Content-Type: multipart/mixed; boundary="-------------1010050442937" This is a multi-part message in MIME format. ---------------1010050442937 Content-Type: text/plain; name="10-164.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="10-164.keywords" q-Laplace transform, Whitney infinitely differentiable function, formal power series, asymptotic expansion in the sense of Poincar ---------------1010050442937 Content-Type: application/postscript; name="qsing_perturbation_cv.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="qsing_perturbation_cv.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.96.1 Copyright 2007 Radical Eye Software %%Title: qsing_perturbation_cv.dvi %%CreationDate: Thu Sep 16 15:23:57 2010 %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMR17 CMMI12 CMR12 CMBX10 CMR10 CMMI10 CMSY7 CMBX12 %%+ CMR8 CMMI8 CMSY10 CMEX10 CMSY8 MSBM10 CMR6 CMMI6 MSBM7 CMTI10 CMSY6 %%+ LASY10 CMBX8 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o qsing_perturbation_cv.ps %+ qsing_perturbation_cv.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2010.09.16:1523 %%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 %! TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]FontType 0 ne{/Metrics exch def dict begin Encoding{exch dup type/integertype ne{ pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def}ifelse}forall Metrics/Metrics currentdict end def}{{1 index type /nametype eq{exit}if exch pop}loop}ifelse[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[ exch{dup CharStrings exch known not{pop/.notdef/Encoding true def}if} forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def}def end %%EndProcSet %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 70 /F put dup 72 /H put dup 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 1D432B0D064B56C598F4338C319309181D78E1629A31ECA5DD8536379B03C383 D10F04E2CB7C8461B10646CD63AFEB7608468CA0FCFC4D3458FB43D22879B515 27DD9CCF44C2BFCD95A4DE911E4915FBC02335E9999FD9B546134081D6DA3792 EC4A76DEBA77635BE52E09986268A919CB48B5EFB1A1301EE0683CB5709BC8CE D819D799020CBA673BA39C911075501395B1FD20EAD392C9D5A8C9FD1198C737 D1A614CF0C0432F29DDEB4BF9DB026908DBE89EE522B7D55DE9BF64AFBE6248E 2E10466655EB9083E7D23E3F0EE26154F191BEBC9987930CD4B4CABE1275BDF9 8755EF3D531FDF91D54954FC53F15A38D1E8F8D1E36447484FA2C09D34813615 838B6330FEAE536D08376E4A0FDDF58CDF5647C9F1FF3A7D1ACAD376DB3CADB6 9459F7A5D4F1864863B79E9F93A1EDE8B99C3138D26227C01F6FE0AAC800F2E5 94DD81CF7B1355B642CE45CB532FC5B535D66EDFFEA076C009E87406D9772D71 848C3C53B7496A5D6B58679EF11E114C5F457C6A0D3CDE50278E4A89D5393B1C F877CF4E2142A4D045C4AA9138105D748903BACC28FD43DFEDB341E1FCDBE2EA D412498FBB5374D6836CFBEB13D4C2B7B9625C25B037FDA9DCC42F5679C4B3C1 6340E341F73A921013790936801297855EDE27C68907C6C87633FE64CBDCFEC9 DD8EC8784857A09FE395D7FE095A87CC113B930AD09643AC6F9F5A44A1154C89 7951C1CEF380FE489886299A1C5A2E1FB2D475FD3478BEDD0EC21A4CBAD46924 9AF698FCD47B9891629762AAD9E00C0C31B4F54AA5E59A575DB50A8961F05D7B D54237D5CE59FDB5C3E83AFEB532CC4F900F7E85358A528D56A6FBE9F3002E56 EA59E9C2459EEA543F906E74846306E52E324E00D300A78B984ADD86E21B5C08 4E19F5F28704CCA9BC6F5B8E11C74674881613A81A1763E4217B4FC14BD6FF18 B37A14375DF5F7CCD561DC7ABC388F4539424239C618F1D92CD37F507FF94133 927748C386DF1C1EA2F61DC341C76A4E72E64AA9723292B83079A9C3FCEA7355 51666A804736DB26A32A5E44562BF0EEC98870D116148D9E17AFD856254CF45F 25F15F27489DC8D77D08639DF8C812A078B0EAA063C385A868E5A6987CFD39FD 6E00B737CEA54752A22AD32F6684586815FCE5A7EC5A35D1200152D50B77F087 7AB9BEE334D8C3FCEBB349C8AE7B73B145A5F58E317ED72296CE06635DC8E750 98E28B9532E1353040DFB3A337B39469CED8B4E6FC73E36348AA349FF9BD1833 8665BBDA964CAE5F114079F83302478F8BA6A8E4C6074E4B522F3E615C2DBC0B 915186260D0D10F0AF06D38B45CD95CEF174CC32124A478AFEC9E2F8311E385D 7A7D8CCABB19C482846BFA20972926FDE8CE74F6EED9AA849898D64E38424049 8C2771CCE5D34AA681721C36006E657B30D83FE7CB55686FDA4C964315C5FECB 859228C261104E7DB8ECE67ED324E255C3B62456FF7F1218C68DDE2C06E6949E B3C16C59F1F9E5FC3CF02E3F90B5190E082044079174C8184D708410E2517E08 F4B54F5F192704C78641D831A478A315EC09DFC700722D37BA304D3B69EFBD2B AC81FA51D129716D6575143F7CBFC7F2F48AEDDFD8F119FDD9CEA49009A70E76 D9B99390E0E40DD3B95E3279FD5552AF1CB6E2E32231EBBA94EE273DDE4A9303 65AE8683734419600A7992500F0DD362D0CC36895877BE0D69C21CAC3681528A 81715D0C77C679BCA3DC4A3927FC060BFA68AAFA87F88AF482B9F694A540C085 121061A514E3D9E0E6826D3B08A4FD01DCAB892A09B96ACA8889737950449FC2 A9024BE39CEC787BE8D305548152E286426B2295409AAF7CF95A9A2953B9841F 46E559C0ABA963C8F111BB2FBD22DB83D30E3DA7A1FD776B0D624042F7F4E2C9 E1DB9D6C71EA88A5C8493595589B290082FE4F021769C2AE4C625250784003FA 7084F7C09773D552143E790196A30E1E123EF2A38BAC41FD51A8E370FE1B5089 CA3BF71254CBC60087B08B2DF62D3B2761B41DA202F47DFAE1137A92BBBFAA76 6EF73BE8832C8AEFC8DB6A388FAD95B3DFDF1270218CE362D976DD5DA0AAEC3E 281C346DA6F96BB0D7C7882D936037033FBCAB8329D4AE6D8A626640BF75388A 6ECDA13A06CEEAE00BD7F514D4AF8D560CFC3CFAD3DFDE2E5A912B2420F1872F 9E79433DC61DBE8B7F80A0713C0C0A0719BDA02C175D96203D5F89CC55F7C21A 58A215D59799FCFB31C0173EF5E2D71D2B3B64FDC993AF3DF3C290FCB39AA0DE 86C8B08A18F3A5C757D781C2F6663F5F85B9BA81169579CF6D30E37538A2D1EC 0EE3051885E394A58AA4691D0B2A9808E79D59EF86183172B1837634BF3BAC9C 42E2DAFFAB1A83EF294573DE641120293A22A953002F205527B4D4602B45D065 EBD91654EB55AD200C64EEFD9B1C249D263075035F334806F07479C9E0778288 8DB81B51CB916E44B6D3642F2A829F9975145CCB9BA421486B3C2343C79CC6B8 B8F68B5A53AD8DF80DDAC274B44F66A245FE83F640F042FEA40DC7B15A05BF51 532E288FB66C7BA33396B99BEE3C45448ADEC75717EC6AED7BF1467718A587A8 50402C732F200D5D2BF2545BCFD61EE21BD0D3A5F8D1EB690884A10C425B4664 6945309349DBF75A8AF331BCC422277707D54839951D03DA59C786FBFA7EABD3 5FC28A462EDC6235C842FE914BBDC6D85539C54ABCA5D03C6B726C8551B90BBA 60D816EF61A1713F52FB9D0B3A315E060662771CCEF62C965A6806834E0ED099 5806C64D92BBC7B7833DEC767BC93CA60A2AF3EEFB323F02810CE25EF717775D 6CD0DB55D282979FC6E27ECFBA99B5A0D13F9B6E1735DEED8760179E8FA11678 691115D12F4ACBF20C86B2B976989E8CD1B11C9593EBBD16B20DFA386ACC4818 C22B101F9561914068C82ECE5754A9905FBC6CB339F4891FDB3063B6AF7B049C CD8B3886127AA89D0A7B230F9E7855BB480865E15447AFD487B5059542CBA4CC 509C7526272372750D85B8EAFAA3D7892D534206AD058558BAAF7F3BE12A389E B4F15A596E60CAF3B5EE73AD9D5540EAC69C5889E5AAB0B09605DDDF2CBFBE91 1801E1E1B88BFD551179D43489B1C6215745C413468D9B6EEE795996656C59AF 3385D6C65DDC87FCA0122C65DA784F6B 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX8 %!PS-AdobeFont-1.1: CMBX8 1.0 %%CreationDate: 1991 Aug 20 16:36:07 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 72 /H put readonly def /FontBBox{-59 -250 1235 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 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 15 /epsilon1 put dup 67 /C put dup 113 /q put readonly def /FontBBox{-30 -250 1026 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 65 /A put dup 66 /B put dup 67 /C put dup 69 /E put dup 73 /I put dup 76 /L put dup 82 /R put dup 83 /S put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-53 -251 1139 750}readonly def currentdict end currentfile eexec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cleartomark %%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 currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 48 /prime put dup 84 /T put readonly def /FontBBox{-4 -948 1329 786}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFB7605D7BA557CC35D6 49F6EB651B83771034BA0C39DB8D426A24543EF4529E2D939125B5157482688E 9045C2242F4AFA4C489D975C029177CD6497EACD181FF151A45F521A4C4043C2 1F3E76EF5B3291A941583E27DFC68B9211105827590393ABFB8AA4D1623D1761 6AC0DF1D3154B0277BE821712BE7B33385E7A4105E8F3370F981B8FE9E3CF3E0 007B8C9F2D934F24D591C330487DDF179CECEC5258C47E4B32538F948AB00673 F9D549C971B0822056B339600FC1E3A5E51844CC8A75B857F15E7276260ED115 C5FD550F53CE5583743B50B0F9B7C4F836DEF1B263DB260571DCE3DFEC496E93 CB244233D14265C75105A52752F6FD65BEB401C8874BC659BC1D3ACB2B494652 F074549FA28873558649A150EC72E3ADBC7803BC5D9544F32F69E37E51E68023 1EFE3D08447DCF002B47F4DFB14D0CAC9645E415F4603CA72D8A49CF19BE2064 2D64A4445D3B47809C65471EA37674EA95D9EEA5C6954D368AC8958D567D9EFA 2DD8E29122705488A35B5127AE50358FAB1F0D1D0836B3D889FF94A28A014D6F 96E6B6B709EF83E78AC13C12DA3C5E87C10E66FB452D9C6C3538FCE5A10259EF B74BF46BF308DB83B7CCA782A8953D96743A40AD72CBD533DEFB7806EE0D964D D397B16E318A0434AB073BE7A6BF447B82C17EB1DC0D3820F24E1E034784DF7D 3B9561F1768387410F1D8B22DC4BD25DCCDD7000455AC46D612DD9F3D29EC305 F8579C2C8F0B20695B97D60031ABBA0E71500C233AB75810BAAF1ED1A5F41622 1E51E2E8081803FC99B82BEFE9CF9013CA96541532E8A86748DC9FA24C02E854 DA6FB1DF39A984B7BCB6ACA752DB8BB8FAA693ABB4F265707A0E69F9E7D4492D 9C12305A3351D626843F1662A17CBE3FA463FC945AAC496CA709030E4DA176A1 A6056BCFF17C4090E1135655A4CC86644B87A6F18A290E550A868FF4DC95A44E 00602FB8606D95EF6156D768B87748FE571A854EE05EFC584189271439D2B220 BF84C69C20F488B7870593A0759D16E212B41EB882EC4040C4A074F71427 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 18 /grave 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 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M 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 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 readonly def /FontBBox{-163 -250 1146 969}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 58 /colon put dup 65 /A put dup 67 /C put dup 68 /D put dup 76 /L put dup 80 /P put dup 82 /R put dup 84 /T put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 121 /y put readonly def /FontBBox{-301 -250 1164 946}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 26 /rho put dup 59 /comma put dup 73 /I put dup 86 /V put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 115 /s put readonly def /FontBBox{11 -250 1241 750}readonly 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 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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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 69 /E put dup 70 /F put dup 72 /H put dup 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 20 /lessequal put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 73 /I put dup 84 /T put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly 