Content-Type: multipart/mixed; boundary="-------------0807070321352" This is a multi-part message in MIME format. ---------------0807070321352 Content-Type: text/plain; name="08-135.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="08-135.keywords" Cauchy problem, partial integro-differential equation, Gevrey estimates, fuchsian singularity, irregular singularity, small divisor ---------------0807070321352 Content-Type: application/postscript; name="fuchsirregular_2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fuchsirregular_2.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.94b Copyright 2004 Radical Eye Software %%Title: fuchsirregular_2.dvi %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMBX10 CMR10 MSBM10 CMR7 CMSY10 CMBX12 %%+ CMMI10 CMMI8 CMR6 CMR8 CMEX10 CMSY8 CMTI10 LASY10 CMMI6 CMSY6 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -D600 -o fuchsirregular_2.ps %+ fuchsirregular_2.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2008.06.18:1442 %%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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%%EndFont %%BeginFont: LASY10 %!PS-AdobeFont-1.1: LASY10 1.001 %%CreationDate: 1992 Oct 23 20:19:17 %%RevisionDate: 2001 Jun 05 20:19:17 % Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.001) readonly def /Notice (Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved) readonly def /FullName (LASY10) readonly def /FamilyName (LaTeX) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /LASY10 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 /a50 put readonly def /FontBBox{-19 -192 944 683}readonly def /UniqueID 5011949 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 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 67 /C put dup 68 /D put dup 72 /H put dup 76 /L put dup 80 /P put dup 82 /R put dup 84 /T put dup 89 /Y 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 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec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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 13 /fl put dup 14 /ffi put dup 19 /acute 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 56 /eight put dup 58 /colon 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 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % 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 (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 readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 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 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 65 /A put dup 66 /B put dup 77 /M put dup 78 /N put dup 83 /S put dup 85 /U put dup 91 /union put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 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 2 /multiply put dup 3 /asteriskmath put dup 20 /lessequal put dup 21 /greaterequal put dup 49 /infinity put dup 50 /element put dup 83 /S put dup 91 /union put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 8 /braceleftbig put dup 9 /bracerightbig put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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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 20 /caron put dup 22 /macron 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 61 /equal put dup 97 /a put dup 103 /g put dup 114 /r put dup 126 /tilde put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 97 /a put dup 103 /g put dup 114 /r put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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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 14 /delta put dup 16 /zeta put dup 18 /theta put dup 20 /kappa put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 28 /tau put dup 59 /comma put dup 61 /slash put dup 68 /D put dup 75 /K put dup 77 /M put dup 78 /N put dup 82 /R put dup 83 /S put dup 84 /T put dup 88 /X put dup 99 /c put dup 100 /d put dup 101 /e 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 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def 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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 28 /tau put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 75 /K put dup 77 /M put dup 78 /N 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 88 /X put dup 89 /Y put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e 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 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 /UniqueID 5087385 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 11 /ff put dup 12 /fi put dup 14 /ffi put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 22 /macron put dup 33 /exclam put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z 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 dup 126 /tilde put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (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 /UniqueID 5000790 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 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 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 56 /eight put dup 57 /nine put dup 64 /at put dup 65 /A put dup 67 /C put dup 70 /F put dup 74 /J put dup 76 /L put dup 77 /M put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 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{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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Fn(\))33 b Fe(in)g Fm(G)p Fn(\()p Fm(T)8 b(;)15 b(X)7 b Fn(\))p Fe(.)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 3809 100 a Fn(4)75 399 y(W)-8 b(e)32 b(start)e(with)h(some)f(preliminary)g(lemma)h(from)f([12)q(].)75 591 y Fd(Lemma)35 b(1)45 b Fe(L)-5 b(et)40 b Fm(U)769 605 y Fj(1)809 591 y Fn(\()p Fm(t;)15 b(x)p Fn(\))p Fe(,)42 b Fm(U)1136 605 y Fj(2)1176 591 y Fn(\()p Fm(t;)15 b(x)p Fn(\))40 b Fe(b)-5 b(e)40 b(in)f Fm(G)p Fn(\()p Fm(T)8 b(;)15 b(X)7 b Fn(\))p Fe(.)65 b(Then,)41 b(the)f(pr)-5 b(o)g(duct)42 b Fm(U)2887 605 y Fj(1)2927 591 y Fn(\()p Fm(t;)15 b(x)p Fn(\))p Fm(U)3184 605 y Fj(2)3224 591 y Fn(\()p Fm(t;)g(x)p Fn(\))41 b Fe(b)-5 b(elongs)40 b(to)75 704 y Fm(G)p Fn(\()p Fm(T)8 b(;)15 b(X)7 b Fn(\))p Fe(.)43 b(Mor)-5 b(e)g(over,)75 888 y Fn(\(4\))672 b Fg(jj)p Fm(U)974 902 y Fj(1)1014 888 y Fn(\()p Fm(t;)15 b(x)p Fn(\))p Fm(U)1271 902 y Fj(2)1311 888 y Fn(\()p Fm(t;)g(x)p Fn(\))p Fg(jj)1556 906 y Fj(\()p Fl(T)6 b(;X)f Fj(\))1772 888 y Fg(\024)25 b(jj)p Fm(U)1980 902 y Fj(1)2020 888 y Fn(\()p Fm(t;)15 b(x)p Fn(\))p Fg(jj)2265 906 y Fj(\()p Fl(T)6 b(;X)f Fj(\))2455 888 y Fg(jj)p Fm(U)2567 902 y Fj(2)2607 888 y Fn(\()p Fm(t;)15 b(x)p Fn(\))p Fg(jj)2852 906 y Fj(\()p Fl(T)6 b(;X)f Fj(\))3042 888 y Fm(:)75 1072 y Fe(In)33 b(other)g(wor)-5 b(ds,)35 b(the)e(norme)-5 b(d)35 b(ve)-5 b(ctor)33 b(sp)-5 b(ac)g(e)34 b Fn(\()p Fm(G)p Fn(\()p Fm(T)8 b(;)15 b(X)7 b Fn(\))p Fm(;)15 b Fg(jj)p Fm(:)p Fg(jj)2221 1091 y Fj(\()p Fl(T)6 b(;X)f Fj(\))2413 1072 y Fn(\))33 b Fe(is)g(in)f(fact)h(a)g(Banach)h(algebr)-5 b(a.)75 1265 y Fn(In)30 b(the)g(follo)m(wing,)i(for)e Fm(u)p Fn(\()p Fm(t;)15 b(x)p Fn(\))27 b Fg(2)d Ff(C)p Fn([[)p Fm(t;)15 b(x)p Fn(]],)32 b(w)m(e)f(denote)g(b)m(y)f Fm(@)2206 1232 y Fh(\000)p Fj(1)2201 1287 y Fl(x)2301 1265 y Fm(u)p Fn(\()p Fm(t;)15 b(x)p Fn(\))31 b(the)f(formal)h(series) 3265 1191 y Fi(R)3325 1218 y Fl(x)3308 1297 y Fj(0)3384 1265 y Fm(u)p Fn(\()p Fm(t;)15 b(\034)10 b Fn(\))p Fm(d\034)g Fn(.)75 1471 y Fd(Lemma)35 b(2)45 b Fe(L)-5 b(et)29 b Fm(h)748 1485 y Fj(0)788 1471 y Fm(;)15 b(h)880 1485 y Fj(1)945 1471 y Fg(2)25 b Ff(N)j Fe(such)h(that)h Fm(h)1562 1485 y Fj(0)1627 1471 y Fg(\024)25 b Fm(h)1775 1485 y Fj(1)1814 1471 y Fe(.)41 b(The)28 b(op)-5 b(er)g(ator)32 b Fm(@)2471 1430 y Fl(h)2512 1439 y Fk(0)2466 1495 y Fl(t)2551 1471 y Fm(@)2604 1438 y Fh(\000)p Fl(h)2700 1447 y Fk(1)2599 1493 y Fl(x)2767 1471 y Fe(is)c(a)h(b)-5 b(ounde)g(d)30 b(line)-5 b(ar)29 b(op)-5 b(er)g(ator)75 1584 y(fr)g(om)46 b Fn(\()p Fm(G)p Fn(\()p Fm(T)8 b(;)15 b(X)7 b Fn(\))p Fm(;)15 b Fg(jj)p Fm(:)p Fg(jj)826 1602 y Fj(\()p Fl(T)6 b(;X)f Fj(\))1018 1584 y Fn(\))45 b Fe(into)f(itself.)77 b(Mor)-5 b(e)g(over,)48 b(ther)-5 b(e)46 b(exists)e(a)h(c)-5 b(onstant)46 b Fm(C)53 b(>)46 b Fn(0)e Fe(such)h(that)g(the)75 1697 y(estimates)34 b(hold)75 1881 y Fn(\(5\))767 b Fg(jj)p Fm(@)1060 1840 y Fl(h)1101 1849 y Fk(0)1055 1905 y Fl(t)1140 1881 y Fm(@)1193 1843 y Fh(\000)p Fl(h)1289 1852 y Fk(1)1188 1903 y Fl(x)1328 1881 y Fm(U)10 b Fn(\()p Fm(t;)15 b(x)p Fn(\))p Fg(jj)1645 1899 y Fj(\()p Fl(T)6 b(;X)f Fj(\))1860 1881 y Fg(\024)25 b Fm(C)7 b(T)2094 1843 y Fh(\000)p Fl(h)2190 1852 y Fk(0)2228 1881 y Fm(X)2310 1843 y Fl(h)2351 1852 y Fk(1)2390 1881 y Fg(jj)p Fm(U)j Fn(\()p Fm(t;)15 b(x)p Fn(\))p Fg(jj)2757 1899 y Fj(\()p Fl(T)6 b(;X)f Fj(\))2947 1881 y Fm(;)75 2065 y Fe(for)33 b(al)5 b(l)33 b Fm(U)10 b Fn(\()p Fm(t;)15 b(x)p Fn(\))26 b Fg(2)f Fm(G)p Fn(\()p Fm(T)8 b(;)15 b(X)7 b Fn(\))p Fe(.)75 2258 y Fd(Lemma)35 b(3)45 b Fe(L)-5 b(et)40 b Fm(h)759 2272 y Fj(1)799 2258 y Fm(;)15 b(h)891 2272 y Fj(2)968 2258 y Fg(2)36 b Ff(N)j Fe(and)h(let)f Fm(U)10 b Fn(\()p Fm(t;)15 b(x)p Fn(\))40 b Fe(b)-5 b(e)39 b(in)g Fm(G)p Fn(\()p Fm(T)2189 2272 y Fj(0)2229 2258 y Fm(;)15 b(X)2344 2272 y Fj(0)2384 2258 y Fn(\))p Fe(.)62 b(Then,)41 b(ther)-5 b(e)40 b(exist)f Fm(T)8 b(;)15 b(X)44 b(>)37 b Fn(0)i Fe(smal)5 b(l)75 2370 y(enough)46 b(\(dep)-5 b(ending)47 b(on)f Fm(h)1065 2384 y Fj(1)1105 2370 y Fm(;)15 b(h)1197 2384 y Fj(2)1237 2370 y Fe(\))46 b(such)g(that)h(the)f(formal)h(series) f Fm(@)2528 2330 y Fl(h)2569 2339 y Fk(1)2523 2395 y Fl(t)2608 2370 y Fm(@)2661 2337 y Fl(h)2702 2346 y Fk(2)2656 2393 y Fl(x)2741 2370 y Fm(U)10 b Fn(\()p Fm(t;)15 b(x)p Fn(\))46 b Fe(b)-5 b(elongs)47 b(to)f Fm(G)p Fn(\()p Fm(T)8 b(;)15 b(X)7 b Fn(\))p Fe(.)75 2483 y(Mor)-5 b(e)g(over,)34 b(ther)-5 b(e)33 b(exists)g(a)g(c)-5 b(onstant)35 b Fm(K)c(>)25 b Fn(0)33 b Fe(such)g(that)75 2668 y Fn(\(6\))696 b Fg(jj)p Fn(\()p Fm(@)1024 2627 y Fl(h)1065 2636 y Fk(1)1019 2692 y Fl(t)1105 2668 y Fm(@)1158 2630 y Fl(h)1199 2639 y Fk(2)1153 2690 y Fl(x)1238 2668 y Fm(U)10 b Fn(\)\()p Fm(t;)15 b(x)p Fn(\))p Fg(jj)1590 2686 y Fj(\()p Fl(T)6 b(;X)f Fj(\))1806 2668 y Fg(\024)25 b Fm(T)1968 2630 y Fh(\000)p Fl(h)2064 2639 y Fk(1)2102 2668 y Fm(X)2184 2630 y Fh(\000)p Fl(h)2280 2639 y Fk(2)2318 2668 y Fm(K)7 b Fg(jj)p Fm(U)j Fn(\()p Fm(t;)15 b(x)p Fn(\))p Fg(jj)2769 2686 y Fj(\()p Fl(T)2837 2695 y Fk(0)2874 2686 y Fl(;X)2952 2695 y Fk(0)2986 2686 y Fj(\))3018 2668 y Fm(;)75 2852 y Fe(for)33 b(al)5 b(l)33 b Fm(U)10 b Fn(\()p Fm(t;)15 b(x)p Fn(\))26 b Fg(2)f Fm(G)p Fn(\()p Fm(T)888 2866 y Fj(0)929 2852 y Fm(;)15 b(X)1044 2866 y Fj(0)1084 2852 y Fn(\))p Fe(.)75 3044 y Fn(In)30 b(the)g(follo)m(wing,)i(w)m(e)f(put) 1530 3238 y Fm(w)r Fn(\()p Fm(t;)15 b(x)p Fn(\))27 b(=)1915 3125 y Fl(S)t Fh(\000)p Fj(1)1917 3152 y Fi(X)1922 3347 y Fl(j)t Fj(=0)2066 3238 y Fm(')2125 3252 y Fl(j)2162 3238 y Fn(\()p Fm(t)p Fn(\))2275 3177 y Fm(x)2327 3144 y Fl(j)p 2275 3217 89 4 v 2285 3301 a Fm(j)5 b Fn(!)2374 3238 y Fm(;)75 3490 y Fn(where)30 b Fm(')397 3504 y Fl(j)434 3490 y Fn(\()p Fm(t)p Fn(\))h(b)s(elong)f(to)h Fm(G)p Fn(\()p Fm(T)1128 3504 y Fj(0)1169 3490 y Fm(;)15 b(X)1284 3504 y Fj(0)1324 3490 y Fn(\).)75 3682 y Fd(De\014nition)35 b(2)46 b Fe(Consider)34 b(the)f(map)h Fm(A)25 b Fn(:)g Ff(C)p Fn([[)p Fm(t;)15 b(x)p Fn(]])27 b Fg(!)e Ff(C)p Fn([[)p Fm(t;)15 b(x)p Fn(]])33 b Fe(de\014ne)-5 b(d)34 b(as)f(fol)5 b(lows)75 3905 y Fn(\(7\))84 b Fm(A)p Fn(\()p Fm(v)s Fn(\()p Fm(t;)15 b(x)p Fn(\)\))28 b(=)883 3818 y Fi(X)778 4020 y Fj(\()p Fl(k)842 4029 y Fk(0)876 4020 y Fl(;k)933 4029 y Fk(1)967 4020 y Fj(\))p Fh(2S)1084 4029 y Fk(1)1134 3905 y Fm(a)1182 3920 y Fl(k)1219 3929 y Fk(0)1254 3920 y Fl(k)1291 3929 y Fk(1)1329 3905 y Fn(\()p Fm(t;)15 b(x)p Fn(\)\()p Fm(@)1612 3864 y Fl(k)1649 3873 y Fk(0)1607 3929 y Fl(t)1689 3905 y Fm(@)1742 3867 y Fl(k)1779 3876 y Fk(1)1814 3867 y Fh(\000)p Fl(S)1737 3927 y(x)1919 3905 y Fm(v)s Fn(\()p Fm(t;)g(x)p Fn(\))22 b(+)e Fm(@)2327 3864 y Fl(k)2364 3873 y Fk(0)2322 3929 y Fl(t)2403 3905 y Fm(@)2456 3867 y Fl(k)2493 3876 y Fk(1)2451 3927 y Fl(x)2531 3905 y Fm(w)r Fn(\()p Fm(t;)15 b(x)p Fn(\)\))448 4174 y(+)20 b(\()p Fm(@)627 4137 y Fh(\000)p Fl(S)622 4197 y(x)733 4174 y Fm(v)s Fn(\(0)p Fm(;)15 b(x)p Fn(\))22 b(+)e Fm(w)r Fn(\(0)p Fm(;)15 b(x)p Fn(\)\))1536 4088 y Fi(X)1424 4289 y Fj(\()p Fl(\024)1492 4298 y Fk(0)1529 4289 y Fl(;\024)1590 4298 y Fk(1)1624 4289 y Fj(\))p Fh(2S)1741 4298 y Fk(2)1791 4174 y Fm(b)1830 4188 y Fl(\024)1871 4197 y Fk(0)1906 4188 y Fl(\024)1947 4197 y Fk(1)1985 4174 y Fn(\()p Fm(t;)g(x)p Fn(\)\()p Fm(@)2268 4134 y Fl(\024)2309 4143 y Fk(0)2263 4198 y Fl(t)2349 4174 y Fm(@)2402 4137 y Fl(\024)2443 4146 y Fk(1)2478 4137 y Fh(\000)p Fl(S)2397 4197 y(x)2583 4174 y Fm(v)s Fn(\()p Fm(t;)g(x)p Fn(\))22 b(+)e Fm(@)2991 4134 y Fl(\024)3032 4143 y Fk(0)2986 4198 y Fl(t)3070 4174 y Fm(@)3123 4137 y Fl(\024)3164 4146 