563 a(2)75 3687 y(for)30 b(all)h(0)26 b Fm(\024)f Fq(\014)31 b Fm(\024)25 b Fq(d)20 b Fm(\000)g Fq(\022)s Fs(.)40 b(It)30 b(turns)f(out)i(that)306 3933 y Fq(J)356 3948 y Fp(a;\014)s Fo(;)p Fp(\033)516 3957 y Fh(0)551 3948 y Fp(;\033)642 3933 y Fm(\024)25 b Fq(\024)p Fs(!)c(sup)830 4012 y Fp(T)10 b Fy(2)p Fd(C)1002 3872 y Fm(j)p Fq(T)j Fm(j)1118 3839 y Fp(s)p 1002 3912 154 4 v 1032 3996 a Fq(a)1080 3969 y Fp(\024)1165 3933 y Fs(\(1)21 b(+)f Fq(a)p Fs(\))1440 3896 y Fp(m)1522 3933 y Fs(exp)1676 3805 y Fn(\022)1743 3933 y Fm(\000)1824 3872 y Fs(1)p 1824 3912 46 4 v 1824 3996 a(2)1879 3933 y(\()p Fm(j)p Fq(T)13 b Fm(j)21 b Fs(+)f Fq(a)p Fs(\))2225 3896 y Fp(q)2263 3805 y Fn(\023)2346 3933 y Fs(exp)15 b(\()p Fq(\033)s(r)2631 3948 y Fp(b)2666 3933 y Fs(\()p Fq(\014)5 b Fs(\))15 b Fm(f)q Fs(\()p Fm(j)p Fq(T)e Fm(j)21 b Fs(+)f Fq(a)p Fs(\))3199 3896 y Fp(q)3257 3933 y Fm(\000)g(j)p Fq(T)13 b Fm(j)3464 3896 y Fp(q)3503 3933 y Fm(g)p Fs(\))j Fq(:)75 4185 y Fs(With)31 b(the)f(same)h(c)m(hoice)h(for)f(the)f (radius)g Fq(a)g Fs(as)h(b)s(efore)1509 4386 y Fq(a)25 b Fs(=)g(\()p Fm(j)p Fq(T)13 b Fm(j)1829 4349 y Fp(q)1888 4386 y Fs(+)20 b(1\))2060 4344 y Fo(1)p Fp(=q)2188 4386 y Fm(\000)g(j)p Fq(T)13 b Fm(j)p Fq(;)75 4588 y Fs(w)m(e)31 b(obtain)g(that)245 4886 y Fq(J)295 4901 y Fp(a;\014)s Fo(;)p Fp(\033)455 4910 y Fh(0)490 4901 y Fp(;\033)582 4886 y Fm(\024)25 b Fq(\024)p Fs(!)15 b(exp)h(\()p Fq(\033)s(r)1056 4901 y Fp(b)1091 4886 y Fs(\()p Fq(\014)5 b Fs(\)\))16 b(exp)1422 4757 y Fn(\022)1489 4886 y Fm(\000)1570 4824 y Fs(1)p 1570 4865 V 1570 4948 a(2)1625 4757 y Fn(\023)1712 4886 y Fs(sup)1707 4964 y Fp(T)10 b Fy(2)p Fd(C)1868 4886 y Fm(j)p Fq(T)j Fm(j)1984 4848 y Fp(s)2031 4690 y Fn(\020)2086 4791 y Fs(1)20 b(+)g(\()q Fm(j)p Fq(T)13 b Fm(j)2394 4758 y Fp(q)2452 4791 y Fs(+)20 b(1\))2624 4749 y Fo(1)p Fp(=q)2753 4791 y Fm(\000)g(j)p Fq(T)13 b Fm(j)2960 4690 y Fn(\021)3014 4713 y Fp(m)p 2031 4865 1050 4 v 2121 4882 a Fn(\020)2175 4982 y Fs(\()p Fm(j)p Fq(T)g Fm(j)2326 4956 y Fp(q)2385 4982 y Fs(+)20 b(1\))2556 4941 y Fo(1)p Fp(=q)2685 4982 y Fm(\000)g(j)p Fq(T)13 b Fm(j)2892 4882 y Fn(\021)2947 4904 y Fp(\024)3106 4886 y Fs(exp)3260 4757 y Fn(\022)3327 4886 y Fm(\000)3408 4824 y Fs(1)p 3408 4865 46 4 v 3408 4948 a(2)3463 4886 y Fm(j)p Fq(T)g Fm(j)3579 4848 y Fp(q)3617 4757 y Fn(\023)75 5201 y Fs(for)30 b(all)h(0)26 b Fm(\024)f Fq(\014)31 b Fm(\024)25 b Fq(d)20 b Fm(\000)g Fq(\022)s Fs(.)75 5314 y(On)30 b(the)g(other)h(hand,)e(there)i(exists)g(a)g(constan)m(t)g Fq(c)1838 5328 y Fo(1)1903 5314 y Fq(>)25 b Fs(0,)31 b(suc)m(h)f(that)881 5630 y(sup)876 5709 y Fp(T)10 b Fy(2)p Fd(C)1037 5630 y Fm(j)p Fq(T)j Fm(j)1153 5592 y Fp(s)1200 5435 y Fn(\020)1255 5535 y Fs(1)20 b(+)g(\()q Fm(j)p Fq(T)13 b Fm(j)1563 5502 y Fp(q)1621 5535 y Fs(+)20 b(1\))1793 5494 y Fo(1)p Fp(=q)1922 5535 y Fm(\000)g(j)p Fq(T)13 b Fm(j)2129 5435 y Fn(\021)2183 5457 y Fp(m)p 1200 5609 1050 4 v 1290 5626 a Fn(\020)1344 5727 y Fs(\()p Fm(j)p Fq(T)g Fm(j)1495 5701 y Fp(q)1554 5727 y Fs(+)20 b(1\))1725 5685 y Fo(1)p Fp(=q)1854 5727 y Fm(\000)g(j)p Fq(T)13 b Fm(j)2061 5626 y Fn(\021)2116 5649 y Fp(\024)2275 5630 y Fs(exp)2429 5502 y Fn(\022)2496 5630 y Fm(\000)2577 5568 y Fs(1)p 2577 5609 46 4 v 2577 5692 a(2)2632 5630 y Fm(j)p Fq(T)g Fm(j)2748 5592 y Fp(q)2786 5502 y Fn(\023)2878 5630 y Fm(\024)25 b Fq(c)3013 5644 y Fo(1)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 3764 100 a Fs(12)75 399 y(and)30 b(therefore,)h(there)f(exists)h(a)g(constan)m(t)h Fq(c)1624 413 y Fo(2)1689 399 y Fq(>)25 b Fs(0,)31 b(indep)s(enden)m(t)e(of)i Fq(\014)5 b Fs(,)30 b(suc)m(h)g(that)1709 598 y Fq(J)1759 613 y Fp(a;\014)s Fo(;)p Fp(\033)1919 622 y Fh(0)1954 613 y Fp(;\033)2046 598 y Fm(\024)25 b Fq(c)2181 612 y Fo(2)75 798 y Fs(for)30 b(all)h(0)26 b Fm(\024)f Fq(\014)31 b Fm(\024)25 b Fq(d)20 b Fm(\000)g Fq(\022)s Fs(.)40 b(This)29 b(pro)m(v)m(es)i(the)g(lemma)g(3.)216 959 y(No)m(w,)h(w)m(e)g (return)e(to)i(the)f(pro)s(of)f(of)h(prop)s(osition)g(4.)43 b(Using)32 b(Lemma)f(3,)h(w)m(e)f(obtain)h(for)e(\(8\))j(that)e(there) 75 1072 y(exists)g(a)g(constan)m(t)g Fq(c)807 1086 y Fo(1)872 1072 y Fq(>)25 b Fs(0,)31 b(indep)s(enden)m(t)f(of)g Fq(\014)5 b Fs(,)31 b(suc)m(h)f(that)949 1272 y Fm(jj)p Fs(\()q Fq(T)1101 1234 y Fp(s)1137 1272 y Fq(@)1190 1234 y Fp(\024)1185 1294 y(T)1241 1272 y Fq(w)1306 1287 y Fp(n)p Fo(+)p Fp(\027;\014)s Fo(+)p Fp(\022)1595 1272 y Fs(\))16 b(\()p Fq(T)d Fs(\))p Fm(jj)1833 1304 y Fp(\014)s Fo(;)p Fp(\033)1967 1272 y Fm(\024)25 b Fq(c)2102 1286 y Fo(1)2157 1272 y Fm(jj)q Fq(w)2273 1287 y Fp(n)p Fo(+)p Fp(\027;\014)s Fo(+)p Fp(\022)2562 1272 y Fs(\()p Fq(T)13 b Fs(\))p Fm(j)q(j)2749 1304 y Fp(\014)s Fo(+)p Fp(\022)r Fo(;)p Fp(\033)2942 1313 y Fh(0)75 1480 y Fs(for)30 b(all)h Fq(n)25 b Fm(\025)g Fs(0)31 b(and)f(for)g(all)h(0)26 b Fm(\024)f Fq(\014)30 b Fm(\024)25 b Fq(d)c Fm(\000)f Fq(\022)s Fs(,)29 b(and)h(hence)h(\(7\))g(b)s(ecomes)248 1598 y Fn(\014)248 1653 y(\014)248 1707 y(\014)278 1598 y(\014)278 1653 y(\014)278 1707 y(\014)308 1602 y(\020)363 1703 y Fq(T)429 1665 y Fp(s)465 1703 y Fq(@)518 1665 y Fp(\027)513 1725 y(t)562 1703 y Fq(@)615 1665 y Fp(\024)610 1725 y(T)665 1703 y Fq(@)718 1665 y Fp(\022)713 1725 y(z)757 1703 y Fq(W)856 1602 y Fn(\021)925 1703 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))1220 1598 y Fn(\014)1220 1653 y(\014)1220 1707 y(\014)1252 1598 y(\014)1252 1653 y(\014)1252 1707 y(\014)1282 1766 y Fp(\016)1313 1775 y Fh(1)1348 1766 y Fp(;\016)1399 1775 y Fh(2)1433 1766 y Fo(;)p Fp(\033)486 2043 y Fm(\024)25 b Fq(c)621 2057 y Fo(1)661 2043 y Fq(\016)704 2004 y Fy(\000)p Fp(\027)701 2069 y Fo(1)803 2043 y Fq(\016)846 2004 y Fy(\000)p Fp(\022)843 2069 y Fo(2)960 1929 y Fp(d)p Fy(\000)p Fp(\022)957 1956 y Fn(X)956 2154 y Fp(\014)s Fo(=0)1105 1956 y Fn(X)1104 2152 y Fp(n)p Fy(\025)p Fo(0)1262 1981 y Fs(\()p Fq(n)c Fs(+)e Fq(\027)26 b Fs(+)20 b Fq(\014)26 b Fs(+)20 b Fq(\022)s Fs(\)!)p 1262 2022 636 4 v 1421 2105 a(\()p Fq(n)h Fs(+)e Fq(\014)5 b Fs(\)!)1923 2043 y Fm(j)q(j)p Fq(w)2039 2058 y Fp(n)p Fo(+)p Fp(\027;\014)s Fo(+)p Fp(\022)2328 2043 y Fs(\()p Fq(T)13 b Fs(\))p Fm(j)q(j)2515 2075 y Fp(\014)s Fo(+)p Fp(\022)r Fo(;)p Fp(\033)2708 2084 y Fh(0)2907 1981 y Fq(\016)2950 1943 y Fp(n)p Fo(+)p Fp(\027)2947 2007 y Fo(1)3092 1981 y Fq(\016)3135 1937 y Fp(\014)s Fo(+)p Fp(\022)3132 2007 y Fo(2)p 2772 2022 V 2772 2105 a Fs(\()p Fq(n)20 b Fs(+)g Fq(\027)26 b Fs(+)20 b Fq(\014)25 b Fs(+)20 b Fq(\022)s Fs(\)!)3418 2043 y Fq(:)75 2304 y Fs(Using)31 b(the)f(estimates)i(\(5\))1373 2403 y(\()p Fq(n)21 b Fs(+)e Fq(\027)26 b Fs(+)20 b Fq(\014)26 b Fs(+)20 b Fq(\022)s Fs(\)!)p 1373 2444 V 1532 2527 a(\()p Fq(n)h Fs(+)e Fq(\014)5 b Fs(\)!)2045 2465 y Fm(\030)25 b Fs(\()p Fq(n)20 b Fs(+)g Fq(\014)5 b Fs(\))2433 2427 y Fp(\027)t Fo(+)p Fp(\022)75 2673 y Fs(as)24 b Fq(n)g Fs(tends)f(to)i(in\014nit)m(y)e(and)h(for)f(0)j Fm(\024)f Fq(\014)31 b Fm(\024)25 b Fq(d)7 b Fm(\000)g Fq(\022)s Fs(,)25 b(there)f(exist)h(t)m(w)m(o)g(constan)m(ts)g Fq(c)2817 2687 y Fo(2)2882 2673 y Fm(\025)g Fs(1,)h Fq(c)3113 2687 y Fo(3)3178 2673 y Fq(>)f Fs(0,)h(indep)s(enden)m(t)75 2786 y(of)31 b Fq(n)e Fs(and)h Fq(\014)5 b Fs(,)31 b(suc)m(h)f(that) 1457 2880 y(\()p Fq(n)20 b Fs(+)g Fq(\027)26 b Fs(+)20 b Fq(\014)25 b Fs(+)20 b Fq(\022)s Fs(\)!)p 1457 2921 V 1616 3004 a(\()p Fq(n)g Fs(+)g Fq(\014)5 b Fs(\)!)2128 2942 y Fm(\024)25 b Fq(c)2263 2956 y Fo(3)2303 2942 y Fq(c)2342 2904 y Fp(n)p Fo(+)p Fp(\027)2342 2968 y Fo(2)75 3150 y Fs(for)30 b(all)h Fq(n)25 b Fm(\025)g Fs(0)31 b(and)f(for)g(all)h(0)26 b Fm(\024)f Fq(\014)30 b Fm(\024)25 b Fq(d)c Fm(\000)f Fq(\022)s Fs(.)39 b(W)-8 b(e)32 b(get)g(that)f (there)f(exists)h(a)g(constan)m(t)h Fq(c)3118 3164 y Fo(4)3183 3150 y Fq(>)25 b Fs(0)30 b(suc)m(h)g(that)445 3327 y Fn(\014)445 3382 y(\014)445 3436 y(\014)475 3327 y(\014)475 3382 y(\014)475 3436 y(\014)505 3331 y(\020)560 3432 y Fq(T)626 3394 y Fp(s)662 3432 y Fq(@)715 3394 y Fp(\027)710 3454 y(t)759 3432 y Fq(@)812 3394 y Fp(\024)807 3454 y(T)862 3432 y Fq(@)915 3394 y Fp(\022)910 3454 y(z)954 3432 y Fq(W)1053 3331 y Fn(\021)1122 3432 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))1417 3327 y Fn(\014)1417 3382 y(\014)1417 3436 y(\014)1449 3327 y(\014)1449 3382 y(\014)1449 3436 y(\014)1479 3495 y Fp(\016)1510 3504 y Fh(1)1545 3495 y Fp(;\016)1596 3504 y Fh(2)1630 3495 y Fo(;)p Fp(\033)1722 3432 y Fm(\024)25 b Fq(c)1857 3446 y Fo(4)1897 3432 y Fq(\016)1940 3393 y Fy(\000)p Fp(\027)1937 3458 y Fo(1)2039 3432 y Fq(\016)2082 3393 y Fy(\000)p Fp(\022)2079 3458 y Fo(2)2240 3318 y Fp(d)2193 3345 y Fn(X)2192 3543 y Fp(\014)s Fo(=0)2341 3345 y Fn(X)2340 3541 y Fp(n)p Fy(\025)p Fo(0)2488 3432 y Fm(jj)q Fq(w)2604 3447 y Fp(n;\014)2713 3432 y Fs(\()p Fq(T)13 b Fs(\))p Fm(j)q(j)2900 3464 y Fp(\014)s Fo(;)p Fp(\033)3003 3473 y Fh(0)3067 3370 y Fs(\()p Fq(c)3141 3384 y Fo(2)3181 3370 y Fq(\016)3221 3384 y Fo(1)3261 3370 y Fs(\))3296 3328 y Fp(n)3359 3370 y Fq(\016)3402 3326 y Fp(\014)3399 3396 y Fo(2)p 3067 3411 383 4 v 3099 3494 a Fs(\()p Fq(n)21 b Fs(+)e Fq(\014)5 b Fs(\)!)3460 3432 y Fq(:)75 3728 y Fs(No)m(w,)31 b(c)m(ho)s(osing)h Fq(\016)717 3742 y Fo(1)782 3728 y Fq(>)25 b Fs(0)30 b(suc)m(h)g(that)h Fq(\016)1395 3742 y Fo(1)1460 3728 y Fm(\024)25 b Fq(\016)1596 3742 y Fo(1)p Fp(;)p Fo(0)1691 3728 y Fq(=c)1775 3742 y Fo(2)1815 3728 y Fs(,)31 b(w)m(e)g(conclude)f(that)694 3850 y Fn(\014)694 3905 y(\014)694 3959 y(\014)724 3850 y(\014)724 3905 y(\014)724 3959 y(\014)754 3854 y(\020)809 3955 y Fq(T)875 3917 y Fp(s)911 3955 y Fq(@)964 3917 y Fp(\027)959 3977 y(t)1008 3955 y Fq(@)1061 3917 y Fp(\024)1056 3977 y(T)1111 3955 y Fq(@)1164 3917 y Fp(\022)1159 3977 y(z)1203 3955 y Fq(W)1302 3854 y Fn(\021)1371 3955 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))1666 3850 y Fn(\014)1666 3905 y(\014)1666 3959 y(\014)1698 3850 y(\014)1698 3905 y(\014)1698 3959 y(\014)1728 4018 y Fp(\016)1759 4027 y Fh(1)1794 4018 y Fp(;\016)1845 4027 y Fh(2)1879 4018 y Fo(;)p Fp(\033)1971 3955 y Fm(\024)25 b Fq(c)2106 3969 y Fo(4)2146 3955 y Fq(\016)2189 3916 y Fy(\000)p Fp(\027)2186 3981 y Fo(1)2288 3955 y Fq(\016)2331 3916 y Fy(\000)p Fp(\022)2328 3981 y Fo(2)2441 3955 y Fm(jj)p Fq(W)13 b Fs(\()p Fq(t;)i(z)t(;)g(T)e Fs(\))p Fm(j)q(j)2937 3981 y Fp(\016)2968 3990 y Fh(1)p Fc(;)p Fh(0)3052 3981 y Fp(;\016)3103 3990 y Fh(2)3137 3981 y Fo(;)p Fp(\033)3197 3990 y Fh(0)75 4194 y Fs(for)30 b(all)h Fq(W)13 b Fs(\()p Fq(t;)i(z)t(;)g(T)e Fs(\))27 b Fm(2)d(G)900 4208 y Fp(q)954 4194 y Fs(\()p Fq(\016)1029 4208 y Fo(1)p Fp(;)p Fo(0)1124 4194 y Fq(;)15 b(\016)1204 4208 y Fo(2)1244 4194 y Fs(;)g Fq(\033)1336 4208 y Fo(0)1376 4194 y Fs(\).)41 b(This)29 b(pro)m(v)m(es)i(the)g(prop)s(osition)f(4.)75 4437 y Fg(2.3)112 b(A)37 b(Cauc)m(h)m(y)h(problem)h(in)e(the)h(w)m(eigh)m(ted) f(Banac)m(h)i(spaces.)75 4608 y Fs(In)30 b(the)g(next)h(t)m(w)m(o)g (de\014nitions,)g(w)m(e)f(in)m(tro)s(duce)h(some)g(linear)f(op)s (erators.)75 4792 y Fe(De\014nition)35 b(2)46 b Fr(L)-5 b(et)33 b Fm(M)g Fr(the)g(line)-5 b(ar)33 b(op)-5 b(er)g(ator)36 b(fr)-5 b(om)34 b Fm(A)p Fs(\()p Fl(C)p Fs(\))p Fm(f)p Fq(t;)15 b(z)t Fm(g)34 b Fr(into)f Fm(A)p Fs(\()p Fl(C)p Fs(\))p Fm(f)p Fq(t;)15 b(z)t Fm(g)34 b Fr(de\014ne)-5 b(d)34 b(as)75 5006 y Fs(\(9\))310 b Fm(M)15 b Fs(\()q Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\)\))27 b(=)e Fq(@)1238 4968 y Fp(S)1233 5028 y(z)1289 5006 y Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))21 b Fm(\000)1950 4919 y Fn(X)1768 5121 y Fi(q)p Fo(=\()p Fp(q)1925 5130 y Fh(0)1959 5121 y Fp(;q)2011 5130 y Fh(1)2045 5121 y Fp(;q)2097 5130 y Fh(2)2131 5121 y Fo(\))p Fy(2Q)2278 5006 y Fq(B)2347 5020 y Fi(q)2394 5006 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))p