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 26 /braceleftbigg put dup 27 /bracerightbigg put dup 80 /summationtext put dup 88 /summationdisplay put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 2 /multiply put dup 14 /openbullet put dup 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 56 /universal put dup 59 /emptyset put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 73 /I put dup 76 /L put dup 84 /T put dup 85 /U put dup 86 /V put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put dup 112 /radical put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 14 /delta put dup 15 /epsilon1 put dup 21 /lambda put dup 23 /nu put dup 26 /rho put dup 28 /tau put dup 59 /comma put dup 61 /slash put dup 68 /D put dup 73 /I put dup 74 /J put dup 75 /K put dup 80 /P put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 88 /X put dup 97 /a put dup 98 /b 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 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly 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 22 /macron 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 126 /tilde put readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec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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 14 /delta put dup 15 /epsilon1 put dup 21 /lambda put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 39 /phi1 put dup 44 /arrowhookleft put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 74 /J put dup 75 /K put dup 77 /M put dup 80 /P 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 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 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D456A09920001A3D2199583EB9B84B4DEE08E3D12939E321 990CD249827D9648574955F61BAAA11263A91B6C3D47A5190165B0C25ABF6D3E 6EC187E4B05182126BB0D0323D943170B795255260F9FD25F2248D04F45DFBFB DEF7FF8B19BFEF637B210018AE02572B389B3F76282BEB29CC301905D388C721 59616893E774413F48DE0B408BC66DCE3FE17CB9F84D205839D58014D6A88823 D9320AE93AF96D97A02C4D5A2BB2B8C7925C4578003959C46E3CE1A2F0EAC4BF 8B9B325E46435BDE60BC54D72BC8ACB5C0A34413AC87045DC7B84646A324B808 6FD8E34217213E131C3B1510415CE45420688ED9C1D27890EC68BD7C1235FAF9 1DAB3A369DD2FC3BE5CF9655C7B7EDA7361D7E05E5831B6B8E2EEC542A7B38EE 03BE4BAC6079D038ACB3C7C916279764547C2D51976BABA94BA9866D79F13909 95AA39B0F03103A07CBDF441B8C5669F729020AF284B7FF52A29C6255FCAACF1 74109050FBA2602E72593FBCBFC26E726EE4AEF97B7632BC4F5F353B5C67FED2 3EA752A4A57B8F7FEFF1D7341D895F0A3A0BE1D8E3391970457A967EFF84F6D8 47750B1145B8CC5BD96EE7AA99DDC9E06939E383BDA41175233D58AD263EBF19 AFC0E2F840512D321166547B306C592B8A01E1FA2564B9A26DAC14256414E4C8 42616728D918C74D13C349F4186EC7B9708B86467425A6FDB3A396562F7EE4D8 40B43621744CF8A23A6E532649B66C2A0002DD04F8F39618E4F572819DD34837 B5A08E643FDCA1505AF6A1FA3DDFD1FA758013CAED8ACDDBBB334D664DFF5B53 9560176672637BD4A2519C2FEF92A9982AEAB5516EB9561607F8874F0EB00695 5ACC280F245C8543983226A1B44FEE78486E6B6DA8DC07D6EE80911A6E67AD4F FD7833FA50B4B2C5D135FE336DECCE8DB17DD4B4EBF52451F392FF5306D7CD21 7D0F48AD21E1C90EAB5999173A657AC2AFF1E9008719CEC5B5B3C8F6FF29AC62 C599CEF7A5C35936DB5857B5EA31B1385CAF0C7943F23D2B938218199E1CA3B7 4D0441B1973A31B1A25D803A0A0A6E7030CCDAE237BE727E867E40A940BD5445 8AD209518E88F045C30A6B7CB59CB4B20AE6375AA517740A9A009BB8FAD26611 A141CD59BDBE339EC12EC9AD357104ED2517456D75F270F507F50BB258F91AAB 719CA6D464ED08D2AD66087861EC1F581642650C284688B189AB6F3D16920B6C EC826D62ABB5BC0FAE8B317FA17337873D45F6A6308525B1EF1F43C7104504B4 063651F18C48EF7EF5287BE2396AC73F14486B702B550C298BF7C389194D7B7F 2ADF3D5504DE5042AC9D4D9BF51CA2EBE01E6FA48F1D55D51E9BF012AEDC85C0 504E533A58BF784DE1EE961ECA3B23E3BDA4BB029A8AC2262258C2513D4E2E2F 322FC86F52B7F0C9AD2BE1B30DA4A2F1F58822726745B7430CC33189836193A1 3CED7A96349E43274ADEE734BA8A13EBFCB7AD7A78A7D3FEB817FF9ECC91DA93 8612CEA48417888A251736F4359AF83086172452EF79586019787D701EF8FE38 6082C13BE5F68BC7C1D95A46D01B2A6C3D719BF755DDFCBA041BDA9FC23030FA 7D8A86D40FDDA84FD67AB7A9344F8A4E5BF12F84F9696D8757565894CDB3FF97 8E877EA6692B5FF34D859E43E8D474CF584C0E9063E917C1E7C624E4F4983093 01D3DAFA9F28C24C38476404789A07E5B3B4E0B135101947D099199C56A47903 46856D8C56C31BF1E1A572E726F2135894CF34D5E9F36B9C5BF4B8DCB3CD434A D40C392C1222703B6411AD6C41386EBABB2AF1882ED8DD2732F4FD76B96ACBBB 133C304C15EC82DD9CD1A3FF6901FE2BC78ADD12F11FD9C3B33502B061E14D6A 0FF78862808A4489A7153F8C81C42A95E96A161F75A6F48212F8B495A6259B30 DE0C3B54DCB0D2079820240C3A9769E5AC9460F5C9260D16DCB9638DCB17C595 CEC23FF4B6E18DAD7E9AE73CEE23E0E919A3EF26DBF70DDED76BF63D7FB9B682 42269E830102400376AAFE990D2E78622FDEFF8207A1A1C38BB1A82C37263952 31CCF9667758ACF1148EFE704F7E0AB66C5015B6B484E174AC68FBE07215E9BE C6E71F71FF8FCBA90EA79C3BBAA81CD58ACCF74849998733F317CD3BF522114B BBEA412B2E745A82211E64F00C9A06CCFF76106302C34A14812C535ABF413644 44A088C3889B06540BC44F5E2CDAFA578508BBD021D72BD11195ECFE48FC2D6C BB921E8C51FA3D14B53A70E51652955315958A7049ADC614D2BC4A8EBBB6D798 37B6249316B7E635F2D6D668604B9DD4721377291AC3E276343AC9DCEC5B50B6 AD4E52148EE729FF2E7C18AE9D2CFAD6B18CCA65C08A2526524875736E0FD591 5EB20F382F9633D18E9734182A1179A014D25DF100FC064DA18084D05EFF709E 4FEF43DF9DCC9D37CA05C7E37A0F624304086FAA05C43AB28F7810357F64EE9A BABBA1348D8E18E4A5197BAB49A66FC7A718FDE1175822F2B8E77B9B2DC10339 2B001F36F458C6EBEBC33CB105201D96650BC08B30236C90D1658E90C7D39541 42A5A88079D1AF7ACF955182F20BC5E98DBC1063B22B4DA27821B4FEB1DFD6F2 86F0CDFF710808CED7E721B45963031F29A19262FE0B931BA603D45C73CE8813 87B49F13057836124BDF34B7C3702EE83AC6161A1198A3F1EB9AD9DBF4E22123 2D0DC61E789EE5F7B6781DA5AE06C7608395A9E212F539B499144375134C4D30 8B330F4C437117BDF20628C630CFA397E2030ADC5F9DB294843CBF3C2810E4DD F3BE0E4B73CB9A955A8C817F20E16C659B4B1E1F65F5BEA98510B6771E66125A 7F2BBCC99818AD5FAD7C7FE4532032A42CDFF3DA3E0BF16F80A7D99A124B3245 97C9F34FB8AA4F310C131B5A23CE6AE144679B60E99B48E5203E7620AD7182DB 1D8A1A78D74F14CE304F7820573D47CEB3950FEC9D892012AB113E0A91C8E2D4 D006CC8C9D1EA82C83E8E03A0CF425AC56AA5F04C3D93CC97653D53F847FF84F 244E32DCDFCB62370E20499C5F21DFB87DEB481A2DBB38DCA54818CEDECF03BA 1713CBD247ABF0885EBF2074721AFACA874C01303E9B3733AFC7126D3890CFEB 18C5B4A4EC0FAE686B35F9A8555DEF40413FA3993A65278CE8E3513D23570970 03109897631A5DA878E880C692548EBBAAF3B804B1CFC8A8BA7E4C3E99378BC1 23DB75837A7BB64673B219574571D07A386D3FE98F4387B6B877AC48946942E0 C00E050F40298522A542B36A892BD5AF18D3A14AEB1538F3D4F66AADE18F3994 7E8CC1E64DA844CC3DFC1B354D4E1B57F6D3A198A505AE769C7BE32ED13521B6 3E9A33CC7CE2AA799846405E02C7A1EA94E98A6DAFF58076BC6867136D0558BD B1794641F8D74F5FF979020BE2FF38EAD283B5607F0748462671C9724BF0DE92 A5932B567A63F83F53C3DC50210256B81F36C76C0624F52AFD94A6C8EE917869 2A6DC7F472F3781E96159CC9815F0A8F8C56B10FDACB0147EFA89735539D375F 2D1EC22908646C8DCCA7C505BC93F4D16092C7F4564E4630087B46983264499E C8A4279957A3A3C75274BA2F7C9973AECECA474D8B80576F68E87A98C8F2DD37 2A499AD560E180742B61401EE9E24340B7FF30C4ACAF24C49ED37FEACF875C83 9D374D951DB4CC32EA65BE256A20FE6A08BDF26AB10B1BB09AEC8E60C942F3FD 0EBF482A7B3CA57910B2B829F0240028F7230C0F80472A4C2B58F309B04E1FB6 76540FF3014EBC2246D47ABBD4980CD391DF666511683576D49D8BD8C178E782 A835FD8E613EB891E6640598B924CCD9F5AE01528C75521330FC6EA2A3DE3CC6 D58C639702CA3040F9CDB8BD60CF1B41996B7721FC8657F7B281E97E39EE11AA 1DAF42D1630B7E5958081CFF385B990AC0403F85E7FE3E29E30CC8995CDA3A52 03475087C3F414D51515BE14B43F99799C58E12A99A760A4F4DD305F1C2B334B BAA2FED5DEC9677E356C24432FEFA5F9AC909F8DE79CAA12F53754B2F7AC3D17 2AC6BE5E32D8990B0487CC6EBDD563BBDB2273AAD504E16C7EAE23DFFDACDE50 2585CF339F35BC4A8260FD0974E31617D341040272267C26FA4E3C1FFA0EEFDA 80BE394536297548F16D78CAD45C518651C63773FF4496BF379AA6E6BBBED754 3A0A221AB9E0ACDC559D862C378DFE60B9DB28DAB40352C88D05500FB6BC3FCB 48A26CCFA47556D39207C5919B3E04993FB04F09093C5C2C1A9796C1CC39B1F1 F349647DA3F65A83EFA9EA4F2803634A285FDA06C03FCC5C031C4F2F40161A46 8B0D26D5FBB893F2B81586BBD609C7C4F150541877B103B9CB0C44F8C5C92552 0B5E99182B769571B50D95181A27E63E0468A0F822EC4985FCC6EF47B21FD32E B6D02EAD691429171C671EB0F0F0F24C88DB28394884B37EA48C433F7CBDD5E8 0587282F113945BAE85AB8149D44D0813B7181835C650DCB7A36C9750F8A99A6 F2F6CC288274EDEA4ED038465DDB55850AB99C057BE04A77AD20C08C5AEDFAD7 4CFF0ED85A8D796C30007B46810404B5011BA79C95AEA0DFA18B314137AB13A9 65F7A7FDED03A3238EF0CAA34EC912B341F761B9194EF8DA7203512703872AB7 093F21C45B2B8D1D4B177A7F16F0E639FA79B86DE507BA5E017E422B26998E2A E6E8EDFA52CBF9A85CF2F347B2FE0DD008B30766E56CFE96ECA269EF80E4D518 9EDC5722B7B24D91003DDBA30032B3470C9849BE16FDCFF598CDE5778F208BEB BD106B89F414DB03953FAE0AFD493A488B6F8766C1640C2442F8E2721567EED3 BE2C840725DC585BC1B191072D74B9C07AF45D77D669AC480B72D9039F0E0B46 85B4E4C82DAA5879505B5951E542A6933B2ADDBE98B5AC984C135A627DD29F6C 6E4675AB27D7892163F8D1AC478CFFA67A2840BA7D077E80712051F24856587D B672DDD7A3B1172C7D52E167883CF9EDC671BB467C37CDC90F5BFF88733F5508 4DC3A190CDD83F7BB684D2DCFE8D649AF8E22E3CF8DD8D89796A77972095630D 847DBE0960C7C16A41626CE1F33877BD7E23C05F481C31C951A0C21E06838116 9D71BB1335FC9E4A4B42A584F311E3179EF4C9ED0F47116B2636557261855297 847590E2CEFE21379825401B5F1018B1D66C0F0F833AFECB1CFF8B5585033AC3 4C93A6A189719AAFBCA2CBEC2F336665D432B2E7F762B995FED421AD2D44A4D5 411FE0AFD2B58C2E05F75B87C25C48B1AE5B152B52FDE45094E035A4C79628A6 E23EFCEC268C08A04108807D3CD30B85D631D641176DEC0441FB368B5F76E27B E49FAEA1C64115F8BD85CA91A104DFDCB2FFDE4FC5CCC94CA1EEFE6FCCB266C0 65D1A90E0AFD28EB965CD42E7CBA0A80BCCBFFE1E06144DFBF680E25695A22FC DBEA91FB818B93460C68764A081BECA1B9144BFE71C24BE7EE2649E13C48E987 6CD0F3C42ED6914185012597187DDA0A8C79C53A339EA8CD9FF651EA6DB10ABE 672518D97A4FACC3FC112D55707B15DE38DCF2CB5A57793968DAC1D161873B11 674B6C0CE182173542E7ED170EB24DDF18F9C8910C509CD9637626F5478E983C 442BA7099872AB8E21A1FAC38768211EEC44E14C2A4D78726B67F1DD6EAACFFB DE9C3AFD1AA7398DA96ACB5087689A244D6F375C3A85B1199BA6A413019BB5CF A193A66D21C0634A4AA17867A8B0E99D9D69161FED5EA3B4C0D4E40938C74B78 1092705BC3AF4F813F14D0C025B3F46C6BBBE7B7B8DFE8059647FA6D34ECAE9E 48E3061A2D3869581020153E38F6A08233770618AB74C9F0C2F85D3D111F77B2 653B6737F763E58967E046460441F59F5550C279BBAF492428DA4DD8D7384ECB 