y Fk(1)3118 4197 y Fl(x)3203 4174 y Fm(w)r Fn(\()p Fm(t;)15 b(x)p Fn(\)\))2533 4431 y(+)20 b Fm(c)p Fn(\()p Fm(t;)15 b(x)p Fn(\)\()p Fm(@)2946 4393 y Fh(\000)p Fl(S)2941 4453 y(x)3054 4431 y Fm(v)s Fn(\()p Fm(t;)g(x)p Fn(\))21 b(+)f Fm(w)r Fn(\()p Fm(t;)15 b(x)p Fn(\)\))3705 4393 y Fj(2)3746 4431 y Fm(:)75 4615 y Fe(By)33 b(c)-5 b(onstruction,)34 b(if)e Fm(v)s Fn(\()p Fm(t;)15 b(x)p Fn(\))27 b Fg(2)d Ff(C)p Fn([[)p Fm(t;)15 b(x)p Fn(]])34 b Fe(is)e(a)h(solution)h(of)f Fm(A)p Fn(\()p Fm(v)s Fn(\()p Fm(t;)15 b(x)p Fn(\)\))27 b(=)e Fm(v)s Fn(\()p Fm(t;)15 b(x)p Fn(\))p Fe(,)34 b(then)1392 4799 y Fm(u)p Fn(\()p Fm(t;)15 b(x)p Fn(\))26 b(=)f Fm(@)1814 4762 y Fh(\000)p Fl(S)1809 4822 y(x)1920 4799 y Fm(v)s Fn(\()p Fm(t;)15 b(x)p Fn(\))21 b(+)f Fm(w)r Fn(\()p Fm(t;)15 b(x)p Fn(\))75 4983 y Fe(is)33 b(a)g(solution)g(of)g(\(2\))h (and)f(\(3\).)75 5176 y Fd(Prop)s(osition)j(1)46 b Fe(Assume)35 b(that)h(the)g(hyp)-5 b(otheses)38 b Fd(i\))p Fe(,)d Fd(ii\))p Fe(,)g Fd(iii\))g Fe(of)g(The)-5 b(or)g(em)38 b(1)d(ar)-5 b(e)36 b(ful\014le)-5 b(d.)50 b(Then,)37 b(ther)-5 b(e)75 5289 y(exist)25 b Fm(T)8 b(;)15 b(X)33 b(>)25 b Fn(0)g Fe(smal)5 b(l)26 b(enough,)h(such)e(that)h(the)g(fol)5 b(lowing)26 b(pr)-5 b(op)g(erties)27 b(ar)-5 b(e)26 b(satis\014e)-5 b(d.)41 b(Ther)-5 b(e)26 b(exists)f(a)g(close)-5 b(d)75 5402 y(b)g(al)5 b(l)40 b Fm(B)318 5416 y Fl(R)413 5402 y Fn(=)d Fg(f)p Fm(v)j Fg(2)d Fm(G)p Fn(\()p Fm(T)8 b(;)15 b(X)7 b Fn(\))39 b(:)e Fg(jj)p Fm(v)s Fg(jj)1320 5420 y Fj(\()p Fl(T)6 b(;X)f Fj(\))1548 5402 y Fg(\024)37 b Fm(R)q Fg(g)p Fe(,)j Fm(R)e(>)f Fn(0)p Fe(,)k(such)f(that)g(the)g (map)g Fm(A)d Fn(:)h Ff(C)p Fn([[)p Fm(t;)15 b(x)p Fn(]])38 b Fg(!)f Ff(C)p Fn([[)p Fm(t;)15 b(x)p Fn(]])75 5514 y Fe(satisfy)75 5627 y Fd(1\))32 b Fm(A)p Fn(\()p Fm(B)372 5641 y Fl(R)431 5627 y Fn(\))25 b Fg(\032)g Fm(B)656 5641 y Fl(R)714 5627 y Fe(,)75 5740 y Fd(2\))32 b Fe(F)-7 b(or)34 b(al)5 b(l)33 b Fm(v)544 5754 y Fj(1)584 5740 y Fm(;)15 b(v)668 5754 y Fj(2)733 5740 y Fg(2)25 b Fm(B)888 5754 y Fl(R)945 5740 y Fe(,)33 b(we)g(have)g Fg(jj)p Fm(A)p Fn(\()p Fm(v)1547 5754 y Fj(2)1587 5740 y Fn(\))21 b Fg(\000)f Fm(A)p Fn(\()p Fm(v)1881 5754 y Fj(1)1921 5740 y Fn(\))p Fg(jj)2006 5759 y Fj(\()p Fl(T)6 b(;X)f Fj(\))2221 5740 y Fg(\024)25 b Fn(\(1)p Fm(=)p Fn(2\))p Fg(jj)p Fm(v)2616 5754 y Fj(2)2678 5740 y Fg(\000)20 b Fm(v)2813 5754 y Fj(1)2852 5740 y Fg(jj)2902 5759 y Fj(\()p Fl(T)6 b(;X)f Fj(\))3092 5740 y Fe(.)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 3809 100 a Fn(5)75 399 y Fd(Pro)s(of)31 b(1\))f Fn(Due)h(to)g(Lemma)g(1,2,3,)h(there)f(exists)g(a)f(constan)m (t)i Fm(C)g(>)25 b Fn(0)31 b(suc)m(h)f(that)75 659 y(\(8\))84 b Fg(jj)p Fm(a)372 674 y Fl(k)409 683 y Fk(0)444 674 y Fl(k)481 683 y Fk(1)520 659 y Fn(\()p Fm(t;)15 b(x)p Fn(\)\()p Fm(@)803 618 y Fl(k)840 627 y Fk(0)798 683 y Fl(t)880 659 y Fm(@)933 621 y Fl(k)970 630 y Fk(1)1004 621 y Fh(\000)p Fl(S)928 681 y(x)1110 659 y Fm(v)s Fn(\()p Fm(t;)g(x)p Fn(\))21 b(+)f Fm(@)1517 618 y Fl(k)1554 627 y Fk(0)1512 683 y Fl(t)1593 659 y Fm(')1652 673 y Fl(j)1689 659 y Fn(\()p Fm(t)p Fn(\))1802 597 y Fm(x)1854 564 y Fl(q)p 1802 638 90 4 v 1812 721 a Fm(q)s Fn(!)1903 659 y(\))p Fg(jj)1988 677 y Fj(\()p Fl(T)6 b(;X)f Fj(\))1831 895 y Fg(\024)25 b Fm(C)7 b(C)2064 909 y Fj(1)2103 895 y Fm(X)g Fn(\()p Fm(T)2286 857 y Fh(\000)p Fl(k)2378 866 y Fk(0)2417 895 y Fm(X)2499 857 y Fl(S)t Fh(\000)p Fl(k)2638 866 y Fk(1)2677 895 y Fm(R)20 b Fn(+)g Fm(T)2923 857 y Fh(\000)p Fl(k)3015 866 y Fk(0)3053 895 y Fg(jj)p Fm(')3162 909 y Fl(j)3200 895 y Fn(\()p Fm(t)p Fn(\))3313 833 y Fm(x)3365 800 y Fl(q)p 3313 874 V 3323 957 a Fm(q)s Fn(!)3413 895 y Fg(jj)3463 913 y Fj(\()p Fl(T)3531 922 y Fk(0)3567 913 y Fl(;X)3645 922 y Fk(0)3679 913 y Fj(\))3711 895 y Fn(\))p Fm(;)75 1294 y Fn(\(9\))84 b Fg(jj)p Fn(\()p Fm(@)412 1256 y Fh(\000)p Fl(S)407 1316 y(x)519 1294 y Fm(v)s Fn(\(0)p Fm(;)15 b(x)p Fn(\))22 b(+)e Fm(w)r Fn(\(0)p Fm(;)15 b(x)p Fn(\)\))p Fm(b)1234 1308 y Fl(\024)1275 1317 y Fk(0)1312 1308 y Fl(\024)1353 1317 y Fk(1)1391 1294 y Fn(\()p Fm(t;)g(x)p Fn(\)\()p Fm(@)1674 1253 y Fl(\024)1715 1262 y Fk(0)1669 1318 y Fl(t)1755 1294 y Fm(@)1808 1256 y Fl(\024)1849 1265 y Fk(1)1884 1256 y Fh(\000)p Fl(S)1803 1316 y(x)1989 1294 y Fm(v)s Fn(\()p Fm(t;)g(x)p Fn(\))22 b(+)e Fm(@)2397 1253 y Fl(\024)2438 1262 y Fk(0)2392 1318 y Fl(t)2476 1294 y Fm(')2535 1308 y Fl(j)2572 1294 y Fn(\()p Fm(t)p Fn(\))2685 1232 y Fm(x)2737 1199 y Fl(q)p 2685 1273 V 2695 1356 a Fm(q)s Fn(!)2786 1294 y(\))p Fg(jj)2871 1312 y Fj(\()p Fl(T)6 b(;X)f Fj(\))907 1530 y Fg(\024)25 b Fm(C)7 b(C)1140 1544 y Fj(1)1179 1530 y Fm(X)g Fn(\()p Fm(X)1378 1492 y Fl(S)1430 1530 y Fm(R)21 b Fn(+)e Fg(jj)p Fm(w)r Fn(\()p Fm(t;)c(x)p Fn(\))p Fg(jj)1972 1548 y Fj(\()p Fl(T)6 b(;X)f Fj(\))2164 1530 y Fn(\)\()p Fm(T)2300 1492 y Fh(\000)p Fl(\024)2396 1501 y Fk(0)2434 1530 y Fm(X)2516 1492 y Fl(S)t Fh(\000)p Fl(\024)2659 1501 y Fk(1)2698 1530 y Fm(R)21 b Fn(+)f Fm(T)2945 1492 y Fh(\000)p Fl(\024)3041 1501 y Fk(0)3079 1530 y Fg(jj)p Fm(')3188 1544 y Fl(j)3225 1530 y Fn(\()p Fm(t)p Fn(\))3338 1468 y Fm(x)3390 1435 y Fl(q)p 3338 1509 V 3348 1592 a Fm(q)s Fn(!)3439 1530 y Fg(jj)3489 1548 y Fj(\()p Fl(T)3557 1557 y Fk(0)3592 1548 y Fl(;X)3670 1557 y Fk(0)3705 1548 y Fj(\))3736 1530 y Fn(\))75 1888 y(\(10\))85 b Fg(jj)p Fm(c)p Fn(\()p Fm(t;)15 b(x)p Fn(\)\()p Fm(@)692 1851 y Fh(\000)p Fl(S)687 1911 y(x)800 1888 y Fm(v)s Fn(\()p Fm(t;)g(x)p Fn(\))21 b(+)f Fm(w)r Fn(\()p Fm(t;)15 b(x)p Fn(\)\))1451 1851 y Fj(2)1492 1888 y Fg(jj)1542 1907 y Fj(\()p Fl(T)6 b(;X)f Fj(\))1492 2039 y Fg(\024)25 b Fm(C)7 b(C)1725 2053 y Fj(1)1763 2039 y Fm(X)g Fn(\()p Fm(X)1962 2002 y Fj(2)p Fl(S)2050 2039 y Fm(R)2120 2002 y Fj(2)2179 2039 y Fn(+)20 b(2)p Fm(X)2397 2002 y Fl(S)2449 2039 y Fm(R)q Fg(jj)p Fm(w)r Fn(\()p Fm(t;)15 b(x)p Fn(\))p Fg(jj)2881 2058 y Fj(\()p Fl(T)6 b(;X)f Fj(\))3092 2039 y Fn(+)20 b Fg(jj)p Fm(w)r Fn(\()p Fm(t;)15 b(x)p Fn(\))p Fg(jj)3545 2002 y Fj(2)3545 2066 y(\()p Fl(T)6 b(;X)f Fj(\))3736 2039 y Fn(\))75 2233 y(for)30 b(all)h Fm(v)s Fn(\()p Fm(t;)15 b(x)p Fn(\))27 b Fg(2)e Fm(B)764 2247 y Fl(R)821 2233 y Fn(,)31 b(all)g(\()p Fm(k)1085 2247 y Fj(0)1125 2233 y Fm(;)15 b(k)1212 2247 y Fj(1)1253 2233 y Fn(\))25 b Fg(2)g Fm(S)1455 2247 y Fj(1)1494 2233 y Fn(,)31 b(all)g(\()p Fm(\024)1763 2247 y Fj(0)1803 2233 y Fm(;)15 b(\024)1895 2247 y Fj(1)1935 2233 y Fn(\))26 b Fg(2)f Fm(S)2138 2247 y Fj(2)2177 2233 y Fn(,)31 b(and)e(all)j(0)25 b Fg(\024)g Fm(j;)15 b(q)29 b Fg(\024)c Fm(S)g Fg(\000)20 b Fn(1.)75 2393 y Fd(Conclusion:)39 b Fn(Let)27 b Fm(R)f(>)f Fn(0.)40 b(If)25 b Fm(X)r(;)15 b(T)40 b Fn(are)26 b(c)m(hosen)h(small)g (enough,)g(the)f(sum)f(o)m(v)m(er)j(\()p Fm(k)3014 2407 y Fj(0)3054 2393 y Fm(;)15 b(k)3141 2407 y Fj(1)3181 2393 y Fn(\))25 b Fg(2)g(S)3382 2407 y Fj(1)3422 2393 y Fn(,)i(\()p Fm(\024)3561 2407 y Fj(0)3601 2393 y Fm(;)15 b(\024)3693 2407 y Fj(1)3733 2393 y Fn(\))26 b Fg(2)75 2506 y(S)130 2520 y Fj(2)204 2506 y Fn(and)34 b(0)f Fg(\024)f Fm(j;)15 b(q)36 b Fg(\024)31 b Fm(S)d Fg(\000)23 b Fn(1,)36 b(of)f(the)g(righ)m(t)g(hand)e(side)i(of)g(the)f(inequalities)i(\(8\),) h(\(9\),)g(\(10\),)h(can)c(b)s(e)g(made)75 2619 y(lo)m(w)m(er)e(than)e Fm(R)h Fn(for)f(all)h Fm(v)s Fn(\()p Fm(t;)15 b(x)p Fn(\))27 b Fg(2)e Fm(B)1314 2633 y Fl(R)1371 2619 y Fn(.)41 b(So)30 b(that)h(if)f Fm(v)f Fg(2)c Fm(B)2071 2633 y Fl(R)2159 2619 y Fn(then)30 b Fm(A)p Fn(\()p Fm(v)s Fn(\))c Fg(2)f Fm(B)2732 2633 y Fl(R)2790 2619 y Fn(.)75 2778 y Fd(2\))30 b Fn(Due)h(to)g(Lemma)f(1,2,)i(there)f(exist)g(a)g(constan)m(t)g Fm(C)h(>)25 b Fn(0)31 b(suc)m(h)f(that)75 2994 y(\(11\))158 3123 y Fg(jj)p Fm(a)256 3138 y Fl(k)293 3147 y Fk(0)328 3138 y Fl(k)365 3147 y Fk(1)403 3123 y Fn(\()p Fm(t;)15 b(x)p Fn(\))p Fm(@)651 3082 y Fl(k)688 3091 y Fk(0)646 3147 y Fl(t)728 3123 y Fm(@)781 3085 y Fl(k)818 3094 y Fk(1)853 3085 y Fh(\000)p Fl(S)776 3145 y(x)958 3123 y Fn(\()p Fm(v)1037 3137 y Fj(2)1077 3123 y Fn(\()p Fm(t;)g(x)p Fn(\))26 b Fg(\000)e Fm(v)1437 3137 y Fj(1)1476 3123 y Fn(\()p Fm(t;)15 b(x)p Fn(\)\))p Fg(jj)1756 3141 y Fj(\()p Fl(T)6 b(;X)f Fj(\))1983 3123 y Fg(\024)35 b Fm(C)7 b(C)2226 3137 y Fj(1)2265 3123 y Fm(X)g(T)2413 3085 y Fh(\000)p Fl(k)2505 3094 y Fk(0)2544 3123 y Fm(X)2626 3085 y Fl(S)t Fh(\000)p Fl(k)2765 3094 y Fk(1)2803 3123 y Fg(jj)p Fm(v)2897 3137 y Fj(2)2937 3123 y Fn(\()p Fm(t;)15 b(x)p Fn(\))26 b Fg(\000)e Fm(v)3297 3137 y Fj(1)3336 3123 y Fn(\()p Fm(t;)15 b(x)p Fn(\))p Fg(jj)3581 3141 y Fj(\()p Fl(T)6 b(;X)f Fj(\))75 3317 y Fn(for)30 b(all)h Fm(v)384 3331 y Fj(1)424 3317 y Fm(;)15 b(v)508 3331 y Fj(2)573 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4025 y Fi(\010)1119 4099 y Fn(\()p Fm(@)1207 4061 y Fh(\000)p Fl(S)1202 4121 y(x)1314 4099 y Fm(v)1358 4113 y Fj(2)1397 4099 y Fn(\(0)p Fm(;)g(x)p Fn(\))22 b Fg(\000)e Fm(@)1770 4061 y Fh(\000)p Fl(S)1765 4121 y(x)1876 4099 y Fm(v)1920 4113 y Fj(1)1959 4099 y Fn(\(0)p Fm(;)15 b(x)p Fn(\)\)\()p Fm(@)2289 4058 y Fl(\024)2330 4067 y Fk(0)2284 4123 y Fl(t)2371 4099 y Fm(@)2424 4061 y Fl(\024)2465 4070 y Fk(1)2500 4061 y Fh(\000)p Fl(S)2419 4121 y(x)2606 4099 y Fm(v)2650 4113 y Fj(2)2689 4099 y Fn(\()p Fm(t;)g(x)p Fn(\))21 b(+)f Fm(@)3049 4058 y Fl(\024)3090 4067 y Fk(0)3044 4123 y Fl(t)3129 4099 y Fm(@)3182 4061 y Fl(\024)3223 4070 y Fk(1)3177 4121 y Fl(x)3262 4099 y Fm(w)r Fn(\()p Fm(t;)15 b(x)p Fn(\)\))1194 4249 y(+)30 b(\()p Fm(@)1383 4211 y Fh(\000)p Fl(S)1378 4271 y(x)1489 4249 y Fm(v)1533 4263 y Fj(1)1572 4249 y Fn(\(0)p Fm(;)15 b(x)p Fn(\))22 b(+)e Fm(w)r Fn(\(0)p Fm(;)15 b(x)p Fn(\)\)\()p Fm(@)2289 4208 y Fl(\024)2330 4217 y Fk(0)2284 4273 y Fl(t)2372 4249 y Fm(@)2425 4211 y 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Fl(S)1859 4813 y Fn(\()p Fm(T)1960 4775 y Fh(\000)p Fl(\024)2056 4784 y Fk(0)2094 4813 y Fm(X)2176 4775 y Fl(S)t Fh(\000)p Fl(\024)2319 4784 y Fk(1)2357 4813 y Fm(R)21 b Fn(+)f Fm(T)2604 4775 y Fh(\000)p Fl(\024)2700 4784 y Fk(0)2738 4813 y Fm(X)2820 4775 y Fh(\000)p Fl(\024)2916 4784 y Fk(1)2955 4813 y Fg(jj)p Fm(w)r Fn(\()p Fm(t;)15 b(x)p Fn(\))p Fg(jj)3317 4831 y Fj(\()p Fl(T)3385 4840 y Fk(0)3423 4831 y Fl(;X)3501 4840 y Fk(0)3535 4831 y Fj(\))3566 4813 y Fn(\))2158 4964 y Fg(\002)30 b Fn(\()p Fm(X)2376 4927 y Fl(S)2427 4964 y Fm(R)21 b Fn(+)f Fg(jj)p Fm(w)r Fn(\()p Fm(t;)15 b(x)p Fn(\))p Fg(jj)2970 4983 y Fj(\()p Fl(T)3038 4992 y Fk(0)3076 4983 y Fl(;X)3154 4992 y Fk(0)3188 4983 y Fj(\))3220 4964 y Fn(\))p Fm(T)3321 4927 y Fh(\000)p Fl(\024)3417 4936 y Fk(0)3455 4964 y Fm(X)3537 4927 y Fl(S)t Fh(\000)p Fl(\024)3680 4936 y Fk(1)3718 4891 y Fi(\011)75 5158 y Fn(for)30 b(all)h Fm(v)384 5172 y Fj(1)424 5158 y Fm(;)15 b(v)508 5172 y Fj(2)573 5158 y Fg(2)25 b Fm(B)728 5172 y Fl(R)786 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Fi(\010)1037 5740 y Fn(\()p Fm(@)1125 5703 y Fh(\000)p Fl(S)1120 5763 y(x)1231 5740 y Fm(v)1275 5754 y Fj(2)1315 5740 y Fn(\()p Fm(t;)g(x)p Fn(\))21 b Fg(\000)f Fm(@)1675 5703 y Fh(\000)p Fl(S)1670 5763 y(x)1781 5740 y Fm(v)1825 5754 y Fj(1)1864 5740 y Fn(\()p Fm(t;)15 b(x)p Fn(\)\)\()p Fm(@)2182 5703 y Fh(\000)p Fl(S)2177 5763 y(x)2290 5740 y Fm(v)2334 5754 y Fj(2)2373 5740 y Fn(\()p Fm(t;)g(x)p Fn(\))21 b(+)f Fm(@)2733 5703 y Fh(\000)p Fl(S)2728 5763 y(x)2839 5740 y Fm(v)2883 5754 y Fj(1)2923 5740 y Fn(\()p Fm(t;)15 b(x)p Fn(\))21 b(+)f(2)p Fm(w)r Fn(\()p Fm(t;)15 b(x)p Fn(\)\))3572 5667 y Fi(\011)3642 5740 y Fm(:)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 3809 100 a Fn(6)75 399 y(F)-8 b(rom)31 b(Lemma)f(1,2,)i(there)f(exists)g(a)f(constan)m(t)i Fm(C)g(>)25 b Fn(0)31 b(suc)m(h)f(that)75 573 y(\(13\))240 b Fg(jjB)585 587 y Fj(2)p Fl(;)p Fj(1)679 573 y Fn(\()p Fm(t;)15 b(x)p Fn(\))p Fg(jj)924 591 y Fj(\()p Fl(T)6 b(;X)f Fj(\))1140 573 y Fg(\024)25 b Fm(C)7 b(C)1373 587 y Fj(1)1412 573 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b(the)g(fact)g(that)g(the)f(op)s(erator)h(\()p Fm(m)20 b Fn(+)f(1\))2252 1857 y Fl(r)2284 1866 y Fk(1)2324 1890 y Fn(\(4)p Fm(t)2437 1857 y Fj(2)2477 1890 y Fm(@)2525 1904 y Fl(t)2575 1890 y Fn(+)g(2)p Fm(t)p Fn(\))2778 1857 y Fl(r)2810 1866 y Fk(2)2870 1890 y Fn(+)g(1)31 b(is)f(in)m(v)m(ertible)i(on)e Ff(C)p Fn([[)p Fm(t)p Fn(]])75 2003 y(for)h(all)h Fm(m)26 b Fg(\025)g Fn(0,)31 b(w)m(e)h(get)g(from)e Fd(\(H2\))h Fn(that)g(all)1745 1980 y(^)1719 2003 y Fm(X)1794 2017 y Fl(m)1861 2003 y Fn(\()p Fm(t)p Fn(\),)h(for)f Fm(m)26 b Fg(\025)g Fm(S)36 b Fn(can)31 b(b)s(e)f(expressed)h(uniquely)f(in)h(terms)75 2116 y(of)d(the)g(formal)g(series)881 2093 y(^)854 2116 y Fm(X)929 2131 y Fl(k)972 2116 y Fn(\()p Fm(t)p Fn(\),)i(0)25 b Fg(\024)g Fm(k)k Fg(\024)c Fm(m)15 b Fg(\000)g Fn(1.)40 b(This)27 b(giv)m(es)i(the)f(existence)i(and)d(unicit)m(y)i(of)3325 2093 y(^)3299 2116 y Fm(X)7 b Fn(\()p Fm(t;)15 b(z)t Fn(\))29 b(formal)75 2229 y(series)i(solution)g(of)f(\(14\).)2805 b Fc(2)75 2385 y Fn(In)30 b(the)g(follo)m(wing)i(de\014nition,)e(w)m(e) h(in)m(tro)s(duce)f(a)h(mo)s(di\014ed)e(Borel)j(transform)e(of)g(order) g(1)h(of)f(formal)h(series.)75 