Fq(@)2742 4961 y Fp(q)2774 4970 y Fh(0)2737 5030 y Fp(t)2814 5006 y Fq(@)2867 4961 y Fp(q)2899 4970 y Fh(1)2862 5034 y Fp(T)2938 5006 y Fq(@)2991 4968 y Fp(q)3023 4977 y Fh(2)2986 5028 y Fp(z)3061 5006 y Fq(U)d Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))75 5310 y Fr(wher)-5 b(e)34 b Fm(Q)p Fr(,)e Fq(S)38 b Fr(and)33 b Fq(B)805 5324 y Fi(q)852 5310 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))34 b Fr(satisfy)g(the)f(fol)5 b(lowing)33 b(assumptions.)44 5518 y Fe(\(A1\))45 b Fq(S)36 b Fs(is)30 b(a)h(p)s(ositiv)m(e)g(in)m (teger,)h Fm(Q)f Fs(is)f(a)h(\014nite)f(subset)g(of)g Fl(N)2175 5485 y Fo(3)2245 5518 y Fs(and)g(w)m(e)g(denote)h(b)m(y)1689 5718 y Fe(q)26 b Fs(=)f(\()p Fq(q)1942 5732 y Fo(0)1981 5718 y Fq(;)15 b(q)2062 5732 y Fo(1)2101 5718 y Fq(;)g(q)2182 5732 y Fo(2)2222 5718 y Fs(\))25 b Fm(2)g(Q)p Fq(:)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 3764 100 a Fs(13)44 399 y Fe(\(A2\))45 b Fs(F)-8 b(or)31 b(all)h Fe(q)25 b Fm(2)g(Q)p Fs(,)31 b Fq(B)957 413 y Fi(q)1004 399 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))31 b(is)g(p)s(olynomial)g(in)f Fq(T)1544 692 y(B)1613 706 y Fi(q)1660 692 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))27 b(=)2078 570 y Fp(\033)r Fo(\()p Fi(q)p Fo(\))2083 606 y Fn(X)2087 801 y Fp(j)t Fo(=0)2234 692 y Fq(b)2273 648 y Fi(q)2273 719 y Fp(j)2320 692 y Fs(\()p Fq(t;)15 b(z)t Fs(\))p Fq(T)2575 655 y Fp(j)302 977 y Fs(where)31 b Fq(\033)s Fs(\()p Fe(q)p Fs(\))h(is)g(the)f(degree)h(of)g Fq(B)1486 991 y Fi(q)1533 977 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))32 b(with)g(resp)s(ect)f(to)h(the)f(v)-5 b(ariable)33 b Fq(T)44 b Fs(and)30 b(the)i(co)s(e\016cien)m(ts)302 1090 y Fq(b)341 1046 y Fi(q)341 1118 y Fp(j)388 1090 y Fs(\()p Fq(t;)15 b(z)t Fs(\))32 b(are)f(holomorphic)f(functions)g (near)g(the)h(origin)g(in)f Fl(C)2462 1057 y Fo(2)2501 1090 y Fs(.)44 1273 y Fe(\(A3\))45 b Fs(F)-8 b(or)31 b(all)h Fe(q)25 b Fs(=)g(\()p Fq(q)844 1287 y Fo(0)883 1273 y Fq(;)15 b(q)964 1287 y Fo(1)1004 1273 y Fq(;)g(q)1085 1287 y Fo(2)1124 1273 y Fs(\))26 b Fm(2)f(Q)p Fs(,)1412 1450 y Fq(b)15 b Fs(\()q Fq(\033)s Fs(\()p Fe(q)p Fs(\))21 b(+)f Fq(q)1835 1464 y Fo(1)1874 1450 y Fs(\()p Fq(q)j Fm(\000)d Fs(1\)\))p 1412 1491 769 4 v 1774 1574 a Fq(q)2210 1512 y Fs(+)g Fq(q)2342 1526 y Fo(0)2402 1512 y Fs(+)f Fq(q)2533 1526 y Fo(2)2598 1512 y Fq(<)25 b(S)302 1746 y Fs(where)30 b Fq(q)e Fm(\025)d Fs(1)31 b(and)f Fq(b)25 b(>)g Fs(1)31 b(are)f(p)s(ositiv)m(e)i(n)m(um)m(b)s(ers)d(de\014ned)g (from)h(our)g(Banac)m(h)h(space)g Fm(G)3385 1760 y Fp(q)3438 1746 y Fs(\()q Fq(\016)3514 1760 y Fo(1)3554 1746 y Fq(;)15 b(\016)3634 1760 y Fo(2)3674 1746 y Fs(;)g Fq(\033)s Fs(\))q(.)75 1945 y Fe(De\014nition)35 b(3)46 b Fr(L)-5 b(et)33 b Fm(N)45 b Fr(b)-5 b(e)33 b(the)g(line)-5 b(ar)34 b(op)-5 b(er)g(ator)35 b(fr)-5 b(om)34 b Fm(A)p Fs(\()p Fl(C)p Fs(\))p Fm(f)p Fq(t;)15 b(z)t Fm(g)34 b Fr(into)f Fm(A)p Fs(\()p Fl(C)p Fs(\))p Fm(f)p Fq(t;)15 b(z)t Fm(g)34 b Fr(de\014ne)-5 b(d)34 b(as)75 2149 y Fs(\(10\))690 b Fm(N)28 b Fs(\()q Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\)\))27 b(=)1597 2063 y Fn(X)1589 2260 y Fi(q)p Fy(2Q)1751 2149 y Fq(B)1820 2163 y Fi(q)1867 2149 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))p Fq(@)2215 2105 y Fp(q)2247 2114 y Fh(0)2210 2173 y Fp(t)2287 2149 y Fq(@)2340 2105 y Fp(q)2372 2114 y Fh(1)2335 2178 y Fp(T)2411 2149 y Fq(@)2464 2112 y Fp(q)2496 2121 y Fh(2)2530 2112 y Fy(\000)p Fp(S)2459 2172 y(z)2636 2149 y Fq(U)d Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))75 2436 y Fr(wher)-5 b(e)34 b Fm(Q)p Fr(,)e Fq(S)38 b Fr(and)33 b Fq(B)805 2450 y Fi(q)852 2436 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))34 b Fr(satisfy)g(the)f(assumptions)i Fs(\()p Fe(A1)p Fs(\))q Fr(,)d Fs(\()p Fe(A2)p Fs(\))i Fr(and)f Fs(\()p Fe(A3)p Fs(\))q Fr(.)75 2635 y Fs(Using)j(the)g (de\014nitions)f(\(9\))i(and)e(\(10\),)k(w)m(e)e(obtain)f(the)g(follo)m (wing)h(relation)g(b)s(et)m(w)m(een)f(the)g(op)s(erators)g Fm(M)75 2748 y Fs(and)30 b Fm(N)75 2939 y Fs(\(11\))1351 b Fm(M)20 b(\016)h Fq(@)1834 2901 y Fy(\000)p Fp(S)1829 2961 y(z)1965 2939 y Fs(=)k(Id)20 b Fm(\000)g(N)75 3129 y Fs(where)30 b(Id)g(is)g(the)g(iden)m(tit)m(y)i(op)s(erator)f(from)f Fm(A)p Fs(\()p Fl(C)p Fs(\))p Fm(f)p Fq(t;)15 b(z)t Fm(g)32 b Fs(in)m(to)f(itself.)75 3287 y(In)23 b(the)i(next)f(prop)s(osition,)h (w)m(e)g(sho)m(w)f(that)h Fm(N)37 b Fs(is)24 b(a)h(linear)f(b)s(ounded) e(op)s(erator)j(on)f(the)g(constructed)g(Banac)m(h)75 3400 y(spaces)31 b(with)f(small)h(norm.)75 3577 y Fe(Prop)s(osition)36 b(5)46 b Fr(L)-5 b(et)27 b Fm(N)40 b Fr(b)-5 b(e)26 b(the)h(op)-5 b(er)g(ator)30 b(de\014ne)-5 b(d)28 b(by)f(\(10\))h(and)f(which)h (satis\014es)g(the)f(assumptions)i Fs(\()p Fe(A1)p Fs(\))q Fr(,)75 3690 y Fs(\()p Fe(A2)p Fs(\))36 b Fr(and)f Fs(\()p Fe(A3)p Fs(\))q Fr(.)48 b(Then,)35 b(ther)-5 b(e)36 b(exists)f(a)g(r)-5 b(e)g(al)36 b(numb)-5 b(er)35 b Fq(\016)2137 3704 y Fo(2)2205 3690 y Fq(>)29 b Fs(0)p Fr(,)35 b(smal)5 b(l)36 b(enough)f(\(which)h (dep)-5 b(ends)36 b(on)f(the)75 3803 y(c)-5 b(o)g(e\016cients)36 b Fq(B)610 3817 y Fi(q)657 3803 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))36 b Fr(for)f Fe(q)29 b Fs(=)g(\()p Fq(q)1395 3817 y Fo(0)1435 3803 y Fq(;)15 b(q)1516 3817 y Fo(1)1555 3803 y Fq(;)g(q)1636 3817 y Fo(2)1676 3803 y Fs(\))29 b Fm(2)g(Q)p Fr(\),)35 b(such)g(that)h Fm(N)48 b Fr(is)35 b(a)g(b)-5 b(ounde)g(d)36 b(line)-5 b(ar)36 b(op)-5 b(er)g(ator)38 b(fr)-5 b(om)75 3841 y Fn(\020)129 3942 y Fm(G)183 3956 y Fp(q)237 3942 y Fs(\()p Fq(\016)312 3956 y Fo(1)352 3942 y Fq(;)15 b(\016)432 3956 y Fo(2)472 3942 y Fs(;)g Fq(\033)s Fs(\))h Fq(;)f Fm(j)q(j)p Fq(:)p Fm(j)q(j)785 3969 y Fp(\016)816 3978 y Fh(1)851 3969 y Fp(;\016)902 3978 y Fh(2)936 3969 y Fo(;)p Fp(\033)1002 3841 y Fn(\021)1093 3942 y Fr(into)36 b(itself.)52 b(Mor)-5 b(e)g(over,)38 b(ther)-5 b(e)37 b(exists)f(a)g(c)-5 b(onstant)38 b Fq(C)2979 3956 y Fo(10)3054 3942 y Fr(,)e Fs(0)c Fq(<)f(C)3362 3956 y Fo(10)3468 3942 y Fq(<)g Fs(1)p Fr(,)37 b(such)75 4072 y(that)1053 4185 y Fm(jjN)28 b Fs(\()q Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\)\))q Fm(j)q(j)1696 4212 y Fp(\016)1727 4221 y Fh(1)1762 4212 y Fp(;\016)1813 4221 y Fh(2)1847 4212 y Fo(;)p Fp(\033)1939 4185 y Fm(\024)25 b Fq(C)2100 4199 y Fo(10)2190 4185 y Fm(jj)p Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))p Fm(j)r(j)2659 4212 y Fp(\016)2690 4221 y Fh(1)2725 4212 y Fp(;\016)2776 4221 y Fh(2)2810 4212 y Fo(;)p Fp(\033)75 4345 y Fr(for)33 b(al)5 b(l)33 b Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))27 b Fm(2)e(G)884 4359 y Fp(q)937 4345 y Fs(\()p Fq(\016)1012 4359 y Fo(1)1052 4345 y Fq(;)15 b(\016)1132 4359 y Fo(2)1172 4345 y Fs(;)g Fq(\033)s Fs(\))q Fr(.)75 4567 y Fe(Pro)s(of)56 b Fs(By)31 b(de\014nition)f(\(10\))i(of)f(the)f (op)s(erator)h Fm(N)43 b Fs(and)30 b(the)g(assumption)g(\()p Fe(A2)p Fs(\))q(,)h(w)m(e)g(get)535 4857 y Fm(j)q(jN)d Fs(\()p Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\)\))r Fm(jj)1179 4884 y Fp(\016)1210 4893 y Fh(1)1244 4884 y Fp(;\016)1295 4893 y Fh(2)1330 4884 y Fo(;)p Fp(\033)1422 4857 y Fm(\024)1526 4771 y Fn(X)1518 4967 y Fi(q)p Fy(2Q)1680 4735 y Fp(\033)r Fo(\()p Fi(q)p Fo(\))1685 4771 y Fn(X)1689 4966 y Fp(j)t Fo(=0)1836 4752 y Fn(\014)1836 4807 y(\014)1836 4861 y(\014)1866 4752 y(\014)1866 4807 y(\014)1866 4861 y(\014)1897 4857 y Fq(b)1936 4813 y Fi(q)1936 4884 y Fp(j)1983 4857 y Fs(\()p Fq(t;)i(z)t Fs(\))p Fq(T)2238 4819 y Fp(j)2275 4857 y Fq(@)2328 4813 y Fp(q)2360 4822 y Fh(0)2323 4881 y Fp(t)2399 4857 y Fq(@)2452 4813 y Fp(q)2484 4822 y Fh(1)2447 4885 y Fp(T)2522 4857 y Fq(@)2575 4819 y Fp(q)2607 4828 y Fh(2)2642 4819 y Fy(\000)p Fp(S)2570 4879 y(z)2747 4857 y Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))3114 4752 y Fn(\014)3114 4807 y(\014)3114 4861 y(\014)3146 4752 y(\014)3146 4807 y(\014)3146 4861 y(\014)3176 4920 y Fp(\016)3207 4929 y Fh(1)3242 4920 y Fp(;\016)3293 4929 y Fh(2)3328 4920 y Fo(;)p Fp(\033)75 5149 y Fs(for)26 b(all)g Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))27 b Fm(2)e(G)865 5163 y Fp(q)918 5149 y Fs(\()p Fq(\016)993 5163 y Fo(1)1033 5149 y Fq(;)15 b(\016)1113 5163 y Fo(2)1153 5149 y Fs(;)g Fq(\033)s Fs(\))q(.)40 b(F)-8 b(rom)26 b(the)g(fact)h(that)f Fq(b)2141 5105 y Fi(q)2141 5176 y Fp(j)2188 5149 y Fs(\()p Fq(t;)15 b(z)t Fs(\),)28 b(for)e Fq(j)31 b Fs(=)25 b(0,)i Fq(:)15 b(:)g(:)q Fs(,)27 b Fq(\033)s Fs(\()p Fe(q)p Fs(\),)h(are)e(holomorphic)75 5273 y(functions)d(near)h(the)g(origin)g(in)f Fl(C)1222 5240 y Fo(2)1261 5273 y Fs(,)i(w)m(e)f(deduce)g(from)f(Prop)s(osition)g (3)i(that)f(there)g(exists)g(a)g(constan)m(t)h Fq(c)3623 5287 y Fo(1)3688 5273 y Fq(>)g Fs(0,)75 5386 y(suc)m(h)30 b(that)403 5653 y Fm(jjN)e Fs(\()q Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\)\))q Fm(j)q(j)1046 5680 y Fp(\016)1077 5689 y Fh(1)1112 5680 y Fp(;\016)1163 5689 y Fh(2)1197 5680 y Fo(;)p Fp(\033)1289 5653 y Fm(\024)25 b Fq(c)1424 5667 y Fo(1)1487 5567 y Fn(X)1479 5764 y Fi(q)p Fy(2Q)1642 5531 y Fp(\033)r Fo(\()p Fi(q)p Fo(\))1646 5567 y Fn(X)1651 5762 y Fp(j)t Fo(=0)1797 5549 y Fn(\014)1797 5603 y(\014)1797 5658 y(\014)1828 5653 y Fq(b)1867 5609 y Fi(q)1867 5681 y Fp(j)1914 5549 y Fn(\014)1914 5603 y(\014)1914 5658 y(\014)1959 5653 y Fs(\()p Fq(\016)2034 5667 y Fo(1)2074 5653 y Fq(;)15 b(\016)2154 5667 y Fo(2)2195 5653 y Fs(\))2245 5576 y Fn(\014)2245 5631 y(\014)2275 5576 y(\014)2275 5631 y(\014)2306 5653 y Fq(T)2372 5616 y Fp(j)2408 5653 y Fq(@)2461 5609 y Fp(q)2493 5618 y Fh(0)2456 5677 y Fp(t)2531 5653 y Fq(@)2584 5609 y Fp(q)2616 5618 y Fh(1)2579 5682 y Fp(T)2655 5653 y Fq(@)2708 5616 y Fp(q)2740 5625 y Fh(2)2774 5616 y Fy(\000)p Fp(S)2703 5676 y(z)2880 5653 y Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))3247 5576 y Fn(\014)3247 5631 y(\014)3279 5576 y(\014)3279 5631 y(\014)3309 5689 y Fp(\016)3340 5698 y Fh(1)3375 5689 y Fp(;\016)3426 5698 y Fh(2)3460 5689 y Fo(;)p Fp(\033)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 3764 100 a Fs(14)75 399 y(for)34 b(all)i Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))33 b Fm(2)f(G)896 413 y Fp(q)950 399 y Fs(\()p Fq(\016)1025 413 y Fo(1)1065 399 y Fq(;)15 b(\016)1145 413 y Fo(2)1185 399 y Fs(;)g Fq(\033)s Fs(\))q(.)53 b(Under)34 b(the)h(assumption)f(\() p Fe(A3)p Fs(\))q(,)i(w)m(e)f(can)g(apply)f(prop)s(osition)g(2)h(and)75 511 y(w)m(e)c(obtain)g(the)f(existence)i(of)f(a)f(constan)m(t)i Fq(c)1620 525 y Fo(2)1685 511 y Fq(>)25 b Fs(0,)31 b(suc)m(h)f(that)341 797 y Fm(jj)q(N)e Fs(\()p Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\)\))r Fm(jj)984 824 y Fp(\016)1015 833 y Fh(1)1050 824 y Fp(;\016)1101 833 y Fh(2)1136 824 y Fo(;)p Fp(\033)1227 797 y Fm(\024)25 b Fq(c)1362 811 y Fo(2)1417 615 y Fn(0)1417 778 y(@)1505 711 y(X)1497 908 y Fi(q)p Fy(2Q)1659 615 y Fn(0)1659 778 y(@)1739 675 y Fp(\033)r Fo(\()p Fi(q)p Fo(\))1744 711 y Fn(X)1748 907 y Fp(j)t Fo(=0)1895 693 y Fn(\014)1895 747 y(\014)1895 802 y(\014)1925 797 y Fq(b)1964 753 y Fi(q)1964 825 y Fp(j)2011 693 y Fn(\014)2011 747 y(\014)2011 802 y(\014)2057 797 y Fs(\()p Fq(\016)2132 811 y Fo(1)2172 797 y Fq(;)15 b(\016)2252 811 y Fo(2)2292 797 y Fs(\))2327 615 y Fn(1)2327 778 y(A)2422 797 y Fq(\016)2465 753 y Fy(\000)p Fp(q)2552 762 y Fh(0)2462 824 y Fo(1)2591 797 y Fq(\016)2634 753 y Fp(S)t Fy(\000)p Fp(q)2768 762 y Fh(2)2631 824 y Fo(2)2807 615 y Fn(1)2807 778 y(A)2901 797 y Fm(j)q(j)p Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))p Fm(j)q(j)3371 824 y Fp(\016)3402 833 y Fh(1)3437 824 y Fp(;\016)3488 833 y Fh(2)3522 824 y Fo(;)p Fp(\033)75 1089 y Fs(for)30 b(all)h Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))27 b Fm(2)e(G)874 1103 y Fp(q)927 