52D6F681EE6C90D4419B9E240E0AD614CA39407BCDD82C7652B4F36E84796F27 E079108F35D8E481DE402897FB1AF01051B8714E62EFB3BEAD690CCB983E507F A6B4EA3E4A674364B427C7C836873C2451E4C30D09FFD9F3C83F3E62994E552A F3726758442D85635AC972033B9029A4272CFC1E3D230D5C11181BE255029710 1BD83CFBC16D7B9E59128B14263EA3C53E001088722F2E73FB2F48DD586C5869 BCACEFB1F7A9A6DAD88CA436E912AE4EB696F11EFC91D086655073FB82A2B826 8DA853754137CCCF03FB39B525F74428DF1BC14EEB19D70A3B220A2D859B42EB C88A060CF47212895CC8517B6D0E54D80BA62783D6F18C05EEFF150578BE0BEB CFA5B1B79B1883CFAC356E6BF30DA7FF9ACBEE45FBE9F5DA1E0E7FBF752EE368 50FD040C56E58A190C9FF5E162A7936FF1205B8DBDB73778A39F839B7A41128B 81C8D3460D20E1352C40A5E06EAB709D3774C3BF8AE0EB6FDB87A8A0B7416E7A A0B390B25A37782D7B35D8CBB29C47E04F1DEAB6B4E5EEEC9A7E286F8E3E1C5C A6025CA960C2C6A10851F6AB26335E1A6A1DD060C55C0CBF49A2D6539AE5D3FB 8614C4F2F61E84FB767C159213727F9D1626049E3626DBA7049FFCBCFF1DB5C1 16A1F7D1F5D35A921956CFA47D56A00027A1E1BC1644790094A9B9558C5BDC30 017A00CB406EDAEC7EA4E32C00F080B56E18B851E0D21C234FF455BD31E3CCBE 2524524FFDA10A054E627B1B51A8B44714E741C244E4C8A0992980E25853CBEA 242BCBC57CCA5CFCECBC5F048B6D5DCFDDA35914C3682AB911E22FA175609DC0 96CAF9F6B2D0C8F100298AB5DB1589E40B40D7C922BFFF746540C73DFCF78CF5 8A39BCE529FDD3FC760D07B4C067E47ED16624C93D4E6AB5C689C0530B8F45CC 4AD459E799B04F01D40EE3302835B95CD1B158A83F3ED36B0910B5330CDD6139 F1320DD56924F57F1B5552CA8B7594EDAC1B65E7502FA75D232B2AEE5FB802F9 D403D5515D5A9B76235CA22AD577214A3E095B52B00EAADE353DBC74E6A9747B 3C32078CD93A1432DA65CFB9E5F6BDB0F4643EFBD787C92E625EF3948413DD76 DE4AE4156789B2A4782CDC2C2066358C784173441EABAA4B791BC595C4F8DC60 3385065D940CC040CBA18428B7D2C04D6EE6C6DE53C1CB47DEE9D353D0FDEB7B 107D7837C7B39C89614614136E11FA36F2A24E7E5BBDEC392B83A093CAD85B64 6B861974B968BE591450667741E9CA8A88B91F99814FF89E5E3207C357BCB535 8A18CBE0BFC53E511BD0A35C49430EF03ED9AF78586116F429F4AE9720C47649 5F2897C4E1789356065F806EAD43E0881E8BF66701BA626BA34B8B03190B7758 4162AD603C5CA72802D8E01D08B0CE1C8C6C4E106E0085B40435173F790479A9 37CA37B60B88B44C9267C0D1DB1C00636C6B8705B3BD4A7F860263AC75D3A2D1 0A5816C12D8402240D2A8DD6B0281B621255C9282B431B48D965F88D2967BD39 0A33A116ABE5034B3DF0CF595E651CB39D56A43EF8116BAA8DF734C0D5056426 FA40E5E824ECA51FDC927F7C04709A87F21D4E8144BFFA9BFAF8D2D6D4FD733B 5792AAF5314A8980165C0A77BAE28D5E10A3C0B4B3CC7E49D06316E60D7767B1 20DFE7921A16FA53C1DC12FFC7E79D4D22D04D12ABFB576A8ABFAB4D38DA0877 224D6F62A9BCD885EF2EC0CA1C953AEA051DB816A7A7FE04470D10B232147876 610CBCBBE22E2514572D6D099648943CFC826E96C09AEBB8415552FC91A8241C 47F505C9BFB5F2649B3D3136356A0761CE5318278C9A27351606AAC44C805CA8 3F90C5BD866749D08D96D8D0486112A75A044174AD1D71CD6C2565D253CA016B 5BE22F2BD27DD99F156C17F4A3EBE9D1E8A5DAF180A9850968D17D6A313E903E A8606EB253A781883FE3BA86D3DC74CB717AE4824F505B94A65A3342F99E8A09 ACF0FA98670A1588330BCF31178F669F6B66C642F8F04AE14076DF530F917AE8 A5B093DC71067CB9D98415481908E92F2131751BC7FE7EFACEFF10F956C8A1BE F63F31994AD118D8F60CEADFF6D7A0E962730F1A0604930CF23B3F56BEE0EB9E 6843741FE33D047CE4693E61BB947BC0D44941F97349F2EF1BE2C36338195272 D660D061678C7FD0EC9921EDD0C5276E18177B725BF7C3727C11403DFB1FFB31 0A78385268C8573E9A642CEE1224C59B665C6E39A81679BFB2D9DD611DA11630 89BF19361F81C35F8F3E8D040B5F8DD0AF69F01DDD531E7B45C6684132AF9745 409D0EB7E7D6F33E2F4E03170DD473725CCB1905EBDB1B824E84450FBBDD4A05 DBE6A23C26C21EE50FBC73EFDA835C8559DFA8E6253CD1FF1D47C28E62B27826 A5FABA79914109DEA3AAFB2936E8DB96D13E1A36B3DDBFD999CCAA0F1E7D5275 CB71ED3E07A8E54A636233948130521E8FCA67E4637053B4F7509DD193FFF7C7 6C73B1D3585823A31A4A215AB58CDD85E69426499A19BAC4786F94CA0F662441 967FEA31AA9D7D23A8B6168DBB4F85BD057889934A752378F9516D139EBDCFE6 DA17C789C139D0833F7B3AAFC60ED23AEA1585E4917D28E2B54557B244C0A8A3 A136C2394C5CC34390691EDFAD3909C5AC8C4F2C4DBB86620FBD9C720D9FBBFD 5E40FE3C06E345095B323340F34546EED921CB584B6117E32F274F5C47E4334D 934078C53DC7B91DEF1EAE6BBFDF1B4F53CA31D157271B0D2E22246751875C28 C7323F94E4654406BFCDC11AE46461F556253ED97ADF2B5B20F89AF9C88B6773 0BB87512DC2B9C970A7AAB6D40E4B9B912F84E1342EB13CD3AB9A7551FCFBBBB 89C99319F13501FB2C09F4E5D35EB96E7A326EE41F7150C0D62C16BAED41F97E 241EDF97A03D503D9051117D94D28EFC687A30AC4515C55189D2D8A3EB3D4124 2762EF27B75A59BB49F02163F18539E2365DDD4995F9681C203552BD693AF0A3 685834AAD2F70952E6F7FF8E7366BF44D130F2BAF93AD7098B29428825C0F3D5 7358C5B2806867F81BFCA1DE50B99FF7E2280AC0A866551BAB5E138294F5C424 5D00C252438F15A66D0FBA2335574D21069ED4CBC3D14B0136398FFF3E109313 5B2B2F23BAD328C3629BD81760CF1282C3B68DBA498B4AE9B0FE7EC4F7F9EBC1 E1A7AFA76182CB2B42D3A78055FBE44C4F74C86B2F3F01563E10AE77AB21DE43 19C722B72D5493542CADF6DB181B996E44E1D621FAB0B0D368B616DABEADDE48 F9EF9232FB43FEEC95D8276B8C6673611C6D68D198D7647DDBEEB6285489095B E5FA5CC25F4F13194A13F3BD11F74C7E3E363403FC7A44B62E7466FB5076A901 027C87E7F77D087265FAF7E0CDC7DE25E1D4A090CBF4CF9297E15A7F37845BD8 663EEED198B2A63B8F60D1C227CD7A6E4FC7FB6C0E20A9A46C7C8791B9DBD92C BBB2F3DD3CAF22B3B89936977669A1A2C371104CFDEBA854F224A4141F14DB7C CF49950C36F833BE40FB7095ACFCE739F393AB80772564D26C94C45145FDCB70 F97300FC92DE8486218F2F41B67721E2615E7E05DD9C00775E06D92BF1307281 83412307BF5DFEB633F31262F1EDF8F7F874BEBF4F09C5BD8029A2717ED41F70 F5EAC25CE0C261D4BD50793471391DDD41A3A1FB6A4E649C9CFC54FBF4AE33A7 A4825F31F876158EF290D412042D2F92E72FA8A43307B83635F4A46EEB610516 09EDFA71B9BE6BD4CF6F3521E8BE6B11545950FB375F2F34A65F738ABD7CC874 09D2B777C15A5FF69A7EB9296E24E0A2575C40D6ECF419B4D47BFD1519F9BFFC 532A2CB38A6B7A569B3DD829F68E616E49E8EED2722B584D421829293CA3684F 4BA158A2020D5A4319E6A677BD0CE8A05A42121C7462676262BEDB7289ED4F03 C169C772650F5D3CBBFC2E40BC1E3195662A30B5E24EF3120F476D3EBC2DDF9A 5743BAB7DA926F4F13D1AED6043793E690E17C025C950C85B329CDF1B71BF5A2 B1E14352A400259E1D0C4F6F15CB1DCC3DED838C3B0CE768570A85C9294D2A59 69B92C6C68739328E862D8FB585D4C17CD8BB255D35F0AED96F9978576DA76FE 3EA9FC8C679268F15279F399F3B6888768B6C4621237FDD40C16B4CE070FFD0F 022F031B75D6C5B17B720D5ABE5FB97253241835F236E36293CF0FA00C2EC97C EE9680CC56CE59F0FA8102DAF950254EA6754029AE3E41B8BF3F0357635E15BB 13E2F123625597B92A0893F165CB4D46816881A82E72AE6153436523C4A360F8 5139637661DB8F144AC64FFBAA3AD47A8B66463B1B7BD931EE3A6BB8138720AC B0E0F8E7BB1C341B6C0845FFD3F4AE9377BECEE69E63F3B420CDE7BFA6AE4019 C73366EFC88575333BEAFF9E9B3B3724CF8BC3B1375C2016B4860F032EE1BB6C 28E5D9F551E95D34845E63C57AC0DACE9787781AD9800DD5F880F1473CCF4A9C DB1B325A6F897CBA8E1A7D70F235639A35099ECA3AA55A694035B3AEB16150B4 FAF9F854127BC7FA2829BF3BE49A57C51968C7DBD799FFAE7169E7FFBEE3705E EA55F341CB973C87EF8E70C684BCF2FFC6BC3A9711BA92380914E0E35E2F7090 2E25B4A9686839BF53259BC5EF1BCE30B4618E697B584DC014381063215CB5E3 3FB24945149E339464AFC22CDEC6D4AE5DADCAD37E99827B8FB8222A0472BF6D A1F6291A6DE9F5930B853E44F3B68DA364963C38883FF0AAE4181546A6250D65 963799E5AFCB7243C0E404C12C266CA04064B1DCFA6FD7463938D558944A874A 8EFA6B2CE95212C4CFFAFFFC8BEEFFB69058060D6D6D49584CEF9DAA9CC0FE77 29B3127F4D00A5AEA439515C40AA8568E77D11B88625E4621FDA8E44C8F0E2D4 5F751722D6ACF51ECDCF27EC80BDEF4AA4BD0A33900D7C79863693D546744E59 E3DABE5BF2911AF0CCAF8EF58D580074FC82E3609D234E1A75D56D1F685D8AC8 AB5BF844D901A2787AE55E8F8F10121B5B619F5B05DF449C5ABA1AA41956C60A DED250EC7C2A8BD8387D413250FEEEB4369A42A403642D8DFE5D0C0A5404B4DC 4FA5D6E881D8560046B5EF83BDCF99093B5EEE6CD5256A479522BF135A54FCB8 5D076F61C285E8A5C41D5C44F941EB64CAFEDE2A166F24E964E73DC30C000E35 E8F60A82BE9F3D5480E7F455ACFCAA4FD5647BE7A29DE87A7D5B43BB4EA7ED22 735A6D274EBC44A30E02E14DADCA408C6499B9ECE0C3166A844EEE6F078D3B31 D7AF3FB9325E3826EEFC0EDF3B46CECE3DC3D74579F6C33A0AF69F967D653745 01682E07114A02D984E0C13A93371D7D7F63D66EDEFE3A92F84557DB16DB63CE 2C0AC43358A85576E0DCAF7324CB1B4B0F2CB25B87AB2915BCFD54D5C2E18C83 FF4E8D47E93393B9CB63BC892DEA5BFFDF5B73A40DF3FCA4138790E9F5B96A40 AD2ED2B0C7EA8B32F06F39065D4BAEE09595276283ACEFD8739A452A32D3DE61 CDD518A82C43EFB9EF98E1B12E84143CB404D37C5EEB797D4469A34A5BE1A5A2 4471504DA7066F4431E7F489229D0FECFD0AF0A538419C49244B23F586E2BFE3 301D062C0400F858538BC9331009CB99F2F320ABB47354ACAD3015A207A027C8 E0A317903DF0B922F261E32B75113101AB38AEEA94D63A54D65FAAC509812F7C CD9BD374FAF8E90578D7A716818B4EF98AED3512CDB352A2CE9091A24E0CDD09 2F1E0AFBA5C2881E3D13B9F6B58C0ED816303C3A0E96F461A90882B42A0372BA 8014E672A079996514206CE6A68123AD39B4895F8B06C67C781AFE80C0BC7F1D A9EE0450E4A1D0965DFB715CB7329749E8803413533F0F230DF40BCF4435E892 ADCF72735C6E8C3D638BAA111D8A7DFE7C4F6816895564D0AE4D5C04BF419846 9606A141EDFAF02DF6294C8893C67373AB8B37A85C4A894611057D01411F7249 331BB7C7687C3E6CCBE85951AA1ADDBD3B13ABC59AA5938158FC66E98A5B87F9 F55E47A308A56B9FA4E20946A66A9EB1CB2C1FB2E21DB1CAB77C5F23D3B790EE A10F5E77A19E339F1D310D04E3396FBF5691ECBB4851F92F368267218B3E4AFD 6A28E1BAADB6B21798558C1B9F62AA188B2EC303D6B2FE91A27E3C7B6BF83765 F4C88815A2AA624F070AF4354B09260AECD1A8D298578102FF3A4E7903AA6975 C8B49CDC2F976A600859410139535C099B063A659022AEA4550B6D71BEB2BAAD 5C3E6D4BD9C438AAA570C4F37E43434C8FA5E196AB6B67B2DBA0C68B688E9AE8 14F563DB50BA8CBB2858D8C2D46B1B64FC07780E9D4F1FCAB5859DE117B3ABAA 57275FBA777C772063D9F3418506B45565790DC1F0E3D014F4C9329589D3A245 8073B72D2E1B53081C2E0F88ED8AD62E98CB27710EFD6C110B9709596E1E1D11 7439F89B74C258272A3F2B48E39C31666B11A3C6432256F79056244B306B4DCC 7E962B644BBDF27D43FFACAD4DB166EC939A72A90F7A9F7FCA773FC1E0746725 9A23A1A04C75894F894C5F0A42E8045ED1FB1F717FFA5751CF343FC7E6A8B67E 9E7F2E4BCA654CAF06C69A741C5148F2FAAFA33D7F3F9B6F9946D8FD9BE2B1EA 679BD3059B7DFBCFCB9E4F50FF0B8DA7DA878EF9640DA807A52D9748D3475211 1393B6562DB152D0327AB5B035257FB48ECD7096744916938F66C5E252B14BEB 70FB9699672221A8204175138B3BDD5BBA982B0B225B4E165DE53CB38F0C1F52 F91DA06D7F36382CCDB3C1AFE746D38797FE27A69F91B4AA5ADAE604BE7C9D45 B1418E8F3BC7FBE73EEC56B401EB363B204E7D37CF4DF1A2033246397D65966C 86B3D4C840FD06BAD1D0AC492BF5C564B607BCF597D402024A667402A758D315 EEB3AD1500483FDF03B73E5536AE14259334589BBC53743DDB40BFDF32CD6321 61354855B27515EC3CD290F278DE3E55392FDC532C7383A014144F2EAFD7A294 EC5182AF6BCA13DB66E45FD1ADCA68DEEB3AE844610CB4EC4C19481D01281C97 7A2DE4BD98EAA22F377F4CFCECB0FF1753B3EBA3816520C4C2D98F9937E5FAE4 601B154DB08DCBB6D25C89EF4E6E1833ABA6B8AB2EFD55BC671D4AED9B217EBB 6C0D72D2B94482351CBDC7B3B58B7725CD8E3D8296E91320D20426A4C809D027 