2556 y Fd(De\014nition)k(3)46 b Fe(L)-5 b(et)1677 2680 y Fn(^)1651 2703 y Fm(X)7 b Fn(\()p Fm(t)p Fn(\))26 b(=)1958 2617 y Fi(X)1963 2812 y Fl(j)t Fh(\025)p Fj(0)2105 2703 y Fm(a)2153 2717 y Fl(j)2199 2641 y Fm(t)2232 2608 y Fl(j)p 2199 2682 70 4 v 2200 2765 a Fm(j)5 b Fn(!)75 2950 y Fe(b)-5 b(e)33 b(a)g(formal)h(series)e(with)i(c)-5 b(o)g(e\016cients)33 b Fm(a)1520 2964 y Fl(j)1582 2950 y Fg(2)25 b Ff(C)p Fe(.)41 b(We)33 b(de\014ne)f(the)h(formal)i(series)1509 3194 y Fn(\()p Fg(B)1604 3208 y Fj(1)1669 3172 y Fn(^)1643 3194 y Fm(X)8 b Fn(\)\()p Fm(t)p Fn(\))26 b(=)1986 3108 y Fi(X)1990 3304 y Fl(j)t Fh(\025)p Fj(0)2132 3194 y Fm(a)2180 3208 y Fl(j)2284 3133 y Fm(t)2317 3100 y Fl(j)p 2227 3174 185 4 v 2227 3257 a Fn(\(2)p Fm(j)5 b Fn(\)!)75 3492 y Fe(as)33 b(the)g(Bor)-5 b(el)33 b(tr)-5 b(ansform)35 b(of)e(or)-5 b(der)34 b(1)f(of)1559 3469 y Fn(^)1533 3492 y Fm(X)7 b Fn(\()p Fm(t)p Fn(\))33 b Fe(c)-5 b(onstructe)g(d)35 b(fr)-5 b(om)33 b(the)g(moment)h(function)f Fm(m)p Fn(\()p Fm(u)p Fn(\))25 b(=)g(\000\(1)c(+)75 3605 y(2)p Fm(u)p Fn(\))p Fm(=)p Fn(\000\(1)h(+)e Fm(u)p Fn(\))33 b Fe(for)g Fn(Re\()p Fm(u)p Fn(\))26 b Fg(\025)f Fn(0)33 b Fe(\(se)-5 b(e)33 b(se)-5 b(ction)34 b(5.5)f(of)g([1])g(for)g(a)g(r)-5 b(efer)g(enc)g(e\).)75 3797 y Fd(Lemma)35 b(5)45 b Fe(A)33 b(formal)h(series)1521 3923 y Fn(^)1495 3946 y Fm(X)7 b Fn(\()p Fm(t;)15 b(z)t Fn(\))27 b(=)1899 3860 y Fi(X)1889 4056 y Fl(m)p Fh(\025)p Fj(0)2083 3923 y Fn(^)2056 3946 y Fm(X)2131 3960 y Fl(m)2198 3946 y Fn(\()p Fm(t)p Fn(\))2311 3885 y Fm(z)2357 3852 y Fl(m)p 2311 3925 113 4 v 2315 4009 a Fm(m)p Fn(!)75 4217 y Fe(wher)-5 b(e)348 4194 y Fn(^)322 4217 y Fm(X)397 4231 y Fl(m)464 4217 y Fn(\()p Fm(t)p Fn(\))26 b Fg(2)f Ff(C)p Fn([[)p Fm(t)p Fn(]])p Fe(,)g Fm(m)g Fg(\025)g Fn(0)p Fe(,)h(satis\014es)e(the)g(pr)-5 b(oblem)25 b(\(14\),)h(with)e(given)f(initial)h(c)-5 b(onditions)25 b Fn(\()p Fm(@)3443 4172 y Fl(j)3438 4228 y(z)3506 4194 y Fn(^)3480 4217 y Fm(X)8 b Fn(\)\()p Fm(t;)15 b Fn(0\))27 b(=)101 4307 y(^)75 4330 y Fm(X)150 4344 y Fl(j)187 4330 y Fn(\()p Fm(t)p Fn(\))p Fe(,)33 b Fn(0)25 b Fg(\024)g Fm(j)31 b Fg(\024)25 b Fm(S)g Fg(\000)20 b Fn(1)p Fe(,)33 b(if)g(and)g(only)h(if)e(the)h(formal)h(series)1402 4559 y Fm(V)21 b Fn(\()p Fm(t;)15 b(z)t Fn(\))26 b(=)1798 4472 y Fi(X)1787 4668 y Fl(m)p Fh(\025)p Fj(0)1940 4559 y Fn(\()p Fg(B)2035 4573 y Fj(1)2100 4536 y Fn(^)2074 4559 y Fm(X)2149 4573 y Fl(m)2216 4559 y Fn(\)\()p Fm(t)2319 4521 y Fj(2)2359 4559 y Fn(\))2404 4497 y Fm(z)2450 4464 y Fl(m)p 2404 4538 V 2408 4621 a Fm(m)p Fn(!)75 4831 y Fe(satis\014es)34 b(the)f(fol)5 b(lowing)33 b(non)g(line)-5 b(ar)34 b(p)-5 b(artial)35 b(di\013er)-5 b(ential)34 b(e)-5 b(quation)75 5041 y Fn(\(16\))85 b(\()p Fm(t)388 5003 y Fj(2)p Fl(r)455 5012 y Fk(2)494 5041 y Fn(\()p Fm(z)t(@)623 5055 y Fl(z)684 5041 y Fn(+)19 b(1\))854 5003 y Fl(r)886 5012 y Fk(1)946 5041 y Fn(+)h(1\))p Fm(@)1170 5003 y Fl(S)1165 5063 y(z)1222 5041 y Fm(V)g Fn(\()p Fm(t;)15 b(z)t Fn(\))1181 5246 y(=)1367 5160 y Fi(X)1277 5361 y Fj(\()p Fl(k)1341 5370 y Fk(0)1376 5361 y Fl(;k)1433 5370 y Fk(1)1467 5361 y Fj(\))p Fh(2S)1604 5246 y Fm(b)1643 5261 y Fl(k)1680 5270 y Fk(0)1714 5261 y Fl(k)1751 5270 y Fk(1)1790 5246 y Fn(\()p Fm(z)t Fn(\))p Fm(@)1959 5205 y Fj(2)p Fl(k)2031 5214 y Fk(0)1954 5270 y Fl(t)2071 5246 y Fm(@)2124 5208 y Fl(k)2161 5217 y Fk(1)2119 5268 y Fl(z)2200 5246 y Fm(V)20 b Fn(\()p Fm(t;)15 b(z)t Fn(\))21 b(+)f Fm(b)p Fn(\()p Fm(z)t Fn(\))p Fm(@)2777 5260 y Fl(t)2823 5122 y Fi(Z)2914 5149 y Fl(t)2874 5328 y Fj(0)2959 5246 y Fm(V)g Fn(\()p Fm(t)g Fg(\000)g Fm(\034)5 b(;)15 b(z)t Fn(\))p Fm(V)22 b Fn(\()p Fm(\034)5 b(;)15 b(z)t Fn(\))p Fm(d\034)5 b(;)75 5529 y Fe(with)33 b(given)f(initial)i(c)-5 b(onditions)75 5714 y Fn(\(17\))718 b(\()p Fm(@)1041 5676 y Fl(j)1036 5736 y(z)1078 5714 y Fm(V)20 b Fn(\)\()p Fm(t;)15 b Fn(0\))27 b(=)e(\()p Fg(B)1592 5728 y Fj(1)1658 5691 y Fn(^)1632 5714 y Fm(X)1707 5728 y Fl(j)1744 5714 y Fn(\)\()p Fm(t)1847 5676 y Fj(2)1887 5714 y Fn(\))g Fg(2)g Ff(C)p Fn([[)p Fm(t)p Fn(]])66 b Fm(;)80 b Fn(0)26 b Fg(\024)f Fm(j)31 b Fg(\024)25 b Fm(S)g Fg(\000)20 b Fn(1)p Fm(:)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 3809 100 a Fn(8)75 399 y Fd(Pro)s(of)31 b Fn(Let)g(for)f(all)h Fm(m)26 b Fg(\025)f Fn(0,)1596 540 y(^)1570 563 y Fm(X)1645 577 y Fl(m)1712 563 y Fn(\()p Fm(t)p Fn(\))g(=)1936 477 y Fi(X)1941 672 y Fl(j)t Fh(\025)p Fj(0)2083 563 y Fm(a)2131 577 y Fl(j;m)2255 501 y Fm(t)2288 468 y Fl(j)p 2255 542 70 4 v 2256 625 a Fm(j)5 b Fn(!)2335 563 y Fm(:)75 816 y Fn(So)30 b(that)1408 962 y(\()p Fg(B)1503 976 y Fj(1)1568 940 y Fn(^)1542 962 y Fm(X)1617 976 y Fl(m)1684 962 y Fn(\)\()p Fm(t)1787 925 y Fj(2)1827 962 y Fn(\))c(=)1984 876 y Fi(X)1988 1072 y Fl(j)t Fh(\025)p Fj(0)2130 962 y Fm(a)2178 976 y Fl(j;m)2342 901 y Fm(t)2375 868 y Fj(2)p Fl(j)p 2303 942 185 4 v 2303 1025 a Fn(\(2)p Fm(j)5 b Fn(\)!)2497 962 y Fm(:)75 1215 y Fn(F)-8 b(rom)31 b(the)f(classical)j(iden)m(tit)m(y)1135 1339 y Fi(Z)1226 1365 y Fl(t)1186 1545 y Fj(0)1256 1463 y Fn(\()p Fm(t)20 b Fg(\000)g Fm(\034)10 b Fn(\))1520 1425 y Fl(j)1549 1434 y Fk(1)1588 1463 y Fm(\034)1638 1425 y Fl(j)1667 1434 y Fk(2)1705 1463 y Fm(d\034)36 b Fn(=)2090 1401 y Fm(j)2127 1415 y Fj(1)2167 1401 y Fn(!)p Fm(j)2229 1415 y Fj(2)2269 1401 y Fn(!)p 1933 1442 518 4 v 1933 1525 a(\(1)21 b(+)f Fm(j)2162 1539 y Fj(1)2222 1525 y Fn(+)g Fm(j)2350 1539 y Fj(2)2390 1525 y Fn(\)!)2461 1463 y Fm(t)2494 1425 y Fl(j)2523 1434 y Fk(1)2557 1425 y Fj(+)p Fl(j)2641 1434 y Fk(2)2675 1425 y Fj(+1)2769 1463 y Fm(;)75 1703 y Fn(w)m(e)31 b(get)g(that)853 1867 y Fm(@)901 1881 y Fl(t)946 1743 y Fi(Z)1037 1770 y Fl(t)996 1949 y Fj(0)1107 1844 y Fn(^)1081 1867 y Fm(X)1156 1881 y Fl(m)1218 1890 y Fk(1)1258 1867 y Fn(\()p Fm(t)20 b Fg(\000)g Fm(\034)10 b Fn(\))1549 1844 y(^)1522 1867 y Fm(X)1597 1881 y Fl(m)1659 1890 y Fk(2)1699 1867 y Fn(\()p Fm(\034)g Fn(\))p Fm(d\034)36 b Fn(=)2038 1781 y Fi(X)2043 1977 y Fl(j)t Fh(\025)p Fj(0)2170 1867 y Fn(\()2273 1781 y Fi(X)2205 1976 y Fl(j)2234 1985 y Fk(1)2268 1976 y Fj(+)p Fl(j)2352 1985 y Fk(2)2386 1976 y Fj(=)p Fl(j)2488 1867 y Fm(a)2536 1881 y Fl(j)2565 1890 y Fk(1)2600 1881 y Fl(;m)2682 1890 y Fk(1)2720 1867 y Fm(a)2768 1881 y Fl(j)2797 1890 y Fk(2)2832 1881 y Fl(;m)2914 1890 y Fk(2)2952 1867 y Fn(\))2997 1806 y Fm(t)3030 1773 y Fl(j)p 2997 1846 70 4 v 2998 1929 a Fm(j)5 b Fn(!)75 2120 y(and)30 b(that)545 2350 y Fm(@)593 2364 y Fl(t)639 2226 y Fi(Z)729 2252 y Fl(t)689 2432 y Fj(0)759 2350 y Fn(\()p Fg(B)854 2364 y Fj(1)920 2327 y Fn(^)894 2350 y Fm(X)969 2364 y Fl(m)1031 2373 y Fk(1)1070 2350 y Fn(\)\(\()p Fm(t)21 b Fg(\000)f Fm(\034)10 b Fn(\))1405 2312 y Fj(2)1445 2350 y Fn(\)\()p Fg(B)1575 2364 y Fj(1)1641 2327 y Fn(^)1615 2350 y Fm(X)1690 2364 y Fl(m)1752 2373 y Fk(2)1791 2350 y Fn(\)\()p Fm(\034)1911 2312 y Fj(2)1951 2350 y Fn(\))p Fm(d\034)36 b Fn(=)2205 2263 y Fi(X)2210 2459 y Fl(j)t Fh(\025)p Fj(0)2337 2350 y Fn(\()2441 2263 y Fi(X)2372 2459 y Fl(j)2401 2468 y Fk(1)2435 2459 y Fj(+)p Fl(j)2519 2468 y Fk(2)2553 2459 y Fj(=)p Fl(j)2656 2350 y Fm(a)2704 2364 y Fl(j)2733 2373 y Fk(1)2767 2364 y Fl(;m)2849 2373 y Fk(1)2887 2350 y Fm(a)2935 2364 y Fl(j)2964 2373 y Fk(2)2999 2364 y Fl(;m)3081 2373 y Fk(2)3119 2350 y Fn(\))3204 2288 y Fm(t)3237 2255 y Fj(2)p Fl(j)p 3164 2329 185 4 v 3164 2412 a Fn(\(2)p Fm(j)5 b Fn(\)!)3359 2350 y Fm(;)75 2636 y Fn(for)30 b(all)h Fm(m)420 2650 y Fj(1)460 2636 y Fm(;)15 b(m)580 2650 y Fj(2)645 2636 y Fg(\025)25 b Fn(0.)41 b(Moreo)m(v)m(er,)32 b(w)m(e)f(ha)m(v)m(e)h(that)1078 2884 y(\(4)p Fm(t)1191 2846 y Fj(2)1231 2884 y Fm(@)1279 2898 y Fl(t)1329 2884 y Fn(+)20 b(2)p Fm(t)p Fn(\))1560 2861 y(^)1533 2884 y Fm(X)1608 2898 y Fl(m)1675 2884 y Fn(\()p Fm(t)p Fn(\))26 b(=)1900 2797 y Fi(X)1905 2993 y Fl(j)t Fh(\025)p Fj(1)2031 2884 y Fn(\(2)p Fm(j)5 b Fn(\)\(2)p Fm(j)29 b Fg(\000)20 b Fn(1\))p Fm(a)2553 2898 y Fl(j)t Fh(\000)p Fj(1)p Fl(;m)2772 2822 y Fm(t)2805 2789 y Fl(j)p 2772 2863 70 4 v 2773 2946 a Fm(j)5 b Fn(!)75 3170 y(and)1095 3317 y Fm(t)1128 3279 y Fj(2)1167 3317 y Fn(\()p Fg(B)1262 3331 y Fj(1)1328 3294 y Fn(^)1302 3317 y Fm(X)1377 3331 y Fl(m)1444 3317 y Fn(\)\()p Fm(t)1547 3279 y Fj(2)1587 3317 y Fn(\))25 b(=)1743 3230 y Fi(X)1748 3426 y Fl(j)t Fh(\025)p Fj(1)1874 3317 y Fn(\(2)p Fm(j)5 b Fn(\)\(2)p Fm(j)29 b Fg(\000)20 b Fn(1\))p Fm(a)2396 3331 y Fl(j)t Fh(\000)p Fj(1)p Fl(;m)2655 3255 y Fm(t)2688 3222 y Fj(2)p Fl(j)p 2615 3296 185 4 v 2615 3379 a Fn(\(2)p Fm(j)5 b Fn(\)!)2809 3317 y Fm(;)75 3569 y Fn(for)30 b(all)h Fm(m)25 b Fg(\025)g Fn(0.)42 b(The)29 b(Lemma)i(follo)m(ws.) 2300 b Fc(2)216 3729 y Fn(Let)29 b Fm(S)433 3744 y Fl(d;\016)552 3729 y Fn(=)c Fg(f)p Fm(t)h Fg(2)f Ff(C)f Fn(:)i Fg(j)p Fm(d)18 b Fg(\000)e Fn(arg)r(\()p Fm(t)p Fn(\))p Fg(j)26 b Fm(<)f(\016)s Fg(g)30 b Fn(b)s(e)e(an)h(op)s(en)f(sector)i(cen)m (tered)g(at)f(0)h(with)e(bisecting)i(direction)75 3842 y Fm(d)h Fg(2)f Ff(R)j Fn(and)g(with)g(op)s(ening)g Fm(\016)i(>)30 b Fn(0.)51 b(F)-8 b(or)34 b(all)g(in)m(tegers)h Fm(m)30 b Fg(\025)h Fn(0,)k(let)f Fm(D)2535 3856 y Fl(m)2635 3842 y Fn(b)s(e)f(the)h(op)s(en)f(disc)g(cen)m(tered)i(at)f(0)75 3955 y(with)27 b(radius)f(1)p Fm(=)p Fn(\(2\()p Fm(m)13 b Fn(+)g(1\))1011 3922 y Fl(r)1043 3931 y Fk(1)1081 3922 y Fl(=)p Fj(2)p Fl(r)1183 3931 y Fk(2)1222 3955 y Fn(\).)40 b(W)-8 b(e)28 b(denote)f(b)m(y)g Fg(A)1960 3970 y Fl(\027;d;\016)o(;m) 2213 3955 y Fn(the)h(v)m(ector)g(space)g(of)f(holomorphic)g(functions) 75 4068 y Fm(G)k Fn(on)f Fm(S)359 4083 y Fl(d;\016)473 4068 y Fg([)20 b Fm(D)629 4082 y Fl(m)695 4068 y Fn(,)31 b(con)m(tin)m(uous)g(on)f(the)h(closure)1808 4045 y(\026)1793 4068 y Fm(S)1849 4083 y Fl(d;\016)1962 4068 y Fg([)2064 4045 y Fn(\026)2043 4068 y Fm(D)2118 4082 y Fl(m)2215 4068 y Fn(suc)m(h)f(that)895 4275 y Fg(jj)p Fm(G)p Fn(\()p Fm(t)p Fn(\))p Fg(jj)1169 4290 y Fl(d;m)1318 4275 y Fn(:=)25 b Fm(M)1527 4289 y Fj(0)1696 4275 y Fn(sup)1582 4363 y Fl(t)p Fh(2)1666 4346 y Fj(\026)1654 4363 y Fl(S)1697 4375 y Fb(d;\016)1783 4363 y Fh([)1846 4346 y Fj(\026)1830 4363 y Fl(D)1888 4371 y Fb(m)1946 4275 y Fn(\(1)c(+)f Fg(j)p Fm(t)p Fg(j)2221 4237 y Fj(2)2261 4275 y Fn(\))p Fm(e)2338 4237 y Fh(\000)p Fl(\027)t Fh(j)p Fl(t)p Fh(j)2501 4275 y Fg(j)p Fm(G)p Fn(\()p Fm(t)p Fn(\))p Fg(j)27 b Fm(<)e Fn(+)p Fg(1)p Fm(;)75 4548 y Fn(where)30 b Fm(\027)h(>)25 b Fn(0)31 b(and)e Fm(M)850 4562 y Fj(0)915 4548 y Fm(>)c Fn(1)31 b(is)f(a)h(constan)m(t)h(suc)m(h)e(that)1167 4676 y Fi(Z)1258 4703 y Fh(j)p Fl(t)p Fh(j)1218 4882 y Fj(0)1749 4739 y Fm(ds)p 1352 4779 884 4 v 1352 4862 a Fn(\(1)21 b(+)f Fm(s)1587 4836 y Fj(2)1626 4862 y Fn(\)\(1)h(+)f(\()p Fg(j)p Fm(t)p Fg(j)h(\000)f Fm(s)p Fn(\))2161 4836 y Fj(2)2201 4862 y Fn(\))2271 4800 y Fg(\024)2488 4739 y Fm(M)2576 4753 y Fj(0)p 2377 4779 351 4 v 2377 4862 a Fn(\(1)h(+)f Fg(j)p Fm(t)p Fg(j)2652 4836 y Fj(2)2692 4862 y Fn(\))2737 4800 y Fm(;)75 5051 y Fn(for)30 b(all)h Fm(t)25 b Fg(2)500 5028 y Fn(\026)484 5051 y Fm(S)540 5066 y Fl(d;\016)654 5051 y Fg([)756 5028 y Fn(\026)735 5051 y Fm(D)810 5065 y Fj(1)850 5051 y Fn(.)40 b(In)30 b(the)g(follo)m(wing,)j(w)m(e)d(will)h(need)f(a)h(lemma)g(from)f([7)q (],)g(Corollary)h(4.9.)75 5256 y Fd(Lemma)k(6)45 b Fe(The)33 b(fol)5 b(lowing)34 b(ine)-5 b(quality)33 b(hold)993 5503 y Fg(jj)1058 5379 y Fi(Z)1150 5406 y Fl(t)1110 5586 y Fj(0)1195 5503 y Fm(F)13 b Fn(\()p Fm(t)20 b Fg(\000)g Fm(\034)10 b Fn(\))p Fm(G)p Fn(\()p Fm(\034)g Fn(\))p Fm(d\034)g Fg(jj)1868 5518 y Fl(d;m)2018 5503 y Fg(\024)25 b(jj)p Fm(F)13 b Fn(\()p Fm(t)p Fn(\))p Fg(jj)2388 5518 y Fl(d;m)2512 5503 y Fg(jj)p Fm(G)p Fn(\()p Fm(t)p Fn(\))p Fg(jj)2786 5518 y Fl(d;m)2911 5503 y Fm(;)75 5740 y Fe(for)33 b(al)5 b(l)33 b Fm(F)s(;)15 b(G)26 b Fg(2)f(A)707 5755 y Fl(\027;d;\016)o(;m)933 5740 y Fe(.)