1089 y Fs(\()p Fq(\016)1002 1103 y Fo(1)1042 1089 y Fq(;)15 b(\016)1122 1103 y Fo(2)1163 1089 y Fs(;)g Fq(\033)s Fs(\))q(.)40 b(No)m(w,)32 b(with)e Fq(\016)1839 1103 y Fo(2)1904 1089 y Fq(>)25 b Fs(0)30 b(small)h(enough,)g(w)m(e)g(ha)m(v)m(e)958 1384 y(0)26 b Fq(<)f(c)1164 1398 y Fo(2)1227 1297 y Fn(X)1219 1494 y Fi(q)p Fy(2Q)1381 1201 y Fn(0)1381 1364 y(@)1461 1261 y Fp(\033)r Fo(\()p Fi(q)p Fo(\))1466 1297 y Fn(X)1470 1493 y Fp(j)t Fo(=0)1617 1279 y Fn(\014)1617 1334 y(\014)1617 1388 y(\014)1647 1384 y Fq(b)1686 1339 y Fi(q)1686 1411 y Fp(j)1733 1279 y Fn(\014)1733 1334 y(\014)1733 1388 y(\014)1779 1384 y Fs(\()p Fq(\016)1854 1398 y Fo(1)1894 1384 y Fq(;)15 b(\016)1974 1398 y Fo(2)2014 1384 y Fs(\))2049 1201 y Fn(1)2049 1364 y(A)2144 1384 y Fq(\016)2187 1339 y Fy(\000)p Fp(q)2274 1348 y Fh(0)2184 1410 y Fo(1)2313 1384 y Fq(\016)2356 1339 y Fp(S)t Fy(\000)p Fp(q)2490 1348 y Fh(2)2353 1410 y Fo(2)2554 1384 y Fs(=:)25 b Fq(C)2740 1398 y Fo(6)2805 1384 y Fq(<)g Fs(1)p Fq(:)75 1670 y Fs(This)30 b(pro)m(v)m(es)h(the)g(prop)s(osition.)41 b(As)30 b(a)h(consequence)g(of)g(the)g(prop)s(osition)f(5,)h(w)m(e)g (deduce)f(the)h(in)m(v)m(ertibilit)m(y)75 1783 y(of)g(the)f(op)s (erator)h Fm(M)20 b(\016)h Fq(@)950 1750 y Fy(\000)p Fp(S)945 1805 y(z)1056 1783 y Fs(.)75 1956 y Fe(Prop)s(osition)36 b(6)46 b Fr(L)-5 b(et)38 b Fm(M)g Fr(the)g(op)-5 b(er)g(ator)40 b(de\014ne)-5 b(d)39 b(by)f(\(9\))g(and)h(which)f(satis\014es)h(the)f (assumptions)i Fs(\()p Fe(A1)p Fs(\))q Fr(,)75 2068 y Fs(\()p Fe(A2)p Fs(\))c Fr(and)f Fs(\()p Fe(A3)p Fs(\))q Fr(.)48 b(Then,)35 b(ther)-5 b(e)36 b(exists)f(a)g(r)-5 b(e)g(al)36 b(numb)-5 b(er)35 b Fq(\016)2137 2082 y Fo(2)2205 2068 y Fq(>)29 b Fs(0)p Fr(,)35 b(smal)5 b(l)36 b(enough)f(\(which)h (dep)-5 b(ends)36 b(on)f(the)75 2181 y(c)-5 b(o)g(e\016cients)43 b Fq(B)617 2195 y Fi(q)664 2181 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))43 b Fr(for)g Fe(q)f Fs(=)g(\()p Fq(q)1443 2195 y Fo(0)1482 2181 y Fq(;)15 b(q)1563 2195 y Fo(1)1603 2181 y Fq(;)g(q)1684 2195 y Fo(2)1723 2181 y Fs(\))43 b Fm(2)e(Q)p Fr(\),)k(such)d(that,)j Fm(M)28 b(\016)f Fq(@)2790 2148 y Fy(\000)p Fp(S)2785 2204 y(z)2938 2181 y Fr(is)42 b(a)g(b)-5 b(ounde)g(d)44 b(invertible)75 2321 y(op)-5 b(er)g(ator)40 b(fr)-5 b(om)656 2220 y Fn(\020)710 2321 y Fm(G)764 2335 y Fp(q)818 2321 y Fs(\()p Fq(\016)893 2335 y Fo(1)933 2321 y Fq(;)15 b(\016)1013 2335 y Fo(2)1053 2321 y Fs(;)g Fq(\033)s Fs(\))h Fq(;)f Fm(j)q(j)p Fq(:)p Fm(j)q(j)1366 2348 y Fp(\016)1397 2357 y Fh(1)1432 2348 y Fp(;\016)1483 2357 y Fh(2)1517 2348 y Fo(;)p Fp(\033)1584 2220 y Fn(\021)1675 2321 y Fr(into)37 b(itself.)55 b(Mor)-5 b(e)g(over,)39 b(ther)-5 b(e)38 b(exists)f(a)g(c)-5 b(onstant)39 b Fq(C)3570 2335 y Fo(11)3678 2321 y Fq(>)32 b Fs(0)p Fr(,)75 2451 y(such)h(that)820 2460 y Fn(\014)820 2514 y(\014)820 2569 y(\014)850 2460 y(\014)850 2514 y(\014)850 2569 y(\014)880 2491 y(\000)922 2564 y Fm(M)20 b(\016)h Fq(@)1170 2527 y Fy(\000)p Fp(S)1165 2587 y(z)1276 2491 y Fn(\001)1318 2508 y Fy(\000)p Fo(1)1427 2564 y Fs(\()p Fq(V)g Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\)\))1867 2460 y Fn(\014)1867 2514 y(\014)1867 2569 y(\014)1898 2460 y(\014)1898 2514 y(\014)1898 2569 y(\014)1928 2627 y Fp(\016)1959 2636 y Fh(1)1994 2627 y Fp(;\016)2045 2636 y Fh(2)2079 2627 y Fo(;)p Fp(\033)2171 2564 y Fm(\024)25 b Fq(C)2332 2578 y Fo(11)2422 2564 y Fm(jj)p Fq(V)20 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))p Fm(j)r(j)2892 2591 y Fp(\016)2923 2600 y Fh(1)2958 2591 y Fp(;\016)3009 2600 y Fh(2)3044 2591 y Fo(;)p Fp(\033)75 2762 y Fr(for)33 b Fq(V)20 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))27 b Fm(2)e(G)755 2776 y Fp(q)808 2762 y Fs(\()q Fq(\016)884 2776 y Fo(1)923 2762 y Fq(;)15 b(\016)1003 2776 y Fo(2)1044 2762 y Fs(;)g Fq(\033)s Fs(\))q Fr(.)75 2935 y Fs(No)m(w,)31 b(w)m(e)g(are)g(able)g(to)g(pro)m(v)m(e)g(the)g(main)f(result)g(of)h (this)f(section.)75 3108 y Fe(Theorem)35 b(1)46 b Fr(Consider)33 b(a)g(p)-5 b(artial)35 b(di\013er)-5 b(ential)34 b(e)-5 b(quation)34 b(of)f(the)g(form)75 3303 y Fs(\(12\))776 b Fq(@)1064 3266 y Fp(S)1059 3326 y(z)1115 3303 y Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))27 b(=)1613 3217 y Fn(X)1604 3414 y Fi(q)p Fy(2Q)1767 3303 y Fq(B)1836 3317 y Fi(q)1883 3303 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))p Fq(@)2231 3259 y Fp(q)2263 3268 y Fh(0)2226 3327 y Fp(t)2303 3303 y Fq(@)2356 3259 y Fp(q)2388 3268 y Fh(1)2351 3332 y Fp(T)2427 3303 y Fq(@)2480 3266 y Fp(q)2512 3275 y Fh(2)2475 3326 y Fp(z)2550 3303 y Fq(U)d Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))75 3586 y Fr(which)37 b(satis\014es)h(the) e(assumptions)j Fs(\()p Fe(A1)p Fs(\))q Fr(,)e Fs(\()p Fe(A2)p Fs(\))h Fr(and)f Fs(\()p Fe(A3)p Fs(\))q Fr(.)53 b(We)36 b(imp)-5 b(ose)38 b(the)f(fol)5 b(lowing)37 b(initial)g(c)-5 b(ondi-)75 3699 y(tions.)43 b(F)-7 b(or)33 b(al)5 b(l)33 b Fs(0)26 b Fm(\024)f Fq(j)31 b Fm(\024)25 b Fq(S)g Fm(\000)20 b Fs(1)p Fr(,)75 3885 y Fs(\(13\))1474 3812 y Fn(\000)1515 3885 y Fq(@)1568 3848 y Fp(j)1563 3908 y(z)1605 3885 y Fq(U)1677 3812 y Fn(\001)1734 3885 y Fs(\()p Fq(t;)15 b Fs(0)p Fq(;)g(T)e Fs(\))27 b(=)e Fq( )2210 3899 y Fp(j)2246 3885 y Fs(\()p Fq(t;)15 b(T)e Fs(\))75 4072 y Fr(wher)-5 b(e)34 b(the)f Fq( )542 4086 y Fp(j)578 4072 y Fs(\()p Fq(t;)15 b(T)e Fs(\))34 b Fr(b)-5 b(elong)33 b(to)g Fm(G)1258 4086 y Fp(q)1311 4072 y Fs(\()q Fq(\016)1387 4086 y Fo(1)p Fp(;)p Fo(0)1481 4072 y Fq(;)15 b(\016)1561 4086 y Fo(2)p Fp(;)p Fo(0)1656 4072 y Fs(;)g Fq(\033)1748 4086 y Fo(0)1788 4072 y Fs(\))q Fr(,)32 b(for)h(given)f Fq(\016)2306 4086 y Fo(1)p Fp(;)p Fo(0)2401 4072 y Fr(,)g Fq(\016)2501 4086 y Fo(2)p Fp(;)p Fo(0)2629 4072 y Fr(and)h Fq(\033)2857 4086 y Fo(0)2922 4072 y Fq(>)25 b Fs(0)p Fr(.)216 4185 y(Then,)i(ther)-5 b(e)26 b(exists)g Fq(\016)976 4199 y Fo(1)1016 4185 y Fr(,)h Fq(\016)1111 4199 y Fo(2)1176 4185 y Fr(and)f Fq(\033)i(>)d Fs(0)p Fr(,)i(such)f(that)g(the)g(pr)-5 b(oblem)27 b(\(12\))g(and)f(\(13\))h(has)f(a)g(unique)e(solution)75 4298 y Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))27 b Fm(2)d(G)608 4312 y Fp(q)662 4298 y Fs(\()p Fq(\016)737 4312 y Fo(1)777 4298 y Fq(;)15 b(\016)857 4312 y Fo(2)897 4298 y Fs(;)g Fq(\033)s Fs(\))q Fr(.)75 4515 y Fe(Pro)s(of)57 b Fs(A)31 b(formal)g(series)h Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))27 b Fm(2)f(A)p Fs(\()p Fl(C)p Fs(\)[[)p Fq(t;)15 b(z)t Fs(]])33 b(whic)m(h)e(satis\014es)g(the)g(equation)h (\(13\))h(can)e(b)s(e)f(written)75 4628 y(in)g(the)h(form)1203 4741 y Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))26 b(=)f Fq(@)1745 4703 y Fy(\000)p Fp(S)1740 4763 y(z)1851 4741 y Fq(V)20 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))22 b(+)e Fq(W)13 b Fs(\()p Fq(t;)i(z)t(;)g(T)e Fs(\))75 4899 y(where)1401 5075 y Fq(W)g Fs(\()p Fq(t;)i(z)t(;)g(T)e Fs(\))27 b(:=)1943 4961 y Fp(S)t Fy(\000)p Fo(1)1945 4989 y Fn(X)1950 5184 y Fp(j)t Fo(=0)2095 5075 y Fq( )2154 5089 y Fp(j)2190 5075 y Fs(\()p Fq(t;)15 b(T)e Fs(\))2409 5014 y Fq(z)2455 4981 y Fp(j)p 2409 5054 83 4 v 2416 5137 a Fq(j)5 b Fs(!)2503 5075 y Fq(:)75 5328 y Fs(F)-8 b(rom)34 b(the)f(fact)h(that)g(the)f(initial)i(conditions)e(\(13\))i(b) s(elong)e(to)h Fm(G)2376 5342 y Fp(q)2429 5328 y Fs(\()q Fq(\016)2505 5342 y Fo(1)p Fp(;)p Fo(0)2599 5328 y Fq(;)15 b(\016)2679 5342 y Fo(2)p Fp(;)p Fo(0)2775 5328 y Fs(;)g Fq(\033)2867 5342 y Fo(0)2906 5328 y Fs(\))34 b(and)e(Prop)s(osition)h (3,)i(w)m(e)75 5441 y(get)k(that)e Fq(W)13 b Fs(\()p Fq(t;)i(z)t(;)g(T)e Fs(\))39 b(b)s(elongs)e(to)h Fm(G)1375 5455 y Fp(q)1428 5441 y Fs(\()q Fq(\016)1504 5455 y Fo(1)p Fp(;)p Fo(0)1598 5441 y Fq(;)15 b(\016)1678 5455 y Fo(2)p Fp(;)p Fo(0)1774 5441 y Fs(;)g Fq(\033)1866 5455 y Fo(0)1905 5441 y Fs(\))q(.)61 b(Moreo)m(v)m(er,)42 b Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))38 b(is)g(a)g(solution)g(of)f (\(12\))i(and)75 5554 y(\(13\))32 b(if)e(and)g(only)h(if)f Fq(V)20 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))32 b(satis\014es)f(the)f (equation)75 5740 y(\(14\))1107 5667 y Fn(\000)1149 5740 y Fm(M)20 b(\016)h Fq(@)1397 5703 y Fy(\000)p Fp(S)1392 5763 y(z)1503 5667 y Fn(\001)1560 5740 y Fs(\()p Fq(V)f Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\)\))27 b(=)e Fm(\000M)15 b Fs(\()q Fq(W)e Fs(\()p Fq(t;)i(z)t(;)g(T)e Fs(\)\))j Fq(:)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 3764 100 a Fs(15)75 399 y(F)-8 b(rom)31 b(the)f(fact)i(that)f Fq(W)13 b Fs(\()p Fq(t;)i(z)t(;)g(T)e Fs(\))31 b(is)f(a)h(p)s(olynomial)g(in)f(the)g(v)-5 b(ariable)32 b Fq(z)t Fs(,)e(and)g(that)75 511 y Fq(W)13 b Fs(\()p Fq(t;)i(z)t(;)g(T)e Fs(\))28 b Fm(2)f(G)639 525 y Fp(q)692 511 y Fs(\()p Fq(\016)767 525 y Fo(1)p Fp(;)p Fo(0)862 511 y Fq(;)15 b(\016)942 525 y Fo(2)p Fp(;)p Fo(0)1037 511 y Fs(;)g Fq(\033)1129 525 y Fo(0)1169 511 y Fs(\),)32 b(using)f(Prop)s(osition)h(3)g(and)f(Prop)s(osition)g(4,)i(w)m(e)f(get) h(that)f(there)f(exist)i Fq(\016)3815 525 y Fo(1)75 624 y Fs(and)d Fq(\033)e(>)d Fs(0)31 b(suc)m(h)f(that)1330 737 y Fm(M)15 b Fs(\()p Fq(W)e Fs(\()p Fq(t;)i(z)t(;)g(T)e Fs(\)\))27 b Fm(2)e(G)2085 751 y Fp(q)2138 737 y Fs(\()p Fq(\016)2213 751 y Fo(1)2253 737 y Fq(;)15 b(\016)2333 751 y Fo(2)p Fp(;)p Fo(0)2428 737 y Fs(;)g Fq(\033)s Fs(\))h Fq(:)75 904 y Fs(Therefore,)35 b(b)m(y)f(Prop)s(osition)g(6,)i (for)e Fq(\016)1429 918 y Fo(2)1501 904 y Fq(>)d Fs(0)j(small)h (enough,)g(there)g(exists)f(a)h(unique)e(solution)i Fq(V)20 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))75 1017 y(of)48 b(\(14\))i(whic)m(h)d(b)s(elongs)h(to)h Fm(G)1210 1031 y Fp(q)1263 1017 y Fs(\()p Fq(\016)1338 1031 y Fo(1)1378 1017 y Fq(;)15 b(\016)1458 1031 y Fo(2)1498 1017 y Fs(;)g Fq(\033)s Fs(\))q(.)94 b(T)-8 b(o)48 b(conclude,)53 b(it)48 b(remains)g(to)h(sho)m(w)f(that)g Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))56 b Fm(2)75 1130 y(G)129 1144 y Fp(q)182 1130 y Fs(\()q Fq(\016)258 1144 y Fo(1)297 1130 y Fq(;)15 b(\016)377 1144 y Fo(2)418 1130 y Fs(;)g Fq(\033)s Fs(\))q(.)72 b(Indeed,)44 b(w)m(e)d(ha)m(v)m(e)h Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))45 b(=)e Fq(@)1924 1097 y Fy(\000)p Fp(S)1919 1152 y(z)2030 1130 y Fq(V)20 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))29 b(+)e Fq(W)13 b Fs(\()p 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Fi(k)2120 2692 y Fs(\()p Fq(t;)15 b(z)t Fs(\))p Fq(@)2362 2652 y Fp(k)2399 2661 y Fh(0)2357 2716 y Fp(t)2439 2692 y Fq(@)2492 2655 y Fp(k)2529 2664 y Fh(1)2487 2715 y Fp(z)2568 2692 y Fq(X)7 b Fs(\()p Fq(t;)15 b(z)t Fs(\))75 2996 y(where)30 b Fq(S)35 b Fs(is)c(a)f(p)s(ositiv)m(e)i(in)m(teger)f(and)f Fm(J)47 b Fs(is)30 b(a)h(\014nite)f(set)h(whic)m(h)f(satis\014es)75 3200 y(\(17\))973 b Fm(J)41 b(\032)1407 3126 y Fn(\010)1460 3200 y Fe(k)26 b Fs(=)f(\()p Fq(k)1719 3214 y Fo(0)1759 3200 y Fq(;)15 b(k)1846 3214 y Fo(1)1886 3200 y Fs(\))26 b Fm(2)e Fl(N)2098 3162 y Fo(2)2153 3200 y Fm(j)h Fq(k)2250 3214 y Fo(1)2315 3200 y Fm(\024)g Fq(S)g Fm(\000)20 b Fs(1)2628 3126 y Fn(\011)2697 3200 y Fq(:)75 3404 y Fs(The)30 b(co)s(e\016cien)m(ts)i Fq(a)770 3419 y Fi(k)817 3404 y Fs(\()p Fq(t;)15 b(z)t Fs(\))32 b(are)f(holomorphic)g(functions)f(on) g(the)h(domain)f Fm(D)j Fs(with)e(resp)s(ect)f(to)h Fq(t)g Fs(and)f(near)75 3517 y(the)j(origin)g(in)f Fl(C)h Fs(with)f(resp)s (ect)h(to)g Fq(z)t Fs(.)48 b(Our)31 b(aim)i(is)g(to)g(lo)s(ok)g(for)g Fr(tr)-5 b(ansseries)34 b Fs(solution)g Fq(X)7 b Fs(\()p Fq(t;)15 b(z)t Fs(\))34 b(of)f(\(16\))h(in)75 3630 y(p)s(o)m(w)m(ers)c (of)h Fq(\036)p Fs(\()p Fq(t)p Fs(\))g(and)f Fq(z)75 3893 y Fs(\(18\))1347 3870 y(^)1321 3893 y Fq(X)8 b Fs(\()p Fq(t;)15 b(z)t Fs(\))26 b(=)1716 3806 y Fn(X)1715 4004 y Fp(\014)s Fy(\025)p Fo(0)1863 3806 y Fn(X)1870 4004 y