E94E48B8953147E2C0B63869D6043C9C9777DCB379B2F97B0F34961536A7FAFB 01893FA03A526484E9CE51E044A3669964E15E5D2A42A2571C55F71DD3CC109F F4DC8293D2644413E4756DC7D937017B350DBBC81C81EA7CD31D4CB8FD206D91 DDD6A882DF3DA7940183FE7E469277299D893B2032A57BF7A3218B3C4F1C969D 798A168AB30B9342F6C0FF9C06ADE54324D6C839A3F613491E52A032EFF15554 6622846C4AB3E4AAAA703E3A66D808341473A6634C290B96828BB10C4CD6E903 13A4550C75907D32C10596414CC3668FA467487AC79208BFAF6F5615CDF50FAA 41E66373933118D32BC280A1430239634508CB4DBF27108BE9F8A8BF4443E9D8 0C9CF4295C25F2D584F34279E9176B4034A3B28ECA323CF3C566F56C1BF209A2 799000AD9A3F9661D073E2F635C6D86CF83C8814B771528B69E5DFA9EED4D861 1D65A25ABA69E27711657A98FE5F97C62ECECD9D32200BC69F1A71436621F33B E162239ECEEF27AB7B1BAEC03C8BFDD1131446CB2BE6A52F5B6383CE563C557B 5920A5E0AC08279E30EA44BA4A653FBEB5DD4C149954F613CBF3FC093A511490 A2EAB88CC6A9B8897A811EA36C7C915246B658AE5B61969CC271277ECD54FD38 CD8AD056C97EC78D8D1B134E590875168244408F8670D4B5C45B2DE293A74353 CFFCA45FEB520318334C25ED8A2A92C433945CE51D8BE0E6F3898FE32BB1F2B7 321623A3197E0E0C96491CA22A57C1ADEF0533B71313A6A248EAEC5B311BBEDB 58657DEF74A6B5C670DEA1BDECD0174A0949283E6AE74936BE8213DF88C4A9A5 176722F39B7A380911B672BDB2A6E94FE3003E65B63CE239B5F1AA91FE8E7DA6 5BA23E5261E3DE4158EDF53FE9D410984E4B33D0D806136A6CD9D54463B9EB2D C3893E1415A88FDCDEE0A466EB57EE875A65258FA6DA80D33D8596DA594257B1 5071A2E19A673FA21087736F7DEDA27ADBA7826CFC699B4902B086275C36A298 3EB833908D32813F20FA0C8921AD142FFEA31F729398131F56CB5BD43E613B70 7B89E04A324A2549AB3B6D75A3760A05312409EDB1F606708DD4AC7F607A0717 887C10946A8E508F5A530F59556A7DF2C945453AA280E0BABFC0CF373016A668 F6E4BC4E50EE377185C8333977D15B089D04A5CF4560CB0845010C0FA04D84C3 2C9DEC58BE9D8525F3E9A846B604D96D52A6CBDD00B14A80504F797D3CDB780E 1F21F68835BCD2DBA352A05CA7476D6143F6795EB36A0655EA4940316FE74FF7 2145B7AA7E25612611B137E118F4816EAD7E6488E9F2C657747CDC53FE14965B E97F98DE8EFE6FEEF6333E33B19E3FD9C66DC8E946604697139A239603B5B302 2323B690119ED54D220F24A136FC8B89224BA5840C31EE8E983F2220E9430D99 50FEF638B472049B481BC2302162388F3E49A4CBEE989FAC207F957AC155D160 3559847603FD499CAB4F7391E31E89282A7CBB91F6EA9BB63C0A4128ACD9B34B A4BFA22277F88038FF1BAA8244620BE014E4D7BB4F0E95397687CC30B6F01C9B CC6CE35BA7FA09AB034F747C1A1F8919D4B67F427E43129B536E58DACBD72078 B6BE8EE326F0C286DFE3852A0BCF66278D3A9D346BEEB9354BD151848234F0AE 4629F9E6846769645514177465B5846C2C84A327EE7B62BE719473FA5C5A9A43 F5C276876BB021E2BCBAAEB9A1A371F72CA9F67DA59C25ADFB13E45B44DAA8B4 15CF90906F1C1FFA44D295BDAE40AC30DCDE59D6A8B441F9941F5D6DA06803A5 D0A5CABA4EA01F2449ABD01746D1E61292FCBA311B867FE622106A79D08B4810 8B25C3542B43407A73A493AF325838BF3620E6E16CE3D81FBD98FD013A920185 EAEE3916F95E16651646FD7DFC1BF13512477EDF8FB2E5D2DB28A85EE959174F 331EFA471689D47682EACEAA0A90C64B33D101247BD446FFFD81CC92AE8525F5 63C8A37F0DA617332BD867047712E949C713C4A13111CFD66371B4BB5E257502 9ED33C86B5808153B4C861A2B0F7C65C081CA6B04B245C6434519B60E15D6BEA A7F68205ABCD3B0ED4C6BF65DA0E4544CB7D502B2D6D65EFC091DC4DC1F9CDD6 DF5D36845B6FF02FDD461DE473B4E1FAF29886FD3A11437DD5B2D38E7208288E 287A6706C7494871FFD038546F7795B7E32C0C3F9E176D74C072FCB57D78FFE5 1DA3BEF477E4105C75A699B579ED76371103F2B73889D4487E3DE350270309E7 8733F4B81D72A90C9EB8276C5E0058CE347F464E3A9733964C89C8CC79BF223F 67AE87B2780A9288AB3B5A5E83D12567B27012256BF69779360932199F058D2F 61872D0A191E8D6FB9A1C03CA573B2D9235701C07D3A31EA343B61A190554B23 399EE838D8C576656779F3E2D0E71F04BB27FCD70E77957DD1254543F719D64C 6F9C694433D3E21B9713B41882234CCCF8F3A2EF3D6B698908ACB01360874E58 B3331428C3500776CF0732C26723ED7C2F3225C92997B8167B9AC053B26C59B5 77209236BAB43682A282DB92DE1854AEED8F9C18114D7D612DBA619195E01F8E 565F7648EE9DBA0A7216A0D032244C95492DFB8692CD5387F158FD7A897977C6 423F85A902B594BA9324F2E0F781A9245323227A45228F0CDB59FED8AAF86976 E62AC9050C1C19A96985E7ACC36F4D28E37F6A252BD60E75EA8F4B34054B69EB 1F447D963C336BEC2F45BDDF800BCE31151769918A5900445135A0F5F9BC7E60 9D560BC10039F2FB8D90059E9ADFC899E360F6C011161C0F74864CCA48081AC2 264FB4570FFCD1BED54A9B4494F7F8A44864859232178E82F0091BA763809F1E 8A843B75A5FE364810D631283EEDAB15A20297660981F75575C5E5EEF0431311 C2266EE1793D18F6423E0E2409740AB7754F97D938ACF4822654BD43544B0502 1D6B513882715DB7E8B24E2DA0AF2A9CA12AFAE87A52C761041AE9CBEC85D253 8944B6C94DA64DEDB567986E97F30B898199DF125273D2C6EC7F040D2DE13C9D FBEED2BDADCE6A6C75D7B6AF20B67ECD40FB5CF506DA6FDDE418B6DAC7158272 DBD1C618A026012D9C78F130582C808ADDFBCA69A5523858B71D67A93233B36D 81E6E4DC74EA7D3E48DDB73A7FCB1521A124644C5EC8917A0BBD6A97F3A029FC 59A4094C38219F2D866CE179A676F1A912076810B1376430FB286386B3E6CD29 84C1A9735B1B5FAA0EECDD2D70AD4F3E5D6289DD99BE727414D30F134B20E6CA BE242BD92F5794823C6C9C8CCE632D86519AD0975A0378491B6C9BE834AC5523 2F6DDFE62B25025F609E40FA5F1701046629539A201EC15EDAFF3A3821C3B067 EB33EEAE440CB100404E75DDA898C10D010613EA03578483CCD2FEF88970BBB5 FF7E3E0CCBF141D2079C2B20FD3152AF2C3C4EC982146B877331D35E45063E92 0392BD67C7727F6748A6713D4E2F1E497CBABEC5F12C906F4FF9FB733BD73FA0 873FD31A48B8CDCFF5FD67CDA91791D4189DBF88DA628E35AC688354886FA784 EA076EB50441A7F071922C472B10191A3E5256731203020FB96B61926E081392 FF778AEBC5AD1D93B044AC98EBD76BE7DA3686D67327FF38832607E423FD6000 CBD4DBDDDCCE5AF491DD3BCC717013C53FD43AFEAFD1DDB9548332D6EC312AA8 BF78D24168A4486E2CEFFFC1BB1073A3E51F4F05EC9D555EFF84C679EFD4540D 163C3D9B8ECF5C865784ADBD7E6F8E2999869FBB37A41A3C76D420D3BA0E6D9C 1971D3D05093ECB842799BE2163E496F13BBF8C9A0FA25F8A0943496353608D5 22F7FAC5C3123AF7D92CCBFBD945D35FC28A46036F4D188BDDD550A721F18340 E29ABE2ECC59149ECF9B120BB94FA1EA644F0B8950633E9EE4E5FC23CEBA5AC0 B800EB5459D8F5085CB1DE8D636839985DA21D4BE56060B12B48FF19F6BE68DF 8731C1C9522C4EC150C23E552146970B9EF41D75F858C16A6EE17B9525F763CA FCB6FE1902D9D5ECFF00FDE1141BE3C9372DE93E90E8DFCD8807089501990068 0C9FA844DD98929FEBAE026914FFEBC65F00B40211420B2DC99B219BCF11E036 0ACC795505CAF89EE4D2104A1C13D779CA79853EF6D25191728E3125935878D8 9A1FE352180F25B83198AEE63DB29314E74051745EC44125DDFAD020B5039369 29C8F3C0173B649665E0AD54862C9C31C6617C4B33A4A8049CCDCF09BBEB1F9B EA8AE17A47AE8C04192A9841826B46040ABAE43F19DD03BFC3D99F1B4EA151AC 162C2DA009304F71D1B4C1D1140431C5D3C52C48713F5D296526F5FAF3075C22 34620B28DF8C8F759E8076687E6936A2EE87170463B9D5631C61BA6497F4C348 4447A62517FFC525602FF954BD60D5A25311FDC97022B2FBB683F90BE011D3BE 2CF107F069ED16B96FF871AAC2740CFC76F31A263BA2C05816468D2648E5ED85 B18444839945589922733796EDB5A92E2A25673D8E542FDC2B2620D19330CE97 F5B772F79492C80A66BB355631E98202A49E2C6E3A1E9EAEB31A3F1A3205F959 1061D0E55F4894AB16E23ED68B7C7B1E66ED62CD35C7F77A8F61A620D7AECF83 C9F4F08904FFED3E0A55921637B1BE35ABE4C6ADECFFA37EB4580EDBDD5FA088 1946B1815CC24E43DE2ACC1046144AD8BC7D77AA1B9F8908208F758618EB1DC8 FDDA26D583369DDB57F069D03F190C1667FEB86300B73FFB4C52A53D15B8AD26 3351E9FC10708C72D12E01812E3743BD6174352E02C73043B4B7D5B742DF6F8E AB568D0DD5D3E8D5B77C2F4083ECAE63E1BC139245C1F92504D5040FE4F6A543 4CEA4ABD85E7CB1C4EF5FA264D57CEDA17325B5CB649BB3EDDB6E64F62863CC7 2674FD129C1919C474694AA910C59BC3CA0C650C186B60F5B9AC67EDAED92D37 8D00D638E0DAB643A548422A512409941AA7D0D543C75507FF067D2F631557BF B87A3E37DDD25F187DF0586DC692D2D397F76ACAEB4D1541322D4FD458654898 BA5796D15ECC578C0C7B148E2A42BEA5320F855DD8048A6B6CEF2419752332A1 99474B199CEEA07D3F03307C6F5A2942FB2F1F42A92BC8BC0AD5DF4AB2168C2C 01B480AF500167774C5F5B48CE08D4F3651EF6EB5B74B554FFC2CDA0850FCED5 6F19677628EABE86D9319C37A4B5171DFE557E9EE170795D495689ADD2F4EFC9 60DAC45B13472A9D9052C64A4F9390AC9EB78788C8AE565C7B08930844B292C4 04883EF0CB5F6B3AA36C790D16F9899C36C833205C3BBC3651ED430067E9EC61 1527AD08EBCCFD75AE1517F8F2573F34DC712A35D6A283A4B6764F2CDA8B5D3C 33F1A362BE415B810C9AD7D01AE3A3D51F6A2D16DEF36525BDFA2FF2D77E1E3C 67CF669D66A0B7AE33A84A55E9F6CE944B824C2787D49CE5A04A04 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 2 /Theta put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 19 /acute 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 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X 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 dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /infinity put readonly def /FontBBox{-15 -951 1252 782}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 19 /acute put dup 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 57 /nine put dup 64 /at put dup 65 /A put dup 67 /C put dup 70 /F 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 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 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 currentdict end currentfile eexec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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. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 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 45 /hyphen put dup 79 /O put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 121 /y put readonly def /FontBBox{-33 -250 945 749}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F075EA0A10A15B0ED05D5039DA41B32B16E95 A3CE9725A429B35BAD796912FC328E3A28F96FCADA20A598E247755E7E7FF801 BDB00E9B9B086BDBE6EDCF841A3EAFC6F5284FED3C634085BA4EE0FC6A026E96 96D55575481B007BF93CA452EE3F71D83FAAB3D9DEDD2A8F96C5840EAE5BE5DC 9322E81DFF5E250DEB386E12A49FC9FBF9B4C25C3283F3CEA74B8278A1B09DA7 E9AE4FBAAF23EDF5A3E07D39385D521547C3AAAB8EB70549756EBA8EF445AF4A 497CA924ACCC3DD5456F8E2C7E36946A5BF14E2E959895F7C94F49137256BE46 4A238684D52792234869EAE1A6D8ADF4E138B79472D2A90A6CA99E2394CC20CD 3841733046175B20CEBE372327BF13428EED6A3E2FDF84C2DBA4B0AD584EE9DF B51828D3B8F385846158C29C9AC3496CB9692DD10219697B2ED4D425C3957FD8 C4600D76E045C561216EF05D38177243C314877A69A1C22E3BEC611A2EE5A216 9B7C264CF6D1839DBBD78A40610F2C0D7C2FE09FFA9822FF55035AD52546970F 83EED2D30EABB1F303091EBC11A5379B12BB3F405E371519A53EA9D66174ED25 A2E55463EC71A97BE4C04B39E68112956117C8252DB6FB14AB64534B4BCD568B 246DB833982B38CDE7268BBF74B6B0C18091E1B1F87D32D66F4DD023D1F10D2A 7736A960F72AC01F733A11023832CD68FB6288A5977743F781214D8FA9C0C3F7 80001321D4397771F728FD9EE57CFE7D9192B887EC883EB1505068261DC40089 7B7D2820F06515CD74513521F6397FEAB3AD3572D9A8269430E407E357422461 1785FC2782047F4C0339D79B16862D939F3A37F78E4E2174E4FBF132539CB760 207999FF86F6A3EBE48EB0A1CA635450FDEEF79EB16D853F3BF4B7AB327313BA 