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 3809 100 a Fn(9)216 399 y(In)25 b(the)h(follo)m(wing)i(theorem,)f(w)m(e)g(estimate)h(the)e(rate)g(of)g (gro)m(wth)h(of)f(the)g(co)s(e\016cien)m(ts)i(in)d Fg(jj)53 b(jj)3418 414 y Fl(d;m)3541 399 y Fg(\000)p Fn(norms)75 511 y(of)38 b(formal)h(series)g(solutions)f(of)h(the)f(problem)g (\(16\))i(for)e(giv)m(en)h(initial)h(conditions)e(\(17\))i(in)e(some)h (v)m(ector)75 624 y(space)31 b Fg(A)391 639 y Fl(\027;d;\016)o(;j)587 624 y Fn(,)g(0)25 b Fg(\024)g Fm(j)31 b Fg(\024)25 b Fm(S)g Fg(\000)20 b Fn(1.)75 810 y Fd(Theorem)35 b(2)46 b Fe(L)-5 b(et)33 b(a)g(formal)h(series)1511 1023 y Fm(V)20 b Fn(\()p Fm(t;)15 b(z)t Fn(\))26 b(=)1906 936 y Fi(X)1895 1132 y Fl(m)p Fh(\025)p Fj(0)2063 1023 y Fm(V)2116 1037 y Fl(m)2183 1023 y Fn(\()p Fm(t)p Fn(\))2296 961 y Fm(z)2342 928 y Fl(m)p 2296 1002 113 4 v 2300 1085 a Fm(m)p Fn(!)75 1293 y Fe(b)-5 b(e)33 b(a)g(solution)g(of)g(\(16\))h(with)f(initial)h (c)-5 b(onditions)1034 1470 y Fn(\()p Fm(@)1122 1433 y Fl(j)1117 1493 y(z)1159 1470 y Fm(V)20 b Fn(\)\()p Fm(t;)15 b Fn(0\))27 b(=)e Fm(V)1631 1484 y Fl(j)1668 1470 y Fn(\()p Fm(t)p Fn(\))h Fg(2)f(A)1956 1485 y Fl(\027;d;\016)o(;j) 2217 1470 y Fm(;)80 b Fn(0)26 b Fg(\024)f Fm(j)31 b Fg(\024)25 b Fm(S)g Fg(\000)20 b Fn(1)p Fm(;)75 1647 y Fe(for)40 b Fm(\027)k(>)38 b Fn(0)p Fe(,)j(wher)-5 b(e)41 b Fm(d)d Fg(2)g Ff(R)h Fe(and)h Fm(\016)i(>)c Fn(0)i Fe(ar)-5 b(e)40 b(such)g(that)h(the)f(se)-5 b(ctor)2563 1624 y Fn(\026)2548 1647 y Fm(S)2604 1662 y Fl(d;\016)2737 1647 y Fe(do)g(es)41 b(not)f(c)-5 b(ontain)41 b(any)f(of)g(the)75 1760 y(c)-5 b(omplex)34 b(numb)-5 b(ers)740 1972 y Fm(r)781 1987 y Fl(k)r(;m)931 1972 y Fn(=)1250 1910 y(1)p 1037 1951 471 4 v 1037 2070 a(2\()p Fm(m)21 b Fn(+)f(1\))1415 1983 y Fb(r)1444 1998 y Fk(1)p 1400 2004 94 3 v 1400 2045 a(2)p Fb(r)1459 2060 y Fk(2)1533 1972 y Fn(exp\()1737 1910 y Fm(i\031)p 1717 1951 126 4 v 1717 2034 a Fn(2)p Fm(r)1803 2048 y Fj(2)1853 1972 y Fn(\(2)p Fm(k)k Fn(+)c(1\)\))66 b Fm(;)81 b(k)28 b Fg(2)d(f)p Fn(0)p Fm(;)15 b(:)g(:)g(:)j(;)d Fn(2)p Fm(r)2922 1986 y Fj(2)2982 1972 y Fg(\000)20 b Fn(1)p Fg(g)p Fm(;)75 2241 y Fe(for)36 b(al)5 b(l)35 b Fm(m)30 b Fg(\025)f Fn(0)p Fe(.)50 b(Note)35 b(that)i Fn(2)p Fm(r)1184 2256 y Fl(k)r(;m)1344 2241 y Fe(ar)-5 b(e)36 b(the)f(r)-5 b(o)g(ots)38 b(of)d(the)h(p)-5 b(olynomial)38 b Fn(\()p Fm(m)22 b Fn(+)g(1\))2912 2208 y Fl(r)2944 2217 y Fk(1)2983 2241 y Fm(t)3016 2208 y Fj(2)p Fl(r)3083 2217 y Fk(2)3144 2241 y Fn(+)g(1)p Fe(.)49 b(Assume)35 b(that)75 2354 y(ther)-5 b(e)34 b(exist)e Fm(C)578 2368 y Fj(1)643 2354 y Fm(>)25 b Fn(0)p Fm(;)15 b(T)39 b(>)25 b Fn(0)33 b Fe(such)f(that)75 2564 y Fn(\(18\))1178 b Fg(jj)p Fm(@)1516 2523 y Fl(n)1559 2532 y Fk(0)1511 2588 y Fl(t)1598 2564 y Fm(V)1651 2578 y Fl(j)1688 2564 y Fn(\()p Fm(t)p Fn(\))p Fg(jj)1841 2579 y Fl(d;j)1959 2564 y Fg(\024)25 b Fm(C)2120 2578 y Fj(1)2160 2564 y Fn(\()2215 2502 y(1)p 2205 2543 66 4 v 2205 2626 a Fm(T)2281 2564 y Fn(\))2316 2526 y Fl(n)2359 2535 y Fk(0)2398 2564 y Fm(n)2453 2578 y Fj(0)2492 2564 y Fn(!)75 2762 y Fe(for)33 b(al)5 b(l)33 b Fn(0)26 b Fg(\024)f Fm(j)31 b Fg(\024)25 b Fm(S)g Fg(\000)20 b Fn(1)p Fe(,)33 b(al)5 b(l)33 b Fm(n)1144 2776 y Fj(0)1208 2762 y Fg(\025)25 b Fn(0)p Fe(.)216 2875 y(Then,)33 b(ther)-5 b(e)33 b(exist)g Fm(C)984 2889 y Fj(2)1024 2875 y Fm(;)15 b(T)8 b(;)15 b(X)33 b(>)25 b Fn(0)33 b Fe(such)g(that)75 3085 y Fn(\(19\))724 b Fg(jj)p Fm(@)1062 3044 y Fl(n)1105 3053 y Fk(0)1057 3109 y Fl(t)1144 3085 y Fm(V)1197 3099 y Fl(m)1264 3085 y Fn(\()p Fm(t)p Fn(\))p Fg(jj)1417 3100 y Fl(d;m)1566 3085 y Fg(\024)25 b Fm(C)1727 3099 y Fj(2)1766 3085 y Fn(\()1830 3023 y(1)p 1811 3064 83 4 v 1811 3147 a Fm(X)1904 3085 y Fn(\))1939 3047 y Fl(m)2006 3085 y Fn(\()2061 3023 y(1)p 2051 3064 66 4 v 2051 3147 a Fm(T)2127 3085 y Fn(\))2162 3047 y Fl(n)2205 3056 y Fk(0)2244 3085 y Fn(\()p Fm(m)20 b Fn(+)g(1\))2576 2998 y Fb(r)2605 3013 y Fk(1)p 2561 3019 94 3 v 2561 3060 a(2)p Fb(r)2620 3075 y Fk(2)2665 3034 y Fl(n)2708 3043 y Fk(0)2746 3085 y Fm(n)2801 3099 y Fj(0)2840 3085 y Fn(!)p Fm(m)p Fn(!)75 3283 y Fe(for)33 b(al)5 b(l)33 b Fm(n)405 3297 y Fj(0)444 3283 y Fm(;)15 b(m)26 b Fg(\025)f Fn(0)p Fe(.)75 3509 y Fd(Pro)s(of)31 b Fn(W)-8 b(e)32 b(expand)d(the)i(functions)1145 3694 y Fm(b)1184 3709 y Fl(k)1221 3718 y Fk(0)1255 3709 y Fl(k)1292 3718 y Fk(1)1331 3694 y Fn(\()p Fm(z)t Fn(\))26 b(=)1569 3608 y Fi(X)1570 3805 y Fl(k)r Fh(\025)p Fj(0)1715 3694 y Fm(b)1754 3709 y Fl(k)1791 3718 y Fk(0)1826 3709 y Fl(k)1863 3718 y Fk(1)1897 3709 y Fl(k)1940 3694 y Fm(z)1986 3657 y Fl(k)2089 3694 y Fm(;)76 b(b)p Fn(\()p Fm(z)t Fn(\))27 b(=)2468 3608 y Fi(X)2469 3805 y Fl(k)r Fh(\025)p Fj(0)2614 3694 y Fm(b)2653 3709 y Fl(k)2696 3694 y Fm(z)2742 3657 y Fl(k)75 3979 y Fn(in)h(p)s(o)m(w)m(er)g(series) g(near)g(0.)41 b(Let)28 b Fm(V)20 b Fn(\()p Fm(t;)15 b(z)t Fn(\))27 b(=)1539 3911 y Fi(P)1635 4006 y Fl(m)p Fh(\025)p Fj(0)1807 3979 y Fm(V)1860 3993 y Fl(m)1926 3979 y Fn(\()p Fm(t)p Fn(\))2039 3943 y Fl(z)2075 3920 y Fb(m)p 2040 3958 95 4 v 2046 4010 a Fl(m)p Fj(!)2172 3979 y Fn(b)s(e)h(a)g(formal)g(series)h(satisfying)f(\(16\).)42 b(W)-8 b(e)29 b(ha)m(v)m(e)75 4092 y(the)i(follo)m(wing)g(recursion)f (relation)i(for)e(the)h(formal)f(series)h Fm(V)2217 4106 y Fl(m)2283 4092 y Fn(\()p Fm(t)p Fn(\),)75 4362 y(\(20\))330 4300 y Fm(V)383 4314 y Fl(m)p Fj(+)p Fl(S)551 4300 y Fn(\()p Fm(t)p Fn(\))p 330 4341 325 4 v 439 4424 a Fm(m)p Fn(!)689 4362 y(=)876 4275 y Fi(X)785 4476 y Fj(\()p Fl(k)849 4485 y Fk(0)884 4476 y Fl(;k)941 4485 y Fk(1)975 4476 y Fj(\))p Fh(2S)1199 4275 y Fi(X)1112 4467 y Fl(q)1144 4476 y Fk(1)1178 4467 y Fj(+)p Fl(q)1265 4476 y Fk(2)1299 4467 y Fj(=)p Fl(m)1652 4298 y Fm(b)1691 4313 y Fl(k)1728 4322 y Fk(0)1763 4313 y Fl(k)1800 4322 y Fk(1)1834 4313 y Fl(q)1866 4322 y Fk(1)p 1442 4341 673 4 v 1442 4424 a Fn(\()p Fm(m)20 b Fn(+)g(1\))1748 4398 y Fl(r)1780 4407 y Fk(1)1820 4424 y Fm(t)1853 4398 y Fj(2)p Fl(r)1920 4407 y Fk(2)1978 4424 y Fn(+)g(1)2135 4298 y Fm(@)2188 4258 y Fj(2)p Fl(k)2260 4267 y Fk(0)2183 4322 y Fl(t)2299 4298 y Fm(V)2352 4313 y Fl(q)2384 4322 y Fk(2)2418 4313 y Fj(+)p Fl(k)2510 4322 y Fk(1)2548 4298 y Fn(\()p Fm(t)p Fn(\))p 2135 4341 518 4 v 2341 4424 a Fm(q)2382 4438 y Fj(2)2421 4424 y Fn(!)1600 4691 y(+)1837 4604 y Fi(X)1690 4796 y Fl(q)1722 4805 y Fk(1)1757 4796 y Fj(+)p Fl(q)1844 4805 y Fk(2)1877 4796 y Fj(+)p Fl(q)1964 4805 y Fk(3)1998 4796 y Fj(=)p Fl(m)2423 4629 y Fm(b)2462 4643 y Fl(q)2494 4652 y Fk(3)p 2141 4670 673 4 v 2141 4753 a Fn(\()p Fm(m)g Fn(+)g(1\))2447 4727 y Fl(r)2479 4736 y Fk(1)2519 4753 y Fm(t)2552 4727 y Fj(2)p Fl(r)2619 4736 y Fk(2)2677 4753 y Fn(+)g(1)2834 4623 y Fm(@)2882 4637 y Fl(t)2927 4550 y Fi(R)2987 4576 y Fl(t)2970 4655 y Fj(0)3032 4623 y Fm(V)3085 4637 y Fl(q)3117 4646 y Fk(1)3155 4623 y Fn(\()p Fm(t)h Fg(\000)f Fm(\034)10 b Fn(\))p Fm(V)3473 4637 y Fl(q)3505 4646 y Fk(2)3543 4623 y Fn(\()p Fm(\034)g Fn(\))p Fm(d\034)p 2834 4670 928 4 v 3192 4753 a(q)3233 4767 y Fj(1)3272 4753 y Fn(!)p Fm(q)3338 4767 y Fj(2)3378 4753 y Fn(!)75 4965 y(for)30 b(all)h Fm(t)25 b Fg(2)500 4942 y Fn(\026)484 4965 y Fm(S)540 4980 y Fl(d;\016)654 4965 y Fg([)756 4942 y Fn(\026)735 4965 y Fm(D)810 4979 y Fl(m)877 4965 y Fn(.)216 5078 y(F)-8 b(rom)31 b(the)g(Leibniz)f(form) m(ula,)h(w)m(e)g(get)g(that)75 5343 y(\(21\))85 b Fm(@)373 5302 y Fl(n)416 5311 y Fk(0)368 5367 y Fl(t)455 5343 y Fn(\()710 5279 y Fm(b)749 5294 y Fl(k)786 5303 y Fk(0)821 5294 y Fl(k)858 5303 y Fk(1)892 5294 y Fl(q)924 5303 y Fk(1)p 500 5322 673 4 v 500 5405 a Fn(\()p Fm(m)20 b Fn(+)g(1\))806 5379 y Fl(r)838 5388 y Fk(1)878 5405 y Fm(t)911 5379 y Fj(2)p Fl(r)978 5388 y Fk(2)1036 5405 y Fn(+)g(1)1193 5279 y Fm(@)1246 5239 y Fj(2)p Fl(k)1318 5248 y Fk(0)1241 5303 y Fl(t)1357 5279 y Fm(V)1410 5294 y Fl(q)1442 5303 y Fk(2)1476 5294 y Fj(+)p Fl(k)1568 5303 y Fk(1)1606 5279 y Fn(\()p Fm(t)p Fn(\))p 1193 5322 518 4 v 1399 5405 a Fm(q)1440 5419 y Fj(2)1479 5405 y Fn(!)1720 5343 y(\))1253 5611 y(=)1452 5524 y Fi(X)1349 5722 y Fl(h)1390 5731 y Fk(1)1424 5722 y Fj(+)p Fl(h)1520 5731 y Fk(2)1555 5722 y Fj(=)p Fl(n)1653 5731 y Fk(0)1769 5549 y Fm(n)1824 5563 y Fj(0)1863 5549 y Fn(!)p 1712 5590 235 4 v 1712 5673 a Fm(h)1764 5687 y Fj(1)1804 5673 y Fn(!)p Fm(h)1881 5687 y Fj(2)1921 5673 y Fn(!)1956 5611 y Fm(b)1995 5626 y Fl(k)2032 5635 y Fk(0)2066 5626 y Fl(k)2103 5635 y Fk(1)2138 5626 y Fl(q)2170 5635 y Fk(1)2208 5611 y Fm(@)2261 5570 y Fl(h)2302 5579 y Fk(2)2256 5635 y Fl(t)2341 5611 y Fn(\()2700 5549 y(1)p 2386 5590 673 4 v 2386 5673 a(\()p Fm(m)g Fn(+)g(1\))2692 5647 y Fl(r)2724 5656 y Fk(1)2764 5673 y Fm(t)2797 5647 y Fj(2)p Fl(r)2864 5656 y Fk(2)2923 5673 y Fn(+)g(1)3069 5611 y(\))3114 5547 y Fm(@)3167 5507 y Fl(h)3208 5516 y Fk(1)3243 5507 y Fj(+2)p Fl(k)3370 5516 y Fk(0)3162 5572 y Fl(t)3409 5547 y Fm(V)3462 5562 y Fl(q)3494 5571 y Fk(2)3528 5562 y Fj(+)p Fl(k)3620 5571 y Fk(1)3658 5547 y Fn(\()p Fm(t)p Fn(\))p 3114 5590 648 4 v 3385 5673 a Fm(q)3426 5687 y Fj(2)3465 5673 y Fn(!)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 3764 100 a Fn(10)75 399 y(and)75 677 y(\(22\))85 b Fm(@)373 636 y Fl(n)416 645 y Fk(0)368 701 y Fl(t)455 677 y Fn(\()782 615 y Fm(b)821 629 y Fl(q)853 638 y Fk(3)p 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begin 11 10 bop 3764 100 a Fn(11)75 399 y(where)28 b Fm(\032)d Fg(2)g Fn([0)p Fm(;)15 b Fn(+)p Fg(1)p Fn(\))30 b(and)e Fm(\034)1046 414 y Fl(k)1113 399 y Fg(2)d Ff(R)p Fn(.)40 b(By)29 b(construction,)h(the)f(direction)g Fm(d)g Fn(and)f(the)g(op)s(ening)h Fm(\016)j Fn(of)d(the)f(sector)90 488 y(\026)75 511 y Fm(S)131 526 y Fl(d;\016)257 511 y Fn(are)k(c)m(hosen)h(in)e(suc)m(h)h(a)h(w)m(a)m(y)g(that)f(for)g(all) h Fm(k)e Fg(2)d(f)p Fn(0)p Fm(;)15 b(:)g(:)g(:)j(;)d Fn(2)p Fm(r)2294 525 y Fj(2)2355 511 y Fg(\000)21 b Fn(1)p Fg(g)p Fn(,)34 b(w)m(e)e(ha)m(v)m(e)i Fm(e)2985 478 y Fl(i\034)3040 490 y Fb(k)3111 511 y Fg(6)p Fn(=)27 b(1.)47 b(So)32 b(that)g(there)75 624 y(exists)f Fm(C)h(>)25 b Fn(0)30 b(suc)m(h)h(that)1661 737 y Fg(j)p Fm(\032e)1775 700 y Fl(i\034)1830 712 y Fb(k)1894 737 y Fg(\000)20 b Fn(1)p Fg(j)26 b(\025)f Fm(C)q(;)75 894 y Fn(for)30 b(all)h Fm(\032)26 b Fg(\025)f Fn(0,)31 b(for)f(all)h Fm(\034)915 909 y Fl(k)988 894 y Fn(with)f Fm(k)e Fg(2)d(f)p Fn(0)p Fm(;)15 b(:)g(:)g(:)j(;)d Fn(2)p Fm(r)1735 908 y Fj(2)1795 894 y Fg(\000)20 b Fn(1)p Fg(g)p Fn(.)42 b(W)-8 b(e)31 b(deduce)f(that)75 1168 y(\(26\))634 b Fg(j)p Fm(t)20 b Fg(\000)1071 1107 y Fm(e)1140 1029 y Fb(i\031)p 1124 1041 94 3 v 1124 1082 a Fk(2)p Fb(r)1183 1097 y Fk(2)1227 1056 y Fj(\(2)p Fl(k)r Fj(+1\))p 1048 1148 426 4 v 1048 1267 a Fn(\()p Fm(m)h Fn(+)f(1\))1381 1180 y Fb(r)1410 1195 y Fk(1)p 1365 1201 94 3 v 1365 1242 a(2)p Fb(r)1424 1257 y Fk(2)1483 1168 y Fg(j)26 b Fn(=)1830 1107 y(1)p 1640 1148 426 4 v 1640 1267 a(\()p Fm(m)20 b Fn(+)g(1\))1972 1180 y Fb(r)2001 1195 y Fk(1)p 1957 1201 94 3 v 1957 1242 a(2)p Fb(r)2016 1257 y Fk(2)2075 1168 y Fg(j)p Fm(\032e)2189 1131 y Fl(i\034)2244 1143 y Fb(k)2307 1168 y Fg(\000)g Fn(1)p Fg(j)26 b(\025)2777 1107 y Fm(C)p 2600 1148 426 4 v 2600 1267 a Fn(\()p Fm(m)20 b Fn(+)g(1\))2932 1180 y Fb(r)2961 1195 y Fk(1)p 2917 1201 94 3 v 2917 1242 a(2)p Fb(r)2976 1257 y Fk(2)3035 1168 y Fm(;)75 1443 y Fn(for)33 b(all)i Fm(t)30 b Fg(2)517 1420 y 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3417 y Fk(2)1842 3408 y Fj(=)p Fl(n)1940 3417 y Fk(0)2072 3210 y Fi(X)1990 3408 y Fl(l)2011 3417 y Fk(1)2045 3408 y Fj(+)p Fl(l)2121 3417 y Fk(2)2156 3408 y Fj(=)p Fl(h)2252 3417 y Fk(1)2301 3297 y Fg(j)p Fm(b)2365 3311 y Fl(q)2397 3320 y Fk(3)2436 3297 y Fg(j)p Fn(\()2519 3235 y(1)p 2506 3276 72 4 v 2506 3359 a Fm(C)2588 3297 y Fn(\))2623 3259 y Fl(h)2664 3268 y Fk(2)2698 3259 y Fj(+1)2802 3234 y Fm(w)2867 3249 y Fl(l)2888 3258 y Fk(1)2923 3249 y Fl(;q)2975 3258 y Fk(1)p 2802 3276 211 4 v 2809 3359 a Fm(l)2836 3373 y Fj(1)2876 3359 y Fn(!)p Fm(q)2942 3373 y Fj(1)2981 3359 y Fn(!)3033 3234 y Fm(w)3098 3249 y Fl(l)3119 3258 y Fk(2)3154 3249 y Fl(;q)3206 3258 y Fk(2)p 3033 3276 V 3040 3359 a Fm(l)3067 3373 y Fj(2)3106 3359 y Fn(!)p Fm(q)3172 3373 y Fj(2)3212 3359 y Fn(!)75 3552 y Fe(and)585 3680 y Fg(N)660 3694 y Fj(3)699 3680 y Fn(\()p Fm(n)789 3694 y Fj(0)829 3680 y Fm(;)15 b(m)p Fn(\))26 b(=)1253 3594 y Fi(X)1106 3785 y Fl(q)1138 3794 y Fk(1)1172 3785 y Fj(+)p Fl(q)1259 3794 y 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3919 67 4 v 569 3972 a Fl(r)601 3981 y Fk(2)645 3940 y Fn(])20 b(+)g(1)33 b Fe(and)38 b Fn(\024)-49 b Fm(r)1077 3954 y Fj(1)1142 3940 y Fn(=)25 b([)1290 3903 y Fl(r)1322 3912 y Fk(1)p 1273 3919 102 4 v 1273 3972 a Fj(2)p Fl(r)1340 3981 y Fk(2)1384 3940 y Fn(])c(+)f(1)p Fe(.)75 4203 y Fd(Pro)s(of)31 b Fn(W)-8 b(e)32 b(rewrite)e(the)h(expressions)f Fg(M)1564 4217 y Fl(j)1601 4203 y Fn(\()p Fm(n)1691 4217 y Fj(0)1730 4203 y Fm(;)15 b(m)p Fn(\),)31 b Fm(j)g Fn(=)25 b(1)p Fm(;)15 b Fn(2)p Fm(;)g Fn(3,)33 b(in)d(the)g(follo)m(wing)i(form)75 4483 y(\(33\))510 4422 y Fg(M)619 4436 y Fj(1)658 4422 y Fn(\()p Fm(n)748 4436 y Fj(0)788 4422 y Fm(;)15 b(m)p Fn(\))p 330 4462 794 4 v 330 4582 a(\()p Fm(m)20 b Fn(+)g(1)h(+)e Fm(S)5 b Fn(\))834 4495 y Fb(r)863 4510 y Fk(1)p 819 4516 94 3 v 819 4557 a(2)p Fb(r)878 4572 y Fk(2)923 4531 y Fl(n)966 4540 y Fk(0)1004 4582 y Fm(n)1059 4596 y Fj(0)1098 4582 y Fn(!)1159 4483 y(=)1341 4397 y Fi(X)1255 4589 y Fl(q)1287 4598 y Fk(1)1321 4589 y Fj(+)p Fl(q)1408 4598 y Fk(2)1442 4589 y Fj(=)p Fl(m)1677 4397 y Fi(X)1574 4595 y Fl(h)1615 4604 y Fk(1)1649 4595 y Fj(+)p Fl(h)1745 4604 y Fk(2)1780 4595 y Fj(=)p Fl(n)1878 4604 y Fk(0)1927 4483 y Fg(j)p Fm(b)1991 4498 y Fl(k)2028 4507 y Fk(0)2063 4498 y Fl(k)2100 4507 y Fk(1)2134 4498 y Fl(q)2166 4507 y Fk(1)2204 4483 y Fg(j)p Fn(\()2288 4422 y(1)p 2274 4462 72 4 v 2274 4546 a Fm(C)2356 4483 y Fn(\))2391 4446 y Fl(h)2432 4455 y Fk(2)2467 4446 y Fj(+1)2561 4483 y Fn(\()2693 4422 y Fm(m)20 b Fn(+)f(1)p 2606 4462 409 4 v 2606 4546 a Fm(m)h Fn(+)g(1)h(+)f Fm(S)3025 4483 y Fn(\))3085 4397 y Fb(r)3114 4412 y Fk(1)p 3070 4417 94 3 v 3070 4459 a(2)p Fb(r)3129 4474 y Fk(2)3174 4433 y Fl(n)3217 4442 y Fk(0)1564 4816 y Fg(\002)1664 4754 y Fn(\()p Fm(q)1740 4768 y Fj(2)1800 4754 y Fn(+)g Fm(k)1938 4768 y Fj(1)1998 4754 y Fn(+)g(1\))2195 4668 y Fb(r)2224 4683 y Fk(1)p 2180 4689 V 2180 4730 a(2)p Fb(r)2239 4745 y Fk(2)2283 4704 y Fj(\()p Fl(h)2351 4713 y Fk(1)2386 4704 y Fj(+2)p Fl(k)2513 4713 y Fk(0)2548 4704 y 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Fl(q)2641 5575 y Fk(1)2679 5552 y Fn(\()p Fm(t)p Fn(\))p Fg(jj)2832 5567 y Fl(d;m)p Fj(+)p Fl(S)p 2386 5594 679 4 v 2386 5713 a Fn(\()p Fm(q)2462 5727 y Fj(1)2521 5713 y Fn(+)g(1\))2718 5626 y Fb(r)2747 5641 y Fk(1)p 2703 5647 94 3 v 2703 5689 a(2)p Fb(r)2762 5704 y Fk(2)2807 5662 y Fl(l)2828 5671 y Fk(1)2867 5713 y Fm(l)2894 5727 y Fj(1)2933 5713 y Fn(!)p Fm(q)2999 5727 y Fj(1)3038 5713 y Fn(!)3090 5552 y Fg(jj)p Fm(@)3193 5511 y Fl(l)3214 5520 y Fk(2)3188 5576 y Fl(t)3254 5552 y Fm(V)3307 5566 y Fl(q)3339 5575 y Fk(2)3377 5552 y Fn(\()p Fm(t)p Fn(\))p Fg(jj)3530 5567 y Fl(d;m)p Fj(+)p Fl(S)p 3084 5594 679 4 v 3084 5713 a Fn(\()p Fm(q)3160 5727 y Fj(2)3219 5713 y Fn(+)g(1\))3416 5626 y Fb(r)3445 5641 y Fk(1)p 3401 5647 94 3 v 3401 5689 a(2)p Fb(r)3460 5704 y Fk(2)3505 5662 y Fl(l)3526 5671 y Fk(2)3564 5713 y Fm(l)3591 5727 y Fj(2)3631 5713 y Fn(!)p Fm(q)3697 5727 y Fj(2)3736 5713 y Fn(!)