Fp(`)p Fy(\025)p Fo(0)2010 3893 y Fq(X)2085 3908 y Fp(`;\014)2181 3893 y Fs(\()p Fq(t)p Fs(\))2294 3831 y Fq(\036)p Fs(\()p Fq(t)p Fs(\))2451 3798 y Fp(`)p 2294 3872 191 4 v 2358 3955 a Fq(`)p Fs(!)2505 3831 y Fq(z)2551 3798 y Fp(\014)p 2505 3872 94 4 v 2511 3955 a Fq(\014)5 b Fs(!)75 4194 y(where)37 b Fq(X)420 4209 y Fp(`;\014)516 4194 y Fs(\()p Fq(t)p Fs(\))h(are)f(holomorphic)h(functions)f(on)g(the) h(domain)f Fm(D)j(\032)c Fl(C)p Fs(.)62 b(T)-8 b(o)37 b(\014x)g(the)h(notations,)i(for)e(all)75 4307 y(in)m(tegers)32 b Fq(n)24 b Fm(\025)h Fs(1,)31 b(for)g(all)g(v)m(ectors)h Fe(p)25 b Fs(=)g(\()p 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b(the)g(left-hand)f(side)g(term)h(of)f(\(16\),)j(w)m(e)d (obtain)1206 1942 y Fq(@)1259 1904 y Fp(S)1254 1964 y(z)1336 1919 y Fs(^)1310 1942 y Fq(X)7 b Fs(\()p Fq(t;)15 b(z)t Fs(\))27 b(=)1705 1855 y Fn(X)1704 2053 y Fp(\014)s Fy(\025)p Fo(0)1852 1855 y Fn(X)1858 2053 y Fp(`)p Fy(\025)p Fo(0)1999 1942 y Fq(X)2074 1957 y Fp(`;\014)s Fo(+)p Fp(S)2271 1942 y Fs(\()p Fq(t)p Fs(\))2384 1880 y Fq(\036)p Fs(\()p Fq(t)p Fs(\))2541 1847 y Fp(`)p 2384 1921 V 2448 2004 a Fq(`)p Fs(!)2595 1880 y Fq(z)2641 1847 y Fp(\014)p 2595 1921 94 4 v 2601 2004 a Fq(\014)5 b Fs(!)2698 1942 y Fq(:)75 2236 y Fs(F)-8 b(or)31 b(the)g(righ)m(t-hand)f(side)g(term,)h (it)g(will)g(b)s(e)e(more)i(complicated.)42 b(A)31 b(\014rst)e (calculus)i(sho)m(ws)f(us)g(that)75 2492 y(\(20\))734 b Fq(@)1022 2452 y Fp(k)1059 2461 y Fh(0)1017 2516 y Fp(t)1098 2492 y Fq(@)1151 2455 y Fp(k)1188 2464 y Fh(1)1146 2515 y Fp(z)1253 2469 y Fs(^)1227 2492 y Fq(X)7 b Fs(\()p Fq(t;)15 b(z)t Fs(\))26 b(=)1621 2406 y Fn(X)1620 2604 y Fp(\014)s 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y(,)i(w)m(e)f(use)g(F)-8 b(a\022)-45 b(a)32 b(di)e(Bruno)g(form)m(ula) 75 4110 y(\(21\))551 b Fq(@)839 4072 y Fp(m)834 4132 y(t)916 4048 y Fq(\036)p Fs(\()p Fq(t)p Fs(\))1073 4015 y Fp(`)p 916 4089 V 980 4172 a Fq(`)p Fs(!)1142 4110 y(=)1361 4023 y Fn(X)1279 4216 y Fb(p)18 b Fk(2)g(P)1439 4226 y Fa(2)1471 4216 y Fh(\()p Fc(m)p Fh(\))1279 4274 y Fk(j)p Fb(p)p Fk(j)1355 4284 y Fa(1)1406 4274 y Fk(\024)f Fc(`)1779 4048 y Fq(m)p Fs(!)p 1640 4089 385 4 v 1640 4172 a Fq(p)1686 4186 y Fo(1)1725 4172 y Fs(!)e Fq(:)g(:)g(:)i(p)1933 4186 y Fp(m)1999 4172 y Fs(!)2058 4048 y Fq(\036)p Fs(\()p Fq(t)p Fs(\))2215 4015 y Fp(`)p Fy(\000j)p Fi(p)p Fy(j)2384 4024 y Fh(1)p 2044 4089 393 4 v 2044 4172 a Fs(\()p Fq(`)k Fm(\000)e(j)p Fe(p)p Fm(j)2336 4186 y Fo(1)2376 4172 y Fs(\)!)2495 3996 y Fp(m)2468 4023 y Fn(Y)2462 4221 y Fp(k)r Fo(=1)2606 3954 y Fn( )2688 4048 y Fq(\036)2742 4015 y Fo(\()p Fp(k)r Fo(\))2840 4048 y Fs(\()p Fq(t)p Fs(\))p 2688 4089 256 4 v 2778 4172 a Fq(k)s Fs(!)2953 3954 y Fn(!)3025 3977 y Fp(p)3061 3989 y Fc(k)3118 4110 y Fq(:)75 4461 y Fs(W)-8 b(e)32 b(in)m(tro)s(duce)e(the)g(notation)632 4664 y(\012)698 4627 y Fi(k)698 4687 y Fp(\014)760 4664 y Fs(\()q Fq(t;)15 b(\036)p Fs(\()p Fq(t)p Fs(\)\))26 b(:=)1208 4578 y Fn(X)1215 4775 y Fp(`)p Fy(\025)p Fo(0)1355 4664 y Fq(D)1433 4627 y Fi(k)1430 4687 y Fp(`;\014)1541 4664 y Fs(\()p Fq(t;)15 b(\036)p Fs(\()p Fq(t)p Fs(\)\))149 5064 y(=)245 4977 y Fn(X)251 5175 y Fp(`)p Fy(\025)p Fo(0)432 4949 y Fp(k)469 4958 y Fh(0)402 4977 y Fn(X)391 5172 y Fp(m)p Fo(=0)752 5002 y Fq(k)799 5016 y Fo(0)839 5002 y Fs(!)p 569 5043 479 4 v 569 5126 a Fq(m)p Fs(!\()p Fq(k)756 5140 y Fo(0)816 5126 y Fm(\000)20 b Fq(m)p Fs(\)!)1073 4963 y Fn(\020)1127 5064 y Fq(@)1180 5023 y Fp(k)1217 5032 y Fh(0)1252 5023 y Fy(\000)p Fp(m)1175 5088 y(t)1373 5064 y Fq(X)1448 5079 y Fp(`;\014)s Fo(+)p Fp(k)1632 5088 y Fh(1)1670 5064 y Fs(\()p Fq(t)p Fs(\))1773 4963 y Fn(\021)1982 4977 y(X)1885 5170 y Fb(p)e Fk(2)f(P)2044 5180 y Fa(2)2076 5170 y Fh(\()p 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Fq(t)p Fs(\))1352 5584 y Fq(\036)p Fs(\()p Fq(t)p Fs(\))1509 5551 y Fp(`)p 1352 5624 191 4 v 1417 5708 a Fq(`)p Fs(!)1563 5584 y Fq(z)1609 5551 y Fp(\014)p 1563 5624 94 4 v 1569 5708 a Fq(\014)5 b Fs(!)1692 5645 y(=)1933 5559 y Fn(X)1788 5760 y Fi(k)p Fo(=\()p Fp(k)1950 5769 y Fh(0)1985 5760 y Fp(;k)2042 5769 y Fh(1)2075 5760 y Fo(\))p Fy(2J)2225 5645 y Fq(a)2273 5660 y Fi(k)2320 5645 y Fs(\()p Fq(t;)15 b(z)t Fs(\))2527 5559 y Fn(X)2524 5757 y Fp(\014)s Fy(\025)p Fo(0)2674 5645 y Fs(\012)2740 5608 y Fi(k)2740 5668 y Fp(\014)2802 5645 y Fs(\()p Fq(t;)g(\036)p Fs(\()p Fq(t)p Fs(\)\))3129 5584 y Fq(z)3175 5551 y Fp(\014)p 3129 5624 V 3135 5708 a Fq(\014)5 b Fs(!)3232 5645 y Fq(:)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 3764 100 a Fs(17)75 399 y(Using)31 b(for)f Fq(a)518 414 y Fi(k)565 399 y Fs(\()p Fq(t;)15 b(z)t Fs(\))31 b(the)g(T)-8 b(a)m(ylor)32 b(expansion)1519 661 y Fq(a)1567 676 y Fi(k)1614 661 y Fs(\()p Fq(t;)15 b(z)t Fs(\))26 b(=)1926 575 y Fn(X)1925 773 y Fp(\014)s Fy(\025)p Fo(0)2074 661 y Fq(a)2122 624 y Fi(k)2122 685 y Fp(\014)2169 661 y Fs(\()p Fq(t)p Fs(\))2282 600 y Fq(z)2328 567 y Fp(\014)p 2282 641 94 4 v 2288 724 a Fq(\014)5 b Fs(!)2386 661 y Fq(;)75 973 y Fs(where)30 b(the)g(co)s(e\016cien)m(ts)j Fq(a)1003 941 y Fi(k)1003 1001 y Fp(\014)1050 973 y Fs(\()p Fq(t)p Fs(\))e(are)f(holomorphic)h (functions)f(on)g Fm(D)s Fs(,)g(w)m(e)h(obtain)g(from)f(\(22\))1381 1214 y Fn(X)1380 1411 y Fp(\014)s Fy(\025)p Fo(0)1529 1117 y Fn(0)1529 1281 y(@)1608 1214 y(X)1614 1411 y Fp(`)p Fy(\025)p Fo(0)1755 1300 y Fq(X)1830 1315 y Fp(`;\014)s Fo(+)p Fp(S)2027 1300 y Fs(\()p Fq(t)p Fs(\))2140 1239 y Fq(\036)p Fs(\()p Fq(t)p Fs(\))2297 1206 y Fp(`)p 2140 1279 191 4 v 2204 1362 a Fq(`)p Fs(!)2341 1117 y Fn(1)2341 1281 y(A)2446 1239 y Fq(z)2492 1206 y Fp(\014)p 2446 1279 94 4 v 2452 1362 a Fq(\014)5 b Fs(!)738 1727 y(=)835 1641 y Fn(X)834 1838 y Fp(\014)s Fy(\025)p Fo(0)983 1544 y Fn(0)983 1708 y(@)1208 1641 y(X)1062 1842 y Fi(k)p Fo(=\()p Fp(k)1224 1851 y Fh(0)1259 1842 y Fp(;k)1316 1851 y Fh(1)1350 1842 y Fo(\))p Fy(2J)1560 1609 y Fp(\014)1516 1641 y Fn(X)1500 1838 y Fp(\014)1540 1847 y Fh(2)1574 1838 y Fo(=0)1883 1665 y Fq(\014)g Fs(!)p 1689 1706 471 4 v 1689 1789 a Fq(\014)1740 1803 y Fo(2)1780 1789 y Fs(!\()p Fq(\014)26 b Fm(\000)20 b Fq(\014)2059 1803 y Fo(2)2099 1789 y Fs(\)!)2169 1727 y Fq(a)2217 1689 y Fi(k)2217 1750 y Fp(\014)s Fy(\000)p Fp(\014)2355 1759 y Fh(2)2394 1727 y Fs(\()p Fq(t)p Fs(\)\012)2563 1689 y Fi(k)2563 1750 y Fp(\014)2603 1759 y Fh(2)2656 1727 y Fs(\()q Fq(t;)15 b(\036)p Fs(\()p Fq(t)p Fs(\)\))2958 1544 y Fn(1)2958 1708 y(A)3063 1665 y Fq(z)3109 1632 y Fp(\014)p 3063 1706 94 4 v 3069 1789 a Fq(\014)5 b Fs(!)3166 1727 y Fq(:)75 2071 y Fs(By)31 b(iden)m(ti\014cation)h(of)e (the)h(co)s(e\016cien)m(ts)h(of)1599 2009 y Fq(z)1645 1976 y Fp(\014)p 1599 2050 V 1605 2133 a Fq(\014)5 b Fs(!)1703 2071 y(,)30 b(w)m(e)h(get)75 2424 y(\(23\))590 2338 y Fn(X)596 2535 y Fp(`)p Fy(\025)p Fo(0)736 2424 y Fq(X)811 2439 y Fp(`;\014)s Fo(+)p Fp(S)1009 2424 y Fs(\()p Fq(t)p Fs(\))1122 2363 y Fq(\036)p Fs(\()p Fq(t)p Fs(\))1279 2330 y Fp(`)p 1122 2403 191 4 v 1186 2486 a Fq(`)p Fs(!)1348 2424 y(=)1589 2338 y Fn(X)1444 2539 y Fi(k)p Fo(=\()p Fp(k)1606 2548 y Fh(0)1641 2539 y Fp(;k)1698 2548 y Fh(1)1732 2539 y Fo(\))p Fy(2J)1942 2306 y Fp(\014)1898 2338 y Fn(X)1881 2535 y Fp(\014)1921 2544 y Fh(2)1956 2535 y Fo(=0)2265 2363 y Fq(\014)5 b Fs(!)p 2071 2403 471 4 v 2071 2486 a Fq(\014)2122 2500 y Fo(2)2162 2486 y Fs(!\()p Fq(\014)26 b Fm(\000)20 b Fq(\014)2441 2500 y Fo(2)2481 2486 y Fs(\)!)2551 2424 y Fq(a)2599 2387 y Fi(k)2599 2447 y Fp(\014)s Fy(\000)p Fp(\014)2737 2456 y Fh(2)2775 2424 y Fs(\()p Fq(t)p Fs(\)\012)2944 2387 y Fi(k)2944 2447 y Fp(\014)2984 2456 y Fh(2)3038 2424 y Fs(\()p Fq(t;)15 b(\036)p Fs(\()p Fq(t)p Fs(\)\))75 2744 y(for)30 b(all)h Fq(\014)g Fm(2)25 b Fl(N)p Fs(.)40 b(On)30 b(the)g(righ)m(t-hand)g(side,)h(the)f(term)h(\012)2042 2711 y Fi(k)2042 2771 y Fp(\014)2082 2780 y Fh(2)2135 2744 y Fs(\()p Fq(t;)15 b(\036)p Fs(\()p Fq(t)p Fs(\)\))32 b(has)e(the)h(form)75 3054 y(\(24\))628 b(\012)929 3016 y Fi(k)929 3077 y Fp(\014)969 3086 y Fh(2)1022 3054 y Fs(\()p Fq(t;)15 b(\036)p Fs(\()p Fq(t)p Fs(\)\))27 b(=)1445 2967 y Fn(X)1451 3165 y Fp(`)p Fy(\025)p Fo(0)1632 2939 y Fp(k)1669 2948 y Fh(0)1602 2967 y Fn(X)1591 3162 y Fp(m)p Fo(=0)1759 2953 y Fn(\020)1813 3054 y Fq(@)1866 3013 y Fp(k)1903 3022 y Fh(0)1938 3013 y Fy(\000)p Fp(m)1861 3078 y(t)2060 3054 y Fq(X)2135 3069 y Fp(`;\014)2224 3078 y Fh(2)2257 3069 y Fo(+)p Fp(k)2349 3078 y Fh(1)2388 3054 y Fs(\()p Fq(t)p Fs(\))2491 2953 y Fn(\021)2561 3054 y Fq(!)2621 3010 y Fi(k)p Fp(;m)2618 3083 y(`)2765 3054 y Fs(\()p Fq(t;)15 b(\036)p Fs(\()p Fq(t)p Fs(\)\))75 3348 y(where)494 3518 y Fq(!)554 3474 y Fi(k)p Fp(;m)551 3547 y(`)698 3518 y Fs(\()q Fq(t;)g(\036)p Fs(\()p Fq(t)p Fs(\)\))26 b(=)1262 3456 y Fq(k)1309 3470 y Fo(0)1349 3456 y Fs(!)p 1131 3497 374 4 v 1131 3580 a(\()p Fq(k)1213 3594 y Fo(0)1273 3580 y Fm(\000)20 b Fq(m)p Fs(\)!)1653 3431 y Fn(X)1571 3624 y Fb(p)e Fk(2)g(P)1731 3634 y Fa(2)1763 3624 y Fh(\()p Fc(m)p Fh(\))1571 3682 y Fk(j)p Fb(p)p Fk(j)1647 3692 y Fa(1)1698 3682 y Fk(\024)f Fc(`)2101 3456 y Fs(1)p 1932 3497 385 4 v 1932 3580 a Fq(p)1978 3594 y Fo(1)2017 3580 y Fs(!)e Fq(:)g(:)g(:)i(p)2225 3594 y Fp(m)2291 3580 y Fs(!)2350 3456 y Fq(\036)p Fs(\()p Fq(t)p Fs(\))2507 3423 y Fp(`)p Fy(\000j)p Fi(p)p Fy(j)2676 3432 y Fh(1)p 2336 3497 393 4 v 2336 3580 a Fs(\()p Fq(`)k Fm(\000)e(j)p Fe(p)p Fm(j)2628 3594 y Fo(1)2668 3580 y Fs(\)!)2787 3404 y Fp(m)2760 3431 y Fn(Y)2754 3629 y Fp(k)r Fo(=1)2898 3362 y Fn( )2980 3456 y Fq(\036)3034 3423 y Fo(\()p Fp(k)r Fo(\))3132 3456 y Fs(\()p Fq(t)p Fs(\))p 2980 3497 256 4 v 3070 3580 a Fq(k)s Fs(!)3245 3362 y Fn(!)3317 3385 y Fp(p)3353 3397 y Fc(k)3410 3518 y Fq(;)75 3853 y Fs(con)m(tains)32 b(p)s(o)m(w)m(ers)e(of)g Fq(\036)p Fs(\()p Fq(t)p Fs(\))h(and)f(p)s(o)m(w)m(ers)g(of)h Fq(\036)1660 3820 y Fo(\()p Fp(k)r Fo(\))1758 3853 y Fs(\()p Fq(t)p Fs(\))f(for)h Fq(k)d Fs(=)d(1)p Fq(;)15 b(:)g(:)g(:)i(;)e(m)p Fs(.)75 4024 y(In)30 b(the)g(next)h(t)m(w)m(o)g (lemmas,)h(w)m(e)e(are)h(going)h(to)f(sp)s(ecify)f(the)g(form)g(of)h Fq(!)2551 3980 y Fi(k)p Fp(;m)2548 4054 y(`)2695 4024 y Fs(\()p Fq(t;)15 b(\036)p Fs(\()p Fq(t)p Fs(\)\))q(.)75 4212 y Fe(Lemma)35 b(4)45 b Fr(L)-5 b(et)33 b(us)g(c)-5 b(onsider)34 b(the)f(or)-5 b(dinary)35 b(di\013er)-5 b(ential)34 b(e)-5 b(quation)33 b(\(15\))h(with)1300 4503 y Fq(P)13 b Fs(\()p Fq(t;)i(X)7 b Fs(\))26 b(=)1766 4390 y Fp(d)1718 4417 y Fn(X)1723 4612 y Fp(j)t Fo(=0)1865 4503 y Fq(p)1911 4517 y Fp(j)1947 4503 y Fs(\()p Fq(t)p Fs(\))2060 4442 y Fq(X)2142 4409 y Fp(j)p 2060 4482 119 4 v 2086 4566 a Fq(j)5 b Fs(!)2190 4503 y Fq(;)201 b(d)25 b Fm(\025)g Fs(2)75 4802 y Fr(wher)-5 b(e)44 b(the)f(c)-5 b(o)g(e\016cients)44 b Fq(p)1023 4816 y Fp(j)1059 4802 y Fs(\()p Fq(t)p Fs(\))f Fr(ar)-5 b(e)44 b(holomorphic)i(on)d(some)g (domain)i Fm(D)h(\032)e Fl(C)e Fr(and)i Fq(d)f Fr(is)f(the)h(de)-5 b(gr)g(e)g(e)44 b(of)75 4915 y Fq(P)13 b Fs(\()p Fq(t;)i(X)7 b Fs(\))31 b Fr(with)g(r)-5 b(esp)g(e)g(ct)31 b(to)g(the)f(variable)h Fq(X)37 b Fr(\(in)30 b(p)-5 b(articular)32 b Fq(p)2200 4930 y Fp(d)2240 4915 y Fs(\()p Fq(t)p Fs(\))26 b Fm(6\021)f Fs(0)p Fr(\).)42 b(Then,)30 b(for)h(al)5 b(l)30 b(p)-5 b(ositive)31 b(inte)-5 b(ger)30 b Fq(n)p Fr(,)75 5028 y(ther)-5 b(e)34 b(exists)e(a)h(function)g Fq(Q)1059 5042 y Fp(n)1106 5028 y Fs(\()p Fq(t;)15 b(X)7 b Fs(\))34 b Fr(such)f(that)75 5232 y Fs(\(25\))1322 b Fq(\036)1611 5195 y Fo(\()p Fp(n)p Fo(\))1713 5232 y Fs(\()p Fq(t)p Fs(\))25 b(=)g Fq(Q)2009 5246 y Fp(n)2071 5232 y Fs(\()q Fq(t;)15 b(\036)p Fs(\()p Fq(t)p Fs(\)\))75 5436 y Fr(with)1467 5614 y Fq(Q)1539 5628 y Fp(n)1586 5614 y Fs(\()p Fq(t;)g(X)7 b Fs(\))27 b(=)1960 5500 y Fp(d)1996 5508 y Fc(n)1934 5528 y Fn(X)1938 5723 y Fp(j)t Fo(=0)2080 5614 y Fq(q)2121 5628 y Fp(n;j)2220 5614 y Fs(\()p Fq(t)p Fs(\))2333 5552 y Fq(X)2415 5519 y Fp(j)p 2333 5593 V 2359 5676 a Fq(j)5 b Fs(!)