9991211FE10DE49402F33CF209D2C418CAECDE152683D499F1BFBC417FCFDF49 7173A632CB6DA4F25CA5C808E531F611057486D33D8476E4F39A3696186EE751 13F205669C8786622894CCB91B0F2A8E94BCF590F771055380D82F866B31AF79 7C460EB271C13215242EEC69B663426C6EB8DF7FEFC3BA5DF8F1190950A0B87B 57EA9B710E0E45E9C032E0C75BBDFD6F18E9B03D5406EDCE8906C5F0DA20C758 BF86980408E4E89936CCA7C687B140F480D3A781C49E6B0310132363FDE21BEB 5E4AF480D2EDE51F371C1640A77964E1E44577379015043590384EBCAAA56FF2 68334FE585F14B77259A199976FF17C5C3AA4C82D68C27C817328BE5B88E9419 D5994F43D3E0F11B7EC733A6E5D4438E832D5CD05C2BCEF578C2F4F7DADCAFC7 EF5B9AA58AD0B8F4ED6CEB4396F09DDDDC6FBA853C3DFE1C18D24A1A7AD025F2 2A4FB5F3BA21A998A7D50DB7FCB908F804983820BC31A57C090B55AD92EF5968 1641163D1735B6BA6AB5373AFD7BC1A4A3B4002CE693CBFD017299113CDCFDDC 3A4EEA6E30071AA9B9F0323555A961E7D3BA35ABEFA5CD795C68A1D7DCF2CCA2 82736A4CF7F6224B23621E107F82F3FC384980F3A6EB8A4CBF91F7CA47C39C41 D4478F9809454BF0991E3E3460B5E1F71A448C0F312BCF408B193DDDE988B1A3 225BA2E454A6A0A35A6B2B5473D2A0056E9B901A27C8AEB460AC844C7D7EBF89 D0D1022CA80B06820E28B789A80BE3CBAC627F9B061E663C32F3CA2D3CC6A90F D1E06D0CB457093E2F6ABBF3AAEF83F34A848D67D4D5C21EC74E2654A4BACB5C 7E9077784725B88F43260B78C510D3B36EF8DADD0C410177AFA116FC4C120AA9 51D0E62C4C837D4A7B9C2FBD878C6D0D77715D30EF4F653BBF77F7196384915F 33772A00C1473C1951E227EBD8E32B93973C71BB8755CE91C041697BE6F1885B D4B5CF6C59E3B93454178E65DED63056CA16FE8AA8E95D9A7219CF5FB714F9A3 FCBC8271C888BD844315EB38BB42082E727C7B2DC46BA15610353941D94BC2A5 9943FD0A39088FB01916733E2AE5AD30D159313A59FED5084363C4D61B34313E 442703D345C9473C9CE85E9AAFFE7113645531D74B85758FF79893338D97D940 ADCDB21FCD2EC6154AB58DC52455007C2A1C3AFB18DB431AFE0495412AC3903B D3D7D6CAADB7B51DA46553A5CFD99A57F54505260AE33CB87BF388C8226BC0E0 E7DF6E15CC7FA08CDA4392FBDBD1C6BA72EC43C05D003AAC5C60526F72B04A7A 827145018AAB5AFF71064CF9898A36D4F76ED1847258E400CC15C0B32D112DF2 15955D83719D0E761923BEB7F75816478694ECFBCEF6F057274CB645699BD119 D00E439713921A5ACE6170698308AF1E152C70CA52E72B9F7D9C91DAA573A010 09806985796C23A06BC8323397A4A539A80060DA25C5FFF8059E8D2C5E0FCD72 374D96250E9D2C48888D3041FF148B5194751FC247934C50157C8BA1D76E3CC2 56D365257D9C2D6860A1F00CC6027913A880D140566CA369F48EF29ECEC257EE DE42345A8AC02F5E58DFD538FC44A4001A048DA91688BAB8B3D73BD377266E91 3CDF7F9ACE4755118DB5B33D356CAAE02B3287873F956F8733E90169A1765F7C C5CE6E6E6BF564758796E2EC6688C5D1D719CA80F78007706A26DDB84EF1B5AF 2E97CCBFF6EEA04C5C754A49429FD8F6CA096F2E07B77004769CB58AA8082871 97DAD8797D41BEDE1A1969F45BF3A11E13408A488CC299FF66A96341232BC039 A7CA6E6CBD3C00BB0B9A7A67790CB994C5C5380DE5FB9F35DCDE2ECCD725D1C6 EE4B7527314C262760A52A9C14C45B4109853BE3632DE946628388CF6CB7B765 FBF297499B390DE5BA897467B634B05908669E671FB7036F19FB6910F830BE46 C0C296DD7151CD6789092030A805792D6147C189BB14B4A480C41EC2D892E216 FFDD93B8FB3780488F7A63328AFC10758D4F8B25EC95FA4E36584911023B5D6D 3C806E6D9F227790AFAFEA7B0F6B3B3FAC3FC87A9CD73CC07D5027032BDC1BA6 23FB371301AF02CCDC5FBEDB2C4C47CA467565757584BBCE960ADCC489BCC91E 458F163B8867B3D797C038220B1DB9968FE0992550E811B321910A05F742E03C 445AEEB7DD98C5DEA11EA383C07259639BFD054888527B9775EA036F3E64A173 778DD7556332FC63AE7A715CA325F530A17F9B01A7298581171861029BC6F6F4 282466EA5D721B118956AC071E94169C856AAE3C936622C9BA4C0839DDB459D5 179E49AC8A08A54F3F0DD2C2EA2D634CAB14CECDED404B0990CD5D491C6FAE91 39B5EC65678B583532F54AD4B36B3A9BB005CA9BCA352D37AFF695DC3FE1DC07 8553DEE8AC4DDF4CD20129CF5873C57A68EE1960A2F482C7189827610868D244 D68B1DA772C1EE8AB35E0753A2DC837EEEC07607AA7F0D647CF2E287D28ADE30 69D7D08AB946EF77336A6A01AF6C7F4FF9B6F02D241EFAC6C7938EE3DC9F40E4 271A526A21EDA97F8A9A76A2535E595F62A8EF45C374358667A3945EDC032250 D8EDBF3B25932C1A0A3F86E90D4BC450BF0AB2268A2C3FE7B71F014051694FB3 F3513C93C7C52A29B0BEE9789C72F49EDABCB523597BE9104351956CD88D9BC4 D7C4E90202E29B46E4A836FAEC1E0DF7CFB85A0FC20C0D4B7E48D8315FA87C5D BD54D495DADA17DBA6D5C4D16334612985DACE3FA03D7110DC5107D6B862E875 D74CA1B100771BC249F04EDF3F90F699ACF5B9C480080D191BB40C352882785F 96DEE9F99850306F0C46371A4500CC37CD62F7B3EF11D6BA71CE8580550225E8 142A80170B8DC05B1D8966570B56FFA90A73FB3A7D6B6D9E119A386E6FD70968 9551697D1DCCA58FEE6A72668C495CD18A4E759EA49F5FBF3525848A2CA63899 3EE82A0A73346F295248184B65DE9A569DBD745DE23D2D4F733DBF6A57E1CC31 4FF08DB4BA408FE66E24866880647D76DC7EC8C941C197D53DEA12C84854262B 34D1F6CEF148D23CBF8BC9DC1C92A84095F4EC23434CBE3AD7BA45ECA6B53D00 2DCED56084C6DF8B02E971B239352C2C669E26A1464426357EA75D837C0A90D5 5F0FF2FCC56C637A74C512B8330E75AFB4C23A2ADD8ADBCD434F22337BE8E785 896B9D15D75B5C5497AFFB4A43C7F28F963381C7176644DBBDE943302DF74D00 637BC91C57BD6BE0682BE277B91DB7E3A999E42F1694C0A437986DEFD7924A8D 5E09702317EEADE077D371B4D5403FDE31E8993F1FB9D329C4110F629CF5521A 5988054FFACBED7882445278F9688358D83A87CC9A6FD5E15A313678AD6810EF 0CBAC896C242E961FF06EAF15BAA011DA07A728E7DFB281C453B68BE84859BF2 C44589586504BDBB00079F98517AB0F2550C21953D74C0F72093BFF9B8A4D40F CF24FB73CD431A9BF0802C79585148F0880E573A742B6A644AC1EFD02E61E671 A6BE20D725C398E043A94A3BF0CE6901D3B1F4C6422BDC34FF5056B530EA8A1B 395CB0B2334B390E6A50739927D787E573F90E7B7EA719192F0E58D1AA6B0D8C A656CE8ACA9E10A8F8542E264C8E6DAA3D68203E3D27AC20D0EE102D3CB12FFE 0E59C1D3310BDB950132EF15F2E6F56C0D7A5EC8E7A8745053EB5DBE52D42D39 7CFC0B851C1018AB09B37F020379CBC99CF0E8417D12EEF881C4DAF614AA3AEB DA7E6914E4EA4077886A5414F8DB7F28C3E90B964278DA5D197C9DCDC4A2AAEE 20AE4C0A45583FD09440F24C0DA167DE83DF4476449EB30AAF3206FDAE2A4AB9 7CA3325CD4D055D34F3334F11710EFA1294DE0B658B3EB4AFA47C462EE1EFB01 61904DDD7751D06DDCE9549C96AD722687899D1CA44C3B931DEAEA50D35A9212 0B428AC266D2DE05FA28418F68E130C94221BED18AE00C40F88BBF8EEA8649B8 F3A62696051CAD3E37B1BDC675488DE769B424BC48E4C7C28E3394CDF705F6F2 BD3557BDC2DF59E5BD9A1B0DCFEEA67E4E076FBCEF64CFEE98A17019536BA0E8 25816550F4DB6EF1040263436F3B43AAE3C764B074865A4C132C35976C885B95 264FBD5C476D989E4D7C3CCD4A4608D38F81F4A18AB5313FFCEC953C4E885D63 0585DB94679589493BDE9EAE78183AE7FE9A2CE65769FB698621634E9F6023D2 41A5A82E0CBF3584520DC56E252C092DD7FE6D903EBD133982BB14A5A6DAEA61 C5FC54450A9A8187D869A740DF0D126BE7981185CAEA85F6AA4B72859B70A587 725AF5E76025396A8E75D1DEC0764F8245293CBBB94981DB6C8EFC3A475A1A87 FE747C3C3B4028D08299702257778AEA96759252EF9BD483E3DDFF2D12601349 295A33A1D83975988D2B47B2B6CED1434EFBDB98BDECB11335AA7E747E0D7142 5B0490F20AB05DF4C7B8ABAB51E8672FF9C4109E638DFB1CF15098FDDED03A8B E0AD9D492D52F17D7ACA704DDCD79218705DD34CB8219D4203F48E2901D1CE7A 7BC8028B6A2238F7EB345B643EE7DE5EDBEF2602E8B5FE8E1BB9575B794C7C82 9EE7C44706B4F876179A5F980E8DC5B1544282AEB1EBC7A41D4F7ADD85B29BA3 456DBD47585B0DA3A692C69D05D7C14C6E2720FDEBB9FDA1F6C92090624B883B 8476B2910CD74083AB30EE2BDCFDA63EBCD02EC0AD2DCE9C36A45BBCB56282B6 286ED7849B981B0983ACC00AE67C64C724A27AE3D784E869960DC5EBCD9D8191 24CECE58234D700193B921DB31DAAC1E0753AF033B5AF0451996CB2109002086 6B01AF78FF7E2BD625E71015180E9095047EFF612C80F4B51D5002792FC24458 CDC8BD51D98C6EA651CE8D953A0AD3FFCF1ECED26D748BB1706DB342936D8234 5025CC022E7E4AD6C3011D07FF655F898A847518BBEABDA81841CDF5D5D6900B 521B76206E8F7CB2EA4BBFF17D4AD6343992D9D94AC4F3E6FF780DC8854836A2 005A8C9542B863684C2F846EFB961F688449A8C4E2787D6053531204AD493627 16F8EA94E778DEB8B3E37ED6272E4FE53E617C754224BB439E4813516FA5E773 412D556590E4DDE11005FF9285 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39139632 55387786 1000 600 600 (qsing_perturbation_cv.dvi) @start /Fa 177[36 78[{}1 49.8132 /MSBM7 rf /Fb 183[63 72[{}1 66.4176 /CMBX8 rf /Fc 142[43 45[70 67[{}2 99.6264 /CMMI12 rf /Fd 134[59 59 1[59 62 44 44 46 1[62 56 62 93 31 2[31 62 56 34 51 62 50 62 54 9[116 3[62 6[67 6[74 1[81 80 85 11[56 56 56 56 56 2[31 37 45[{}35 99.6264 /CMBX12 rf /Fe 142[52 97[47 15[{}2 119.552 /CMMI12 rf /Ff 205[68 50[{}1 90.9091 /LASY10 rf /Fg 171[35 35[18 47[48{}3 49.8132 /CMSY6 rf /Fh 134[44 42 60 42 49 30 37 38 42 46 46 51 74 23 42 28 28 46 42 28 42 46 42 42 46 2[46 6[91 1[68 65 51 2[62 70 1[82 57 1[48 35 1[70 59 62 69 1[64 68 6[28 46 46 46 1[46 46 46 46 46 46 1[28 33 28 2[37 37 20[46 46 3[80 53 51 56 11[{}63 90.9091 /CMTI10 rf /Fi 134[55 3[58 41 41 43 1[58 52 58 87 29 55 1[29 58 1[32 48 3[51 9[108 2[73 1[78 1[71 3[63 7[80 76 1[79 6[29 2[52 52 52 52 52 52 52 7[41 41 27[58 12[{}35 90.9091 /CMBX10 rf /Fj 165[44 7[48 3[48 5[52 1[41 70[{}5 66.4176 /MSBM7 rf /Fk 140[29 6[20 33 1[22 36 17[36 12[27 13[19 32[32 26[{}9 49.8132 /CMMI6 rf /Fl 203[30 30 30 30 30 48[{}5 49.8132 /CMR6 rf /Fm 165[61 7[66 3[66 5[71 1[56 61 1[66 67[{}7 90.9091 /MSBM10 rf /Fn 149[20 21[39 10[39 22[47 71 19 14[71 11[55 55 16[35 2[55{}11 66.4176 /CMSY8 rf /Fo 167[131 7[96 52[68 68 26[{}4 90.9091 /CMEX10 rf /Fp 143[76 1[45 3[25 2[45 45 9[61 61 4[56 57 50 7[63 2[50 3[48 70 48 1[73 5[45 2[51 0 0 3[61 91 15[91 6[71 4[71 71 5[45 11[71 1[71{}29 90.9091 /CMSY10 rf /Fq 133[33 35 40 1[34 41 25 33 32 32 36 1[43 62 21 37 29 24 41 5[30 37 8[58 1[41 48 41 43 2[45 4[60 39 31 4[58 6[35 1[20 30[31 1[36 2[35 1[41 5[29 31 14[{}37 66.4176 /CMMI8 rf /Fr 129[35 64[55 8[35 35 35 35 35 4[55 1[27 27 17[35 22[{}11 66.4176 /CMR8 rf /Fs 133[42 45 52 65 44 52 33 43 41 41 46 1[55 80 27 47 37 31 52 43 45 42 1[39 39 48 7[53 75 86 53 62 53 56 2[58 2[88 1[77 50 40 1[71 58 67 75 65 69 68 48 1[71 45 71 25 25 13[25 4[59 8[54 1[40 52 47 52 40 45 1[53 5[37 40 14[{}61 90.9091 /CMMI10 rf /Ft 128[45 45 2[45 40 48 48 66 48 51 35 36 36 48 51 45 51 76 25 48 28 25 51 45 28 40 51 40 51 45 2[45 25 1[25 56 68 68 93 68 68 66 51 67 1[62 71 68 83 57 1[47 33 68 71 59 62 69 66 64 68 3[71 1[25 25 45 45 45 45 45 45 45 45 45 45 45 25 30 25 71 1[35 35 25 5[25 10[45 2[45 4[76 1[51 53 8[71 2[{}84 90.9091 /CMR10 rf /Fu 134[71 71 97 1[75 52 53 55 1[75 67 75 112 37 2[37 75 67 41 61 75 60 75 65 9[139 4[100 8[50 3[88 1[97 96 102 11[67 67 67 67 67 3[45 45[{}34 119.552 /CMBX12 rf /Fv 206[66 49[{}1 58.1154 /CMSY7 rf /Fw 142[37 45[59 67[{}2 83.022 /CMMI10 rf /Fx 134[44 44 60 44 46 32 33 33 44 46 42 46 69 23 2[23 46 42 25 37 46 37 46 42 6[51 2[85 1[62 1[46 61 1[57 2[76 52 65 43 6[60 8[23 4[42 1[42 42 42 42 1[23 28 23 24[42 19[{}44 83.022 /CMR10 rf /Fy 139[37 38 39 14[42 53 46 31[72 65[{}7 83.022 /CMBX10 rf /Fz 135[51 1[51 54 38 38 38 51 54 1[54 81 27 51 1[27 54 1[30 43 54 43 54 49 10[73 73 70 