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 3764 100 a Fn(13)75 399 y(and)75 622 y(\(35\))510 561 y Fg(M)619 575 y Fj(3)658 561 y Fn(\()p Fm(n)748 575 y Fj(0)788 561 y Fm(;)15 b(m)p Fn(\))p 330 601 794 4 v 330 720 a(\()p Fm(m)20 b Fn(+)g(1)h(+)e Fm(S)5 b Fn(\))834 634 y Fb(r)863 649 y Fk(1)p 819 654 94 3 v 819 696 a(2)p Fb(r)878 711 y Fk(2)923 670 y Fl(n)966 679 y Fk(0)1004 720 y Fm(n)1059 734 y Fj(0)1098 720 y Fn(!)1159 622 y(=)1402 536 y Fi(X)1255 727 y Fl(q)1287 736 y Fk(1)1321 727 y Fj(+)p Fl(q)1408 736 y Fk(2)1442 727 y Fj(+)p Fl(q)1529 736 y Fk(3)1563 727 y Fj(=)p Fl(m)1798 536 y Fi(X)1695 733 y Fl(h)1736 742 y Fk(1)1770 733 y Fj(+)p Fl(h)1866 742 y Fk(2)1901 733 y Fj(=)p Fl(n)1999 742 y Fk(0)2048 622 y Fg(j)p Fm(b)2112 636 y Fl(q)2144 645 y Fk(3)2183 622 y Fg(j)p Fn(\()2266 561 y(1)p 2253 601 72 4 v 2253 684 a Fm(C)2335 622 y Fn(\))2370 585 y Fl(h)2411 594 y Fk(2)2445 585 y Fj(+1)2539 622 y Fn(\()2671 561 y Fm(m)20 b Fn(+)g(1)p 2584 601 409 4 v 2584 684 a Fm(m)h Fn(+)f(1)g(+)g Fm(S)3003 622 y Fn(\))3064 535 y Fb(r)3093 550 y Fk(1)p 3049 556 94 3 v 3049 597 a(2)p Fb(r)3108 612 y Fk(2)3152 571 y Fl(n)3195 580 y Fk(0)1530 955 y Fg(\002)1631 893 y Fn(\()p Fm(q)1707 907 y Fj(1)1767 893 y Fn(+)g(1\))1964 806 y Fb(r)1993 821 y Fk(1)p 1948 827 V 1948 868 a(2)p Fb(r)2007 883 y Fk(2)2052 842 y Fj(\()p Fl(h)2120 851 y Fk(1)2155 842 y Fj(+1\))p 1631 934 646 4 v 1704 1053 a Fn(\()p Fm(m)g Fn(+)g(1\))2036 966 y Fb(r)2065 981 y Fk(1)p 2021 987 94 3 v 2021 1028 a(2)p Fb(r)2080 1043 y Fk(2)2125 1002 y Fl(h)2166 1011 y Fk(1)2343 891 y Fg(jj)p Fm(@)2446 851 y Fl(h)2487 860 y Fk(1)2522 851 y Fj(+1)2441 916 y Fl(t)2617 891 y Fm(V)2670 905 y Fl(q)2702 914 y Fk(1)2740 891 y Fn(\()p Fm(t)p Fn(\))p Fg(jj)2893 906 y Fl(d;m)p Fj(+)p Fl(S)p 2297 934 868 4 v 2297 1053 a Fn(\()p Fm(q)2373 1067 y Fj(1)2432 1053 y Fn(+)g(1\))2629 966 y Fb(r)2658 981 y Fk(1)p 2614 987 94 3 v 2614 1028 a(2)p Fb(r)2673 1043 y Fk(2)2718 1002 y Fj(\()p Fl(h)2786 1011 y Fk(1)2820 1002 y Fj(+1\))2942 1053 y Fm(h)2994 1067 y Fj(1)3034 1053 y Fn(!)p Fm(q)3100 1067 y Fj(1)3139 1053 y Fn(!)3184 891 y Fg(jj)p Fm(V)3287 905 y Fl(q)3319 914 y Fk(2)3358 891 y Fn(\()p Fm(t)p Fn(\))p Fg(jj)3511 906 y Fl(d;m)p Fj(+)p Fl(S)p 3184 934 552 4 v 3408 1017 a Fm(q)3449 1031 y Fj(2)3488 1017 y Fn(!)3746 955 y Fm(;)75 1208 y Fn(for)30 b(all)h Fm(n)395 1222 y Fj(0)434 1208 y Fm(;)15 b(m)26 b Fg(\025)f Fn(0.)41 b(On)29 b(the)i(other)g(hand,)e(w)m(e)i(ha)m(v)m(e)h(the)e (inequalities)75 1381 y(\(36\))1236 b Fg(jj)p Fm(F)13 b Fn(\()p Fm(t)p Fn(\))p Fg(jj)1745 1396 y Fl(d;m)1863 1405 y Fk(2)1929 1381 y Fg(\024)25 b(jj)p Fm(F)13 b Fn(\()p Fm(t)p Fn(\))p Fg(jj)2299 1396 y Fl(d;m)2417 1405 y Fk(1)75 1555 y Fn(for)35 b(all)h Fm(m)430 1569 y Fj(1)503 1555 y Fg(\024)d Fm(m)687 1569 y Fj(2)726 1555 y Fn(,)k(all)f(functions)f Fm(F)13 b Fn(\()p Fm(t)p Fn(\))34 b Fg(2)f(A)1691 1570 y Fl(\027;d;\016)o(;m)1914 1579 y Fk(1)1952 1555 y Fn(,)k(whic)m(h)e (results)g(from)g(the)g(inclusion)3360 1532 y(\026)3339 1555 y Fm(D)3414 1569 y Fl(m)3476 1578 y Fk(2)3549 1555 y Fg(\032)3674 1532 y Fn(\026)3653 1555 y Fm(D)3728 1569 y Fl(m)3790 1578 y Fk(1)3829 1555 y Fn(.)75 1668 y(Moreo)m(v)m(er,)e(w) m(e)e(ha)m(v)m(e)g(also)h(the)e(inequalities)75 1879 y(\(37\))963 b(\()1243 1818 y Fm(q)1284 1832 y Fj(2)1343 1818 y Fn(+)20 b Fm(k)1481 1832 y Fj(1)1541 1818 y Fn(+)g(1)p 1243 1858 435 4 v 1342 1941 a Fm(m)g Fn(+)g(1)1687 1879 y(\))1748 1792 y Fb(r)1777 1807 y Fk(1)p 1733 1813 94 3 v 1733 1854 a(2)p Fb(r)1792 1869 y Fk(2)1837 1828 y Fl(h)1878 1837 y Fk(1)1941 1879 y Fg(\024)25 b Fn(\()2082 1818 y Fm(m)c Fn(+)f Fm(S)25 b Fn(+)20 b(1)p 2082 1858 409 4 v 2169 1941 a Fm(m)g Fn(+)g(1)2501 1879 y(\))2562 1792 y Fb(r)2591 1807 y Fk(1)p 2546 1813 94 3 v 2546 1854 a(2)p Fb(r)2605 1869 y Fk(2)2650 1828 y Fl(n)2693 1837 y Fk(0)75 2086 y Fn(for)35 b(0)e Fg(\024)g Fm(h)453 2100 y Fj(1)526 2086 y Fg(\024)g Fm(n)685 2100 y Fj(0)724 2086 y Fn(,)j(0)e Fg(\024)f Fm(q)1009 2100 y Fj(2)1081 2086 y Fg(\024)f Fm(m)j Fn(and)g(0)e Fg(\024)g Fm(k)1710 2100 y Fj(1)1783 2086 y Fg(\024)g Fm(S)5 b Fn(.)54 b(Finally)-8 b(,)38 b(starting)e(from)f(the)g(inequalit)m(y)h(\(31\))h(and)75 2199 y(using)d(the)h(equalities)i(\(33\),)h(\(34\),)f(\(35\))g(and)d (the)h(inequalities)h(\(36\))h(and)d(\(37\),)k(w)m(e)d(get)h(the)f (inequalities)75 2312 y(\(32\).)3496 b Fc(2)216 2465 y Fn(W)-8 b(e)32 b(consider)e(no)m(w)g(the)h(follo)m(wing)h(auxiliary)f (Cauc)m(h)m(y)g(problem)75 2672 y(\(38\))85 b Fm(@)373 2634 y Fl(S)368 2694 y(x)424 2672 y Fm(U)10 b Fn(\()p Fm(t;)15 b(x)p Fn(\))26 b(=)903 2585 y Fi(X)813 2787 y Fj(\()p Fl(k)877 2796 y Fk(0)912 2787 y Fl(;k)969 2796 y Fk(1)1003 2787 y Fj(\))p Fh(2S)1140 2672 y Fm(R)q Fn(\()p Fm(t)p Fn(\))p Fg(j)p Fm(b)p Fg(j)1402 2687 y Fl(k)1439 2696 y Fk(0)1474 2687 y Fl(k)1511 2696 y Fk(1)1549 2672 y Fn(\()p Fm(x)p Fn(\))p Fm(@)1724 2631 y Fj(2)p Fl(k)1796 2640 y Fk(0)1719 2696 y Fl(t)1836 2672 y Fm(@)1889 2634 y Fl(k)1926 2643 y Fk(1)1884 2694 y Fl(x)1965 2672 y Fn(\()p Fm(x@)2100 2686 y Fl(x)2165 2672 y Fn(+)20 b(1\))2339 2634 y Fj(~)-38 b Fl(r)2368 2643 y Fk(1)2403 2634 y Fl(k)2440 2643 y Fk(0)2478 2672 y Fm(U)10 b Fn(\()p Fm(t;)15 b(x)p Fn(\))1667 2926 y(+)20 b Fm(R)q Fn(\()p Fm(t)p Fn(\))p Fg(j)p Fm(b)p Fg(j)p Fn(\()p Fm(x)p Fn(\)\(\()p Fm(U)10 b Fn(\()p Fm(t;)15 b(x)p Fn(\)\))2514 2888 y Fj(2)2577 2926 y Fn(+)20 b Fm(U)10 b Fn(\(0)p Fm(;)15 b(x)p Fn(\))p Fm(@)2995 2940 y Fl(t)3026 2926 y Fn(\()p Fm(x@)3161 2940 y Fl(x)3226 2926 y Fn(+)20 b(1\))3400 2888 y Fj(\024)-38 b Fl(r)3429 2897 y Fk(1)3468 2926 y Fm(U)10 b Fn(\()p Fm(t;)15 b(x)p Fn(\)\))75 3099 y(with)30 b(initial)i(conditions)75 3291 y(\(39\))669 b(\()p Fm(@)992 3254 y Fl(j)987 3314 y(x)1031 3291 y Fm(U)10 b Fn(\)\()p Fm(t;)15 b Fn(0\))27 b(=)1468 3205 y Fi(X)1449 3401 y Fl(n)1492 3410 y Fk(0)1527 3401 y Fh(\025)p Fj(0)1632 3291 y Fm(w)1697 3305 y Fl(n)1740 3314 y Fk(0)1775 3305 y Fl(;j)1843 3230 y Fm(t)1876 3197 y Fl(n)1919 3206 y Fk(0)p 1841 3271 120 4 v 1841 3354 a Fm(n)1896 3368 y Fj(0)1935 3354 y Fn(!)1995 3291 y(=)e Fm(')2150 3305 y Fl(j)2187 3291 y Fn(\()p Fm(t)p Fn(\))61 b Fm(;)76 b Fn(0)26 b Fg(\024)f Fm(j)31 b Fg(\024)25 b Fm(S)g Fg(\000)20 b Fn(1)p Fm(;)75 3558 y Fn(where)34 b(~)-49 b Fm(r)379 3572 y Fj(1)444 3558 y Fn(=)25 b([)575 3521 y Fl(r)607 3530 y Fk(1)p 575 3538 67 4 v 575 3590 a Fl(r)607 3599 y Fk(2)651 3558 y Fn(])c(+)f(1,)35 b(\024)-49 b Fm(r)930 3572 y Fj(1)994 3558 y Fn(=)25 b([)1143 3521 y Fl(r)1175 3530 y Fk(1)p 1125 3538 102 4 v 1125 3590 a Fj(2)p Fl(r)1192 3599 y Fk(2)1237 3558 y Fn(])c(+)e(1,)1331 3788 y Fm(R)q Fn(\()p Fm(t)p Fn(\))26 b(=)1644 3702 y Fi(X)1626 3898 y Fl(n)1669 3907 y Fk(0)1703 3898 y Fh(\025)p Fj(0)1793 3788 y Fn(\()1852 3727 y(1)p 1838 3767 72 4 v 1838 3850 a Fm(C)1920 3788 y Fn(\))1955 3751 y Fl(n)1998 3760 y Fk(0)2033 3751 y Fj(+1)2127 3788 y Fm(t)2160 3751 y Fl(n)2203 3760 y Fk(0)2267 3788 y Fn(=)2458 3727 y(1)p 2373 3767 216 4 v 2373 3850 a Fm(C)g Fg(\000)20 b Fm(t)75 4050 y Fn(and)951 4163 y Fg(j)p Fm(b)p Fg(j)1040 4178 y Fl(k)1077 4187 y Fk(0)1112 4178 y Fl(k)1149 4187 y Fk(1)1187 4163 y Fn(\()p Fm(x)p Fn(\))26 b(=)1442 4077 y Fi(X)1431 4272 y Fl(m)p Fh(\025)p Fj(0)1599 4163 y Fg(j)p Fm(b)1663 4178 y Fl(k)1700 4187 y Fk(0)1734 4178 y Fl(k)1771 4187 y Fk(1)1806 4178 y Fl(m)1872 4163 y Fg(j)p Fm(x)1949 4125 y Fl(m)2077 4163 y Fm(;)76 b Fg(j)p Fm(b)p Fg(j)p Fn(\()p Fm(x)p Fn(\))26 b(=)2522 4077 y Fi(X)2511 4272 y Fl(m)p Fh(\025)p Fj(0)2679 4163 y Fg(j)p Fm(b)2743 4177 y Fl(m)2810 4163 y Fg(j)p Fm(x)2887 4125 y Fl(m)2953 4163 y Fm(:)216 4408 y Fn(F)-8 b(rom)40 b(the)g (assumptions)f Fd(\(H1\))p Fn(,)i Fd(\(H2\))p Fn(,)g(w)m(e)f(get)h (that)f(the)f(h)m(yp)s(otheses)h Fd(i\))p Fn(,)h Fd(ii\))e Fn(of)h(Theorem)f(1)h(are)75 4521 y(satis\014ed)29 b(for)f(the)g (equation)i(\(38\).)41 b(F)-8 b(rom)29 b(the)g(assumption)f(\(18\),)i (w)m(e)f(deduce)f(that)h(the)g(h)m(yp)s(othesis)f Fd(iii\))g Fn(of)75 4634 y(Theorem)33 b(1)g(are)h(satis\014ed)f(for)g(the)g (problem)g(\(38\),)i(\(39\).)51 b(W)-8 b(e)34 b(apply)f(Theorem)g(1)g (to)h(the)f(problem)g(\(38\),)75 4747 y(\(39\).)42 b(W)-8 b(e)32 b(get)f(that)g(the)g(unique)e(formal)i(solution)1386 4961 y Fm(U)10 b Fn(\()p Fm(t;)15 b(x)p Fn(\))26 b(=)1834 4875 y Fi(X)1775 5071 y Fl(n)1818 5080 y Fk(0)1853 5071 y Fl(;m)p Fh(\025)p Fj(0)2040 4961 y Fm(U)2102 4975 y Fl(n)2145 4984 y Fk(0)2180 4975 y Fl(;m)2278 4900 y Fm(t)2311 4867 y Fl(n)2354 4876 y Fk(0)p 2276 4941 120 4 v 2276 5024 a Fm(n)2331 5038 y Fj(0)2370 5024 y Fn(!)2415 4900 y Fm(x)2467 4867 y Fl(m)p 2415 4941 119 4 v 2422 5024 a Fm(m)p Fn(!)75 5230 y(of)g(the)f(problem)g(\(38\),)j(\(39\))f(b)s (elongs)e(to)h(the)g(space)g Fm(G)p Fn(\()p Fm(T)8 b(;)15 b(X)7 b Fn(\))27 b(for)e Fm(T)8 b(;)15 b(X)33 b(>)25 b Fn(0)h(small)f(enough.)39 b(In)25 b(particular,)75 5343 y(there)31 b(exists)g Fm(C)624 5357 y Fj(2)688 5343 y Fm(>)25 b Fn(0)31 b(suc)m(h)f(that)75 5546 y(\(40\))1050 b Fg(j)p Fm(U)1372 5560 y Fl(n)1415 5569 y Fk(0)1449 5560 y Fl(;m)1536 5546 y Fg(j)25 b(\024)g Fm(C)1747 5560 y Fj(2)1786 5546 y Fn(\()1842 5484 y(1)p 1831 5525 66 4 v 1831 5608 a Fm(T)1907 5546 y Fn(\))1942 5508 y Fl(n)1985 5517 y Fk(0)2024 5546 y Fn(\()2088 5484 y(1)p 2069 5525 83 4 v 2069 5608 a Fm(X)2162 5546 y Fn(\))2197 5508 y Fl(m)2264 5546 y Fn(\()p Fm(n)2354 5560 y Fj(0)2414 5546 y Fn(+)19 b Fm(m)p Fn(\)!)75 5740 y(for)30 b(all)h Fm(n)395 5754 y Fj(0)434 5740 y Fm(;)15 b(m)26 b Fg(\025)f Fn(0.)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 3764 100 a Fn(14)75 399 y Fd(Lemma)35 b(10)46 b Fe(The)33 b(p)-5 b(ositive)33 b(numb)-5 b(ers)33 b Fm(U)1532 413 y Fl(n)1575 422 y Fk(0)1610 413 y Fl(;m)1729 399 y Fe(satisfy)g(the)g(fol)5 b(lowing)34 b(r)-5 b(e)g(cursion)34 b(r)-5 b(elation)75 649 y Fn(\(41\))866 587 y Fm(U)928 601 y Fl(n)971 610 y Fk(0)1006 601 y Fl(;m)p Fj(+)p Fl(S)p 866 628 328 4 v 918 712 a Fm(n)973 726 y Fj(0)1012 712 y Fn(!)p Fm(m)p Fn(!)1229 649 y(=)1415 563 y Fi(X)1325 764 y Fj(\()p Fl(k)1389 773 y Fk(0)1423 764 y Fl(;k)1480 773 y Fk(1)1514 764 y Fj(\))p Fh(2S)1636 649 y Fn(\()p Fg(U)1728 663 y Fj(1)1768 649 y Fn(\()p Fm(n)1858 663 y Fj(0)1898 649 y Fm(;)15 b(m)p Fn(\))20 b(+)g Fg(U)2221 663 y Fj(2)2261 649 y Fn(\()p Fm(n)2351 663 y Fj(0)2390 649 y Fm(;)15 b(m)p Fn(\))21 b(+)f Fg(U)2714 663 y Fj(3)2753 649 y Fn(\()p Fm(n)2843 663 y Fj(0)2882 649 y Fm(;)15 b(m)p Fn(\)\))75 953 y Fe(for)33 b(al)5 b(l)33 b Fm(n)405 967 y Fj(0)444 953 y Fm(;)15 b(m)26 b Fg(\025)f Fn(0)p Fe(,)33 b(wher)-5 b(e)457 1195 y Fg(U)514 1209 y Fj(1)553 1195 y Fn(\()p Fm(n)643 1209 y Fj(0)682 1195 y Fm(;)15 b(m)p Fn(\))26 b(=)1046 1109 y Fi(X)959 1301 y Fl(q)991 1310 y Fk(1)1025 1301 y Fj(+)p Fl(q)1112 1310 y Fk(2)1146 1301 y Fj(=)p Fl(m)1382 1109 y Fi(X)1279 1307 y Fl(h)1320 1316 y Fk(1)1354 1307 y Fj(+)p Fl(h)1450 1316 y Fk(2)1484 1307 y Fj(=)p Fl(n)1582 1316 y Fk(0)1632 1195 y Fg(j)p Fm(b)1696 1210 y Fl(k)1733 1219 y Fk(0)1767 1210 y Fl(k)1804 1219 y Fk(1)1838 1210 y Fl(q)1870 1219 y Fk(1)1909 1195 y Fg(j)p Fn(\()1992 1134 y(1)p 1979 1174 72 4 v 1979 1258 a Fm(C)2061 1195 y Fn(\))2096 1158 y Fl(h)2137 1167 y Fk(2)2171 1158 y Fj(+1)2266 1195 y Fn(\()p Fm(q)2342 1209 y Fj(2)2401 1195 y Fn(+)20 b Fm(k)2539 1209 y Fj(1)2599 1195 y Fn(+)g(1\))2773 1158 y Fj(~)-38 b Fl(r)2802 1167 y Fk(1)2837 1158 y Fl(k)2874 1167 y Fk(0)2923 1132 y Fm(U)2985 1147 y Fl(h)3026 1156 y Fk(1)3060 1147 y Fj(+2)p Fl(k)3187 1156 y Fk(0)3222 1147 y Fl(;q)3274 1156 y Fk(2)3307 1147 y Fj(+)p Fl(k)3399 1156 y Fk(1)p 2923 1174 515 4 v 3069 1258 a Fm(h)3121 1272 y Fj(1)3161 1258 y Fn(!)p Fm(q)3227 1272 y Fj(2)3266 1258 y Fn(!)3448 1195 y Fm(;)688 1550 y Fg(U)745 1564 y Fj(2)784 1550 y Fn(\()p Fm(n)874 1564 y Fj(0)914 1550 y Fm(;)15 b(m)p Fn(\))25 b(=)1337 1463 y Fi(X)1190 1655 y Fl(q)1222 1664 y Fk(1)1256 1655 y Fj(+)p Fl(q)1343 1664 y Fk(2)1377 1655 y Fj(+)p