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 3764 100 a Fs(18)75 399 y Fr(wher)-5 b(e)34 b(the)f(c)-5 b(o)g(e\016cients)33 b Fq(q)987 413 y Fp(n;j)1086 399 y Fs(\()p Fq(t)p Fs(\))g Fr(ar)-5 b(e)33 b(holomorphic)j(on)d Fm(D)s Fr(.)42 b(Mor)-5 b(e)g(over,)1258 572 y Fq(d)1305 586 y Fp(n)1378 572 y Fs(:=)25 b(deg)1637 594 y Fp(X)1719 572 y Fq(Q)1791 586 y Fp(n)1838 572 y Fs(\()p Fq(t;)15 b(X)7 b Fs(\))27 b(=)e Fq(n)p Fs(\()p Fq(d)20 b Fm(\000)g Fs(1\))h(+)f(1)75 746 y Fr(is)33 b(the)g(de)-5 b(gr)g(e)g(e)33 b(of)g Fq(Q)773 760 y Fp(n)820 746 y Fs(\()p Fq(t;)15 b(X)7 b Fs(\))34 b Fr(with)f(r)-5 b(esp)g(e)g(ct)35 b(to)e(the)g(variable)g Fq(X)40 b Fr(and)1301 988 y Fq(q)1342 1003 y Fp(n;d)1441 1011 y Fc(n)1487 988 y Fs(\()p Fq(t)p Fs(\))26 b(=)f(\()p Fq(p)1793 1003 y Fp(d)1834 988 y Fs(\()p Fq(t)p Fs(\)\))1972 947 y Fp(n)2074 875 y(n)2038 902 y Fn(Y)2035 1098 y Fp(j)t Fo(=1)2333 926 y Fq(d)2380 940 y Fp(j)2417 926 y Fs(!)p 2182 968 411 4 v 2182 1051 a Fq(d)p Fs(!\()p Fq(d)2336 1065 y Fp(j)2394 1051 y Fm(\000)20 b Fq(d)p Fs(\)!)2603 988 y Fq(:)75 1291 y Fe(Pro)s(of)70 b Fs(W)-8 b(e)45 b(pro)m(v)m(e)g(this)f (lemma)h(b)m(y)f(induction)g(on)g Fq(n)p Fs(.)81 b(The)43 b(assertions)i(are)f(true)g(for)g Fq(n)k Fs(=)g(1,)g(with)75 1404 y Fq(Q)147 1418 y Fo(1)186 1404 y Fs(\()p Fq(t;)15 b(X)7 b Fs(\))27 b(=)e Fq(P)13 b Fs(\()p Fq(t;)i(X)7 b Fs(\),)32 b(therefore)552 1661 y Fq(d)599 1675 y Fo(1)664 1661 y Fs(=)25 b Fq(d)h Fs(=)f(\()p Fq(d)20 b Fm(\000)g Fs(1\))h(+)f(1)183 b(and)e Fq(q)1911 1676 y Fo(1)p Fp(;d)2002 1685 y Fh(1)2040 1661 y Fs(\()p Fq(t)p Fs(\))26 b(=)f Fq(p)2311 1676 y Fp(d)2351 1661 y Fs(\()p Fq(t)p Fs(\))h(=)f Fq(p)2622 1676 y Fp(d)2662 1661 y Fs(\()p Fq(t)p Fs(\))2825 1548 y Fo(1)2784 1575 y Fn(Y)2780 1770 y Fp(j)t Fo(=1)3079 1600 y Fq(d)3126 1614 y Fo(1)3166 1600 y Fs(!)p 2929 1640 414 4 v 2929 1724 a Fq(d)p Fs(!\()p Fq(d)3083 1738 y Fo(1)3143 1724 y Fm(\000)20 b Fq(d)p Fs(\)!)3352 1661 y Fq(:)75 1929 y Fs(W)-8 b(e)32 b(supp)s(ose,)d(that)i(the)f (assertions)h(are)g(true)f(un)m(til)h Fq(n)25 b Fm(\025)g Fs(1.)41 b(Di\013eren)m(tating)32 b(\(25\))g(giv)m(es)75 2190 y(\(26\))660 b Fq(\036)949 2153 y Fo(\()p Fp(n)p Fo(+1\))1142 2190 y Fs(\()p Fq(t)p Fs(\))25 b(=)1393 2076 y Fp(d)1429 2084 y Fc(n)1366 2104 y Fn(X)1371 2299 y Fp(j)t Fo(=0)1513 2190 y Fq(q)1557 2153 y Fy(0)1554 2213 y Fp(n;j)1652 2190 y Fs(\()p Fq(t)p Fs(\))1765 2129 y Fq(\036)p Fs(\()p Fq(t)p Fs(\))1922 2096 y Fp(j)p 1765 2169 195 4 v 1829 2252 a Fq(j)5 b Fs(!)1990 2190 y(+)20 b Fq(\036)2135 2153 y Fy(0)2159 2190 y Fs(\()p Fq(t)p Fs(\))2277 2076 y Fp(d)2313 2084 y Fc(n)2356 2076 y Fy(\000)p Fo(1)2296 2104 y Fn(X)2301 2299 y Fp(j)t Fo(=0)2461 2190 y Fq(q)2502 2204 y Fp(n;j)t Fo(+1)2691 2190 y Fs(\()p Fq(t)p Fs(\))2804 2129 y Fq(\036)p Fs(\()p Fq(t)p Fs(\))2961 2096 y Fp(j)p 2804 2169 V 2868 2252 a Fq(j)5 b Fs(!)3009 2190 y Fq(:)75 2458 y Fs(By)31 b(\(15\),)h(the)e(higest)i(p)s(o)m(w)m (er)e(in)g(\(26\))i(of)e Fq(\036)p Fs(\()p Fq(t)p Fs(\))h(is)g(in)f (the)h(second)f(sum,)g(it)h(is)666 2691 y Fq(p)712 2706 y Fp(d)752 2691 y Fs(\()p Fq(t)p Fs(\))p Fq(q)896 2706 y Fp(n;d)995 2714 y Fc(n)1042 2691 y Fs(\()p Fq(t)p Fs(\))1155 2629 y Fq(\036)p Fs(\()p Fq(t)p Fs(\))1312 2596 y Fp(d)p Fy(\000)p Fo(1+)p Fp(d)1529 2604 y Fc(n)p 1155 2670 423 4 v 1156 2753 a Fq(d)p Fs(!\()p Fq(d)1310 2767 y Fp(n)1379 2753 y Fm(\000)20 b Fs(1\)!)1613 2691 y(=)1719 2629 y(\()p Fq(d)g Fm(\000)g Fs(1)h(+)f Fq(d)2116 2643 y Fp(n)2163 2629 y Fs(\)!)p 1719 2670 506 4 v 1761 2753 a Fq(d)p Fs(!\()p Fq(d)1915 2767 y Fp(n)1984 2753 y Fm(\000)g Fs(1\)!)2234 2691 y Fq(p)2280 2706 y Fp(d)2320 2691 y Fs(\()p Fq(t)p Fs(\))p Fq(q)2464 2706 y Fp(n;d)2563 2714 y Fc(n)2609 2691 y Fs(\()p Fq(t)p Fs(\))2765 2629 y Fq(\036)p Fs(\()p Fq(t)p Fs(\))2922 2596 y Fp(d)p Fy(\000)p Fo(1+)p Fp(d)3139 2604 y Fc(n)p 2723 2670 V 2723 2753 a Fs(\()p Fq(d)h Fm(\000)f Fs(1)g(+)g Fq(d)3120 2767 y Fp(n)3168 2753 y Fs(\)!)3238 2691 y Fq(:)75 2913 y Fs(Therefore,)30 b(the)h(righ)m(t-hand)f(side)g(of)h(\(26\))h(can)f(b)s(e)e(written)i (as)1269 3181 y Fq(Q)1341 3195 y Fp(n)p Fo(+1)1494 3181 y Fs(\()p Fq(t;)15 b(\036)p Fs(\()p Fq(t)p Fs(\)\))27 b(:=)1942 3060 y Fp(d)1978 3069 y Fc(n)p Fh(+1)1954 3095 y Fn(X)1958 3290 y Fp(j)t Fo(=0)2113 3181 y Fq(q)2154 3195 y Fp(n)p Fo(+1)p Fp(;j)2342 3181 y Fs(\()p Fq(t)p Fs(\))2455 3120 y Fq(\036)p Fs(\()p Fq(t)p Fs(\))2612 3087 y Fp(j)p 2455 3160 195 4 v 2519 3244 a Fq(j)5 b Fs(!)75 3449 y(where)28 b(the)g(co)s(e\016cien)m(ts)j Fq(q)990 3463 y Fp(n)p Fo(+1)p Fp(;j)1178 3449 y Fs(\()p Fq(t)p Fs(\))e(are)g(holomorphic)f(on)h Fm(D)h Fs(whic)m(h)f(are)f (sums)g(and)f(pro)s(ducts)g(of)i(the)f Fq(p)3689 3463 y Fp(j)3726 3449 y Fs(\()p Fq(t)p Fs(\),)75 3562 y Fq(q)116 3576 y Fp(n;j)215 3562 y Fs(\()p Fq(t)p Fs(\))i(and)g Fq(q)569 3529 y Fy(0)566 3588 y Fp(n;j)665 3562 y Fs(\()p Fq(t)p Fs(\).)41 b(Moreo)m(v)m(er,)789 3736 y Fq(d)836 3750 y Fp(n)p Fo(+1)999 3736 y Fs(:=)25 b(deg)1258 3758 y Fp(X)1340 3736 y Fq(Q)1412 3750 y Fp(n)p Fo(+1)1549 3736 y Fs(\()p Fq(t;)15 b(X)7 b Fs(\))27 b(=)e Fq(d)20 b Fm(\000)g Fs(1)h(+)f Fq(d)2259 3750 y Fp(n)2331 3736 y Fs(=)25 b(\()p Fq(n)20 b Fs(+)g(1\)\()p Fq(d)i Fm(\000)e Fs(1\))h(+)f(1)75 3909 y(and)601 4083 y Fq(q)642 4098 y Fp(n)p Fo(+1)p Fp(;d)831 4107 y Fc(n)p Fh(+1)955 4083 y Fs(\()p Fq(t)p Fs(\))25 b(=)1340 4022 y Fq(d)1387 4036 y Fp(n)p Fo(+1)1525 4022 y Fs(!)p 1189 4062 512 4 v 1189 4146 a Fq(d)p Fs(!\()p Fq(d)1343 4160 y Fp(n)p Fo(+1)1502 4146 y Fm(\000)20 b Fq(d)p Fs(\)!)1711 4083 y Fq(p)1757 4098 y Fp(d)1797 4083 y Fs(\()p Fq(t)p Fs(\))p Fq(q)1941 4098 y Fp(n;d)2040 4106 y Fc(n)2086 4083 y Fs(\()p Fq(t)p Fs(\))26 b(=)f(\()q Fq(p)2393 4098 y Fp(d)2433 4083 y Fs(\()p Fq(t)p Fs(\)\))2572 4041 y Fp(n)p Fo(+1)2724 3970 y Fp(n)p Fo(+1)2732 3997 y Fn(Y)2729 4192 y Fp(j)t Fo(=1)3033 4021 y Fq(d)3080 4035 y Fp(j)3117 4021 y Fs(!)p 2882 4062 411 4 v 2882 4146 a Fq(d)p Fs(!\()p Fq(d)3036 4160 y Fp(j)3094 4146 y Fm(\000)20 b Fq(d)p Fs(\)!)3303 4083 y Fq(:)75 4324 y Fs(This)30 b(pro)m(v)m(es)h(our)f(lemma.)75 4486 y Fe(Lemma)35 b(5)45 b Fr(L)-5 b(et)35 b Fq(m)29 b Fm(2)f Fl(N)34 b Fr(with)h Fq(m)28 b Fm(\025)h Fs(1)34 b Fr(and)i Fe(p)28 b Fm(2)h(P)1904 4500 y Fo(2)1943 4486 y Fs(\()p Fq(m)p Fs(\))p Fr(.)48 b(Under)35 b(the)f(assumptions)j(of)e (L)-5 b(emma)35 b(4,)h(ther)-5 b(e)75 4599 y(exist)33 b(a)g(p)-5 b(ositive)33 b(inte)-5 b(ger)33 b Fq(D)1069 4613 y Fp(m)1169 4599 y Fr(and)g(holomorphic)j(c)-5 b(o)g(e\016cients) 34 b Fq(Q)2398 4566 y Fp(m)2398 4625 y(j)2464 4599 y Fs(\()p Fq(t)p Fs(\))f Fr(on)g Fm(D)s Fr(,)f(such)h(that)1316 4761 y Fp(m)1289 4788 y Fn(Y)1282 4986 y Fp(k)r Fo(=1)1426 4719 y Fn( )1508 4813 y Fq(\036)1562 4780 y Fo(\()p Fp(k)r Fo(\))1660 4813 y Fs(\()p Fq(t)p Fs(\))p 1508 4854 256 4 v 1598 4937 a Fq(k)s Fs(!)1773 4719 y Fn(!)1845 4742 y Fp(p)1881 4754 y Fc(k)1948 4875 y Fs(=)2052 4761 y Fp(D)2110 4769 y Fc(m)2044 4788 y Fn(X)2049 4984 y Fp(j)t Fo(=0)2191 4875 y Fq(Q)2263 4837 y Fp(m)2263 4897 y(j)2329 4875 y Fs(\()p Fq(t)p Fs(\))2442 4813 y Fq(\036)p Fs(\()p Fq(t)p Fs(\))2599 4780 y Fp(j)p 2442 4854 195 4 v 2506 4937 a Fq(j)5 b Fs(!)75 5138 y Fr(wher)-5 b(e)1526 5251 y Fq(D)1601 5265 y Fp(m)1694 5251 y Fs(=)25 b Fq(m)p Fs(\()p Fq(d)20 b Fm(\000)g Fs(1\))h(+)f Fm(j)p Fe(p)p Fm(j)2363 5265 y Fo(1)75 5402 y Fr(and)1004 5599 y Fq(Q)1076 5562 y Fp(m)1076 5622 y(D)1134 5630 y Fc(m)1197 5599 y Fs(\()p Fq(t)p Fs(\))26 b(=)f Fq(D)1497 5613 y Fp(m)1563 5599 y Fs(!)15 b(\()q Fq(p)1685 5614 y Fp(d)1725 5599 y Fs(\()p Fq(t)p Fs(\)\))1864 5557 y Fp(m)1946 5417 y Fn(0)1946 5580 y(@)2059 5486 y Fp(m)2032 5513 y Fn(Y)2026 5711 y Fp(k)r Fo(=1)2179 5443 y Fn(Q)2265 5470 y Fp(k)2265 5538 y(j)t Fo(=1)2518 5464 y Fp(d)2554 5474 y Fc(j)2587 5464 y Fo(!)p 2417 5491 291 4 v 2417 5544 a Fp(d)p Fo(!\()p Fp(d)2536 5554 y Fc(j)2569 5544 y Fy(\000)p Fp(d)p Fo(\)!)p 2179 5578 539 4 v 2353 5662 a Fq(k)s Fs(!)p Fq(d)2475 5677 y Fp(k)2519 5662 y Fs(!)2727 5417 y Fn(1)2727 5580 y(A)2807 5439 y Fp(p)2843 5451 y Fc(k)2900 5599 y Fq(:)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 3764 100 a Fs(19)75 399 y Fe(Pro)s(of)56 b Fs(Using)31 b(Lemma)g(4,)g(w)m(e)f(ha)m(v)m(e)1051 591 y Fp(m)1025 618 y Fn(Y)1018 816 y Fp(k)r Fo(=1)1162 549 y Fn( )1244 643 y Fq(\036)1298 610 y Fo(\()p Fp(k)r Fo(\))1396 643 y Fs(\()p Fq(t)p Fs(\))p 1244 684 256 4 v 1334 767 a Fq(k)s Fs(!)1509 549 y Fn(!)1581 572 y Fp(p)1617 584 y Fc(k)1684 704 y Fs(=)1813 591 y Fp(m)1787 618 y Fn(Y)1780 816 y Fp(k)r Fo(=1)1924 522 y Fn(0)1924 685 y(@)2029 643 y Fs(1)p 2014 684 76 4 v 2014 767 a Fq(k)s Fs(!)2143 587 y Fp(d)2179 599 y Fc(k)2114 618 y Fn(X)2119 814 y Fp(j)t Fo(=0)2261 704 y Fq(q)2302 719 y Fp(k)r(;j)2396 704 y Fs(\()p Fq(t)p Fs(\))2509 643 y Fq(\036)p Fs(\()p Fq(t)p Fs(\))2666 610 y Fp(j)p 2509 684 195 4 v 2572 767 a Fq(j)5 b Fs(!)2714 522 y Fn(1)2714 685 y(A)2793 544 y Fp(p)2829 556 y Fc(k)2886 704 y Fq(:)75 1013 y Fs(In)30 b(the)g(righ)m(t)h(handside,)f(the)g(higest)h(p)s(o)m (w)m(er)g(of)f Fq(\036)p Fs(\()p Fq(t)p Fs(\))h(is)308 1287 y Fq(D)383 1301 y Fp(m)475 1287 y Fs(:=)631 1173 y Fp(m)596 1201 y Fn(X)598 1398 y Fp(k)r 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Fq(;)15 b(r)s Fs(\))9 b Fm(n)g(f)p Fq(t)3342 5641 y Fo(0)3383 5627 y Fm(g)p Fs(,)27 b(otherwise)75 5740 y(the)k(p)s(oin)m(t)f Fq(t)502 5754 y Fo(0)572 5740 y Fs(is)g(called)h(an)g(algebraic)h (branc)m(h)e(p)s(oin)m(t.)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 3764 100 a Fs(21)75 399 y Fg(4.2)112 b(An)38 b(auxiliary)h(linear)f(Cauc)m(h)m(y)g(problem.)75 570 y Fs(Let)1467 683 y Fq(v)1511 698 y Fp(n)1554 707 y Fh(0)1589 698 y Fp(;`;\014)1729 683 y Fs(:=)25 b(sup)1854 761 y Fp(t)p Fy(2D)2002 683 y Fm(j)q Fq(@)2081 642 y Fp(n)2124 651 y Fh(0)2076 707 y Fp(t)2163 683 y Fq(X)2238 698 y Fp(`;\014)2333 683 y Fs(\()p Fq(t)p Fs(\))p Fm(j)75 905 y Fs(for)30 b(all)h Fq(n)395 919 y Fo(0)434 905 y Fs(,)f Fq(`)p Fs(,)g Fq(\014)h Fm(\025)25 b Fs(0,)31 b(where)e(\()p Fq(X)1233 920 y Fp(`;\014)1329 905 y Fs(\()p Fq(t)p Fs(\)\))1468 937 y Fp(`)p Fy(\025)p Fo(0)p Fp(;\014)s Fy(\025)p Fo(0)1774 905 y Fs(is)h(the)h(sequence)f(constructed)g(in)g (Prop)s(osition)g(7.)41 b(In)30 b(this)75 1018 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5555 y Fo(1)2477 5541 y Fs(\)!)26 b(=:)f Fq(A)2752 5556 y Fp(n)2795 5565 y Fh(0)2829 5556 y Fy(\000)p Fp(n)2927 5565 y Fh(1)2962 5556 y Fp(;j;\014)s Fy(\000)p Fp(\014)3169 5565 y Fh(2)3206 5541 y Fq(:)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 3764 100 a Fs(22)75 399 y(Therefore,)30 b(w)m(e)h(obtain)g(from)f(\(32\))75 656 y(\(34\))352 b Fq(v)631 671 y Fp(n)674 680 y Fh(0)709 671 y Fp(;`;\014)s Fo(+)p Fp(S)951 656 y Fm(\024)1192 570 y Fn(X)1047 771 y Fi(k)p Fo(=\()p Fp(k)1209 780 y Fh(0)1244 771 y Fp(;k)1301 780 y Fh(1)1334 771 y Fo(\))p Fy(2J)1545 538 y Fp(\014)1501 570 y Fn(X)1484 768 y Fp(\014)1524 777 y Fh(2)1558 768 y Fo(=0)1704 542 y Fp(k)1741 551 y Fh(0)1674 570 y Fn(X)1664 764 y Fp(m)p Fo(=0)1980 570 y Fn(X)1832 771 y Fo(\()p Fp(L;)p Fi(p)p Fp(;j)t Fo(\))p Fy(2)p Fo(\001)2158 783 y Fc(`;m)2321 542 y Fp(n)2364 551 y Fh(0)2294 570 y Fn(X)2276 764 y Fp(n)2319 773 y Fh(1)2353 764 y Fo(=0)2663 595 y Fq(\014)5 b Fs(!)p 2468 636 471 4 v 2468 719 a Fq(\014)2519 733 y Fo(2)2559 719 y Fs(!\()p Fq(\014)26 b Fm(\000)20 b Fq(\014)2838 733 y Fo(2)2878 719 y Fs(\)!)3089 595 y Fq(k)3136 609 y Fo(0)3176 595 y Fs(!)p 2958 636 374 4 v 2958 719 a(\()p Fq(k)3040 733 y Fo(0)3101 719 y Fm(\000)g Fq(m)p Fs(\)!)683 988 y Fm(\002)933 927 y Fs(1)p 764 967 385 4 v 764 1050 a Fq(p)810 1064 y Fo(1)849 1050 y Fs(!)15 b Fq(:)g(:)g(:)i(p)1057 1064 y Fp(m)1123 1050 y Fs(!)1314 927 y Fq(`)p Fs(!)p 1168 967 356 4 v 1168 1050 a Fq(j)5 b Fs(!