54 72 4[89 61 5[64 2[70 1[73 76 6[49 2[49 49 49 49 49 49 49 1[27 33 27 4[27 19[49 19[{}44 99.6264 /CMR12 rf /FA 142[62 113[{}1 143.462 /CMMI12 rf /FB 134[70 1[96 1[73 51 52 51 70 73 66 73 1[36 2[36 73 66 1[58 73 58 73 66 17[103 33[43 33[75 11[{}22 143.462 /CMR17 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 end %%EndSetup %%Page: 1 1 TeXDict begin 1 0 bop 152 763 a FB(On)44 b(singularly)e(p)t(erturb)t (ed)i FA(q)5 b FB(-di\013erence-di\013eren)l(tial)46 b(equations)d(with)1397 946 y(irregular)g(singularit)l(y)1623 1198 y Fz(St)m(\023)-46 b(ephane)34 b(Malek)1534 1314 y(Univ)m(ersit)m(\023)-46 b(e)35 b(de)e(Lille)g(1)1455 1431 y(UFR)f(de)h(math)m(\023)-46 b(ematiques)1140 1547 y(59655)31 b(Villeneuv)m(e)36 b(d'Ascq)e(Cedex)g(F)-8 b(rance)1206 1663 y(Stephane.Malek@math.univ-lille1.fr)1573 1779 y(T)g(el)34 b(03)e(20)g(43)g(42)g(40)1547 1983 y(Septem)m(b)s(er,) j(16)d(2009)1782 2318 y Fy(Abstract)75 2567 y Fx(W)-7 b(e)27 b(study)f(a)g Fw(q)s Fx(-analog)e(of)i(a)g(singularly)f(p)r (erturb)r(ed)h(Cauc)n(h)n(y)g(problem)f(with)i(irregular)d(singularit)n (y)h(in)h(the)h(complex)75 2667 y(domain.)57 b(W)-7 b(e)34 b(construct)g(solutions)g(of)g(this)h(problem)f(whic)n(h)g(are)f (holomorphic)g(on)h(op)r(en)h(half)f Fw(q)s Fx(-spirals.)56 b(Using)75 2767 y(a)38 b(v)n(ersion)g(of)g(a)h Fw(q)s Fx(-analog)e(of)h(the)h(Malgrange-Sibuy)n(a)d(theorem)i(obtained)h(b)n (y)f(J-P)-7 b(.)38 b(Ramis,)k(J.)c(Saulo)n(y)g(and)h(C.)75 2866 y(Zhang,)31 b(w)n(e)g(sho)n(w)f(the)i(existence)f(of)g(a)f(formal) h(p)r(o)n(w)n(er)f(series)g(solution)g(in)i(the)f(p)r(erturbation)g (parameter)f(whic)n(h)h(is)75 2966 y(the)d Fw(q)s Fx(-asymptotic)f (expansion)f(of)i(these)g(holomorphic)e(solutions.)75 3129 y(Key)k(w)n(ords:)42 b(q-Laplace)29 b(transform,)i(Whitney)g Fw(C)1726 3098 y Fv(1)1828 3129 y Fx(functions,)h(formal)e(p)r(o)n(w)n (er)g(series,)g(P)n(oincar)n(\023)-39 b(e)28 b(asymptotic)i(ex-)75 3241 y(pansions.)36 b(2000)26 b(MSC:)i(35C10,)d(35C20.)75 3727 y Fu(1)135 b(In)l(tro)t(duction)75 3930 y Ft(In)30 b(the)g(pap)s(er)g([7],)h(w)m(e)g(ha)m(v)m(e)g(studied)f(singular)g(p)s (erturbations)g(of)g(the)h(follo)m(wing)g(form)75 4134 y(\(1\))993 b Fs(\017t)1253 4097 y Fr(2)1292 4134 y Fs(@)1340 4148 y Fq(t)1370 4134 y Fs(@)1423 4097 y Fq(S)1418 4157 y(z)1474 4134 y Fs(u)p Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))27 b(=)e Fs(F)13 b Ft(\()p Fs(\017;)i(z)t(;)g(@)2232 4148 y Fq(t)2263 4134 y Fs(;)g(@)2351 4148 y Fq(z)2392 4134 y Ft(\))p Fs(u)p Ft(\()p Fs(\017;)g(t;)g(z)t Ft(\))75 4339 y(where)32 b Fs(F)45 b Ft(is)32 b(some)h(di\013eren)m(tial)g(op)s (erator)g(with)f(p)s(olynomial)g(co)s(e\016cien)m(ts.)48 b(These)32 b(equations)h(\(1\))g(b)s(elong)75 4451 y(to)d(a)h(class)f (of)g(partial)g(di\013eren)m(tial)i(equations)e(with)f(so-called)j (irregular)e(singularit)m(y)g(at)h Fs(t)25 b Ft(=)g(0,)30 b(for)g Fs(\017)25 b Fp(6)p Ft(=)g(0.)75 4564 y(W)-8 b(e)35 b(constructed)g(formal)f(p)s(o)m(w)m(er)g(series)40 b(^)-51 b Fs(u)p Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))33 b(=)1988 4496 y Fo(P)2084 4591 y Fq(k)r Fn(\025)p Fr(0)2232 4564 y Fs(h)2284 4579 y Fq(k)2327 4564 y Ft(\()p Fs(t;)15 b(z)t Ft(\))p Fs(\017)2553 4531 y Fq(k)2631 4564 y Ft(with)34 b(co)s(e\016cien)m(ts)i Fs(h)3358 4579 y Fq(k)3401 4564 y Ft(\()p Fs(t;)15 b(z)t Ft(\))35 b(whic)m(h)75 4677 y(are)e(holomorphic)g(functions)f(with)g(resp)s(ect)h(to)g Fs(t)f Ft(in)h(some)g(sector)g(cen)m(tered)h(at)f(0)g(in)f Fm(C)h Ft(and)f Fs(z)k Ft(uniformly)75 4790 y(on)26 b(a)g(small)h(disc) f(cen)m(tered)h(at)g(0.)39 b(W)-8 b(e)28 b(sho)m(w)m(ed)e(that)h(there) f(exist)h(actual)g(holomorphic)f(solutions)h Fs(u)3559 4804 y Fq(i)3587 4790 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))75 4903 y(of)30 b(\(1\),)h(ha)m(ving)k(^)-50 b Fs(u)29 b Ft(as)h(Gevrey)g(asymptotic)g(of)g(order)f(1)h(on)f(some)h (sectors)g Fs(S)2733 4917 y Fq(i)2790 4903 y Ft(with)f(resp)s(ect)h(to) g Fs(\017)p Ft(,)g(where)f Fs(S)3827 4917 y Fq(i)75 5016 y Ft(ha)m(v)m(e)34 b(op)s(ening)e(larger)h(than)f Fs(\031)j Ft(and)d(whose)g(union)g(co)m(v)m(ers)i(a)e(punctured)f(neigh)m(b)s (orho)s(o)s(d)g(of)h(the)h(origin)g(in)75 5129 y Fm(C)p Ft(.)216 5242 y(In)28 b(this)g(pap)s(er,)g(w)m(e)h(study)e(a)i Fs(q)s Ft(-analog)h(of)e(the)h(equation)g(\(1\))g(with)f(resp)s(ect)h (to)g(the)f(v)-5 b(ariable)29 b Fs(t)p Ft(,)g(namely)75 5355 y(w)m(e)i(will)g(consider)f(the)g(follo)m(wing)i Fs(q)s Ft(-di\013erence-di\013eren)m(tial)h(equations)75 5640 y(\(2\))372 b Fs(\017t@)685 5603 y Fq(S)680 5663 y(z)736 5640 y Fs(X)7 b Ft(\()p Fs(\017;)15 b(q)s(t;)g(z)t Ft(\))22 b(+)e Fs(@)1294 5603 y Fq(S)1289 5663 y(z)1345 5640 y Fs(X)7 b Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))27 b(=)1816 5527 y Fq(S)t Fn(\000)p Fr(1)1819 5554 y Fo(X)1820 5751 y Fq(k)r Fr(=0)1968 5640 y Fs(p)2014 5655 y Fq(k)2056 5640 y Ft(\()p Fs(\017;)15 b(z)t Ft(\)\()p Fs(t\033)2369 5654 y Fq(q)2409 5640 y Ft(\))2444 5603 y Fq(m)2506 5615 y Fl(0)p Fk(;k)2598 5640 y Ft(\()p Fs(@)2686 5603 y Fq(k)2681 5663 y(z)2729 5640 y Fs(X)7 b Ft(\)\()p Fs(\017;)15 b(t;)g(z)t(q)3121 5603 y Fn(\000)p Fq(m)3238 5615 y Fl(1)p Fk(;k)3332 5640 y Ft(\))p eop end %%Page: 2 2 TeXDict begin 2 1 bop 3809 100 a Ft(2)75 399 y(where)42 b Fs(q)47 b Fp(2)e Fm(C)d Ft(suc)m(h)f(that)i Fp(j)p Fs(q)s Fp(j)i Fs(>)g Ft(1,)h Fs(m)1528 414 y Fr(0)p Fq(;k)1625 399 y Fs(;)15 b(m)1745 414 y Fr(1)p Fq(;k)1885 399 y Ft(are)42 b(p)s(ositiv)m(e)i(in)m(tegers,)i Fs(p)2819 414 y Fq(k)2862 399 y Ft(\()p Fs(\017;)15 b(z)t Ft(\))43 b(are)f(p)s(olynomials)h(in)75 511 y Fs(z)j Ft(with)41 b(holomorphic)g(co)s(e\016cien)m(ts)i(in)e Fs(\017)h Ft(on)f(some)h(neigh)m(b)s(orho)s(o)s(d)e(of)h(0)h(in)f Fm(C)g Ft(and)g Fs(\033)3202 525 y Fq(q)3281 511 y Ft(is)h(the)f (dilation)75 624 y(op)s(erator)g(de\014ned)e(b)m(y)h(\()p Fs(\033)998 638 y Fq(q)1037 624 y Fs(X)7 b Ft(\)\()p Fs(\017;)15 b(t;)g(z)t Ft(\))44 b(=)e Fs(X)7 b Ft(\()p Fs(\017;)15 b(q)s(t;)g(z)t Ft(\).)72 b(As)41 b(in)f(previous)g(w)m (orks,)j(see)e([5)q(],)i([6)q(],)g(the)e(maps)75 737 y(\()p Fs(t;)15 b(z)t Ft(\))26 b Fp(7!)g Ft(\()p Fs(q)486 704 y Fq(m)548 716 y Fl(0)p Fk(;k)640 737 y Fs(t;)15 b(z)t(q)803 704 y Fn(\000)p Fq(m)920 716 y Fl(1)p Fk(;k)1012 737 y Ft(\))31 b(are)f(assumed)g(to)h(b)s(e)e(v)m(olume)i(shrinking)e (maps,)h(meaning)h(that)g(the)f(mo)s(dulus)75 850 y(of)h(the)f (Jacobian)h(determinan)m(t)g Fp(j)p Fs(q)s Fp(j)1322 817 y Fq(m)1384 829 y Fl(0)p Fk(;k)1472 817 y Fn(\000)p Fq(m)1589 829 y Fl(1)p Fk(;k)1711 850 y Ft(is)g(less)g(than)f(1,)h(for) f(all)h(0)26 b Fp(\024)f Fs(k)j Fp(\024)d Fs(S)g Fp(\000)20 b Ft(1.)216 963 y(W)-8 b(e)25 b(will)e(construct)h(actual)g (holomorphic)g(solutions)f Fs(X)7 b Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))25 b(of)f(\(2\))g(in)f(the)g(v)-5 b(ariable)24 b Fs(\017)f Ft(on)h(some)f(op)s(en)75 1076 y Fs(q)s Ft(-spirals)33 b(with)g(holomorphic)h(co)s(e\016cien)m(ts)h(on)e(a)h(domain)f(in)g Fm(C)2342 1043 y Fr(2)2414 1076 y Ft(whic)m(h)h(is)f(a)h(pro)s(duct)e (of)h(a)h(half)f Fs(q)s Ft(-spiral)75 1189 y(and)f(a)i(disc,)g(for)f (giv)m(en)h(initial)g(conditions)f(\()p Fs(@)1716 1145 y Fq(j)1711 1201 y(z)1754 1189 y Fs(X)7 b Ft(\)\()p Fs(\017;)15 b(t;)g Ft(0\),)36 b(0)30 b Fp(\024)g Fs(j)35 b Fp(\024)29 b Fs(S)e Fp(\000)22 b Ft(1.)49 b(Our)32 b(goal)i(is)g(to)f(in)m(v)m (estigate)75 1302 y(the)e(asymptotic)g(of)g(these)g(solutions)f(with)g (resp)s(ect)h(to)g Fs(\017)f Ft(near)g(the)h(origin)g(on)f(the)h(giv)m (en)g(op)s(en)f Fs(q)s Ft(-spirals.)216 1415 y(The)f(metho)s(dology)h (of)f(pro)s(of)g(will)g(b)s(e)g(the)g(same)h(as)f(the)g(one)h(in)m(tro) s(duced)e(b)m(y)i(M.)f(Canalis-Durand,)h(J.)75 1528 y(Mozo-F)-8 b(ernandez)25 b(and)c(R.)h(Sc)m(h\177)-45 b(afk)m(e)23 b(in)f(the)g(pap)s(er)f([1)q(].)38 b(The)22 b(idea)g(consists)h(in)f (transforming)f(the)h(singularly)75 1641 y(p)s(erturb)s(ed)34 b(equation)j(\(2\))g(using)f(the)h(linear)g(map)f Fs(t)f Fp(7!)g Fs(t=\017)h Ft(in)m(to)i(an)e(auxiliary)h(regularly)g(p)s (erturb)s(ed)c Fs(q)s Ft(-)75 1753 y(di\013erence)28 b(di\013eren)m(tial)g(equation)h(whic)m(h)e(has)g(a)g(so-called)j (irregular)d(singularit)m(y)h(at)g Fs(t)d Ft(=)g(0,)k(see)f(\(47\).)41 b(The)75 1866 y(approac)m(h)31 b(w)m(e)g(will)f(follo)m(w)i(is)e(based) g(on)h(a)f(resummation)h(pro)s(cess)e(of)i(formal)g(p)s(o)m(w)m(er)f (series)1492 2056 y(^)1478 2079 y Fs(Y)20 b Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))27 b(=)1940 1992 y Fo(X)1950 2190 y Fq(l)q Fn(\025)p Fr(0)2086 2079 y Fs(W)2172 2094 y Fq(l)2198 2079 y Ft(\()p Fs(\017;)15 b(z)t Ft(\))p Fs(t)2424 2041 y Fq(l)75 2372 y Ft(with)35 b(resp)s(ect)g(to)h(the)f(v)-5 b(ariable)36 b Fs(t)p Ft(,)g(whic)m(h)f(are)g(solutions)h(of)f(the)g (constructed)g(auxiliary)h(problem.)54 b(This)75 2485 y(resummation)31 b(metho)s(d)f(w)m(as)i(in)m(tro)s(duced)e(b)m(y)h(J-P) -8 b(.)32 b(Ramis)f(and)f(C.)h(Zhang)g(in)f(the)i(pap)s(ers)d([11)r(],) i([12)q(],)h(and)75 2598 y(is)e(a)g Fs(q)s Ft(-analog)i(of)e(the)g (classical)i(Borel-Laplace)g(summabilit)m(y)f(pro)s(cedure.)39 b(This)29 b(metho)s(d)g(has)h(b)s(een)f(used)75 2711 y(with)34 b(success)h(to)h(study)d(the)i(asymptotic)h(prop)s(erties)e (of)h(analytic)h(solutions)f(to)h Fs(q)s Ft(-di\013erence)f(equations) 75 2824 y(with)30 b(irregular)h(singularit)m(y)-8 b(,)31 b(see)g([2)q(],)g([3)q(].)216 2937 y(W)-8 b(e)32 b(study)d(a)i Fs(q)s Ft(-Borel)h(transform)d(of)1550 2914 y(^)1536 2937 y Fs(Y)20 b Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))32 b(with)e(resp)s(ect)g(to)h Fs(t)p Ft(,)1352 3176 y(^)1325 3199 y Fs(W)13 b Ft(\()p Fs(\017;)i(\034)5 b(;)15 b(z)t Ft(\))27 b(=)1825 3113 y Fo(X)1834 3311 y Fq(l)q Fn(\025)p Fr(0)1971 3199 y Fs(W)2057 3214 y Fq(l)2083 3199 y Ft(\()p Fs(\017;)15 b(z)t Ft(\))2403 3138 y Fs(\034)2453 3105 y Fq(l)p 2286 3178 308 4 v 2286 3266 a Fs(q)2330 3239 y Fq(l)q Fr(\()p Fq(l)q Fn(\000)p Fr(1\))p Fq(=)p Fr(2)75 3493 y Ft(whic)m(h)31 b(satis\014es)h(a)f(regular)g(di\013eren)m(tial)i (equation)f(with)f(con)m(traction)i(in)e(the)g(v)-5 b(ariable)32 b Fs(z)t Ft(,)g(rational)h(co)s(e\016-)75 3606 y(cien)m(ts)28 b(in)f Fs(\034)10 b Ft(,)28 b(and)f(holomorphic)g(in)f(\()p Fs(\017;)15 b(z)t Ft(\))29 b(near)e(the)g(origin)h(in)e