Fl(q)1464 1664 y Fk(3)1498 1655 y Fj(=)p Fl(m)1734 1463 y Fi(X)1631 1661 y Fl(h)1672 1670 y Fk(1)1706 1661 y Fj(+)p Fl(h)1802 1670 y Fk(2)1836 1661 y Fj(=)p Fl(n)1934 1670 y Fk(0)2066 1463 y Fi(X)1984 1661 y Fl(l)2005 1670 y Fk(1)2039 1661 y Fj(+)p Fl(l)2115 1670 y Fk(2)2150 1661 y Fj(=)p Fl(h)2246 1670 y Fk(1)2295 1550 y Fg(j)p Fm(b)2359 1564 y Fl(q)2391 1573 y Fk(3)2430 1550 y Fg(j)p Fn(\()2513 1488 y(1)p 2500 1529 72 4 v 2500 1612 a Fm(C)2582 1550 y Fn(\))2617 1512 y Fl(h)2658 1521 y Fk(2)2692 1512 y Fj(+1)2797 1487 y Fm(U)2859 1502 y Fl(l)2880 1511 y Fk(1)2914 1502 y Fl(;q)2966 1511 y Fk(1)p 2797 1529 208 4 v 2802 1612 a Fm(l)2829 1626 y Fj(1)2868 1612 y Fn(!)p Fm(q)2934 1626 y Fj(1)2974 1612 y Fn(!)3024 1487 y Fm(U)3086 1502 y Fl(l)3107 1511 y Fk(2)3142 1502 y Fl(;q)3194 1511 y Fk(2)p 3024 1529 V 3029 1612 a Fm(l)3056 1626 y Fj(2)3096 1612 y Fn(!)p Fm(q)3162 1626 y Fj(2)3201 1612 y Fn(!)75 1805 y Fe(and)597 1938 y Fg(U)654 1952 y Fj(3)694 1938 y Fn(\()p Fm(n)784 1952 y Fj(0)823 1938 y Fm(;)15 b(m)p Fn(\))26 b(=)1247 1852 y Fi(X)1100 2044 y Fl(q)1132 2053 y Fk(1)1166 2044 y Fj(+)p Fl(q)1253 2053 y Fk(2)1287 2044 y Fj(+)p Fl(q)1374 2053 y Fk(3)1408 2044 y Fj(=)p Fl(m)1644 1852 y Fi(X)1540 2050 y Fl(h)1581 2059 y Fk(1)1616 2050 y Fj(+)p Fl(h)1712 2059 y Fk(2)1746 2050 y Fj(=)p Fl(n)1844 2059 y Fk(0)1893 1938 y Fg(j)p Fm(b)1957 1952 y Fl(q)1989 1961 y Fk(3)2028 1938 y Fg(j)p Fn(\()2111 1877 y(1)p 2098 1917 72 4 v 2098 2001 a Fm(C)2180 1938 y Fn(\))2215 1901 y Fl(h)2256 1910 y Fk(2)2290 1901 y Fj(+1)2385 1938 y Fn(\()p Fm(q)2461 1952 y Fj(1)2520 1938 y Fn(+)20 b(1\))2694 1901 y Fj(\024)-38 b Fl(r)2723 1910 y Fk(1)2772 1875 y Fm(U)2834 1890 y Fl(h)2875 1899 y Fk(1)2910 1890 y Fj(+1)p Fl(;q)3052 1899 y Fk(1)p 2772 1917 318 4 v 2820 2001 a Fm(h)2872 2015 y Fj(1)2912 2001 y Fn(!)p Fm(q)2978 2015 y Fj(1)3017 2001 y Fn(!)3110 1876 y Fm(U)3172 1890 y Fj(0)p Fl(;q)3259 1899 y Fk(2)p 3110 1917 188 4 v 3151 2001 a Fm(q)3192 2015 y Fj(2)3231 2001 y Fn(!)3307 1938 y Fm(;)75 2193 y Fe(with)33 b(initial)h(c)-5 b(onditions)1348 2306 y Fm(U)1410 2320 y Fl(n)1453 2329 y Fk(0)1487 2320 y Fl(;j)1569 2306 y Fn(=)25 b Fm(w)1730 2320 y Fl(n)1773 2329 y Fk(0)1807 2320 y Fl(;j)1928 2306 y Fm(;)81 b Fn(0)25 b Fg(\024)g Fm(j)31 b Fg(\024)25 b Fm(S)g Fg(\000)20 b Fn(1)75 2473 y Fe(for)33 b(al)5 b(l)33 b Fm(n)405 2487 y Fj(0)470 2473 y Fg(\025)25 b Fn(0)p Fe(.)75 2711 y Fd(Pro)s(of)31 b Fn(A)g(direct)g(calculation)h (sho)m(ws)e(that)649 2915 y Fm(R)q Fn(\()p Fm(t)p Fn(\))p Fg(j)p Fm(b)p Fg(j)911 2930 y Fl(k)948 2939 y Fk(0)983 2930 y Fl(k)1020 2939 y Fk(1)1059 2915 y Fn(\()p Fm(x)p Fn(\))p Fm(@)1234 2874 y Fj(2)p Fl(k)1306 2883 y Fk(0)1229 2939 y Fl(t)1345 2915 y Fm(@)1398 2877 y Fl(k)1435 2886 y Fk(1)1393 2937 y Fl(x)1474 2915 y Fn(\()p Fm(x@)1609 2929 y Fl(x)1674 2915 y Fn(+)20 b(1\))1848 2877 y Fj(~)-38 b Fl(r)1877 2886 y Fk(1)1912 2877 y Fl(k)1949 2886 y Fk(0)1987 2915 y Fm(U)10 b Fn(\()p Fm(t;)15 b(x)p Fn(\))26 b(=)2436 2828 y Fi(X)2376 3024 y Fl(n)2419 3033 y Fk(0)2454 3024 y Fl(;m)p Fh(\025)p Fj(0)2641 2915 y Fg(U)2698 2929 y Fj(1)2738 2915 y Fn(\()p Fm(n)2828 2929 y Fj(0)2867 2915 y Fm(;)15 b(m)p Fn(\))p Fm(t)3055 2877 y Fl(n)3098 2886 y Fk(0)3137 2915 y Fm(x)3189 2877 y Fl(m)3255 2915 y Fm(;)333 3271 y(@)386 3233 y Fl(S)381 3293 y(x)437 3271 y Fm(U)10 b Fn(\()p Fm(t;)15 b(x)p Fn(\))26 b(=)885 3184 y Fi(X)826 3380 y Fl(n)869 3389 y Fk(0)904 3380 y Fl(;m)p Fh(\025)p Fj(0)1101 3208 y Fm(U)1163 3222 y Fl(n)1206 3231 y Fk(0)1241 3222 y Fl(;m)p Fj(+)p Fl(S)p 1101 3250 328 4 v 1152 3333 a Fm(n)1207 3347 y Fj(0)1246 3333 y Fn(!)p Fm(m)p Fn(!)1438 3271 y Fm(t)1471 3233 y Fl(n)1514 3242 y Fk(0)1553 3271 y Fm(x)1605 3233 y Fl(m)1732 3271 y Fm(;)76 b(R)q Fn(\()p Fm(t)p Fn(\))p Fg(j)p Fm(b)p Fg(j)p Fn(\()p Fm(x)p Fn(\)\()p Fm(U)10 b Fn(\()p Fm(t;)15 b(x)p Fn(\)\))2554 3233 y Fj(2)2621 3271 y Fn(=)2777 3184 y Fi(X)2717 3380 y Fl(n)2760 3389 y Fk(0)2795 3380 y Fl(;m)p Fh(\025)p Fj(0)2982 3271 y Fg(U)3039 3285 y Fj(2)3079 3271 y Fn(\()p Fm(n)3169 3285 y Fj(0)3208 3271 y Fm(;)g(m)p Fn(\))p Fm(t)3396 3233 y Fl(n)3439 3242 y Fk(0)3478 3271 y Fm(x)3530 3233 y Fl(m)75 3528 y Fn(and)725 3640 y Fm(R)q Fn(\()p Fm(t)p Fn(\))p Fg(j)p Fm(b)p Fg(j)p Fn(\()p Fm(x)p Fn(\))p Fm(U)10 b Fn(\(0)p Fm(;)15 b(x)p Fn(\))p Fm(@)1436 3654 y Fl(t)1469 3640 y Fn(\()p Fm(x@)1604 3654 y Fl(x)1669 3640 y Fn(+)20 b(1\))1843 3603 y Fj(\024)-38 b Fl(r)1872 3612 y Fk(1)1911 3640 y Fm(U)10 b Fn(\()p Fm(t;)15 b(x)p Fn(\))26 b(=)2359 3554 y Fi(X)2300 3750 y Fl(n)2343 3759 y Fk(0)2378 3750 y Fl(;m)p Fh(\025)p Fj(0)2565 3640 y Fg(U)2622 3654 y Fj(3)2661 3640 y Fn(\()p Fm(n)2751 3654 y Fj(0)2791 3640 y Fm(;)15 b(m)p Fn(\))p Fm(t)2979 3603 y Fl(n)3022 3612 y Fk(0)3061 3640 y Fm(x)3113 3603 y Fl(m)3179 3640 y Fm(;)75 3897 y Fn(whic)m(h)30 b(giv)m(es)i(the)e(lemma.)2755 b Fc(2)75 4060 y Fn(F)-8 b(rom)31 b(the)f(inequalities)i(\(32\))g(and)e (\(41\),)i(w)m(e)f(deduce)f(that)75 4264 y(\(42\))1442 b Fm(w)1742 4278 y Fl(n)1785 4287 y Fk(0)1819 4278 y Fl(;m)1931 4264 y Fg(\024)25 b Fm(U)2089 4278 y Fl(n)2132 4287 y Fk(0)2166 4278 y Fl(;m)75 4469 y Fn(for)34 b(all)i Fm(n)404 4483 y Fj(0)443 4469 y Fm(;)15 b(m)32 b Fg(\025)f Fn(0.)54 b(Finally)-8 b(,)37 b(from)d(this)g(latter)i(inequalit)m(y)g (\(42\))g(and)d(the)i(estimates)h(\(40\),)h(w)m(e)e(get)h(that)75 4582 y(the)31 b(estimates)h(\(19\))f(hold.)41 b(This)29 b(yields)i(Theorem)f(2.)1778 b Fc(2)75 4918 y Fo(4)135 b(Borel-Laplace)55 b(transform)f(of)g(formal)h(series)f(solutions)h(of) f(partial)277 5067 y(di\013eren)l(tial)47 b(equations)75 5270 y Fn(W)-8 b(e)38 b(recall)g(the)e(classical)j(de\014nition)d(of)h Fm(k)s Fg(\000)p Fn(summabilit)m(y)g(of)f(formal)h(series)g(with)f (complex)h(co)s(e\016cien)m(ts,)75 5383 y(see)31 b([1)q(].)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 3764 100 a Fn(15)75 399 y Fd(De\014nition)35 b(4)46 b Fe(A)32 b(formal)i(series)1687 553 y Fn(^)1661 576 y Fm(X)7 b Fn(\()p Fm(t)p Fn(\))26 b(=)1999 462 y Fh(1)1968 489 y Fi(X)1973 685 y Fl(j)t Fj(=0)2115 576 y Fm(a)2163 590 y Fl(j)2199 576 y Fm(t)2232 538 y Fl(j)75 826 y Fe(with)35 b(c)-5 b(o)g(e\016cients)36 b Fm(a)788 840 y Fl(j)853 826 y Fg(2)28 b Ff(C)p Fe(,)34 b(is)h(said)g(to)g(b)-5 b(e)34 b Fm(k)s Fg(\000)p Fe(summable)i(with)f(r)-5 b(esp)g(e)g(ct)36 b(to)f Fm(t)f Fe(in)h(the)f(dir)-5 b(e)g(ction)36 b Fm(d)29 b 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Fi(Z)1805 2774 y Fh(1)p Fl(e)1909 2750 y Fb(i\015)1765 2954 y Fj(0)1990 2871 y Fm(W)13 b Fn(\()p Fm(\034)d Fn(\))p Fm(e)2251 2834 y Fh(\000)p Fj(\()p Fl(\034)e(=t)p Fj(\))2459 2810 y Fb(k)2503 2871 y Fm(d\034)2600 2834 y Fl(k)2643 2871 y Fm(;)75 3108 y Fn(where)33 b(the)i(in)m(tegration)g(is)f(made)g (along)h(the)f(ra)m(y)h(arg)r(\()p Fm(\034)10 b Fn(\))31 b(=)g Fm(\015)39 b Fn(with)34 b Fg(j)p Fm(d)23 b Fg(\000)f Fm(\015)5 b Fg(j)32 b Fm(<)f(\016)s Fn(,)k(is)f(called)h(the)g Fm(k)s Fg(\000)p Fn(sum)75 3221 y(of)g(the)g(formal)g(series)908 3198 y(^)881 3221 y Fm(X)8 b Fn(\()p Fm(t)p Fn(\).)54 b(In)34 b(particular,)i(the)f(function)f Fm(X)7 b Fn(\()p Fm(t)p Fn(\))36 b(is)f(asymptotic)h(to)3165 3198 y(^)3139 3221 y Fm(X)7 b Fn(\()p Fm(t)p Fn(\))35 b(with)g(Gevrey)75 3334 y(\014ltration)c(of)g(order)e(1)p Fm(=k)s Fn(,)j(with)e(resp)s (ect)h(to)g Fm(t)f Fn(on)g(a)h(\014nite)f(sector)1005 3525 y Fm(S)1061 3540 y Fl(d;\022)r(;R)1254 3525 y Fn(=)25 b Fg(f)p Fm(t)h Fg(2)e Ff(C)1605 3488 y Fh(\003)1670 3525 y Fn(:)h Fg(j)p Fm(t)p Fg(j)h(\024)f Fm(R)61 b(;)76 b Fg(j)p Fm(d)21 b Fg(\000)f Fn(arg)r(\()p Fm(t)p Fn(\))p Fg(j)26 b Fm(<)f(\022)s(=)p Fn(2)p Fg(g)p Fm(;)75 3716 y Fn(where)30 b Fm(\022)d(>)e(\031)s(=k)34 b Fn(and)c Fm(R)c(>)f Fn(0.)41 b(This)30 b(means)g(that)h(for)f(all)h Fm(\022)2155 3730 y Fj(1)2220 3716 y Fm(<)25 b(\022)s Fn(,)k(there)i(exist)g Fm(C)q(;)15 b(M)36 b(>)25 b Fn(0,)31 b(suc)m(h)f(that)1271 3991 y Fg(j)p Fm(X)7 b Fn(\()p Fm(t)p Fn(\))21 b Fg(\000)1593 3877 y Fl(n)p Fh(\000)p Fj(1)1594 3905 y Fi(X)1597 4099 y Fl(p)p Fj(=0)1741 3991 y Fm(a)1789 4005 y Fl(p)1829 3991 y Fm(t)1862 3953 y Fl(p)1902 3991 y Fg(j)k(\024)g Fm(C)7 b(M)2218 3953 y Fl(n)2265 3991 y Fn(\()p Fm(n)p Fn(!\))2415 3953 y Fj(1)p Fl(=k)2528 3991 y Fg(j)p Fm(t)p Fg(j)2611 3953 y Fl(n)75 4271 y Fn(for)30 b(all)h Fm(n)25 b Fg(\025)g Fn(1)31 b(and)f Fm(t)25 b Fg(2)g Fm(S)969 4286 y Fl(d;\022)1058 4295 y Fk(1)1092 4286 y Fl(;R)1169 4271 y Fn(.)216 4384 y(In)38 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b(exists)f Fm(\032)25 b Fg(2)g Ff(R)912 5408 y Fj(+)1003 5394 y Fe(such)33 b(that)g(the)g(mo)-5 b(di\014e)g(d)35 b(Bor)-5 b(el)34 b(tr)-5 b(ansform)35 b(of)e(or)-5 b(der)34 b(1,)1509 5655 y Fn(\()p Fg(B)1604 5669 y Fj(1)1669 5632 y Fn(^)1643 5655 y Fm(X)8 b Fn(\)\()p Fm(t)p Fn(\))26 b(=)2016 5541 y Fh(1)1986 5568 y Fi(X)1990 5764 y Fl(j)t Fj(=0)2132 5655 y Fm(a)2180 5669 y Fl(j)2284 5593 y Fm(t)2317 5560 y Fl(j)p 2227 5634 185 4 v 2227 5717 a Fn(\(2)p Fm(j)5 b Fn(\)!)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 3764 100 a Fn(16)75 399 y Fe(is)33 b(absolutely)g(c)-5 b(onver)g(gent)34 b(for)f(al)5 b(l)33 b Fm(t)25 b Fg(2)g Fm(D)1532 413 y Fl(\032)1573 399 y Fe(.)75 557 y Fd(ii\))30 b Fe(Ther)-5 b(e)31 b(exists)g Fm(\016)e(>)c Fn(0)31 b Fe(such)f(that)i(the)f(series)g Fn(\()p Fg(B)1838 571 y Fj(1)1903 534 y Fn(^)1877 557 y Fm(X)7 b Fn(\)\()p Fm(t)p Fn(\))32 b Fe(c)-5 b(an)31 b(b)-5 b(e)30 b(analytic)-5 b(al)5 b(ly)33 b(c)-5 b(ontinue)g(d)32 b(with)f(r)-5 b(esp)g(e)g(ct)32 b(to)75 670 y 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y Fl(d;\022)r(;R)695 4174 y Fn(,)h(all)f Fm(m)g Fg(\025)g Fn(0.)39 b(So)24 b(that,)i(from)d(the)i(estimates)g(ii\))g(and)e(iii\)) j(of)e(Remark)g(1,)i(w)m(e)e(get)h(constan)m(ts)75 4287 y Fm(C)q(;)15 b(X)34 b(>)24 b Fn(0)31 b(suc)m(h)f(that)1516 4433 y Fg(j)p Fm(X)1616 4447 y Fl(m)1683 4433 y Fn(\()p Fm(t)1751 4396 y Fj(2)1791 4433 y Fn(\))p Fg(j)25 b(\024)g Fm(C)7 b Fn(\()2108 4372 y(1)p 2089 4412 83 4 v 2089 4496 a Fm(X)2182 4433 y Fn(\))2217 4396 y Fl(m)2283 4433 y Fm(m)p Fn(!)p Fm(;)75 4626 y Fn(for)30 b(all)h Fm(t)25 b Fg(2)500 4603 y Fn(\026)484 4626 y Fm(S)540 4641 y Fl(d;\022)r(;R)708 4626 y Fn(,)31 b(all)g Fm(m)25 b Fg(\025)g Fn(0.)41 b(The)30 b(estimates)i(\(44\))g(then)e(hold)g(for)g Fm(n)2584 4640 y Fj(0)2648 4626 y Fn(=)25 b(0.)216 4739 y(F)-8 b(rom)30 b(the)g(Cauc)m(h)m(y)g(form)m(ula,)g(this)f(in)m (tegral)i(\(46\))g(do)s(es)e(not)h(dep)s(end)e(on)h(small)h(c)m(hanges) g(of)g(the)f(ra)m(y)h(of)75 4852 y(in)m(tegration)i Fg(f)p Fm(v)d Fg(2)c Ff(C)807 4819 y Fh(\003)846 4852 y Fm(=)15 b Fn(arg)j Fm(v)28 b Fn(=)d Fm(d)c Fg(\000)f Fn(arg)r(\()p Fm(t)p Fn(\))p Fg(g)p Fn(.)41 b(W)-8 b(e)32 b(deduce)e(that)h(the)f Fm(n)2587 4866 y Fj(0)2626 4852 y Fg(\000)p Fn(deriv)-5 b(ativ)m(e)32 b(of)f Fm(X)3293 4866 y Fl(m)3359 4852 y Fn(\()p Fm(t)3427 4819 y Fj(2)3467 4852 y Fn(\))g(writes)871 5119 y Fm(@)924 5079 y Fl(n)967 5088 y Fk(0)919 5144 y Fl(t)1006 5119 y Fn(\()p Fm(X)1116 5133 y Fl(m)1183 5119 y Fn(\()p Fm(t)1251 5082 y Fj(2)1291 5119 y Fn(\)\))26 b(=)f(2)1543 4996 y Fi(Z)1634 5022 y Fh(1)p Fl(e)1738 4999 y Fb(i)p Fk(\()p Fb(d)p Fa(\000)p Fk(arg)r(\()p Fb(t)p Fk(\)\))1594 5202 y Fj(0)2069 5119 y Fm(e)p Fn(\()p Fm(v)2193 5082 y Fj(2)2233 5119 y Fn(\))p Fm(v)2315 5082 y Fl(n)2358 5091 y Fk(0)2393 5082 y Fh(\000)p Fj(1)2488 5119 y Fn(\()p Fm(@)2576 5079 y Fl(n)2619 5088 y Fk(0)2571 5144 y Fl(t)2658 5119 y Fm(V)2711 5133 y Fl(m)2777 5119 y Fn(\)\()p Fm(v)s(t)p Fn(\))p Fm(dv)75 5357 y Fn(for)31 b(all)h Fm(n)397 5371 y Fj(0)462 5357 y Fg(\025)26 b Fn(1.)44 b(Using)31 b(the)g(estimates)i(\(19\))f(and)f(iii\))h(of)f (Remark)g(1,)h(w)m(e)g(get)g(constan)m(ts)g Fm(C)3303 5371 y Fj(2)3343 5357 y Fm(;)15 b(K)q(;)g(X)r(;)g(T)41 b(>)26 b Fn(0)75 5470 y(suc)m(h)k(that)75 5701 y(\(47\))238 b Fg(j)p Fm(@)551 5660 y Fl(n)594 5669 y Fk(0)546 5725 y Fl(t)633 5701 y Fn(\()p Fm(X)743 5715 y Fl(m)811 5701 y Fn(\()p Fm(t)879 5663 y Fj(2)918 5701 y Fn(\)\))p Fg(j)27 b(\024)d Fm(C)1200 5715 y Fj(2)1240 5701 y Fn(\()1304 5639 y(1)p 1285 5680 V 1285 5763 a Fm(X)1378 5701 y Fn(\))1413 5663 y Fl(m)1480 5701 y Fn(\()1535 5639 y(1)p 1525 5680 66 4 v 1525 5763 a Fm(T)1601 5701 y Fn(\))1636 5663 y Fl(n)1679 5672 y Fk(0)1717 5701 y Fn(\()p Fm(m)d Fn(+)f(1\))2034 5614 y Fb(r)2063 5629 y Fk(1)2098 5614 y Fb(n)2136 5629 y Fk(0)p 2035 5635 137 3 v 2056 5676 a(2)p Fb(r)2115 5691 y Fk(2)2185 5701 y Fm(n)2240 5715 y Fj(0)2279 5701 y Fn(!)p Fm(m)p Fn(!)2424 5577 y Fi(Z)2515 5603 y Fj(+)p Fh(1)2475 5783 y Fj(0)2660 5701 y Fm(e)2702 5663 y Fh(\000)p Fl(r)2791 5640 y Fk(2)2826 5663 y Fl(=K)2929 5701 y Fm(r)2973 5663 y Fl(n)3016 5672 y Fk(0)3050 5663 y Fh(\000)p Fj(1)3244 5639 y Fm(e)3286 5606 y Fl(\027)t(r)r Fh(j)p Fl(t)p Fh(j)p 3155 5680 363 4 v 3155 5763 a Fn(1)g(+)g Fm(r)3355 5737 y Fj(2)3394 5763 y Fg(j)p Fm(t)p Fg(j)3477 5737 y Fj(2)3527 5701 y Fm(dr)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 3764 100 a Fn(18)75 399 y(for)30 b(all)h Fm(n)395 413 y Fj(0)460 399 y Fg(\025)25 b Fn(1,)31 b(for)f(all)h