\()p Fq(`)21 b Fm(\000)f Fq(j)5 b Fs(\)!)1741 927 y Fq(n)1796 941 y Fo(0)1835 927 y Fs(!)p 1544 967 515 4 v 1544 1050 a Fq(n)1599 1064 y Fo(1)1638 1050 y Fs(!\()p Fq(n)1753 1064 y Fo(0)1812 1050 y Fm(\000)20 b Fq(n)1958 1064 y Fo(1)1997 1050 y Fs(\)!)2068 988 y Fq(A)2136 1003 y Fp(n)2179 1012 y Fh(0)2213 1003 y Fy(\000)p Fp(n)2311 1012 y Fh(1)2346 1003 y Fp(;j;\014)s Fy(\000)p Fp(\014)2553 1012 y Fh(2)2590 988 y Fq(v)2634 1003 y Fp(k)2671 1012 y Fh(0)2705 1003 y Fy(\000)p Fp(m)p Fo(+)p Fp(n)2920 1012 y Fh(1)2955 1003 y Fp(;L;\014)3083 1012 y Fh(2)3116 1003 y Fo(+)p Fp(k)3208 1012 y Fh(1)75 1185 y Fs(for)30 b(all)h Fq(n)395 1199 y Fo(0)434 1185 y Fs(,)g Fq(`)p Fs(,)f Fq(\014)h Fm(\025)25 b Fs(0.)41 b(No)m(w,)31 b(let)h(\()p Fq(u)1323 1200 y Fp(n)1366 1209 y Fh(0)1400 1200 y Fp(;`;\014)1516 1185 y Fs(\))1551 1216 y Fp(n)1594 1225 y Fh(0)1628 1216 y Fy(\025)p Fo(0)p Fp(;`)p Fy(\025)p Fo(0)p Fp(;\014)s Fy(\025)p Fo(0)2044 1185 y Fs(the)f(sequence)g (satisfying)75 1465 y(\(35\))227 b Fq(u)514 1480 y Fp(n)557 1489 y Fh(0)592 1480 y Fp(;`;\014)s Fo(+)p Fp(S)834 1465 y Fs(=)1075 1378 y Fn(X)930 1580 y Fi(k)p Fo(=\()p Fp(k)1092 1589 y Fh(0)1126 1580 y Fp(;k)1183 1589 y Fh(1)1217 1580 y Fo(\))p Fy(2J)1428 1346 y Fp(\014)1383 1378 y Fn(X)1367 1576 y Fp(\014)1407 1585 y Fh(2)1441 1576 y Fo(=0)1587 1350 y Fp(k)1624 1359 y Fh(0)1557 1378 y Fn(X)1546 1573 y Fp(m)p Fo(=0)1863 1378 y Fn(X)1714 1580 y Fo(\()p Fp(L;)p Fi(p)p Fp(;j)t Fo(\))p Fy(2)p Fo(\001)2040 1592 y Fc(`;m)2203 1350 y Fp(n)2246 1359 y Fh(0)2176 1378 y Fn(X)2158 1573 y Fp(n)2201 1582 y Fh(1)2236 1573 y Fo(=0)2545 1403 y Fq(\014)5 b Fs(!)p 2351 1444 471 4 v 2351 1527 a Fq(\014)2402 1541 y Fo(2)2442 1527 y Fs(!\()p Fq(\014)26 b Fm(\000)20 b Fq(\014)2721 1541 y Fo(2)2761 1527 y Fs(\)!)2972 1403 y Fq(k)3019 1417 y Fo(0)3059 1403 y Fs(!)p 2841 1444 374 4 v 2841 1527 a(\()p Fq(k)2923 1541 y Fo(0)2983 1527 y Fm(\000)g Fq(m)p Fs(\)!)3404 1403 y(1)p 3235 1444 385 4 v 3235 1527 a Fq(p)3281 1541 y Fo(1)3320 1527 y Fs(!)15 b Fq(:)g(:)g(:)h(p)3527 1541 y Fp(m)3594 1527 y Fs(!)1063 1796 y Fm(\002)1290 1735 y Fq(`)p Fs(!)p 1144 1775 356 4 v 1144 1859 a Fq(j)5 b Fs(!\()p Fq(`)21 b Fm(\000)f Fq(j)5 b Fs(\)!)1717 1735 y Fq(n)1772 1749 y Fo(0)1811 1735 y Fs(!)p 1519 1775 515 4 v 1519 1859 a Fq(n)1574 1873 y Fo(1)1613 1859 y Fs(!\()p Fq(n)1728 1873 y Fo(0)1788 1859 y Fm(\000)20 b Fq(n)1934 1873 y Fo(1)1973 1859 y Fs(\)!)2044 1796 y Fq(A)2112 1811 y Fp(n)2155 1820 y Fh(0)2189 1811 y Fy(\000)p Fp(n)2287 1820 y Fh(1)2321 1811 y Fp(;j;\014)s Fy(\000)p Fp(\014)2528 1820 y Fh(2)2565 1796 y Fq(u)2617 1811 y Fp(k)2654 1820 y Fh(0)2689 1811 y Fy(\000)p Fp(m)p Fo(+)p Fp(n)2904 1820 y Fh(1)2938 1811 y Fp(;L;\014)3066 1820 y Fh(2)3100 1811 y Fo(+)p Fp(k)3192 1820 y Fh(1)75 1988 y Fs(for)30 b(all)h Fq(n)395 2002 y Fo(0)434 1988 y Fs(,)g Fq(`)p Fs(,)f Fq(\014)h Fm(\025)25 b Fs(0.)75 2101 y(F)-8 b(rom)38 b(\(17\),)k(w)m(e)c(ha)m(v)m (e)h(0)f Fm(\024)g Fq(k)1145 2115 y Fo(1)1222 2101 y Fm(\024)g Fq(S)30 b Fm(\000)25 b Fs(1)38 b(for)g(all)g Fe(k)g Fs(=)g(\()p Fq(k)2160 2115 y Fo(0)2200 2101 y Fq(;)15 b(k)2287 2115 y Fo(1)2327 2101 y Fs(\))38 b Fm(2)f(J)17 b Fs(.)63 b(W)-8 b(e)39 b(deduce)f(that)g(the)g(sequence)75 2214 y(\()p Fq(u)162 2229 y Fp(n)205 2238 y Fh(0)240 2229 y Fp(;`;\014)355 2214 y Fs(\))390 2245 y Fp(n)433 2254 y Fh(0)468 2245 y Fy(\025)p Fo(0)p Fp(;`)p Fy(\025)p Fo(0)p Fp(;\014)s Fy(\025)p Fo(0)884 2214 y Fs(is)30 b(uniquely)g(determined)g(b)m(y)g(\(35\))i(and)e(the)g(initial)i (conditions)75 2389 y(\(36\))1049 b Fq(u)1336 2404 y Fp(n)1379 2413 y Fh(0)1414 2404 y Fp(;`;\014)1554 2389 y Fs(:=)25 b Fq(v)1719 2404 y Fp(n)1762 2413 y Fh(0)1797 2404 y Fp(;`;\014)1937 2389 y Fs(=)g(sup)2037 2467 y Fp(t)p Fy(2D)2185 2389 y Fm(j)q Fq(@)2264 2348 y Fp(n)2307 2357 y Fh(0)2259 2413 y Fp(t)2345 2389 y Fq(X)2420 2404 y Fp(`;\014)2516 2389 y Fs(\()p Fq(t)p Fs(\))p Fm(j)75 2610 y Fs(for)30 b Fq(\014)h Fs(=)25 b(0)p Fq(;)15 b(:)g(:)g(:)i(;)e(S) 25 b Fm(\000)20 b Fs(1)31 b(and)f(for)g(all)h Fq(n)1384 2624 y Fo(0)1423 2610 y Fs(,)g Fq(`)25 b Fm(\025)g Fs(0.)41 b(W)-8 b(e)31 b(deduce)f(from)g(\(34\),)i(\(35\))g(and)e(\(36\))i(that) 75 2776 y(\(37\))1428 b Fq(v)1707 2791 y Fp(n)1750 2800 y Fh(0)1785 2791 y Fp(;`;\014)1925 2776 y Fm(\024)25 b Fq(u)2073 2791 y Fp(n)2116 2800 y Fh(0)2151 2791 y Fp(;`;\014)75 2943 y Fs(for)30 b(all)h Fq(n)395 2957 y Fo(0)434 2943 y Fs(,)g Fq(`)p Fs(,)f Fq(\014)h Fm(\025)25 b Fs(0.)216 3093 y(No)m(w,)31 b(w)m(e)g(de\014ne)f(the)g(formal)h (series)75 3312 y(\(38\))938 b Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))27 b(=)1664 3226 y Fn(X)1663 3424 y Fp(\014)s Fy(\025)p Fo(0)1811 3226 y Fn(X)1817 3424 y Fp(`)p Fy(\025)p Fo(0)1976 3226 y Fn(X)1958 3422 y Fp(n)2001 3431 y Fh(0)2035 3422 y Fy(\025)p Fo(0)2140 3312 y Fq(u)2192 3327 y Fp(n)2235 3336 y Fh(0)2270 3327 y Fp(;`;\014)2397 3251 y Fq(t)2430 3218 y Fp(n)2473 3227 y Fh(0)p 2395 3292 120 4 v 2395 3375 a Fq(n)2450 3389 y Fo(0)2489 3375 y Fs(!)2534 3251 y Fq(T)2600 3218 y Fp(`)p 2534 3292 99 4 v 2552 3375 a Fq(`)p Fs(!)2653 3251 y Fq(z)2699 3218 y Fp(\014)p 2653 3292 94 4 v 2659 3375 a Fq(\014)5 b Fs(!)75 3575 y(where)35 b(the)g(co)s(e\016cien)m(ts) i Fq(u)1021 3590 y Fp(n)1064 3599 y Fh(0)1099 3590 y Fp(;`;\014)1249 3575 y Fs(satisfy)f(\(35\))h(and)d(\(36\))j(for)e(all)i Fq(n)2444 3589 y Fo(0)2483 3575 y Fs(,)f Fq(`)p Fs(,)h Fq(\014)h Fm(\025)c Fs(0.)56 b(W)-8 b(e)36 b(ha)m(v)m(e)h(the)e(follo)m (wing)75 3688 y(prop)s(osition.)75 3844 y Fe(Prop)s(osition)h(8)46 b Fr(The)33 b(formal)h(series)f(\(38\))h(is)e(the)h(unique)f(solution)i (of)f(the)g(Cauchy)g(pr)-5 b(oblem)75 4024 y Fs(\(39\))602 b Fq(@)890 3986 y Fp(S)885 4047 y(z)941 4024 y Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))27 b(=)1613 3938 y Fn(X)1431 4139 y Fi(q)p Fo(=\()p Fp(q)1588 4148 y Fh(0)1622 4139 y Fp(;q)1674 4148 y Fh(1)1708 4139 y Fp(;q)1760 4148 y Fh(2)1794 4139 y Fo(\))p Fy(2Q)1941 4024 y Fq(B)2010 4038 y Fi(q)2057 4024 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))p Fq(@)2405 3980 y Fp(q)2437 3989 y Fh(0)2400 4048 y Fp(t)2477 4024 y Fq(@)2530 3980 y Fp(q)2562 3989 y Fh(1)2525 4053 y Fp(T)2600 4024 y Fq(@)2653 3986 y Fp(q)2685 3995 y Fh(2)2648 4047 y Fp(z)2724 4024 y Fq(U)d Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))75 4289 y Fr(for)33 b(given)f(initial)h(c) -5 b(onditions)75 4509 y Fs(\(40\))666 b(\()p Fq(@)989 4472 y Fp(j)984 4532 y(z)1026 4509 y Fq(U)10 b Fs(\)\()p Fq(t;)15 b Fs(0)p Fq(;)g(T)e Fs(\))27 b(=)1602 4423 y Fn(X)1550 4620 y Fp(`;n)1642 4629 y Fh(0)1676 4620 y Fp(;)p Fy(\025)p Fo(0)1801 4509 y Fq(v)1845 4524 y Fp(n)1888 4533 y Fh(0)1922 4524 y Fp(;`;j)2039 4448 y Fq(t)2072 4415 y Fp(n)2115 4424 y Fh(0)p 2037 4488 120 4 v 2037 4572 a Fq(n)2092 4586 y Fo(0)2131 4572 y Fs(!)2176 4448 y Fq(T)2242 4415 y Fp(`)p 2176 4488 99 4 v 2194 4572 a Fq(`)p Fs(!)2350 4509 y Fq(;)80 b Fs(0)26 b Fm(\024)f Fq(j)31 b Fm(\024)25 b Fq(S)g Fm(\000)20 b Fs(1)p Fq(;)75 4767 y Fr(wher)-5 b(e)335 4933 y Fm(Q)25 b Fs(=)530 4860 y Fn(\010)583 4933 y Fe(q)h Fs(=)f(\()p Fq(q)836 4947 y Fo(0)875 4933 y Fq(;)15 b(q)956 4947 y Fo(1)996 4933 y Fq(;)g(q)1077 4947 y Fo(2)1116 4933 y Fs(\))26 b Fm(2)e Fl(N)1328 4896 y Fo(3)1383 4933 y Fm(j)h Fe(k)g Fs(=)g(\()p Fq(k)1691 4947 y Fo(0)1732 4933 y Fq(;)15 b(k)1819 4947 y Fo(1)1859 4933 y Fs(\))25 b Fm(2)g(J)17 b Fs(;)40 b Fq(m)25 b Fs(=)g(0)p Fq(;)15 b(:)g(:)g(:)i(;)e(k)2644 4947 y Fo(0)2685 4933 y Fs(;)40 b Fe(p)26 b Fm(2)e(P)2982 4947 y Fo(2)3022 4933 y Fs(\()p Fq(m)p Fs(\);)1509 5100 y Fq(q)1550 5114 y Fo(0)1614 5100 y Fs(=)h Fq(k)1757 5114 y Fo(0)1817 5100 y Fm(\000)20 b Fq(m)p Fs(;)41 b Fq(q)2095 5114 y Fo(1)2159 5100 y Fs(=)25 b Fm(j)p Fe(p)p Fm(j)2363 5114 y Fo(1)2403 5100 y Fs(;)41 b Fq(q)2510 5114 y Fo(2)2574 5100 y Fs(=)25 b Fq(k)2717 5114 y Fo(1)2757 5100 y Fm(g)75 5247 y Fr(and)75 5477 y Fs(\(41\))685 b Fq(B)989 5491 y Fi(q)1036 5477 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))27 b(=)1464 5416 y Fq(k)1511 5430 y Fo(0)1551 5416 y Fs(!)p 1464 5456 113 4 v 1467 5540 a Fq(q)1508 5554 y Fo(0)1547 5540 y Fs(!)1765 5416 y(1)p 1596 5456 385 4 v 1596 5540 a Fq(p)1642 5554 y Fo(1)1681 5540 y Fs(!)15 b Fq(:)g(:)g(:)i(p)1889 5554 y Fp(m)1955 5540 y Fs(!)2006 5391 y Fn(X)2005 5589 y Fp(\014)s Fy(\025)p Fo(0)2161 5363 y Fp(D)2219 5371 y Fc(m)2154 5391 y Fn(X)2158 5586 y Fp(j)t Fo(=0)2301 5391 y Fn(X)2300 5587 y Fp(n)p Fy(\025)p Fo(0)2448 5477 y Fq(A)2516 5492 y Fp(n;j;\014)2684 5416 y Fq(t)2717 5383 y Fp(n)p 2684 5456 80 4 v 2684 5540 a Fq(n)p Fs(!)2784 5416 y Fq(T)2850 5383 y Fp(j)p 2784 5456 103 4 v 2801 5540 a Fq(j)5 b Fs(!)2906 5416 y Fq(z)2952 5383 y Fp(\014)p 2906 5456 94 4 v 2912 5540 a Fq(\014)g Fs(!)75 5740 y Fr(wher)-5 b(e)34 b(the)f(c)-5 b(o)g(e\016cients)33 b Fq(A)1014 5755 y Fp(n;j;\014)1205 5740 y Fr(ar)-5 b(e)33 b(de\014ne)-5 b(d)34 b(by)e(\(33\).)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 3764 100 a Fs(23)75 399 y Fe(Pro)s(of)63 b Fs(W)-8 b(e)38 b(plug)f Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))38 b(in)m(to)g(\(39\))h(using)e(the)g(expansion)g (\(38\).)62 b(F)-8 b(or)38 b(the)f(left-hand)g(side)g(term)g(of)75 511 y(\(39\),)32 b(w)m(e)f(obtain)75 774 y(\(42\))823 b Fq(@)1111 737 y Fp(S)1106 797 y(z)1162 774 y Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))27 b(=)1652 688 y Fn(X)1652 886 y Fp(\014)s Fy(\025)p Fo(0)1800 688 y Fn(X)1806 886 y Fp(`)p Fy(\025)p Fo(0)1964 688 y Fn(X)1946 884 y Fp(n)1989 893 y Fh(0)2024 884 y Fy(\025)p Fo(0)2129 774 y Fq(u)2181 789 y Fp(n)2224 798 y Fh(0)2259 789 y Fp(;`;\014)s Fo(+)p Fp(S)2488 713 y Fq(t)2521 680 y Fp(n)2564 689 y Fh(0)p 2485 753 120 4 v 2485 837 a Fq(n)2540 851 y Fo(0)2579 837 y Fs(!)2624 713 y Fq(T)2690 680 y Fp(`)p 2624 753 99 4 v 2642 837 a Fq(`)p Fs(!)2743 713 y Fq(z)2789 680 y Fp(\014)p 2743 753 94 4 v 2749 837 a Fq(\014)5 b Fs(!)2846 774 y Fq(:)75 1070 y Fs(F)-8 b(or)31 b(the)g(righ)m(t-hand) f(side)g(term,)h(a)g(\014rst)e(calculus)i(sho)m(ws)f(us)g(that)794 1328 y Fq(@)847 1284 y Fp(q)879 1293 y Fh(0)842 1352 y Fp(t)917 1328 y Fq(@)970 1284 y Fp(q)1002 1293 y Fh(1)965 1357 y Fp(T)1041 1328 y Fq(@)1094 1291 y Fp(q)1126 1300 y Fh(2)1089 1351 y Fp(z)1165 1328 y Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))26 b(=)1655 1242 y Fn(X)1654 1440 y Fp(\014)s Fy(\025)p Fo(0)1802 1242 y Fn(X)1809 1440 y Fp(`)p Fy(\025)p Fo(0)1967 1242 y Fn(X)1949 1438 y Fp(n)1992 1447 y Fh(0)2026 1438 y Fy(\025)p Fo(0)2132 1328 y Fq(u)2184 1343 y Fp(n)2227 1352 y Fh(0)2261 1343 y Fo(+)p Fp(q)2348 1352 y Fh(0)2382 1343 y Fp(;`)p Fo(+)p Fp(q)2518 1352 y Fh(1)2551 1343 y Fp(;\014)s Fo(+)p Fp(q)2701 1352 y Fh(2)2752 1267 y Fq(t)2785 1234 y Fp(n)2828 1243 y Fh(0)p 2749 1307 120 4 v 2749 1391 a Fq(n)2804 1405 y Fo(0)2843 1391 y Fs(!)2888 1267 y Fq(T)2954 1234 y Fp(`)p 2888 1307 99 4 v 2906 1391 a Fq(`)p Fs(!)3007 1267 y Fq(z)3053 1234 y Fp(\014)p 3007 1307 94 4 v 3013 1391 a Fq(\014)5 b Fs(!)3110 1328 y Fq(:)75 1629 y Fs(Using)31 b(for)f Fq(B)539 1643 y Fi(q)586 1629 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))31 b(the)g(expansion)f(\(41\),)i(w)m(e)f(obtain)75 1942 y(\(43\))299 b Fq(B)603 1956 y Fi(q)650 1942 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))p Fq(@)998 1898 y Fp(q)1030 1907 y Fh(0)993 1966 y Fp(t)1070 1942 y Fq(@)1123 1898 y Fp(q)1155 1907 y Fh(1)1118 1971 y Fp(T)1194 1942 y Fq(@)1247 1904 y Fp(q)1279 1913 y Fh(2)1242 1964 y Fp(z)1317 1942 y Fq(U)d Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))27 b(=)1808 1856 y Fn(X)1807 2053 y Fp(\014)s Fy(\025)p Fo(0)1955 1856 y Fn(X)1961 2053 y Fp(`)p Fy(\025)p Fo(0)2120 1856 y Fn(X)2102 2052 y Fp(n)2145 2061 y Fh(0)2179 2052 y Fy(\025)p Fo(0)2285 1759 y Fn(0)2285 1923 y(@)2425 1824 y Fp(\014)2381 1856 y Fn(X)2364 2053 y Fp(\014)2404 2062 y Fh(2)2438 2053 y Fo(=0)2606 1828 y Fp(`)2555 1856 y Fn(X)2544 2053 y Fp(`)2573 2062 y Fh(1)2607 2053 y Fo(=0)2757 1827 y Fp(n)2800 1836 y Fh(0)2730 1856 y Fn(X)2712 2050 y Fp(n)2755 2059 y Fh(1)2790 2050 y Fo(=0)3099 1880 y Fq(\014)5 b Fs(!)p 2905 1921 471 4 v 2905 2004 a Fq(\014)2956 2018 y Fo(2)2996 2004 y Fs(!