Fm(C)2323 3573 y Fr(2)2363 3606 y Ft(,)h(see)h(\(20\).)41 b(Under)27 b(suitable)g(initial)i(data,)75 3719 y(w)m(e)h(sho)m(w)f (that)655 3696 y(^)628 3719 y Fs(W)42 b Ft(con)m(v)m(erges)32 b(near)d(the)h(origin)g(in)f Fm(C)1948 3686 y Fr(3)2016 3719 y Ft(\(Theorem)h(3\))g(and)f(can)h(b)s(e)e(analytically)k (extended)75 3832 y(to)e(functions)f Fs(W)662 3846 y Fq(i)690 3832 y Ft(\()p Fs(\017;)15 b(\034)5 b(;)15 b(z)t Ft(\))31 b(de\014ned)d(on)h(pro)s(ducts)f Fs(U)1874 3846 y Fq(i)1902 3832 y Fs(q)1946 3799 y Fn(\000)p Fj(N)2071 3832 y Fp(\002)17 b Fs(V)2212 3846 y Fq(i)2241 3832 y Fs(q)2285 3799 y Fj(N)2354 3832 y Fp(\002)h Fs(D)s Ft(\(0)p Fs(;)d(\032)p Ft(\))31 b(where)d Fp(f)p Fs(U)3122 3846 y Fq(i)3151 3832 y Fs(q)3195 3799 y Fn(\000)p Fj(N)3302 3832 y Fp(g)3347 3846 y Fq(i)p Fn(2)p Fq(I)3488 3832 y Ft(is)h(a)h(\014nite)75 3945 y(set)25 b(of)f(op)s(en)g(half)g Fs(q)s Ft(-spirals)g(whose)g(union)f(forms)h(a)g(so-called)i(go)s(o)s (d)e(co)m(v)m(ering)i(\(in)e(the)h(terminology)g(of)g([10)q(]\))75 4058 y(and)g Fs(V)300 4072 y Fq(i)328 4058 y Ft(,)i Fs(i)f Fp(2)f Fs(I)7 b Ft(,)27 b(are)f(w)m(ell)h(c)m(hosen)f(op)s(en)f(sets)i (of)f Fm(C)1789 4025 y Fn(\003)1828 4058 y Ft(.)39 b(The)25 b(functions)h Fs(W)2548 4072 y Fq(i)2576 4058 y Ft(,)h Fs(i)e Fp(2)g Fs(I)7 b Ft(,)27 b(ha)m(v)m(e)g(also)g(a)f Fs(q)s Ft(-exp)s(onen)m(tial)75 4171 y(gro)m(wth)31 b(with)f(resp)s (ect)g(to)h Fs(\034)10 b Ft(,)31 b(meaning)g(that)g(there)f(exist)h (constan)m(ts)h Fs(C)q(;)15 b(K)33 b(>)25 b Ft(0)30 b(suc)m(h)g(that) 1364 4374 y(sup)1202 4457 y Fq(x)p Fn(2)p Fq(V)1330 4467 y Fk(i)1356 4457 y Fq(;z)s Fn(2)p Fq(D)r Fr(\(0)p Fq(;\032)p Fr(\))1679 4374 y Fp(j)p Fs(W)1790 4388 y Fq(i)1818 4374 y Ft(\()p Fs(\017;)15 b(xq)2026 4337 y Fq(l)2053 4374 y Fs(;)g(z)t Ft(\))p Fp(j)26 b(\024)f Fs(C)7 b Fp(j)p Fs(q)s Fp(j)2487 4337 y Fq(l)2509 4313 y Fl(2)2544 4337 y Fq(=)p Fr(2)2618 4374 y Fs(K)2702 4337 y Fq(l)75 4661 y Ft(for)46 b(all)g Fs(l)53 b Fp(\025)e Ft(0,)g(all)46 b Fs(\017)51 b Fp(2)g Fs(U)1097 4675 y Fq(i)1125 4661 y Fs(q)1169 4628 y Fn(\000)p Fj(N)1276 4661 y Ft(,)f Fs(i)h Fp(2)g Fs(I)i Ft(\(Theorem)46 b(2\).)88 b(T)-8 b(o)46 b(get)h(these)g(estimates,)k(w)m(e)c(reduce)e(the)75 4773 y(problem)38 b(to)h(the)g(study)f(of)g(the)h(gro)m(wth)g (estimates)h(of)f(the)g(co)s(e\016cien)m(ts)h(of)f(a)g(formal)g(series) 3479 4750 y(^)3465 4773 y Fs(V)20 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))40 b(=)75 4818 y Fo(P)171 4913 y Fq(l)q(;h)p Fn(\025)p Fr(0)363 4886 y Fs(v)407 4901 y Fq(l)q(;h)493 4886 y Fs(\030)537 4853 y Fq(l)563 4886 y Fs(x)615 4853 y Fq(h)660 4886 y Fs(=h)p Ft(!)26 b Fp(2)f Fm(C)p Ft([[)p Fs(\030)t(;)15 b(x)p Ft(]],)31 b(solution)f(of)g(a)h(di\013eren)m(tial) g(equation)g(with)e(con)m(traction)j(in)e(the)g(v)-5 b(ariable)75 4999 y Fs(x)p Ft(,)26 b(dilation)f(in)f Fs(\030)k Ft(and)23 b(p)s(olynomial)i(co)s(e\016cien)m(ts)h(in)e Fs(x)p Ft(,)h(see)g(\(17\).)40 b(W)-8 b(e)26 b(construct)e(the)h (formal)f(series)h(solution)75 5112 y(in)32 b(a)h(Banac)m(h)h(space)f (of)f(formal)h(series)1487 5089 y(^)1472 5112 y Fs(U)10 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))29 b(=)1879 5044 y Fo(P)1975 5139 y Fq(l)q(;h)p Fn(\025)p Fr(0)2166 5112 y Fs(u)2218 5127 y Fq(l)q(;h)2305 5112 y Fs(\030)2349 5079 y Fq(l)2375 5112 y Fs(x)2427 5079 y Fq(h)2471 5112 y Fs(=h)p Ft(!)34 b(for)e(whic)m(h)g(there)h(exist)g(constan)m(ts)75 5225 y Fs(C)q(;)15 b(K)258 5239 y Fr(1)298 5225 y Fs(;)g(K)415 5239 y Fr(2)481 5225 y Fs(>)25 b Ft(0)30 b(suc)m(h)h(that)1420 5348 y Fp(j)p Fs(u)1497 5363 y Fq(l)q(;h)1583 5348 y Fp(j)26 b(\024)f Fs(C)7 b Fp(j)p Fs(q)s Fp(j)1906 5283 y Fl(1)p 1906 5295 31 3 v 1906 5336 a(2)1946 5310 y Fq(l)1968 5287 y Fl(2)2002 5310 y Fn(\000)2067 5283 y Fl(1)p 2067 5295 V 2067 5336 a(4)2108 5310 y Fq(h)2149 5287 y Fl(2)2187 5348 y Fs(K)2271 5310 y Fq(l)2264 5370 y Fr(1)2304 5348 y Fs(K)2388 5310 y Fq(h)2381 5370 y Fr(2)2432 5348 y Fs(h)p Ft(!)75 5514 y(for)30 b(all)h Fs(l)r(;)15 b(h)26 b Fp(\025)f Ft(0)31 b(\(Theorem)g(1\).)216 5627 y(W)-8 b(e)33 b(construct)f(actual)h(solutions)f Fs(Y)1483 5641 y Fq(i)1511 5627 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\),)34 b Fs(i)27 b Fp(2)g Fs(I)7 b Ft(,)33 b(of)f(the)f(auxiliary)i(equation)f (\(47\))h(as)f(a)g(discret)g Fs(q)s Ft(-)75 5740 y(Laplace)25 b(transform)d(of)i(the)f(functions)g Fs(W)1530 5754 y Fq(i)1558 5740 y Ft(\()p Fs(\017;)15 b(\034)5 b(;)15 b(z)t Ft(\),)27 b(with)c(resp)s(ect)g(to)h Fs(\034)10 b Ft(,)25 b(in)e(directions)h Fs(\025)3156 5754 y Fq(i)3184 5740 y Fs(q)3228 5707 y Fj(Z)3300 5740 y Ft(where)f Fs(\025)3609 5754 y Fq(i)3662 5740 y Fp(2)i Fs(V)3801 5754 y Fq(i)3829 5740 y Ft(.)p eop end %%Page: 3 3 TeXDict begin 3 2 bop 3809 100 a Ft(3)75 399 y(These)28 b(functions)g Fs(Y)779 413 y Fq(i)807 399 y Ft(,)h Fs(i)d Fp(2)f Fs(I)7 b Ft(,)29 b(de\014ne)f(holomorphic)h(functions)f(on)g (domains)g Fs(U)2818 413 y Fq(i)2847 399 y Fs(q)2891 366 y Fn(\000)p Fj(N)3014 399 y Fp(\002)16 b Fs(T)3154 413 y Fq(i)3199 399 y Fp(\002)g Fs(D)s Ft(\(0)p Fs(;)f(\032)p Ft(\),)31 b(where)75 511 y Fs(T)128 525 y Fq(i)187 511 y Ft(are)f(complemen)m(ts)i(of)e(closed)i Fs(q)s Ft(-spirals)e(in)g (some)h(neigh)m(b)s(orho)s(o)s(d)e(of)h(0)h(\(Theorem)f(4\).)216 624 y(W)-8 b(e)23 b(observ)m(e)f(that)g(the)g(functions)f Fs(X)1473 638 y Fq(i)1501 624 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))27 b(:=)e Fs(Y)1968 638 y Fq(i)1996 624 y Ft(\()p Fs(\017;)15 b(\017t;)g(z)t Ft(\),)25 b Fs(i)g Fp(2)g Fs(I)7 b Ft(,)24 b(are)e(actual)g(holomorphic)g(solutions)75 737 y(of)g(the)h(initial)g(singularly)g(p)s(erturb)s(ed)c(equation)k (\(2\))h(on)e(domains)g(of)g(the)h(form)f Fs(U)2883 751 y Fq(i)2911 737 y Fs(q)2955 704 y Fn(\000)p Fj(N)3066 737 y Fp(\002)t(T)k(\002)t Fs(D)s Ft(\(0)p Fs(;)15 b(\032)p Ft(\),)25 b(where)75 850 y Fp(T)56 b Ft(is)33 b(some)h(w)m(ell)g(c)m (hosen)g(op)s(en)e(half)h Fs(q)s Ft(-spiral)h(in)f(a)g(small)h(neigh)m (b)s(orho)s(o)s(d)e(of)h(0.)50 b(F)-8 b(rom)34 b(the)f(fact)h(that)g (the)75 963 y(functions)c Fs(Y)520 977 y Fq(i)548 963 y Ft(,)h Fs(i)26 b Fp(2)f Fs(I)7 b Ft(,)31 b(all)g(ha)m(v)m(e)h(the)f (formal)f(series)1886 940 y(^)1872 963 y Fs(Y)51 b Ft(as)31 b Fs(q)s Ft(-Gevrey)g(asymptotic)h(expansion)e(with)g(resp)s(ect)75 1076 y(to)37 b Fs(t)f Ft(\(Theorem)g(4\),)j(w)m(e)e(get)g(that)g(the)g (functions)e Fs(G)1961 1090 y Fq(i;j)2042 1076 y Ft(\()p Fs(\017)p Ft(\))h(:=)f Fs(X)2391 1090 y Fq(j)2428 1076 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))25 b Fp(\000)f Fs(X)2889 1090 y Fq(i)2918 1076 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))37 b(ha)m(v)m(e)h(the)e(formal)75 1189 y(series)315 1169 y(^)315 1189 y(0)27 b(as)g Fs(q)s Ft(-asymptotic)h (expansion)e(\(in)h(the)f(P)m(oincar)m(\023)-43 b(e)29 b(sense\))e(on)g(the)f(in)m(tersections)i Fs(U)3232 1203 y Fq(j)3269 1189 y Fs(q)3313 1156 y Fn(\000)p Fj(N)3432 1189 y Fp(\\)13 b Fs(U)3568 1203 y Fq(i)3596 1189 y Fs(q)3640 1156 y Fn(\000)p Fj(N)3773 1189 y Ft(as)75 1302 y Fm(H)146 1316 y Fn(T)k Fq(;\032)262 1302 y Ft(-v)-5 b(alued)28 b(functions,)g(where)f Fm(H)1321 1316 y Fn(T)17 b Fq(;\032)1464 1302 y Ft(denotes)28 b(the)g(Banac)m(h)h(space)f(of)g(holomorphic)f(b)s (ounded)f(functions)75 1415 y(on)g Fp(T)36 b(\002)13 b Fs(D)s Ft(\(0)p Fs(;)i(\032)p Ft(\))27 b(equipp)s(ed)e(with)h(the)h (suprem)m(um)e(norm.)39 b(The)26 b(tuple)g(of)h(functions)f(\()p Fs(G)3131 1429 y Fq(i;j)3212 1415 y Ft(\()p Fs(\017)p Ft(\)\))3354 1434 y Fr(\()p Fq(i;j)t Fr(\))p Fn(2)p Fq(I)3568 1415 y Fl(2)36 b Ft(forms)75 1528 y(what)41 b(is)h(called)g(a)g Fm(H)842 1542 y Fn(T)17 b Fq(;\032)958 1528 y Ft(-v)-5 b(alued)42 b(co)s(cyle)g(\(Prop)s(osition)g(4\).)75 b(F)-8 b(ollo)m(wing)43 b(the)f(approac)m(h)g(of)f(J.P)-8 b(.)42 b(Ramis,)75 1641 y(J.)35 b(Saulo)m(y)h(and)f(C.)g(Zhang)g(in)g([10)q (],)i(w)m(e)f(sho)m(w)g(that)g(this)f(co)s(cycle)i(splits)e(in)g(the)h (space)g(of)g Fs(C)3402 1608 y Fn(1)3511 1641 y Ft(Whitney)75 1753 y(functions)30 b(on)h(closed)h(sets.)42 b(More)32 b(precisely)-8 b(,)32 b(using)e(the)h(Whitney)g(extension)h(theorem)f (\(see)h([8],)g([9]\),)g(w)m(e)75 1866 y(construct)d(a)f(family)h(of)g Fs(C)996 1833 y Fn(1)1098 1866 y Ft(functions)f Fs(f)1533 1880 y Fq(i)1589 1866 y Ft(in)g(the)g(sense)g(of)h(Whitney)g(on)f (closed)h(sets)g Fs(K)3195 1880 y Fq(i)3223 1866 y Fs(q)3267 1833 y Fn(\000)p Fj(N)3390 1866 y Fp([)16 b(f)p Ft(0)p Fp(g)26 b(\032)f Fm(R)3790 1833 y Fr(2)3829 1866 y Ft(,)75 1979 y(where)c Fs(K)406 1993 y Fq(i)455 1979 y Ft(are)g(compact)i(sets) e(of)g Fs(U)1276 1993 y Fq(i)1325 1979 y Ft(suc)m(h)g(that)h Fs(G)1780 1993 y Fq(i;j)1860 1979 y Ft(\()p Fs(\017)p Ft(\))k(=)f Fs(f)2134 1993 y Fq(j)2170 1979 y Ft(\()p Fs(\017)p Ft(\))q Fp(\000)q Fs(f)2395 1993 y Fq(i)2424 1979 y Ft(\()p Fs(\017)p Ft(\),)f(for)c(all)i Fs(\017)j Fp(2)g Ft(\()p Fs(K)3086 1993 y Fq(i)3115 1979 y Fs(q)3159 1946 y Fn(\000)p Fj(N)3267 1979 y Fp(\\)q Fs(K)3406 1993 y Fq(j)3443 1979 y Fs(q)3487 1946 y Fn(\000)p Fj(N)3594 1979 y Ft(\))q Fp([)q(f)p Ft(0)p Fp(g)p Ft(,)75 2092 y(all)31 b(\()p Fs(i;)15 b(j)5 b Ft(\))28 b Fp(2)c Fs(I)544 2059 y Fr(2)614 2092 y Ft(\(Prop)s(osition)31 b(5\).)216 2205 y(Solving)j(a)630 2181 y(\026)619 2205 y Fs(@)5 b Ft(-problem)33 b(for)g Fm(H)1272 2219 y Fn(T)17 b Fq(;\032)1388 2205 y Ft(-v)-5 b(alued)34 b Fs(C)1779 2172 y Fn(1)1886 2205 y Ft(functions)f(on)g(op)s(en)g(sets)h(in)f Fm(R)2989 2172 y Fr(2)3028 2205 y Ft(,)h(w)m(e)g(sho)m(w)f(that)h(there)75 2318 y(exists)d(a)g Fs(C)474 2285 y Fn(1)579 2318 y Ft(function)f Fs(u)h Ft(on)f(a)h(neigh)m(b)s(orho)s(o)s(d)e(of)i(the)g(origin)g(in)f Fm(R)2482 2285 y Fr(2)2552 2318 y Ft(suc)m(h)g(that)h Fs(X)3029 2332 y Fq(i)3084 2318 y Ft(=)25 b Fs(f)3225 2332 y Fq(i)3273 2318 y Ft(+)20 b Fs(u)31 b Ft(on)f Fs(K)3650 2332 y Fq(i)3678 2318 y Fs(q)3722 2285 y Fn(\000)p Fj(N)3829 2318 y Ft(,)75 2431 y Fs(i)c Fp(2)e Fs(I)37 b Ft(\(Prop)s(osition)29 b(6\).)41 b(Finally)-8 b(,)32 b(w)m(e)d(deduce)g(the)h(main)f(result)g (of)g(this)h(pap)s(er)e(whic)m(h)h(is)g(the)h(existence)g(of)75 2544 y(a)h(formal)f(series)1603 2634 y(^)1577 2657 y Fs(X)7 b Ft(\()p Fs(\017)p Ft(\))26 b(=)1888 2570 y Fo(X)1889 2768 y Fq(k)r Fn(\025)p