Fm(t)25 b Fg(2)g Fm(S)1122 414 y Fl(d;\022)r(;R)1290 399 y Fn(.)216 511 y(Let)k Fm(M)465 525 y Fj(1)530 511 y Fm(>)c Fn(0)30 b(b)s(e)e(suc)m(h)g(that)i Fm(M)1310 525 y Fj(1)1375 511 y Fg(\025)25 b Fm(\021)20 b Fn(+)d Fm(\027)6 b(R)q Fn(,)29 b(for)f(a)h(giv)m(en)h Fm(\021)f(>)c Fn(0.)40 b(W)-8 b(e)31 b(c)m(ho)s(ose)e Fm(M)3056 525 y Fj(2)3121 511 y Fm(>)c Fn(0)k(whic)m(h)g(satis\014es)75 624 y Fm(K)7 b(M)257 591 y Fj(2)247 649 y(1)322 624 y Fm(<)24 b Fn(4)p Fm(M)550 638 y Fj(2)591 624 y Fn(.)40 b(W)-8 b(e)32 b(get)f(that)1171 589 y Fl(r)1205 565 y Fk(2)p 1171 604 69 4 v 1173 656 a Fl(K)1275 624 y Fg(\025)25 b Fm(M)1459 638 y Fj(1)1498 624 y Fm(r)e Fg(\000)d Fm(M)1741 638 y Fj(2)1811 624 y Fn(for)30 b(all)h Fm(r)d Fg(\025)d Fn(0.)41 b(W)-8 b(e)32 b(deduce)e(the)g(estimates)75 883 y(\(48\))391 759 y Fi(Z)482 785 y Fj(+)p Fh(1)442 965 y Fj(0)627 883 y Fm(e)669 845 y Fh(\000)p Fl(r)758 822 y Fk(2)793 845 y Fl(=K)896 883 y Fm(r)940 845 y Fl(n)983 854 y Fk(0)1017 845 y Fh(\000)p Fj(1)1211 821 y Fm(e)1253 788 y Fl(\027)t(r)r Fh(j)p Fl(t)p Fh(j)p 1121 862 363 4 v 1121 945 a Fn(1)21 b(+)f Fm(r)1322 919 y Fj(2)1361 945 y Fg(j)p Fm(t)p Fg(j)1444 919 y Fj(2)1494 883 y Fm(dr)28 b Fg(\024)d Fm(e)1748 845 y Fl(M)1816 854 y Fk(2)1870 759 y Fi(Z)1961 785 y Fj(+)p Fh(1)1920 965 y Fj(0)2105 883 y Fm(e)2147 845 y Fh(\000)p Fj(\()p Fl(M)2297 854 y Fk(1)2333 845 y Fh(\000)p Fl(\027)t Fh(j)p Fl(t)p Fh(j)p Fj(\))p Fl(r)2557 883 y Fm(r)2601 845 y Fl(n)2644 854 y Fk(0)2677 845 y Fh(\000)p Fj(1)2772 883 y Fm(dr)j Fg(\024)d Fm(e)3026 845 y Fl(M)3094 854 y Fk(2)3133 883 y Fn(\()3179 821 y(1)p 3178 862 49 4 v 3178 945 a Fm(\021)3236 883 y Fn(\))3271 845 y Fl(n)3314 854 y Fk(0)3353 883 y Fn(\()p Fm(n)3443 897 y Fj(0)3503 883 y Fg(\000)20 b Fn(1\)!)75 1119 y(for)30 b(all)h Fm(t)25 b Fg(2)g Fm(S)540 1134 y Fl(d;\022)r(;R)708 1119 y Fn(.)41 b(Finally)-8 b(,)32 b(the)e(estimates)i(\(44\))g(follo)m(ws)g(from)e(\(47\),)i(\(48\),)g (for)e(all)h Fm(n)3117 1133 y Fj(0)3182 1119 y Fg(\025)25 b Fn(1.)408 b Fc(2)75 1359 y Fd(De\014nition)35 b(5)46 b Fe(L)-5 b(et)34 b Fm(X)7 b Fn(\()p Fm(t;)15 b(z)t Fn(\))35 b Fe(b)-5 b(e)34 b(a)g(holomorphic)j(function)d(on)g(a)g(domain)h Fm(S)2788 1374 y Fl(d;\022)r(;R)2977 1359 y Fg(\002)21 b Fm(D)3144 1373 y Fl(K)3212 1359 y Fe(,)34 b(wher)-5 b(e)35 b Fm(S)3588 1374 y Fl(d;\022)r(;R)3789 1359 y Fe(is)75 1472 y(an)i(op)-5 b(en)38 b(se)-5 b(ctor)37 b(c)-5 b(enter)g(e)g(d)38 b(at)g(0)f(with)g(dir)-5 b(e)g(ction)38 b Fm(d)33 b Fg(2)f Ff(R)p Fe(,)37 b(op)-5 b(ening)37 b Fm(\022)e Fg(2)d Fn(\(0)p Fm(;)15 b Fn(2)p Fm(\031)s Fn(\))p Fe(,)40 b(r)-5 b(adius)38 b Fm(R)f Fe(and)g Fm(D)3601 1486 y Fl(K)3707 1472 y Fe(is)f(a)75 1585 y(disc)d(c)-5 b(enter)g(e)g(d)34 b(at)f(0)g(with)g(r)-5 b(adius)34 b Fm(K)e(>)25 b Fn(0)p Fe(.)43 b(L)-5 b(et)1775 1562 y Fn(^)1748 1585 y Fm(X)8 b Fn(\()p Fm(t;)15 b(z)t Fn(\))34 b Fe(b)-5 b(e)32 b(a)h(formal)h(series)1535 1763 y Fn(^)1509 1786 y Fm(X)8 b Fn(\()p Fm(t;)15 b(z)t Fn(\))26 b(=)1934 1700 y Fi(X)1903 1897 y Fl(l)q(;m)p Fh(\025)p Fj(0)2112 1786 y Fm(a)2160 1801 y Fl(l)q(m)2249 1786 y Fm(t)2282 1748 y Fl(l)2308 1786 y Fm(z)2354 1748 y Fl(m)75 2065 y Fe(in)38 b(the)h(variables)h Fm(t;)15 b(z)t Fe(.)59 b(We)38 b(make)h(the)g(assumption)h(that)g(ther)-5 b(e)39 b(exists)g Fn(0)d Fm(<)g(\032)g Fg(\024)f Fm(K)45 b Fe(such)39 b(that,)i(for)e(al)5 b(l)75 2178 y Fm(N)35 b Fg(\025)25 b Fn(1)p Fe(,)33 b(the)g(formal)h(series)1393 2349 y Fn(^)1367 2372 y Fm(X)1449 2335 y Fl(N)7 b Fh(\000)p Fj(1)1607 2372 y Fn(\()p Fm(t;)15 b(z)t Fn(\))26 b(=)1929 2286 y Fi(X)1918 2482 y Fl(m)p Fh(\025)p Fj(0)2086 2259 y Fl(N)7 b Fh(\000)p Fj(1)2097 2286 y Fi(X)2107 2484 y Fl(l)q Fj(=0)2254 2372 y Fm(a)2302 2387 y Fl(l)q(m)2391 2372 y Fm(t)2424 2335 y Fl(l)2450 2372 y Fm(z)2496 2335 y Fl(m)75 2631 y Fe(is)35 b(absolutely)h(c)-5 b(onver)g(gent)36 b(for)f Fm(z)f Fg(2)1379 2608 y Fn(\026)1358 2631 y Fm(D)1433 2645 y Fl(\032)1508 2631 y Fe(and)i(is)f(a)g(p)-5 b(olynomial)38 b(in)c Fm(t)p Fe(.)49 b(We)34 b(say)i(that)g(the)f(function)g Fm(X)7 b Fn(\()p Fm(t;)15 b(z)t Fn(\))75 2744 y Fe(is)30 b(Gevr)-5 b(ey)31 b(asymptotic)h(of)e(or)-5 b(der)32 b Fm(\015)j Fe(to)c(the)f(formal)i(series)2174 2721 y Fn(^)2148 2744 y Fm(X)8 b Fn(\()p Fm(t;)15 b(z)t Fn(\))31 b Fe(with)g(r)-5 b(esp)g(e)g(ct)31 b(to)g Fm(t)25 b Fg(2)g Fm(S)3246 2759 y Fl(d;\022)r(;R)3414 2744 y Fe(,)30 b(uniformly)75 2857 y(in)j Fm(z)j Fe(on)417 2834 y Fn(\026)395 2857 y Fm(D)470 2871 y Fl(\032)511 2857 y Fe(,)c(if)h(ther)-5 b(e)33 b(exist)g Fm(C)q(;)15 b(K)1281 2871 y Fj(1)1347 2857 y Fm(>)25 b Fn(0)33 b Fe(such)f(that)1000 3044 y Fn(sup)985 3127 y Fh(j)p Fl(z)s Fh(j\024)p Fl(\032)1166 3044 y Fg(j)p Fm(X)7 b Fn(\()p Fm(t;)15 b(z)t Fn(\))23 b Fg(\000)1602 3021 y Fn(^)1575 3044 y Fm(X)1657 3006 y Fl(N)7 b Fh(\000)p Fj(1)1815 3044 y Fn(\()p Fm(t;)15 b(z)t Fn(\))p Fg(j)27 b(\024)e Fm(C)7 b(K)2308 3006 y Fl(N)2301 3066 y Fj(1)2374 3044 y Fg(j)p Fm(t)p Fg(j)2457 3006 y Fl(N)2525 3044 y Fn(\000\(1)21 b(+)f Fm(\015)5 b(N)10 b Fn(\))75 3298 y Fe(for)33 b(al)5 b(l)33 b Fm(t)25 b Fg(2)g Fm(S)550 3313 y Fl(d;\022)r(;R)718 3298 y Fe(,)32 b(for)i(al)5 b(l)33 b Fm(N)i Fg(\025)25 b Fn(1)p Fe(.)216 3503 y Fn(In)c(the)h(next)h(theorem,)h(w)m(e)e(giv)m(e)i(su\016cien)m (t)e(conditions)g(under)f(whic)m(h)h(for)f(a)i(giv)m(en)f(formal)h (series)3608 3480 y(^)3582 3503 y Fm(X)7 b Fn(\()p Fm(t;)15 b(z)t Fn(\))75 3616 y(solution)38 b(of)f(\(14\))i(there)e(exists)h(an)f (actual)h(solution)g Fm(X)7 b Fn(\()p Fm(t;)15 b(z)t Fn(\))39 b(of)e(\(14\))i(whic)m(h)d(is)i(Gevrey)f(asymptotic)i(to)101 3706 y(^)75 3729 y Fm(X)7 b Fn(\()p Fm(t;)15 b(z)t Fn(\))32 b(with)e(resp)s(ect)g(to)i Fm(t)e Fn(in)g(a)g(sector,)i(uniformly)d(in) i Fm(z)j Fn(in)c(a)h(disc.)75 3925 y Fd(Theorem)k(3)46 b Fe(L)-5 b(et)1521 4033 y Fn(^)1495 4056 y Fm(X)7 b Fn(\()p Fm(t;)15 b(z)t Fn(\))27 b(=)1899 3970 y Fi(X)1889 4166 y Fl(m)p Fh(\025)p Fj(0)2083 4033 y Fn(^)2056 4056 y Fm(X)2131 4070 y Fl(m)2198 4056 y Fn(\()p Fm(t)p Fn(\))2311 3995 y Fm(z)2357 3962 y Fl(m)p 2311 4036 113 4 v 2315 4119 a Fm(m)p Fn(!)75 4326 y Fe(b)-5 b(e)35 b(a)g(formal)i(series)e (wher)-5 b(e)36 b(the)f(c)-5 b(o)g(e\016cients)1725 4303 y Fn(^)1699 4326 y Fm(X)1774 4340 y Fl(m)1840 4326 y Fn(\()p Fm(t)p Fn(\))30 b Fg(2)f Ff(C)p Fn([[)p Fm(t)p Fn(]])p Fe(,)36 b(solution)g(of)f(\(14\))h(with)g(initial)g(c)-5 b(onditions)75 4439 y Fn(\()p Fm(@)163 4395 y Fl(j)158 4451 y(z)226 4416 y Fn(^)200 4439 y Fm(X)8 b Fn(\)\()p Fm(t;)15 b Fn(0\))27 b(=)655 4416 y(^)629 4439 y Fm(X)704 4453 y Fl(j)740 4439 y Fn(\()p Fm(t)p Fn(\))f Fg(2)f Ff(C)p Fn([[)p Fm(t)p Fn(]])p Fe(,)33 b Fn(0)25 b Fg(\024)g Fm(j)31 b Fg(\024)25 b Fm(S)f Fg(\000)19 b Fn(1)p Fe(,)33 b(c)-5 b(onstructe)g(d)34 b(fr)-5 b(om)33 b(L)-5 b(emma)34 b(4.)42 b(We)31 b(make)i(the)f(fol)5 b(lowing)75 4552 y(assumptions.)75 4665 y(1\))39 b(F)-7 b(or)39 b Fn(0)d Fg(\024)f Fm(j)42 b Fg(\024)35 b Fm(S)29 b Fg(\000)24 b Fn(1)p Fe(,)40 b(the)f(Bor)-5 b(el)39 b(tr)-5 b(ansforms)41 b Fn(\()p Fg(B)1997 4679 y Fj(1)2063 4642 y Fn(^)2037 4665 y Fm(X)2112 4679 y Fl(j)2148 4665 y Fn(\)\()p Fm(t)p Fn(\))e Fe(ar)-5 b(e)40 b(holomorphic)h(functions)e(on)g(the)f(disc)75 4778 y Fm(D)s Fn(\(0)p Fm(;)15 b Fn(1)p Fm(=)p Fn(\(4\()p Fm(j)29 b Fn(+)20 b(1\))715 4745 y Fl(r)747 4754 y Fk(1)782 4745 y Fl(=r)849 4754 y Fk(2)888 4778 y Fn(\)\))p Fe(.)75 4891 y(2\))31 b(F)-7 b(or)31 b Fn(0)25 b Fg(\024)g Fm(j)31 b Fg(\024)25 b Fm(S)20 b Fg(\000)15 b Fn(1)p Fe(,)30 b(the)g(functions)g Fm(t)c Fg(7!)f Fm(V)1719 4905 y Fl(j)1755 4891 y Fn(\()p Fm(t)p Fn(\))h(=)f(\()p Fg(B)2075 4905 y Fj(1)2141 4868 y Fn(^)2115 4891 y Fm(X)2190 4905 y Fl(j)2226 4891 y Fn(\)\()p Fm(t)2329 4858 y Fj(2)2369 4891 y Fn(\))31 b Fe(b)-5 b(elong)30 b(to)h(the)f(sp)-5 b(ac)g(e)32 b Fg(A)3269 4906 y Fl(\027;d;\016)o(;j)3495 4891 y Fe(for)e Fm(\027)h(>)25 b Fn(0)75 5004 y Fe(and)34 b Fm(d)25 b Fg(2)g Ff(R)32 b Fe(such)h(that)h(the)f(se)-5 b(ctor)1331 4981 y Fn(\026)1315 5004 y Fm(S)1371 5019 y Fl(d;\016)1498 5004 y Fe(do)g(es)33 b(not)h(c)-5 b(ontain)34 b(any)f(of)g(the)g(c)-5 b(omplex)34 b(numb)-5 b(ers)740 5236 y Fm(r)781 5251 y Fl(k)r(;m)931 5236 y Fn(=)1250 5174 y(1)p 1037 5215 471 4 v 1037 5334 a(2\()p Fm(m)21 b Fn(+)f(1\))1415 5247 y Fb(r)1444 5262 y Fk(1)p 1400 5268 94 3 v 1400 5309 a(2)p Fb(r)1459 5324 y Fk(2)1533 5236 y Fn(exp\()1737 5174 y Fm(i\031)p 1717 5215 126 4 v 1717 5298 a Fn(2)p Fm(r)1803 5312 y Fj(2)1853 5236 y Fn(\(2)p Fm(k)k Fn(+)c(1\)\))66 b Fm(;)81 b(k)28 b Fg(2)d(f)p Fn(0)p Fm(;)15 b(:)g(:)g(:)j(;)d Fn(2)p Fm(r)2922 5250 y Fj(2)2982 5236 y Fg(\000)20 b Fn(1)p Fg(g)p Fm(;)75 5503 y Fe(for)33 b(al)5 b(l)33 b Fm(m)25 b Fg(\025)g Fn(0)p Fe(.)42 b(Mor)-5 b(e)g(over,)34 b(assume)f(that)h(ther)-5 b(e)34 b(exists)f Fm(C)2139 5517 y Fj(1)2203 5503 y Fm(>)25 b Fn(0)p Fm(;)15 b(T)39 b(>)25 b Fn(0)33 b Fe(such)g(that)75 5724 y Fn(\(49\))1178 b Fg(jj)p Fm(@)1516 5683 y Fl(n)1559 5692 y Fk(0)1511 5748 y Fl(t)1598 5724 y Fm(V)1651 5738 y Fl(j)1688 5724 y Fn(\()p Fm(t)p Fn(\))p Fg(jj)1841 5739 y Fl(d;j)1959 5724 y Fg(\024)25 b Fm(C)2120 5738 y Fj(1)2160 5724 y Fn(\()2215 5662 y(1)p 2205 5703 66 4 v 2205 5786 a Fm(T)2281 5724 y Fn(\))2316 5686 y Fl(n)2359 5695 y Fk(0)2398 5724 y Fm(n)2453 5738 y Fj(0)2492 5724 y Fn(!)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 3764 100 a Fn(19)75 399 y Fe(for)33 b(al)5 b(l)33 b Fn(0)26 b Fg(\024)f Fm(j)31 b Fg(\024)25 b Fm(S)g Fg(\000)20 b Fn(1)p Fe(,)33 b(al)5 b(l)33 b Fm(n)1144 413 y Fj(0)1208 399 y Fg(\025)25 b Fn(0)p Fe(.)216 511 y(Then,)33 b(the)g(function)75 750 y Fn(\(50\))1260 b Fm(X)7 b Fn(\()p Fm(t;)15 b(z)t Fn(\))27 b(=)1899 664 y Fi(X)1889 859 y Fl(m)p Fh(\025)p Fj(0)2056 750 y Fm(X)2131 764 y Fl(m)2198 750 y Fn(\()p Fm(t)p Fn(\))2311 688 y Fm(z)2357 655 y Fl(m)p 2311 729 113 4 v 2315 812 a Fm(m)p Fn(!)75 1064 y Fe(wher)-5 b(e)31 b Fm(X)404 1078 y Fl(m)471 1064 y Fn(\()p Fm(t)p Fn(\))p Fe(,)g Fm(m)25 b Fg(\025)g Fn(0)p Fe(,)30 b(ar)-5 b(e)31 b(the)f Fn(1)p Fg(\000)p Fe(sums)h(of)f(the)g(formal)i(series)2400 1041 y Fn(^)2374 1064 y Fm(X)2449 1078 y Fl(m)2515 1064 y Fn(\()p Fm(t)p Fn(\))p Fe(,)f(is)f(holomorphic)j(on)e(the)f(domain)75 1176 y Fm(S)131 1196 y Fj(2)p Fl(d;\020)5 b(;R)331 1177 y Fk(2)382 1176 y Fg(\002)13 b Fm(D)541 1190 y Fl(X)637 1176 y Fe(wher)-5 b(e)30 b Fm(S)946 1196 y Fj(2)p Fl(d;\020)5 b(;R)1146 1177 y Fk(2)1214 1176 y Fe(is)29 b(a)g(se)-5 b(ctor)30 b(with)g(op)-5 b(ening)30 b Fn(0)c Fm(<)f(\020)32 b Fg(\024)25 b Fm(\031)15 b Fg(\000)e Fm(\016)s Fe(,)30 b(for)g Fm(\016)f(>)c Fn(0)k Fe(and)h(r)-5 b(adius)31 b Fm(R)3649 1143 y Fj(2)3713 1176 y Fm(>)25 b Fn(0)75 1289 y Fe(and)37 b Fm(D)330 1303 y Fl(X)435 1289 y Fe(is)f(a)h(disc)g (c)-5 b(enter)g(e)g(d)38 b(at)f(0)g(with)g(r)-5 b(adius)38 b Fm(X)i(>)32 b Fn(0)p Fe(.)54 b(The)37 b(function)f Fm(X)7 b Fn(\()p Fm(t;)15 b(z)t Fn(\))38 b Fe(is)f(a)g(solution)g(of)g (\(14\))75 1402 y(on)f Fm(S)264 1422 y Fj(2)p Fl(d;\020)5 b(;R)464 1403 y Fk(2)524 1402 y Fg(\002)22 b Fm(D)692 1416 y Fl(X)759 1402 y Fe(.)49 b(Mor)-5 b(e)g(o)g(ever,)37 b(the)f(function)f Fm(X)7 b Fn(\()p Fm(t;)15 b(z)t Fn(\))36 b Fe(is)f(Gevr)-5 b(ey)36 b(asymptotic)h(of)e(or)-5 b(der)37 b Fn(2\(1)23 b(+)3614 1365 y Fl(r)3646 1374 y Fk(1)p 3596 1381 102 4 v 3596 1434 a Fj(2)p Fl(r)3663 1443 y Fk(2)3708 1402 y Fn(\))35 b Fe(to)101 1514 y Fn(^)75 1537 y Fm(X)7 b Fn(\()p Fm(t;)15 b(z)t Fn(\))34 b Fe(on)f Fm(S)566 1557 y Fj(2)p Fl(d;\020)5 b(;R)766 1538 y Fk(2)805 1537 y Fe(,)32 b(uniformly)i(in)e Fm(z)37 b Fe(on)1621 1514 y Fn(\026)1600 1537 y Fm(D)1675 1551 y Fl(\032)1748 1537 y Fe(for)d Fm(\032)e Fe(smal)5 b(l)34 b(enough.)75 1797 y Fd(Pro)s(of)i Fn(F)-8 b(rom)36 b(the)g(estimates)h(\(44\))g(for) f Fm(n)1572 1811 y Fj(0)1645 1797 y Fn(=)d(0,)38 b(w)m(e)e(get)g(that)h (the)e(series)h(\(50\))h(con)m(v)m(erge)h(uniformly)d(on)75 1910 y(ev)m(ery)30 b(compact)g(set)g Fm(K)24 b Fg(\002)1027 1887 y Fn(\026)1006 1910 y Fm(D)1081 1924 y Fl(\032)1150 1910 y Fn(of)29 b Fm(S)1308 1930 y Fj(2)p Fl(d;\020)5 b(;R)1508 1911 y Fk(2)1564 1910 y Fg(\002)17 b Fm(D)1727 1924 y Fl(X)1824 1910 y Fn(where)29 b Fm(K)35 b Fn(is)29 b(a)h(compact)g(set)f(of)h(the)f(sector)h Fm(S)3441 1930 y Fj(2)p Fl(d;\020)5 b(;R)3641 1911 y Fk(2)3708 1910 y Fn(and)75 2023 y(0)26 b Fm(<)f(\032)g(<)g(X)7 b Fn(,)30 b(for)g Fm(X)j(>)25 b Fn(0)30 b(from)f(\(44\).)42 b(So)30 b(that)h Fm(X)7 b Fn(\()p Fm(t;)15 b(z)t Fn(\))31 b(de\014nes)e(a)h (holomorphic)g(function)g(on)g(the)g(domain)75 2136 y Fm(S)131 2155 y Fj(2)p Fl(d;\020)5 b(;R)331 2137 y Fk(2)390 2136 y Fg(\002)20 b Fm(D)556 2150 y Fl(X)623 2136 y Fn(.)216 2249 y(Moreo)m(v)m(er)26 b(from)d(the)h(general)g(results)g(on)f(deriv) -5 b(ation,)26 b(pro)s(duct,)e(con)m(v)m(olution)h(of)f(1)p Fg(\000)p Fn(summable)f(formal)75 2362 y(series,)31 b(see)f([1)q(],)g (section)i(6.3,)f(the)f(1)p Fg(\000)p Fn(sums)f Fm(X)1691 2376 y Fl(m)1758 2362 y Fn(\()p Fm(t)p Fn(\))h(satisfy)h(the)f(same)g (recursion)g(relation)h(\(15\))h(as)3610 2339 y(^)3584 2362 y Fm(X)3659 2376 y Fl(m)3726 2362 y Fn(\()p