\()p Fq(\014)26 b Fm(\000)20 b Fq(\014)3275 2018 y Fo(2)3315 2004 y Fs(\)!)293 2323 y Fm(\002)555 2262 y Fq(`)p Fs(!)p 374 2302 425 4 v 374 2385 a Fq(`)412 2399 y Fo(1)451 2385 y Fs(!\()p Fq(`)h Fm(\000)f Fq(`)699 2399 y Fo(1)738 2385 y Fs(\)!)1016 2262 y Fq(n)1071 2276 y Fo(0)1110 2262 y Fs(!)p 819 2302 515 4 v 819 2385 a Fq(n)874 2399 y Fo(1)913 2385 y Fs(!\()p Fq(n)1028 2399 y Fo(0)1087 2385 y Fm(\000)g Fq(n)1233 2399 y Fo(1)1272 2385 y Fs(\)!)1353 2262 y Fq(k)1400 2276 y Fo(0)1440 2262 y Fs(!)p 1353 2302 113 4 v 1356 2385 a Fq(q)1397 2399 y Fo(0)1436 2385 y Fs(!)1654 2262 y(1)p 1485 2302 385 4 v 1485 2385 a Fq(p)1531 2399 y Fo(1)1570 2385 y Fs(!)15 b Fq(:)g(:)g(:)i(p)1778 2399 y Fp(m)1844 2385 y Fs(!)1879 2323 y Fq(A)1947 2338 y Fp(n)1990 2347 y Fh(0)2025 2338 y Fy(\000)p Fp(n)2123 2347 y Fh(1)2157 2338 y Fp(;`)p Fy(\000)p Fp(`)2290 2347 y Fh(1)2324 2338 y Fp(;\014)s Fy(\000)p Fp(\014)2482 2347 y Fh(2)2520 2323 y Fq(u)2572 2338 y Fp(n)2615 2347 y Fh(1)2649 2338 y Fo(+)p Fp(q)2736 2347 y Fh(0)2770 2338 y Fp(;`)2819 2347 y Fh(1)2853 2338 y Fo(+)p Fp(q)2940 2347 y Fh(1)2974 2338 y Fp(;\014)3034 2347 y Fh(2)3068 2338 y Fo(+)p Fp(q)3155 2347 y Fh(2)3193 2195 y Fn(\023)3287 2262 y Fq(t)3320 2229 y Fp(n)3363 2238 y Fh(0)p 3285 2302 120 4 v 3285 2385 a Fq(n)3340 2399 y Fo(0)3379 2385 y Fs(!)3424 2262 y Fq(T)3490 2229 y Fp(`)p 3424 2302 99 4 v 3442 2385 a Fq(`)p Fs(!)3543 2262 y Fq(z)3589 2229 y Fp(\014)p 3543 2302 94 4 v 3549 2385 a Fq(\014)5 b Fs(!)75 2535 y(where)30 b Fq(A)406 2550 y Fp(n)449 2559 y Fh(0)483 2550 y Fy(\000)p Fp(n)581 2559 y Fh(1)616 2550 y Fp(;`)p Fy(\000)p Fp(`)749 2559 y Fh(1)783 2550 y Fp(;\014)s Fy(\000)p Fp(\014)941 2559 y Fh(2)1004 2535 y Fs(=)25 b(0)30 b(for)g Fq(`)21 b Fm(\000)e Fq(`)1501 2549 y Fo(1)1566 2535 y Fq(>)25 b(D)1737 2549 y Fp(m)1804 2535 y Fs(.)75 2712 y(By)31 b(iden)m(ti\014cation)h(of)e(the)h(co)s (e\016cien)m(ts)h(of)1602 2650 y Fq(t)1635 2617 y Fp(n)1678 2626 y Fh(0)p 1599 2691 120 4 v 1599 2774 a Fq(n)1654 2788 y Fo(0)1693 2774 y Fs(!)1739 2650 y Fq(T)1805 2617 y Fp(`)p 1739 2691 99 4 v 1756 2774 a Fq(`)p Fs(!)1857 2650 y Fq(z)1903 2617 y Fp(\014)p 1857 2691 94 4 v 1863 2774 a Fq(\014)5 b Fs(!)1991 2712 y(in)30 b(\(39\),)i(using)e(\(42\))i (and)e(\(43\),)i(w)m(e)f(get)294 3065 y Fq(u)346 3080 y Fp(n)389 3089 y Fh(0)424 3080 y Fp(;`;\014)s Fo(+)p Fp(S)666 3065 y Fs(=)943 2979 y Fn(X)762 3180 y Fi(q)p Fo(=\()p Fp(q)919 3189 y Fh(0)953 3180 y Fp(;q)1005 3189 y Fh(1)1039 3180 y Fp(;q)1091 3189 y Fh(2)1124 3180 y Fo(\))p Fy(2Q)1332 2946 y Fp(\014)1288 2979 y Fn(X)1272 3176 y Fp(\014)1312 3185 y Fh(2)1346 3176 y Fo(=0)1513 2951 y Fp(`)1462 2979 y Fn(X)1451 3176 y Fp(`)1480 3185 y Fh(1)1514 3176 y Fo(=0)1665 2950 y Fp(n)1708 2959 y Fh(0)1638 2979 y Fn(X)1620 3173 y Fp(n)1663 3182 y Fh(1)1697 3173 y Fo(=0)2007 3003 y Fq(\014)5 b Fs(!)p 1813 3044 471 4 v 1813 3127 a Fq(\014)1864 3141 y Fo(2)1903 3127 y Fs(!\()p Fq(\014)26 b Fm(\000)20 b Fq(\014)2182 3141 y Fo(2)2222 3127 y Fs(\)!)2484 3003 y Fq(`)p Fs(!)p 2303 3044 425 4 v 2303 3127 a Fq(`)2341 3141 y Fo(1)2380 3127 y Fs(!\()p Fq(`)h Fm(\000)f Fq(`)2628 3141 y Fo(1)2667 3127 y Fs(\)!)2945 3003 y Fq(n)3000 3017 y Fo(0)3039 3003 y Fs(!)p 2747 3044 515 4 v 2747 3127 a Fq(n)2802 3141 y Fo(1)2841 3127 y Fs(!\()p Fq(n)2956 3141 y Fo(0)3016 3127 y Fm(\000)g Fq(n)3162 3141 y Fo(1)3201 3127 y Fs(\)!)1277 3430 y Fm(\002)1358 3368 y Fq(k)1405 3382 y Fo(0)1445 3368 y Fs(!)p 1358 3409 113 4 v 1361 3492 a Fq(q)1402 3506 y Fo(0)1441 3492 y Fs(!)1659 3368 y(1)p 1490 3409 385 4 v 1490 3492 a Fq(p)1536 3506 y Fo(1)1575 3492 y Fs(!)15 b Fq(:)g(:)g(:)i(p)1783 3506 y Fp(m)1849 3492 y Fs(!)1884 3430 y Fq(A)1952 3445 y Fp(n)1995 3454 y Fh(0)2030 3445 y Fy(\000)p Fp(n)2128 3454 y Fh(1)2162 3445 y Fp(;`)p Fy(\000)p Fp(`)2295 3454 y Fh(1)2329 3445 y Fp(;\014)s Fy(\000)p Fp(\014)2487 3454 y Fh(2)2525 3430 y Fq(u)2577 3445 y Fp(n)2620 3454 y Fh(1)2654 3445 y Fo(+)p Fp(q)2741 3454 y Fh(0)2775 3445 y Fp(;`)2824 3454 y Fh(1)2858 3445 y Fo(+)p Fp(q)2945 3454 y Fh(1)2979 3445 y Fp(;\014)3039 3454 y Fh(2)3073 3445 y Fo(+)p Fp(q)3160 3454 y Fh(2)75 3635 y Fs(for)30 b(all)h Fq(n)395 3649 y Fo(0)434 3635 y Fs(,)g Fq(`)p Fs(,)g Fq(\014)f Fm(\025)25 b Fs(0)31 b(and)f(where)g Fq(A)1345 3650 y Fp(n)1388 3659 y Fh(0)1422 3650 y Fy(\000)p Fp(n)1520 3659 y Fh(1)1554 3650 y Fp(;`)p Fy(\000)p Fp(`)1687 3659 y Fh(1)1721 3650 y Fp(;\014)s Fy(\000)p Fp(\014)1879 3659 y Fh(2)1942 3635 y Fs(=)25 b(0)31 b(for)f Fq(`)20 b Fm(\000)g Fq(`)2440 3649 y Fo(1)2505 3635 y Fq(>)25 b(D)2676 3649 y Fp(m)2743 3635 y Fs(.)40 b(Using)31 b(the)g(de\014nition)f(of)g(the)75 3748 y(set)h Fm(Q)p Fs(,)g(w)m(e)g(ha)m(v)m(e)75 4039 y Fq(u)127 4054 y Fp(n)170 4063 y Fh(0)204 4054 y Fp(;`;\014)s Fo(+)p Fp(S)446 4039 y Fs(=)688 3953 y Fn(X)542 4154 y Fi(k)p Fo(=\()p Fp(k)704 4163 y Fh(0)739 4154 y Fp(;k)796 4163 y Fh(1)830 4154 y Fo(\))p Fy(2J)1046 3924 y Fp(k)1083 3933 y Fh(0)1016 3953 y Fn(X)1005 4147 y Fp(m)p Fo(=0)1279 3953 y Fn(X)1198 4154 y Fi(p)p Fy(2P)1339 4163 y Fh(2)1374 4154 y Fo(\()p Fp(m)p Fo(\))1567 3921 y Fp(\014)1523 3953 y Fn(X)1507 4150 y Fp(\014)1547 4159 y Fh(2)1581 4150 y Fo(=0)1749 3925 y Fp(`)1697 3953 y Fn(X)1686 4150 y Fp(`)1715 4159 y Fh(1)1750 4150 y Fo(=0)1900 3924 y Fp(n)1943 3933 y Fh(0)1873 3953 y Fn(X)1855 4147 y Fp(n)1898 4156 y Fh(1)1932 4147 y Fo(=0)2242 3978 y Fq(\014)5 b Fs(!)p 2048 4018 471 4 v 2048 4101 a Fq(\014)2099 4115 y Fo(2)2138 4101 y Fs(!\()p Fq(\014)26 b Fm(\000)20 b Fq(\014)2417 4115 y Fo(2)2457 4101 y Fs(\)!)2719 3978 y Fq(`)p Fs(!)p 2538 4018 425 4 v 2538 4101 a Fq(`)2576 4115 y Fo(1)2615 4101 y Fs(!\()p Fq(`)h Fm(\000)f Fq(`)2863 4115 y Fo(1)2902 4101 y Fs(\)!)3180 3978 y Fq(n)3235 3992 y Fo(0)3274 3978 y Fs(!)p 2983 4018 515 4 v 2983 4101 a Fq(n)3038 4115 y Fo(1)3077 4101 y Fs(!\()p Fq(n)3192 4115 y Fo(0)3251 4101 y Fm(\000)g Fq(n)3397 4115 y Fo(1)3436 4101 y Fs(\)!)1056 4404 y Fm(\002)1267 4343 y Fq(k)1314 4357 y Fo(0)1354 4343 y Fs(!)p 1137 4383 374 4 v 1137 4466 a(\()p Fq(k)1219 4480 y Fo(0)1279 4466 y Fm(\000)g Fq(m)p Fs(\)!)1700 4343 y(1)p 1530 4383 385 4 v 1530 4466 a Fq(p)1576 4480 y Fo(1)1615 4466 y Fs(!)15 b Fq(:)g(:)g(:)i(p) 1823 4480 y Fp(m)1889 4466 y Fs(!)1925 4404 y Fq(A)1993 4419 y Fp(n)2036 4428 y Fh(0)2070 4419 y Fy(\000)p Fp(n)2168 4428 y Fh(1)2202 4419 y Fp(;`)p Fy(\000)p Fp(`)2335 4428 y Fh(1)2369 4419 y Fp(;\014)s Fy(\000)p Fp(\014)2527 4428 y Fh(2)2565 4404 y Fq(u)2617 4423 y Fp(n)2660 4432 y Fh(1)2694 4423 y Fo(+)p Fp(k)2786 4432 y Fh(0)2821 4423 y Fy(\000)p Fp(m;`)2987 4432 y Fh(1)3021 4423 y Fo(+)p Fy(j)p Fi(p)p Fy(j)3161 4432 y Fh(1)3195 4423 y Fp(;\014)3255 4432 y Fh(2)3289 4423 y Fo(+)p Fp(k)3381 4432 y Fh(1)75 4615 y Fs(for)22 b(all)g Fq(n)378 4629 y Fo(0)417 4615 y Fs(,)i Fq(`)p Fs(,)f Fq(\014)31 b Fm(\025)25 b Fs(0)d(and)f(where)g Fq(A)1287 4630 y Fp(n)1330 4639 y Fh(0)1365 4630 y Fy(\000)p Fp(n)1463 4639 y Fh(1)1497 4630 y Fp(;`)p Fy(\000)p Fp(`)1630 4639 y Fh(1)1664 4630 y Fp(;\014)s Fy(\000)p Fp(\014)1822 4639 y Fh(2)1885 4615 y Fs(=)k(0)d(for)g Fq(`)s Fm(\000)s Fq(`)2332 4629 y Fo(1)2396 4615 y Fq(>)j(D)2567 4629 y Fp(m)2634 4615 y Fs(.)37 b(Or)21 b(with)h(an)f(other)h(form)m(ulation,)75 4727 y(w)m(e)31 b(obtain)75 5001 y Fq(u)127 5016 y Fp(n)170 5025 y Fh(0)204 5016 y Fp(;`;\014)s Fo(+)p Fp(S)446 5001 y Fs(=)688 4914 y Fn(X)542 5116 y Fi(k)p Fo(=\()p Fp(k)704 5125 y Fh(0)739 5116 y Fp(;k)796 5125 y Fh(1)830 5116 y Fo(\))p Fy(2J)1046 4886 y Fp(k)1083 4895 y Fh(0)1016 4914 y Fn(X)1005 5109 y Fp(m)p Fo(=0)1279 4914 y Fn(X)1198 5116 y Fi(p)p Fy(2P)1339 5125 y Fh(2)1374 5116 y Fo(\()p Fp(m)p Fo(\))1567 4882 y Fp(\014)1523 4914 y Fn(X)1507 5112 y Fp(\014)1547 5121 y Fh(2)1581 5112 y Fo(=0)1705 4887 y Fp(D)1763 4895 y Fc(m)1697 4914 y Fn(X)1686 5112 y Fp(`)1715 5121 y Fh(2)1750 5112 y Fo(=0)1900 4886 y Fp(n)1943 4895 y Fh(0)1873 4914 y Fn(X)1855 5109 y Fp(n)1898 5118 y Fh(1)1932 5109 y Fo(=0)2242 4939 y Fq(\014)5 b Fs(!)p 2048 4980 471 4 v 2048 5063 a Fq(\014)2099 5077 y Fo(2)2138 5063 y Fs(!\()p Fq(\014)26 b Fm(\000)20 b Fq(\014)2417 5077 y Fo(2)2457 5063 y Fs(\)!)2719 4939 y Fq(`)p Fs(!)p 2538 4980 425 4 v 2538 5063 a Fq(`)2576 5077 y Fo(2)2615 5063 y Fs(!\()p Fq(`)h Fm(\000)f Fq(`)2863 5077 y Fo(2)2902 5063 y Fs(\)!)3180 4939 y Fq(n)3235 4953 y Fo(0)3274 4939 y Fs(!)p 2983 4980 515 4 v 2983 5063 a Fq(n)3038 5077 y Fo(1)3077 5063 y Fs(!\()p Fq(n)3192 5077 y Fo(0)3251 5063 y Fm(\000)g Fq(n)3397 5077 y Fo(1)3436 5063 y Fs(\)!)1056 5366 y Fm(\002)1267 5304 y Fq(k)1314 5318 y Fo(0)1354 5304 y Fs(!)p 1137 5345 374 4 v 1137 5428 a(\()p Fq(k)1219 5442 y Fo(0)1279 5428 y Fm(\000)g Fq(m)p Fs(\)!)1700 5304 y(1)p 1530 5345 385 4 v 1530 5428 a Fq(p)1576 5442 y Fo(1)1615 5428 y Fs(!)15 b Fq(:)g(:)g(:)i(p) 1823 5442 y Fp(m)1889 5428 y Fs(!)1925 5366 y Fq(A)1993 5381 y Fp(n)2036 5390 y Fh(0)2070 5381 y Fy(\000)p Fp(n)2168 5390 y Fh(1)2202 5381 y Fp(;`)2251 5390 y Fh(2)2285 5381 y Fp(;\014)s Fy(\000)p Fp(\014)2443 5390 y Fh(2)2481 5366 y Fq(u)2533 5384 y Fp(n)2576 5393 y Fh(1)2611 5384 y Fo(+)p Fp(k)2703 5393 y Fh(0)2737 5384 y Fy(\000)p Fp(m;`)p Fy(\000)p Fp(`)2987 5393 y Fh(2)3021 5384 y Fo(+)p Fy(j)p Fi(p)p Fy(j)3161 5393 y Fh(1)3195 5384 y Fp(;\014)3255 5393 y Fh(2)3289 5384 y Fo(+)p Fp(k)3381 5393 y Fh(1)75 5581 y Fs(for)30 b(all)h Fq(n)395 5595 y Fo(0)434 5581 y Fs(,)g Fq(`)p Fs(,)f Fq(\014)h Fm(\025)25 b Fs(0,)31 b(whic)m(h)f(is)g(exactly)i(equation)g(\(35\).)42 b(This)30 b(pro)m(v)m(es)h(our)e(prop)s(osition.)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 3764 100 a Fs(24)75 399 y Fg(4.3)112 b(The)38 b(main)h(result.)75 570 y Fs(W)-8 b(e)32 b(are)e(no)m(w)h(in)f (p)s(osition)g(to)h(state)h(the)f(main)f(result)g(of)h(the)f(pap)s(er.) 75 758 y Fe(Theorem)35 b(2)46 b Fr(Consider)33 b(an)h(or)-5 b(dinary)35 b(di\013er)-5 b(ential)33 b(e)-5 b(quation)75 962 y Fs(\(44\))1385 b Fq(\036)1674 924 y Fy(0)1697 962 y Fs(\()p Fq(t)p Fs(\))26 b(=)f Fq(P)j Fs(\()p Fq(t;)15 b(\036)p Fs(\()p Fq(t)p Fs(\)\))75 1166 y Fr(wher)-5 b(e)27 b Fq(P)13 b Fs(\()p Fq(t;)i(X)7 b Fs(\))26 b Fr(is)g(a)f (holomorphic)k(function)d(on)g(some)g(domain)h Fm(D)h(\032)d Fl(C)g Fr(with)h(r)-5 b(esp)g(e)g(ct)27 b(to)f Fq(t)f Fr(and)h(p)-5 b(olynomial)75 1279 y(in)29 b(the)h(variable)g Fq(X)37 b Fr(of)30 b(de)-5 b(gr)g(e)g(e)30 b Fq(d)c Fm(\025)f Fs(2)p Fr(.)41 b(L)-5 b(et)30 b Fq(\036)p Fs(\()p Fq(t)p Fs(\))g Fr(b)-5 b(e)29 b(a)h(holomorphic)j(solution)e(of)e(\(44\))i(on) f(some)g(punctur)-5 b(e)g(d)75 1392 y(disc)33 b Fq(D)336 1407 y Fp(\022)375 1392 y Fs(\()p Fq(t)443 1406 y Fo(0)483 1392 y Fq(;)15 b(r)564 1406 y Fo(0)604 1392 y Fs(\))25 b Fm(\032)g(D)s Fr(,)32 b(wher)-5 b(e)34 b(an)f(exp)-5 b(ansion)35 b(\(29\))e(holds.)44 b(Consider)34 b(the)f(p)-5 b(artial)35 b(di\013er)-5 b(ential)33 b(e)-5 b(quation)75 1610 y Fs(\(45\))854 b Fq(@)1142 1572 y Fp(S)1137 1632 y(z)1193 1610 y Fq(X)7 b Fs(\()p Fq(t;)15 b(z)t Fs(\))27 b(=)1733 1523 y Fn(X)1587 1725 y Fi(k)p Fo(=\()p Fp(k)1749 1734 y Fh(0)1784 1725 y Fp(;k)1841 1734 y Fh(1)1875 1725 y Fo(\))p Fy(2J)2025 1610 y Fq(a)2073 1625 y Fi(k)2120 1610 y Fs(\()p Fq(t;)15 b(z)t Fs(\))p Fq(@)2362 1569 y Fp(k)2399 1578 y Fh(0)2357 1634 y Fp(t)2439 1610 y Fq(@)2492 1572 y Fp(k)2529 1581 y Fh(1)2487 1632 y Fp(z)2568 1610 y Fq(X)7 b Fs(\()p Fq(t;)15 b(z)t Fs(\))75 1913 y Fr(wher)-5 b(e)34 b Fq(S)j Fr(is)c(a)g(p)-5 b(ositive)33 b(inte)-5 b(ger)33 b(and)h Fm(J)49 b Fr(is)32 b(a)h(\014nite)g(set)g (which)g(satis\014es)1208 2118 y Fm(J)41 b(\032)1407 2044 y Fn(\010)1460 2118 y Fe(k)26 b Fs(=)f(\()p Fq(k)1719 2132 y Fo(0)1759 2118 y Fq(;)15 b(k)1846 2132 y Fo(1)1886 2118 y Fs(\))26 b Fm(2)e Fl(N)2098 2080 y Fo(2)2153 2118 y Fm(j)h Fq(k)2250 2132 y Fo(1)2315 2118 y Fm(\024)g Fq(S)g Fm(\000)20 b Fs(1)2628 2044 y Fn(\011)2697 2118 y Fq(:)75 2322 y Fr(The)34 b(c)-5 b(o)g(e\016cients)35 b Fq(a)775 2337 y Fi(k)822 2322 y Fs(\()p Fq(t;)15 b(z)t Fs(\))35 b Fr(ar)-5 b(e)34 b(holomorphic)j(functions)d(on)g(the)h (domain)g Fm(D)h Fr(with)f(r)-5 b(esp)g(e)g(ct)35 b(to)f Fq(t)g Fr(and)g(ne)-5 b(ar)75 2435 y(the)33 b(origin)g(in)g Fl(C)f Fr(with)h(r)-5 b(esp)g(e)g(ct)34 b(to)g Fq(z)t Fr(.)42 b(We)32 b(make)h(the)g(fol)5 b(lowing)33 b(assumptions.)44 2622 y Fe(\(A4\))45 b Fr(Ther)-5 b(e)34 b(exist)e(two)i(r)-5 b(e)g(al)34 b(c)-5 b(onstants)35 b Fq(b)25 b(>)g Fs(1)33 b Fr(and)g Fq(q)c Fm(\025)24 b Fs(1)33 b Fr(such)g(that)694 2874 y Fs(max)639 2936 y Fo(0)p Fy(\024)p Fp(m)p Fy(\024)p Fp(k)883 2945 y Fh(0)1131 2874 y Fs(max)933 2939 y Fo(\()p Fp(p)996 2948 y Fh(1)1030 2939 y Fp(;:::)o(;p)1165 2947 y Fc(m)1223 2939 y Fo(\))p Fy(2P)1346 2948 y Fh(2)1381 2939 y Fo(\()p Fp(m)p Fo(\))1523 2813 y Fq(b)p Fs(\()p Fq(m)p Fs(\()p Fq(d)22 b Fm(\000)d Fs(1\))i(+)f Fq(q)s Fs(\()p Fq(p)2188 2827 y Fo(1)2248 2813 y Fs(+)g Fq(:)15 b(:)g(:)21 b Fs(+)f Fq(p)2602 2827 y Fp(m)2668 2813 y Fs(\)\))p 1523 2853 1216 4 v 2109 2937 a Fq(q)2769 2874 y Fs(+)g Fq(k)2907 2888 y Fo(0)2967 2874 y Fm(\000)g Fq(m)g Fs(+)g Fq(k)3296 2888 y Fo(1)3361 2874 y Fq(<)25 b(S)302 3133 y Fr(for)34 b(al)5 b(l)33 b Fe(k)26 b Fs(=)g(\()p Fq(k)838 3147 y Fo(0)878 3133 y Fq(;)15 b(k)965 3147 y Fo(1)1005 3133 y Fs(\))26 b Fm(2)g(J)16 b Fr(,)33 b(wher)-5 b(e)34 b Fm(P)1612 3147 y Fo(2)1652 3133 y Fs(\()p Fq(m)p Fs(\))26 b(=)g Fm(f)p Fs(\()p Fq(p)2051 3147 y Fo(1)2091 3133 y Fq(;)15 b(:)g(:)g(:)h(;)f(p)2338 3147 y Fp(m)2405 3133 y Fs(\))26 b Fm(2)g Fl(N)2619 3100 y Fp(m)2685 3133 y Fq(=p)2776 3147 y Fo(1)2836 3133 y Fs(+)20 b(2)p Fq(p)3018 3147 y Fo(2)3078 3133 y Fs(+)g Fq(:)15 b(:)g(:)22 b Fs(+)e Fq(mp)3513 3147 y Fp(m)3605 3133 y Fs(=)25 b Fq(m)p Fm(g)p Fr(,)302 3246 y(for)33 b(al)5 b(l)34 b Fq(m)25 b Fm(\025)g Fs(1)p Fr(,)32 b(and)i Fm(P)1124 3260 y Fo(2)1164 3246 y Fs(\(0\))26 b(=)f Fm(f)p Fs(0)p Fm(g)34 b Fr(by)e(c)-5 b(onvention.)44 3433 y Fe(\(A5\))45 b Fr(L)-5 b(et)34 b Fq(X)535 3448 y Fp(`;\014)631 3433 y Fs(\()p Fq(t)p Fs(\))p Fr(,)g Fq(`)27 b Fm(\025)f Fs(0)p Fr(,)34 b Fs(0)27 b Fm(\024)g Fq(\014)32 b Fm(\024)26 b Fq(S)g Fm(\000)20 b Fs(1)p Fr(,)34 b(b)-5 b(e)34 b(b)-5 b(ounde)g(d)35 b(holomorphic)h(fonctions)f(on)f Fm(D)s Fr(,)f(satisfying)h(the)302 3546 y(estimates)655 3832 y Fs(for)c(all)h Fq(\014)f Fs(=)25 b(0)p Fq(;)15 b(:)g(:)g(:)j(;)d(S)25 b Fm(\000)20 b Fs(1)1764 3746 y Fn(X)1763 3942 y Fp(n)p Fy(\025)p Fo(0)1912 3646 y Fn(\014)1912 3700 y(\014)1912 3755 y(\014)1912 3809 y(\014)1912 3864 y(\014)1912 3919 y(\014)1942 3646 y(\014)1942 3700 y(\014)1942 3755 y(\014)1942 3809 y(\014)1942 3864 y(\014)1942 3919 y(\014)1972 3746 y(X)1978 3944 y Fp(`)p Fy(\025)p Fo(0)2119 3832 y Fs(sup)2123 3910 y Fp(t)p Fy(2D)2271 3832 y Fm(j)p Fq(@)2349 3795 y Fp(n)2344 3855 y(t)2396 3832 y Fq(X)2471 3847 y Fp(`;\014)2567 3832 y Fs(\()p Fq(t)p Fs(\))p Fm(j)2721 3771 y Fq(T)2787 3738 y Fp(`)p 2721 3811 99 4 v 2739 3895 a Fq(`)p Fs(!)2830 3646 y Fn(\014)2830 3700 y(\014)2830 3755 y(\014)2830 3809 y(\014)2830 3864 y(\014)2830 3919 y(\014)2860 3646 y(\014)2860 3700 y(\014)2860 3755 y(\014)2860 3809 y(\014)2860 3864 y(\014)2860 3919 y(\014)2890 3977 y Fo(0;)p Fp(\033)2985 3986 y Fh(0)3049 3759 y Fq(\016)3092 3727 y Fp(n)3089 3784 y Fo(1)p Fp(;)p Fo(0)p 3049 3811 135 4 v 3077 3895 a Fq(n)p Fs(!)3219 3832 y Fq(<)25 b Fs(+)p Fm(1)p Fq(:)75 4195 y Fr(Then,)33 b(the)g(formal)h(series)f(\(18\))1321 4458 y Fq(X)7 b Fs(\()p Fq(t;)15 b(z)t Fs(\))27 b(=)1716 4372 y Fn(X)1715 4570 y Fp(\014)s Fy(\025)p Fo(0)1863 4372 y Fn(X)1870 4570 y Fp(`)p Fy(\025)p Fo(0)2010 4458 y Fq(X)2085 4473 y Fp(`;\014)2181 4458 y Fs(\()p Fq(t)p Fs(\))2294 4397 y Fq(\036)p Fs(\()p Fq(t)p Fs(\))2451 4364 y Fp(`)p 2294 4437 191 4 v 2358 4521 a Fq(`)p Fs(!)2505 4397 y Fq(z)2551 4364 y Fp(\014)p 2505 4437 94 4 v 2511 4521 a Fq(\014)5 b Fs(!)75 4759 y Fr(c)-5 b(onstructe)g(d)39 b(in)e(Pr)-5 b(op)g(osition)39 b(7,)f(solution)h(of)e(\(45\),)i (de\014nes)e(a)g(holomorphic)k(function)c(on)g(the)g(op)-5 b(en)38 b(set)75 4895 y Fq(D)150 4910 y Fp(\022)189 4895 y Fs(\()p Fq(t)257 4909 y Fo(0)297 4895 y Fq(;)15 b(r)378 4909 y Fo(0)418 4895 y Fs(\))24 b Fm(\002)f Fq(D)664 4794 y Fn(\020)719 4895 y Fs(0)p Fq(;)814 4858 y Fp(\016)r(\016)878 4867 y Fh(2)p 814 4874 100 4 v 846 4926 a Fo(2)924 4794 y Fn(\021)978 4895 y Fr(,)39 b(for)f(a)g(given)e Fq(\016)i Fm(2)c Fs(\(0)p Fq(;)15 b Fs(1\))39 b Fr(and)g Fq(\016)2154 4909 y Fo(2)2228 4895 y Fq(>)33 b Fs(0)p Fr(.)57 b(Mor)-5 b(e)g(over,)40 b(the)e(function)f Fq(X)7 b Fs(\()p Fq(t;)15 b(z)t Fs(\))39 b Fr(has)75 5067 y(the)32 b(fol)5 b(lowing)33 b(r)-5 b(ate)32 b(of)g(gr)-5 b(owth)34 b(on)e(the)g(set)f Fq(D)1684 5082 y Fp(\022)1724 5067 y Fs(\()p Fq(t)1792 5081 y Fo(0)1831 5067 y Fq(;)15 b(r)1912 5081 y Fo(0)1952 5067 y Fs(\))j Fm(\002)g Fq(D)2187 4966 y Fn(\020)2241 5067 y Fs(0)p Fq(;)2337 5030 y Fp(\016)r(\016)2401 5039 y Fh(2)p 2337 5046 V 2369 5098 a Fo(2)2447 4966 y Fn(\021)2501 5067 y Fr(.)41 b(If)32 b Fq(r)2706 5081 y Fo(0)2777 5067 y Fr(is)f(smal)5 b(l)33 b(enough,)f(ther)-5 b(e)32 b(exist)75 5197 y(p)-5 b(ositive)34 b(c)-5 b(onstants)34 b Fq(C)867 5211 y Fo(11)974 5197 y Fr(and)g Fq(C)1216 5211 y Fo(12)1323 5197 y Fr(such)f(that)75 5402 y Fs(\(46\))468 b(sup)650 5505 y Fy(j)p Fp(z)s Fy(j\024)791 5469 y Fc(\016)r(\016)849 5484 y Fh(2)p 790 5490 93 3 v 821 5531 a(2)907 5402 y Fm(j)q Fq(X)7 b Fs(\()p Fq(t;)15 b(z)t Fs(\))p Fm(j)27 b(\024)e Fq(C)1417 5416 y Fo(11)1492 5402 y Fs(\(1)c Fm(\000)e Fq(\016)s Fs(\))1761 5364 y Fy(\000)p Fo(1)1857 5402 y Fm(j)p Fq(t)h Fm(\000)g Fq(t)2059 5416 y Fo(0)2098 5402 y Fm(j)2123 5364 y Fy(\000)p Fp(n)2221 5373 y Fh(0)2256 5364 y Fp(m=\026)2415 5402 y Fs(exp\()p Fq(C)2654 5416 y Fo(12)2729 5402 y Fm(j)p Fq(t)h Fm(\000)f Fq(t)2932 5416 y Fo(0)2971 5402 y Fm(j)2996 5364 y Fy(\000)p Fp(n)3094 5373 y Fh(0)3128 5364 y Fp(q)r(=\026)3244 5402 y Fs(\))75 5708 y Fr(for)33 b(al)5 b(l)33 b Fq(t)25 b Fm(2)g(D)564 5723 y Fp(\022)603 5708 y Fs(\()p Fq(t)671 5722 y Fo(0)711 5708 y Fq(;)15 b(r)792 5722 y Fo(0)832 5708 y Fs(\))p Fr(.)p eop end %%Page: 25 25 TeXDict begin 25 24 bop 3764 100 a Fs(25)75 399 y Fe(Pro)s(of)56 b Fs(F)-8 b(rom)31 b(Prop)s(osition)g(8,)g(the)f(formal)h(series)g (\(38\))1173 651 y Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))27 b(=)1664 564 y Fn(X)1663 762 y Fp(\014)s Fy(\025)p Fo(0)1811 564 y Fn(X)1817 762 y Fp(`)p Fy(\025)p Fo(0)1976 564 y Fn(X)1958 760 y Fp(n)2001 769 y Fh(0)2035 760 y Fy(\025)p Fo(0)2140 651 y Fq(u)2192 666 y Fp(n)2235 675 y Fh(0)2270 666 y Fp(;`;\014)2397 589 y Fq(t)2430 556 y Fp(n)2473 565 y Fh(0)p 2395 630 120 4 v 2395 713 a Fq(n)2450 727 y Fo(0)2489 713 y Fs(!)2534 589 y Fq(T)2600 556 y Fp(`)p 2534 630 99 4 v 2552 713 a Fq(`)p Fs(!)2653 589 y Fq(z)2699 556 y Fp(\014)p 2653 630 94 4 v 2659 713 a Fq(\014)5 b Fs(!)75 941 y(is)30 b(solution)h(of)g(the)f (auxiliary)h(partial)h(di\013eren)m(tial)f(equation)h(\(39\))837 1148 y Fq(@)890 1110 y Fp(S)885 1170 y(z)941 1148 y Fq(U)10 b Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))27 b(=)1613 1062 y Fn(X)1431 1263 y Fi(q)p Fo(=\()p 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Fs(\);)1505 1827 y Fq(q)1546 1841 y Fo(0)1610 1827 y Fs(=)h Fq(k)1753 1841 y Fo(0)1813 1827 y Fm(\000)20 b Fq(m)p Fs(;)41 b Fq(q)2091 1841 y Fo(1)2155 1827 y Fs(=)25 b Fm(j)p Fe(p)p Fm(j)2359 1841 y Fo(1)2399 1827 y Fs(;)41 b Fq(q)2506 1841 y Fo(2)2570 1827 y Fs(=)25 b Fq(k)2713 1841 y Fo(1)2753 1827 y Fm(g)75 1989 y Fs(and)908 2166 y Fq(B)977 2180 y Fi(q)1024 2166 y Fs(\()p Fq(t;)15 b(z)t(;)g(T)e Fs(\))26 b(=)1451 2104 y Fq(k)1498 2118 y Fo(0)1538 2104 y Fs(!)p 1451 2145 113 4 v 1454 2228 a Fq(q)1495 2242 y Fo(0)1535 2228 y Fs(!)1753 2104 y(1)p 1583 2145 385 4 v 1583 2228 a Fq(p)1629 2242 y Fo(1)1668 2228 y Fs(!)15 b Fq(:)g(:)g(:)i(p)1876 2242 y Fp(m)1942 2228 y Fs(!)1994 2079 y Fn(X)1993 2277 y Fp(\014)s Fy(\025)p Fo(0)2148 2052 y Fp(D)2206 2060 y Fc(m)2141 2079 y Fn(X)2145 2275 y Fp(j)t Fo(=0)2288 2079 y Fn(X)2287 2275 y Fp(n)p Fy(\025)p Fo(0)2436 2166 y Fq(A)2504 2181 y Fp(n;j;\014)2671 2104 y Fq(t)2704 2071 y Fp(n)p 2671 2145 80 4 v 2671 2228 a Fq(n)p Fs(!)2771 2104 y Fq(T)2837 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b(141.)e(CR)m(C)e(Press,)h(Bo)s(ca)g(Raton,)i(FL,)d(2009.)i(xiv+250)262 3212 y(pp.)120 3399 y([6])47 b(O.)26 b(Costin,)g(R.)g(Costin,)h Fr(On)h(the)h(formation)h(of)f(singularities)g(of)g(solutions)g(of)g (nonline)-5 b(ar)30 b(di\013er)-5 b(ential)262 3512 y(systems)33 b(in)g(antistokes)h(dir)-5 b(e)g(ctions.)32 b Fs(In)m(v)m(en)m(t.)f (Math.)h(145)f(\(2001\),)i(no.)e(3,)g(425{485.)120 3700 y([7])47 b(J.)28 b(Dubinskii,)g Fr(A)n(nalytic)j(pseudo-di\013er)-5 b(ential)33 b(op)-5 b(er)g(ators)34 b(and)d(their)g(applic)-5 b(ations.)31 b Fs(T)-8 b(ranslated)29 b(from)262 3813 y(the)h(Russian.)g(Mathematics)i(and)e(its)g(Applications)i(\(So)m (viet)g(Series\),)e(68.)i(Klu)m(w)m(er)e(Academic)h(Pub-)262 3926 y(lishers)f(Group,)g(Dordrec)m(h)m(t,)i(1991.)g(xii+252)g(pp.)120 4113 y([8])47 b(J.)373 4090 y(\023)365 4113 y(Ecalle,)33 b Fr(Intr)-5 b(o)g(duction)36 b(aux)d(fonctions)i(analysables)g(et)e (pr)-5 b(euve)34 b(c)-5 b(onstructive)34 b(de)f(la)h(c)-5 b(onje)g(ctur)g(e)34 b(de)262 4226 y(Dulac.)c Fs(Actualit)m(\023)-43 b(es)33 b(Math)m(\023)-43 b(ematiques,)33 b(Hermann,)d(P)m(aris,)i (1992.)120 4414 y([9])47 b(E.)22 b(Hille,)k Fr(Or)-5 b(dinary)27 b(di\013er)-5 b(ential)26 b(e)-5 b(quations)26 b(in)f(the)h(c)-5 b(omplex)27 b(domain.)d Fs(Reprin)m(t)e(of)g(the)h (1976)h(original.)262 4527 y(Do)m(v)m(er)32 b(Publications,)f(Inc.,)g (Mineola,)h(NY,)f(1997.)h(xii+484)g(pp.)75 4715 y([10])47 b(S.)35 b(Malek,)k Fr(On)e(c)-5 b(omplex)39 b(singularity)g(analysis)g (for)f(line)-5 b(ar)39 b(q-di\013er)-5 b(enc)g(e-di\013er)g(ential)38 b(e)-5 b(quations,)38 b Fs(J.)262 4827 y(Dynam.)31 b(Con)m(trol.)g (Syst)f(15)h(\(2009\),)i(no.)e(1.)75 5015 y([11])47 b(W.)27 b(Mal\015iet,)i Fr(Solitary)i(wave)e(solutions)h(of)g(nonline)-5 b(ar)30 b(wave)g(e)-5 b(quations.)27 b Fs(Amer.)g(J.)f(Ph)m(ys.)g(60)i (\(1992\),)262 5128 y(no.)i(7,)h(650{654.)75 5316 y([12])47 b(W.)29 b(W)-8 b(alter,)31 b Fr(A)n(n)f(elementary)i(pr)-5 b(o)g(of)32 b(of)f(the)h(Cauchy-Kowalevsky)g(the)-5 b(or)g(em)37 b Fs(Amer.)29 b(Math.)g(Mon)m(thly)262 5429 y(92)i(\(1985\),)i(no.)e (2,)g(115{126.)75 5616 y([13])47 b(A-M.)33 b(W)-8 b(azw)m(az,)35 b Fr(The)f(sine-c)-5 b(osine)34 b(and)h(the)f(tanh)h(metho)-5 b(ds:)47 b(r)-5 b(eliable)35 b(to)-5 b(ols)36 b(for)e(analytic)h(tr)-5 b(e)g(atment)262 5729 y(of)33 b(nonline)-5 b(ar)34 b(disp)-5 b(ersive)34 b(e)-5 b(quations.)31 b Fs(Appl.)f(Math.)h(Comput.)f(173)i (\(2006\),)h(no.)d(1,)h(150{164.)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 3764 100 a Fs(27)75 399 y([14])47 b(P)-8 b(.)30 b(P)m(ong)m(\023)-43 b(erard,)31 b(C.)f(W)-8 b(agsc)m(hal,)32 b Fr(Pr)-5 b(obl)n(\022)-44 b(eme)33 b(de)f(Cauchy)g(dans)h(des)g(esp)-5 b(ac)g(es)33 b(de)f(fonctions)h (enti)n(\022)-44 b(er)-5 b(es.)30 b Fs(J.)262 511 y(Math.)h(Pures)f (Appl.)g(\(9\))h(75)g(\(1996\),)i(no.)e(4,)g(409{418.)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------1103251122648--