Fm(t)p Fn(\),)75 2475 y(for)h(all)g Fm(m)c Fg(\025)g Fn(0,)34 b(in)f(Lemma)g(4.)48 b(So)33 b(that)g Fm(X)7 b Fn(\()p Fm(t;)15 b(z)t Fn(\))35 b(is)e(a)g(solution)g(of)g(\(14\))i(on)d(the)h (domain)g Fm(S)3334 2494 y Fj(2)p Fl(d;\020)5 b(;R)3534 2475 y Fk(2)30 b Fg(\002)22 b Fm(D)3762 2489 y Fl(X)3829 2475 y Fn(.)75 2587 y(The)30 b(rest)g(of)h(the)f(pro)s(of)g(will)h(b)s (e)f(divided)f(in)h(three)h(parts.)75 2750 y Fd(P)m(art)40 b(1.)55 b Fn(In)34 b(this)h(step)g(w)m(e)g(giv)m(e)i(uniform)c (estimates)k(on)e(the)g Fm(n)2378 2764 y Fj(0)2417 2750 y Fg(\000)p Fn(deriv)-5 b(ativ)m(e)36 b(with)f(resp)s(ect)g(to)g Fm(t)g Fn(of)g(the)75 2862 y(function)30 b Fm(X)7 b Fn(\()p Fm(t)581 2829 y Fj(2)621 2862 y Fm(;)15 b(z)t Fn(\).)216 2975 y(W)-8 b(e)32 b(consider)e(the)g(holomorphic)h(function)f Fm(X)7 b Fn(\()p Fm(t)1910 2942 y Fj(2)1950 2975 y Fm(;)15 b(z)t Fn(\))31 b(for)f(all)h(\()p Fm(t;)15 b(z)t Fn(\))26 b Fg(2)f Fm(S)2724 2990 y Fl(d;\022)r(;R)2912 2975 y Fg(\002)20 b Fm(D)3078 2989 y Fl(X)3145 2975 y Fn(,)31 b(where)f Fm(S)3520 2990 y Fl(d;\022)r(;R)3718 2975 y Fn(is)g(a)75 3088 y(sector)k(with)f(direction)g Fm(d)p Fn(,)h(op)s(ening)f Fm(\022)i Fn(with)d(0)e Fm(<)f(\022)j Fg(\024)d Fn(\()p Fm(\031)d Fg(\000)21 b Fm(\016)s Fn(\))p Fm(=)p Fn(2,)36 b(for)d Fm(\016)g(>)c Fn(0,)34 b(and)f(radius)f Fm(R)e(>)f Fn(0.)49 b(F)-8 b(rom)75 3201 y(the)31 b(estimates)h (\(44\),)g(w)m(e)f(get)g(that)g(there)g(exist)g Fm(C)q(;)15 b(X)r(;)g(T)39 b(>)25 b Fn(0)31 b(suc)m(h)f(that)75 3445 y(\(51\))506 b(sup)547 3533 y Fj(\()p Fl(t;z)s Fj(\))p Fh(2)p Fl(S)772 3545 y Fb(d;\022)r(;R)923 3533 y Fh(\002)994 3516 y Fj(\026)978 3533 y Fl(D)1036 3541 y Fb(\032)1087 3445 y Fg(j)p Fm(@)1165 3405 y Fl(n)1208 3414 y Fk(0)1160 3469 y Fl(t)1247 3445 y Fn(\()p Fm(X)7 b Fn(\()p Fm(t)1432 3408 y Fj(2)1473 3445 y Fm(;)15 b(z)t Fn(\)\))p Fg(j)27 b(\024)1787 3384 y Fm(C)7 b(X)p 1787 3424 154 4 v 1840 3508 a(\032)1950 3445 y Fn(\()2006 3384 y(1)p 1995 3424 66 4 v 1995 3508 a Fm(T)2071 3445 y Fn(\))2106 3408 y Fl(n)2149 3417 y Fk(0)2188 3445 y Fn(\()p Fm(n)2278 3459 y Fj(0)2318 3445 y Fn(!\))2378 3408 y Fj(2)2444 3359 y Fi(X)2433 3555 y Fl(m)p Fh(\025)p Fj(0)2586 3445 y Fn(\()p Fm(m)20 b Fn(+)g(1\))2902 3358 y Fb(r)2931 3373 y Fk(1)2966 3358 y Fb(n)3004 3373 y Fk(0)p 2903 3379 137 3 v 2924 3420 a(2)p Fb(r)2983 3435 y Fk(2)3053 3445 y Fn(\()3116 3384 y Fm(\032)p 3098 3424 83 4 v 3098 3508 a(X)3191 3445 y Fn(\))3226 3408 y Fl(m)p Fj(+1)75 3741 y Fn(for)27 b(all)i Fm(n)390 3755 y Fj(0)454 3741 y Fg(\025)c Fn(1,)k(all)f(0)e Fm(<)f(\032)g(<)g(X)7 b Fn(.)40 b(F)-8 b(rom)28 b(the)g(estimates)h(\(6.9\))g(in)e([4)q(])h(page)g(625,)h(w)m 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Fg(\025)25 b Fn(1.)40 b(Using)31 b(classical)g(estimates)h(on)e(the)g(Gamma)g(function,)g(w)m (e)g(deduce)g(from)f(\(51\))j(and)d(\(52\))75 4528 y(that)i(there)g (exists)g(0)25 b Fm(<)g(\032)g(<)g(X)7 b Fn(,)31 b(there)g(exist)g Fm(C)q(;)15 b(K)33 b(>)25 b Fn(0)31 b(suc)m(h)f(that)75 4744 y(\(53\))755 b(sup)796 4832 y Fj(\()p Fl(t;z)s Fj(\))p Fh(2)p Fl(S)1021 4844 y Fb(d;\022)r(;R)1173 4832 y Fh(\002)1244 4815 y Fj(\026)1228 4832 y Fl(D)1286 4840 y Fb(\032)1337 4744 y Fg(j)p Fm(@)1415 4703 y Fl(n)1458 4712 y Fk(0)1410 4768 y Fl(t)1497 4744 y Fn(\()p Fm(X)7 b Fn(\()p Fm(t)1682 4706 y Fj(2)1722 4744 y Fm(;)15 b(z)t Fn(\)\))p Fg(j)27 b(\024)e Fm(C)7 b(K)2182 4706 y Fl(n)2225 4715 y Fk(0)2263 4744 y Fm(n)2318 4758 y Fj(0)2357 4744 y Fn(!\000\(1)21 b(+)f(\(1)h(+)2856 4682 y Fm(r)2897 4696 y Fj(1)p 2833 4723 126 4 v 2833 4806 a Fn(2)p Fm(r)2919 4820 y Fj(2)2969 4744 y Fn(\))p Fm(n)3059 4758 y Fj(0)3098 4744 y Fn(\))75 5023 y(for)30 b(all)h Fm(n)395 5037 y Fj(0)460 5023 y Fg(\025)25 b Fn(1.)75 5185 y Fd(P)m(art)41 b(2.)56 b Fn(In)34 b(this)i(step,)h(w)m(e)e(get)i(uniform)d(estimates)j(on)f(the) f Fm(n)2375 5199 y Fj(0)2414 5185 y Fg(\000)p Fn(deriv)-5 b(ativ)m(e)37 b(with)e(resp)s(ect)g(to)i Fm(t)e Fn(of)g(the)75 5298 y(function)30 b Fm(X)7 b Fn(\()p Fm(t;)15 b(z)t Fn(\).)216 5411 y(W)-8 b(e)32 b(will)f(follo)m(w)h(a)e(metho)s(d)g(in)m (tro)s(duced)g(in)h(the)f(pap)s(er)g([16)q(].)41 b(W)-8 b(e)32 b(consider)e(the)h(di\013eren)m(tial)h(op)s(erator)75 5524 y Fd(Y)c Fn(=)301 5488 y Fj(1)p 288 5503 61 4 v 288 5555 a(2)p Fl(t)359 5524 y Fm(@)407 5538 y Fl(t)467 5524 y Fn(de\014ned)h(on)h Ff(C)972 5491 y Fh(\003)1011 5524 y Fn(.)41 b(W)-8 b(e)31 b(ha)m(v)m(e)h(that)75 5740 y(\(54\))1165 b(\()p Fm(@)1488 5703 y Fl(l)1483 5763 y(t)1515 5740 y Fm(X)7 b Fn(\)\()p Fm(t)1700 5703 y Fj(2)1741 5740 y Fm(;)15 b(z)t Fn(\))26 b(=)f Fd(Y)2066 5703 y Fj(\()p Fl(l)q Fj(\))2146 5740 y Fn(\()p Fm(X)7 b Fn(\()p Fm(t)2331 5703 y Fj(2)2372 5740 y Fm(;)15 b(z)t Fn(\)\))p eop end %%Page: 20 20 TeXDict begin 20 19 bop 3764 100 a Fn(20)75 399 y(where)30 b Fd(Y)420 366 y Fj(\()p Fl(l)q Fj(\))531 399 y Fn(denotes)h(the)f(op)s (erator)h Fd(Y)i Fn(comp)s(osed)d Fm(l)i Fn(times,)f(for)f(all)i Fm(l)27 b Fg(\025)e Fn(0,)31 b(all)g(\()p Fm(t;)15 b(z)t Fn(\))27 b Fg(2)d Fm(S)3239 414 y Fl(d;\022)r(;R)3427 399 y Fg(\002)3540 376 y Fn(\026)3518 399 y Fm(D)3593 413 y Fl(\032)3634 399 y Fn(.)216 511 y(F)-8 b(rom)31 b(the)g(estimates)h(\(45\))g(in)e([16)q(],)h(page)g(183,)h(w)m(e)e(kno) m(w)h(that)75 803 y(\(55\))629 b Fd(Y)946 765 y Fj(\()p Fl(l)q Fj(\))1052 803 y Fn(=)1203 689 y Fl(l)1148 716 y Fi(X)1149 914 y Fl(k)r Fj(=1)1294 803 y Fm(Y)1347 818 y Fl(l)q(;k)1431 803 y Fn(\()p Fm(t)p Fn(\))p Fm(@)1587 765 y Fl(k)1582 825 y(t)1691 803 y Fn(with)61 b Fg(j)p Fm(Y)2007 818 y Fl(l)q(;k)2091 803 y Fn(\()p Fm(t)p Fn(\))p Fg(j)26 b(\024)f Fn(2)2386 765 y Fj(2)p Fl(l)q Fh(\000)p Fl(k)2541 803 y Fn(\()p Fm(l)e Fg(\000)c Fm(k)s Fn(\)!)p Fg(j)p Fm(t)p Fg(j)2909 765 y Fl(k)r Fh(\000)p Fj(2)p Fl(l)75 1087 y Fn(for)30 b(all)h Fm(l)d Fg(\025)d Fn(1,)31 b Fm(t)25 b Fg(2)f Ff(C)801 1054 y Fh(\003)841 1087 y Fn(.)40 b(W)-8 b(e)32 b(consider)e(the)h(function)1322 1373 y Fm(P)1380 1388 y Fl(l)1406 1373 y Fn(\()p Fm(t;)15 b(z)t Fn(\))27 b(=)1718 1259 y Fj(2)p Fl(l)q Fh(\000)p Fj(1)1726 1286 y Fi(X)1728 1481 y Fl(p)p Fj(=0)1865 1373 y Fn(\()p Fm(@)1953 1328 y Fl(p)1948 1397 y(t)1993 1373 y Fn(\()p Fm(X)7 b Fn(\()p Fm(t)2178 1335 y Fj(2)2219 1373 y Fm(;)15 b(z)t Fn(\)\))2375 1391 y Fh(j)p Fl(t)p Fj(=0)2525 1311 y Fm(t)2558 1278 y Fl(p)p 2525 1352 73 4 v 2526 1435 a Fm(p)p Fn(!)75 1665 y(whic)m(h)31 b(is)h(a)g(p)s (olynomial)g(in)f Fm(t)p Fn(,)h(with)f(holomorphic)h(co)s(e\016cien)m (ts)h(on)e Fm(D)2567 1679 y Fl(\032)2608 1665 y Fn(,)h(for)f(all)i Fm(l)c Fg(\025)e Fn(1.)44 b(F)-8 b(rom)32 b(the)g(T)-8 b(a)m(ylor)75 1778 y(form)m(ula)31 b(with)f(in)m(tegral)i(remainder,) 962 2036 y Fm(X)7 b Fn(\()p Fm(t)1112 1999 y Fj(2)1152 2036 y Fm(;)15 b(z)t Fn(\))26 b(=)f Fm(P)1453 2051 y Fl(l)1480 2036 y Fn(\()p Fm(t;)15 b(z)t Fn(\))21 b(+)1781 1912 y Fi(Z)1872 1939 y Fl(t)1831 2119 y Fj(0)1927 1975 y Fn(\()p Fm(t)f Fg(\000)g Fm(s)p Fn(\))2184 1942 y Fj(2)p Fl(l)q Fh(\000)p Fj(1)p 1927 2015 409 4 v 1967 2099 a Fn(\(2)p Fm(l)j Fg(\000)d Fn(1\)!)2345 2036 y Fm(@)2398 1999 y Fj(2)p Fl(l)2393 2059 y(s)2460 2036 y Fn(\()p Fm(X)7 b Fn(\()p Fm(s)2655 1999 y Fj(2)2695 2036 y Fm(;)15 b(z)t Fn(\)\))p Fm(ds;)75 2284 y Fn(w)m(e)31 b(get)g(that)702 2542 y(sup)682 2630 y Fl(z)s Fh(2)781 2613 y Fj(\026)765 2630 y Fl(D)823 2638 y Fb(\032)874 2542 y Fg(j)p Fm(@)952 2504 y Fl(k)947 2564 y(t)995 2542 y Fn(\()p Fm(X)7 b Fn(\()p Fm(t)1180 2504 y Fj(2)1220 2542 y Fm(;)15 b(z)t Fn(\))21 b Fg(\000)f Fm(P)1511 2557 y Fl(l)1538 2542 y Fn(\()p Fm(t;)15 b(z)t Fn(\)\))p Fg(j)27 b(\024)162 b Fn(sup)1910 2630 y Fl(z)s Fh(2)2009 2613 y Fj(\026)1993 2630 y Fl(D)2051 2638 y Fb(\032)2086 2630 y Fl(;\030)s Fh(2)p Fj(\(0)p Fl(;t)p Fj(\))2337 2542 y Fg(j)p Fm(@)2415 2504 y Fj(2)p Fl(l)2410 2565 y(\030)2477 2542 y Fn(\()p Fm(X)7 b Fn(\()p Fm(\030)2673 2504 y Fj(2)2713 2542 y Fm(;)15 b(z)t Fn(\)\))p Fg(j)2953 2480 y(j)p Fm(t)p Fg(j)3036 2447 y Fj(2)p Fl(l)q Fh(\000)p Fl(k)p 2904 2521 332 4 v 2904 2604 a Fn(\(2)p Fm(l)25 b Fg(\000)20 b Fm(k)s Fn(\)!)75 2825 y(for)32 b(all)h(1)28 b Fg(\024)g Fm(k)j Fg(\024)d Fm(l)r Fn(,)33 b(all)g Fm(t)28 b Fg(2)f Fm(S)1113 2840 y Fl(d;\022)r(;R)1281 2825 y Fn(.)46 b(F)-8 b(rom)33 b(the)f(estimates)i(\(53\))f(w)m(e)g(get)g(that)g(there)f(exist)h Fm(C)q(;)15 b(K)36 b(>)27 b Fn(0)33 b(suc)m(h)75 2937 y(that)602 3075 y(sup)582 3163 y Fl(z)s Fh(2)681 3146 y Fj(\026)665 3163 y Fl(D)723 3171 y Fb(\032)774 3075 y Fg(j)p Fm(@)852 3038 y Fl(k)847 3098 y(t)895 3075 y Fn(\()p Fm(X)7 b Fn(\()p Fm(t)1080 3038 y Fj(2)1120 3075 y Fm(;)15 b(z)t Fn(\))22 b Fg(\000)e Fm(P)1412 3090 y Fl(l)1438 3075 y Fn(\()p Fm(t;)15 b(z)t Fn(\)\))p Fg(j)27 b(\024)e Fm(C)7 b(K)1966 3038 y Fj(2)p Fl(l)2117 3014 y Fn(\(2)p Fm(l)r Fn(\)!)p 2036 3055 V 2036 3138 a(\(2)p Fm(l)23 b Fg(\000)d Fm(k)s 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Fn(\))2161 3611 y Fj(2)p Fl(l)2223 3649 y Fm(k)s Fn(!\000\(1)21 b(+)f(2\(1)h(+)2817 3587 y Fm(r)2858 3601 y Fj(1)p 2794 3628 V 2794 3711 a Fn(2)p Fm(r)2880 3725 y Fj(2)2930 3649 y Fn(\))p Fm(l)r Fn(\))p Fg(j)p Fm(t)p Fg(j)3112 3611 y Fj(2)p Fl(l)q Fh(\000)p Fl(k)75 3931 y Fn(for)33 b(all)h(1)c Fg(\024)f Fm(k)k Fg(\024)c Fm(l)r Fn(,)k(all)h Fm(t)c Fg(2)f Fm(S)1126 3946 y Fl(d;\022)r(;R)1294 3931 y Fn(.)48 b(Using)33 b(the)g(estimates)i(\()p Fm(l)24 b Fg(\000)e Fm(k)s Fn(\)!)p Fm(k)s Fn(!)31 b Fg(\024)e Fm(l)r Fn(!,)34 b(for)e(1)e Fg(\024)g Fm(k)i Fg(\024)e Fm(l)r Fn(,)j(w)m(e)h(get)g(from)75 4044 y(the)d(estimates)h(\(55\))f (and)f(\(56\))i(that)f(there)g(exist)g Fm(C)q(;)15 b(K)33 b(>)25 b Fn(0)30 b(suc)m(h)g(that)75 4267 y(\(57\))543 b(sup)758 4355 y Fl(z)s Fh(2)857 4338 y Fj(\026)841 4355 y Fl(D)899 4363 y Fb(\032)950 4267 y Fg(j)p Fd(Y)1057 4230 y Fj(\()p Fl(l)q Fj(\))1138 4267 y Fn(\()p Fm(X)7 b Fn(\()p Fm(t)1323 4230 y Fj(2)1363 4267 y Fm(;)15 b(z)t Fn(\))22 b Fg(\000)d Fm(P)1654 4282 y Fl(l)1681 4267 y Fn(\()p Fm(t;)c(z)t Fn(\)\))p Fg(j)27 b(\024)e Fm(C)7 b Fn(\(4)p Fm(K)g Fn(\))2324 4230 y Fj(2)p Fl(l)2385 4267 y Fn(\000\(1)21 b(+)f(2\(1)i(+)2904 4206 y Fm(r)2945 4220 y Fj(1)p 2882 4246 V 2882 4330 a Fn(2)p Fm(r)2968 4344 y Fj(2)3017 4267 y Fn(\))p Fm(l)r Fn(\))p Fm(l)r Fn(!)75 4545 y(for)30 b(all)h Fm(l)d Fg(\025)d Fn(1,)31 b(for)f(all)h Fm(t)25 b Fg(2)g Fm(S)1057 4560 y Fl(d;\022)r(;R)1225 4545 y Fn(.)216 4658 y(On)30 b(the)g(other)h(hand,)e(from)h(the)h(F)-8 b(a\022)-45 b(a)32 b(di)e(Bruno)g(form)m(ula,)g(w)m(e)h(get)h(that)847 4900 y Fm(@)900 4855 y Fl(p)895 4924 y(t)940 4900 y Fn(\()p Fm(X)7 b Fn(\()p Fm(t)1125 4862 y Fj(2)1166 4900 y Fm(;)15 b(z)t Fn(\)\))26 b(=)1566 4813 y Fi(X)1444 5008 y Fl(m)1506 5017 y Fk(1)1541 5008 y Fj(+2)p Fl(m)1693 5017 y Fk(2)1728 5008 y Fj(=)p Fl(p)1953 4838 y Fm(p)p Fn(!)p 1844 4879 290 4 v 1844 4962 a Fm(m)1924 4976 y Fj(1)1963 4962 y Fn(!)p Fm(m)2068 4976 y Fj(2)2108 4962 y Fn(!)2143 4900 y(\()p Fm(@)2231 4859 y Fl(m)2293 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5318 y Fn(\()p Fm(t;)15 b(z)t Fn(\))26 b(=)f Fm(Q)3599 5333 y Fl(l)3625 5318 y Fn(\()p Fm(t)3693 5285 y Fj(2)3732 5318 y Fm(;)15 b(z)t Fn(\))75 5431 y(where)1237 5608 y Fm(Q)1309 5623 y Fl(l)1335 5608 y Fn(\()p Fm(t;)g(z)t Fn(\))26 b(=)1656 5494 y Fl(l)q Fh(\000)p Fj(1)1646 5522 y Fi(X)1650 5716 y Fl(q)r Fj(=0)1778 5608 y Fn(\()p Fm(@)1866 5564 y Fj(2)p Fl(q)1861 5632 y(t)1940 5608 y Fn(\()p Fm(X)7 b Fn(\()p Fm(t)2125 5571 y Fj(2)2165 5608 y Fm(;)15 b(z)t Fn(\)\))2321 5627 y Fh(j)p Fl(t)p Fj(=0)2529 5547 y Fm(t)2562 5514 y Fl(q)p 2472 5587 186 4 v 2472 5670 a Fn(\(2)p Fm(q)s Fn(\)!)2667 5608 y Fm(:)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 3764 100 a Fn(21)75 399 y(By)40 b(construction)g(of)g(the)g(op)s(erator)g Fd(Y)s Fn(,)h(w)m(e)f(ha)m(v) m(e)h(the)f(iden)m(tit)m(y)h Fd(Y)2515 366 y Fj(\()p Fl(l)q Fj(\))2596 399 y Fn(\()p Fm(P)2689 414 y Fl(l)2715 399 y Fn(\()p Fm(t;)15 b(z)t Fn(\)\))43 b(=)d(\()p Fm(@)3181 366 y Fl(l)3176 421 y(t)3208 399 y Fm(Q)3280 414 y Fl(l)3306 399 y 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%%Page: 23 23 TeXDict begin 23 22 bop 3764 100 a Fn(23)75 399 y([14])47 b(S.)25 b(Ouc)m(hi,)i Fe(Multisummability)j(of)e(formal)j(p)-5 b(ower)30 b(series)e(solutions)i(of)f(nonline)-5 b(ar)30 b(p)-5 b(artial)31 b(di\013er)-5 b(ential)262 511 y(e)g(quations)33 b(in)g(c)-5 b(omplex)34 b(domains.)e Fn(Asymptot.)f(Anal.)g(47)g (\(2006\),)i(no.)e(3-4,)h(187{225.)75 699 y([15])47 b(J.P)-8 b(.)39 b(Ramis,)h Fe(D)n(\023)-44 b(evissage)39 b(Gevr)-5 b(ey.)39 b Fn(Journ)m(\023)-43 b(ees)39 b(Singuli)m(\022)-43 b(eres)39 b(de)f(Dijon)g(\(Univ.)h(Dijon,)i(1978\),)h(pp.)c(4,)262 812 y(173-204,)33 b(Ast)m(\023)-43 b(erisque,)32 b(59-60,)h(So)s(c.)e (Math.)g(F)-8 b(rance,)31 b(P)m(aris,)h(1978.)75 1000 y([16])47 b(V.)37 b(Thilliez,)i Fe(Division)f(by)g(\015at)i(ultr)-5 b(adi\013er)g(entiable)40 b(functions)f(and)g(se)-5 b(ctorial)40 b(extensions.)d Fn(Results)262 1112 y(Math.)31 b(44)g(\(2003\),)i(no.)e (1-2,)h(169{188.)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF 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