Fr(0)2034 2657 y Fs(X)2109 2672 y Fq(k)2152 2657 y Fs(\017)2189 2619 y Fq(k)2232 2657 y Fs(=k)s Ft(!)75 2929 y(where)i Fs(X)411 2944 y Fq(k)479 2929 y Fp(2)d Fm(H)636 2943 y Fn(T)17 b Fq(;\032)752 2929 y Ft(,)29 b Fs(k)f Fp(\025)d Ft(0,)30 b(whic)m(h)e(is)h(the)f Fs(q)s Ft(-asymptotic)i(expansion)f(of)f Fs(X)2716 2943 y Fq(i)2773 2929 y Ft(on)g Fs(K)2974 2943 y Fq(i)3003 2929 y Fs(q)3047 2896 y Fn(\000)p Fj(N)3154 2929 y Ft(,)h Fs(i)c Fp(2)g Fs(I)7 b Ft(,)29 b(and)f(solv)m(es)75 3042 y(the)j(equation)g(\(2\))g(\(Theorem)g(5\).)75 3205 y(The)f(pap)s(er)f (is)i(organized)g(as)g(follo)m(ws.)75 3318 y(In)49 b(section)h(2,)55 b(w)m(e)50 b(in)m(tro)s(duce)g(Banac)m(h)g(spaces)g(of)g(formal)g(p)s (o)m(w)m(er)f(series)h(and)f(w)m(e)h(solv)m(e)h(a)f(con)m(trac-)75 3431 y(tion/dilation)32 b(di\013eren)m(tial)g(equation)f(with)f(p)s (olynomial)h(co)s(e\016cien)m(ts)h(in)e(these)h(spaces.)75 3544 y(In)i(section)j(3,)f(w)m(e)g(solv)m(e)g(a)g(parameter)g(dep)s (ending)d(di\013eren)m(tial)k(equation)f(with)f(rational)h(co)s (e\016cien)m(ts)h(in)75 3657 y(analytic)c(spaces)f(of)f Fs(q)s Ft(-exp)s(onen)m(tial)i(gro)m(wing)f(functions)f(on)g(op)s(en)g (half)g Fs(q)s Ft(-spirals.)75 3769 y(In)e(section)j(4.1,)f(w)m(e)g (recall)g(the)g(de\014nition)f(and)f(prop)s(erties)h(of)g(the)g Fs(q)s Ft(-Laplace)i(transform)d(and)h(the)g(de\014ni-)75 3882 y(tion)i(of)f(the)h Fs(q)s Ft(-asymptotic)h(expansion.)75 3995 y(In)k(section)i(4.2,)i(w)m(e)d(construct)g(actual)h(solutions)f Fs(Y)1949 4009 y Fq(i)1977 3995 y Ft(,)i Fs(i)d Fp(2)f Fs(I)7 b Ft(,)39 b(of)e(an)f(auxiliary)i Fs(q)s Ft(-di\013erence)f (di\013eren)m(tial)75 4108 y(equation)31 b(with)f(irregular)h (singularit)m(y)g(\(47\))h(and)e(w)m(e)h(giv)m(e)g(their)g(formal)f (asymptotic)i(expansion)3569 4085 y(^)3555 4108 y Fs(Y)20 b Ft(.)75 4221 y(In)30 b(section)h(4.3,)h(w)m(e)f(construct)f(actual)i (solutions)f Fs(X)1925 4235 y Fq(i)1953 4221 y Ft(,)g Fs(i)26 b Fp(2)e Fs(I)7 b Ft(,)31 b(of)g(our)f(initial)h(equation)g (\(2\).)75 4334 y(In)f(section)h(5.1,)h(w)m(e)f(build)e(a)i Fm(H)1177 4348 y Fn(T)17 b Fq(;\032)1293 4334 y Ft(-v)-5 b(alued)30 b(co)s(cyle)i(with)e(the)h(help)f(of)g(the)h(functions)f Fs(X)3169 4348 y Fq(i)3197 4334 y Ft(,)h Fs(i)25 b Fp(2)g Fs(I)7 b Ft(.)75 4447 y(In)30 b(section)h(5.2,)h(w)m(e)f(sho)m(w)f (that)h(this)f(latter)i(co)s(cyle)g(splits)e(in)g(the)h(space)g(of)f (Whitney)h Fs(C)3211 4414 y Fn(1)3315 4447 y Ft(functions.)75 4560 y(In)k(section)i(5.3,)i(w)m(e)e(state)g(the)f(main)g(result)g(of)g (this)g(pap)s(er,)g(namely)h(the)f(existence)h(of)g(a)f(formal)g (series)101 4650 y(^)75 4673 y Fs(X)7 b Ft(\()p Fs(\017)p Ft(\))26 b Fp(2)f Fm(H)447 4687 y Fn(T)17 b Fq(;\032)563 4673 y Ft([[)p Fs(\017)p Ft(]])31 b(solving)g(\(2\))h(whic)m(h)e(is)g (the)h Fs(q)s Ft(-asymptotic)h(expansion)e(of)g(the)h(functions)f Fs(X)3383 4687 y Fq(i)3411 4673 y Ft(,)h Fs(i)25 b Fp(2)g Fs(I)7 b Ft(.)75 4959 y Fu(2)135 b(W)-11 b(eigh)l(ted)46 b(Banac)l(h)e(space)h(of)h(formal)g(p)t(o)l(w)l(er)f(series)75 5162 y Fi(De\014nition)35 b(1)46 b Fh(We)32 b(c)-5 b(onsider)34 b(the)f(ve)-5 b(ctor)33 b(sp)-5 b(ac)g(e)34 b Fm(E)1902 5181 y Fq(q)r(;)p Fr(\()p Fq(T)6 b(;X)f Fr(\))2177 5162 y Fh(of)33 b(formal)h(p)-5 b(ower)34 b(series)75 5437 y Ft(\(3\))1070 b Fs(V)20 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))26 b(=)1727 5351 y Fo(X)1661 5549 y Fq(l)q Fn(\025)p Fr(0)p Fq(;h)p Fn(\025)p Fr(0)1939 5437 y Fs(v)1983 5452 y Fq(l)q(;h)2069 5437 y Fs(\030)2113 5400 y Fq(l)2149 5376 y Fs(x)2201 5343 y Fq(h)p 2149 5417 97 4 v 2159 5500 a Fs(h)p Ft(!)2281 5437 y Fp(2)f Fm(C)p Ft([[)p Fs(\030)t(;)15 b(x)p Ft(]])p eop end %%Page: 4 4 TeXDict begin 4 3 bop 3809 100 a Ft(4)75 399 y Fh(such)33 b(that)1100 547 y Fp(jj)p Fs(V)21 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))p Fp(jj)1480 566 y Fr(\()p Fq(T)6 b(;X)f Fr(\))1695 547 y Ft(:=)1882 461 y Fo(X)1816 659 y Fq(l)q Fn(\025)p Fr(0)p Fq(;h)p Fn(\025)p Fr(0)2133 484 y Fp(j)p Fs(v)2202 499 y Fq(l)q(;h)2289 484 y Fp(j)p 2104 527 240 4 v 2104 614 a Fs(q)2148 587 y Fq(P)10 b Fr(\()p Fq(l)q(;h)p Fr(\))2353 547 y Fs(T)2419 510 y Fq(l)2455 486 y Fs(X)2537 453 y Fq(h)p 2455 527 128 4 v 2480 610 a Fs(h)p Ft(!)2618 547 y Fs(<)25 b Fp(1)p Fs(;)75 786 y Fh(wher)-5 b(e)34 b Fs(T)8 b(;)15 b(X)33 b(>)24 b Ft(0)p Fh(,)33 b Fs(q)28 b(>)d Ft(1)33 b Fh(ar)-5 b(e)34 b(p)-5 b(ositive)33 b(r)-5 b(e)g(al)35 b(numb)-5 b(ers)33 b(and)g(wher)-5 b(e)1443 983 y Fs(P)13 b Ft(\()p Fs(l)r(;)i(h)p Ft(\))26 b(=)1837 922 y(1)p 1837 962 46 4 v 1837 1046 a(4)1893 983 y Fs(l)1922 946 y Fr(2)1981 983 y Ft(+)2082 922 y(1)p 2082 962 V 2082 1046 a(2)2137 983 y Fs(l)r(h)21 b Fp(\000)2340 922 y Ft(1)p 2340 962 V 2340 1046 a(2)2395 983 y Fs(h)2447 946 y Fr(2)75 1173 y Fh(for)33 b(al)5 b(l)33 b Fs(l)r(;)15 b(h)26 b Fp(\025)f Ft(0)p Fh(.)42 b(The)33 b(sp)-5 b(ac)g(e)34 b Ft(\()p Fm(E)1227 1192 y Fq(q)r(;)p Fr(\()p Fq(T)6 b(;X)f Fr(\))1469 1173 y Fs(;)15 b Fp(jj)p Fs(:)p Fp(jj)1634 1192 y Fr(\()p Fq(T)6 b(;X)f Fr(\))1825 1173 y Ft(\))33 b Fh(is)f(a)h(Banach)h(sp)-5 b(ac)g(e.)75 1346 y Fi(Remark:)36 b Ft(Notice)22 b(that)f(w)m(e)g(ha)m(v)m(e)g(a)g(con)m(tin)m(uous)g (inclusion)f(\()p Fm(E)2263 1364 y Fq(q)r(;)p Fr(\()p Fq(T)6 b(;X)f Fr(\))2505 1346 y Fs(;)15 b Fp(jj)p Fs(:)p Fp(jj)2670 1364 y Fr(\()p Fq(T)6 b(;X)f Fr(\))2861 1346 y Ft(\))25 b Fs(,)-15 b Fp(!)26 b Ft(\()p Fm(E)3144 1364 y Fq(q)r(;)p Fr(\()p Fq(T)3276 1346 y Fg(0)3298 1364 y Fq(;X)3381 1346 y Fg(0)3403 1364 y Fr(\))3435 1346 y Fs(;)15 b Fp(jj)p Fs(:)p Fp(jj)3600 1364 y Fr(\()p Fq(T)3678 1346 y Fg(0)3702 1364 y Fq(;X)3785 1346 y Fg(0)3807 1364 y Fr(\))3839 1346 y Ft(\))75 1459 y(when)29 b(0)d Fs(<)f(X)561 1426 y Fn(0)610 1459 y Fp(\024)g Fs(X)7 b Ft(,)31 b(0)26 b Fs(<)f(T)1077 1426 y Fn(0)1125 1459 y Fp(\024)g Fs(T)13 b Ft(.)75 1608 y(W)-8 b(e)32 b(consider)e(the)g(in) m(tegration)j(op)s(erator)d Fs(@)1624 1575 y Fn(\000)p Fr(1)1619 1631 y Fq(x)1749 1608 y Ft(de\014ned)g(on)g Fm(C)p Ft([[)p Fs(\030)t(;)15 b(x)p Ft(]])31 b(b)m(y)1093 1831 y Fs(@)1146 1794 y Fn(\000)p Fr(1)1141 1854 y Fq(x)1240 1831 y Ft(\()p Fs(V)21 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\)\))27 b(:=)1803 1745 y Fo(X)1738 1942 y Fq(l)q Fn(\025)p Fr(0)p Fq(;h)p Fn(\025)p Fr(1)2015 1831 y Fs(v)2059 1846 y Fq(l)q(;h)p Fn(\000)p Fr(1)2236 1831 y Fs(\030)2280 1794 y Fq(l)2316 1770 y Fs(x)2368 1737 y Fq(h)p 2316 1810 97 4 v 2326 1894 a Fs(h)p Ft(!)2448 1831 y Fp(2)e Fm(C)p Ft([[)p Fs(\030)t(;)15 b(x)p Ft(]])75 2095 y Fi(Lemma)35 b(1)45 b Fh(L)-5 b(et)40 b Fs(s;)15 b(h)842 2109 y Fr(1)882 2095 y Fs(;)g(h)974 2109 y Fr(2)1051 2095 y Fp(\025)37 b Ft(0)i Fh(b)-5 b(e)39 b(p)-5 b(ositive)41 b(inte)-5 b(gers.)61 b(L)-5 b(et)40 b Fs(T)8 b(;)15 b(X)45 b(>)36 b Ft(0)p Fh(.)62 b(Assume)39 b(that)i(the)e(ine)-5 b(qualities)75 2208 y(hold)75 2373 y Ft(\(4\))1182 b Fs(s)20 b Ft(+)g Fs(h)1578 2387 y Fr(2)1643 2373 y Fp(\025)25 b Ft(2)p Fs(h)1836 2387 y Fr(1)1941 2373 y Fs(;)80 b(m)2126 2387 y Fr(1)2191 2373 y Fp(\025)25 b Fs(s)20 b Ft(+)g Fs(h)2493 2387 y Fr(2)2532 2373 y Fs(:)75 2537 y Fh(Then,)33 b(ther)-5 b(e)33 b(exists)g Fs(C)f(>)25 b Ft(0)33 b Fh(\(dep)-5 b(ending)34 b Fs(q)s Fh(,)e Fs(s;)15 b(h)1785 2551 y Fr(1)1825 2537 y Fs(;)g(h)1917 2551 y Fr(2)1957 2537 y Fs(;)g(m)2077 2551 y Fr(1)2117 2537 y Fh(\))33 b(such)f(that)75 2720 y Ft(\(5\))665 b Fp(jj)p Fs(x)957 2682 y Fq(s)994 2720 y Ft(\()p Fs(@)1082 2682 y Fn(\000)p Fq(h)1178 2691 y Fl(2)1077 2742 y Fq(x)1217 2720 y Fs(V)20 b Ft(\)\()p Fs(q)1404 2682 y Fq(h)1445 2691 y Fl(1)1484 2720 y Fs(\030)t(;)1625 2658 y(x)p 1578 2699 146 4 v 1578 2782 a(q)1622 2756 y Fq(m)1684 2765 y Fl(1)1733 2720 y Ft(\))p Fp(jj)1818 2738 y Fr(\()p Fq(T)6 b(;X)f Fr(\))2033 2720 y Fp(\024)25 b Fs(C)7 b(X)2283 2682 y Fr(\()p Fq(s)p Fr(+)p Fq(h)2439 2691 y Fl(2)2473 2682 y Fr(\))2505 2720 y Fp(jj)p Fs(V)21 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))p Fp(jj)2885 2738 y Fr(\()p Fq(T)6 b(;X)f Fr(\))75 2927 y Fh(for)33 b(al)5 b(l)33 b Fs(V)21 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))26 b Fp(2)f Fm(E)803 2946 y Fq(q)r(;)p Fr(\()p Fq(T)6 b(;X)f Fr(\))1045 2927 y Fh(.)75 3136 y Fi(Pro)s(of)31 b Ft(Let)g Fs(V)20 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))27 b Fp(2)d Fm(C)p Ft([[)p Fs(\030)t(;)15 b(x)p Ft(]].)42 b(W)-8 b(e)32 b(ha)m(v)m(e)f(that)650 3359 y Fs(x)702 3322 y Fq(s)739 3359 y Ft(\()p Fs(@)827 3322 y Fn(\000)p Fq(h)923 3331 y Fl(2)822 3382 y Fq(x)962 3359 y Fs(V)20 b Ft(\)\()p Fs(q)1149 3322 y Fq(h)1190 3331 y Fl(1)1229 3359 y Fs(\030)t(;)1370 3298 y(x)p 1323 3338 V 1323 3422 a(q)1367 3395 y Fq(m)1429 3404 y Fl(1)1478 3359 y Ft(\))26 b(=)1764 3273 y Fo(X)1635 3471 y Fq(l)q Fn(\025)p Fr(0)p Fq(;h)p Fn(\025)p Fq(h)1904 3480 y Fl(2)1937 3471 y Fr(+)p Fq(s)2040 3359 y Fs(v)2084 3378 y Fq(l)q(;h)p Fn(\000)p Fr(\()p Fq(s)p Fr(+)p Fq(h)2378 3387 y Fl(2)2411 3378 y Fr(\))2656 3298 y Fs(q)2700 3265 y Fq(h)2741 3274 y Fl(1)2776 3265 y Fq(l)2802 3298 y Fs(h)p Ft(!)p 2453 3338 631 4 v 2453 3425 a Fs(q)2497 3399 y Fq(m)2559 3408 y Fl(1)2594 3399 y Fr(\()p Fq(h)p Fn(\000)p Fq(s)p Fr(\))2781 3425 y Ft(\()p Fs(h)21 b Fp(\000)f Fs(s)p Ft(\)!)3093 3359 y Fs(\030)3137 3322 y Fq(l)3173 3298 y Fs(x)3225 3265 y Fq(h)p 3173 3338 97 4 v 3183 3422 a Fs(h)p Ft(!)75 3620 y(F)-8 b(rom)31 b(the)f(de\014nition)g(of)h(the)g(norm)e Fp(jj)p Fs(:)p Fp(jj)1497 3639 y Fr(\()p Fq(T)6 b(;X)f Fr(\))1687 3620 y Ft(,)31 b(w)m(e)g(get)g(that)75 3884 y(\(6\))84 b Fp(jj)p Fs(x)376 3847 y Fq(s)413 3884 y Ft(\()p Fs(@)501 3847 y Fn(\000)p Fq(h)597 3856 y Fl(2)496 3907 y Fq(x)636 3884 y Fs(V)21 b Ft(\)\()p Fs(q)824 3847 y Fq(h)865 3856 y Fl(1)904 3884 y Fs(\030)t(;)1044 3823 y(x)p 998 3863 146 4 v 998 3946 a(q)1042 3920 y Fq(m)1104 3929 y Fl(1)1153 3884 y Ft(\))p Fp(jj)1238 3903 y Fr(\()p Fq(T)6 b(;X)f Fr(\))1453 3884 y Ft(=)1678 3798 y Fo(X)1549 3995 y Fq(l)q Fn(\025)p Fr(0)p Fq(;h)p Fn(\025)p Fq(h)1818 4004 y Fl(2)1851 3995 y Fr(+)p Fq(s)1993 3814 y Fp(j)p Fs(v)2062 3832 y Fq(l)q(;h)p Fn(\000)p Fr(\()p Fq(s)p Fr(+)p Fq(h)2356 3841 y Fl(2)2390 3832 y Fr(\))2421 3814 y Fp(j)p 1964 3863 513 4 v 1964 3950 a Fs(q)2008 3924 y Fq(P)10 b Fr(\()p Fq(l)q(;h)p Fn(\000)p Fr(\()p Fq(s)p Fr(+)p Fq(h)2384 3933 y Fl(2)2417 3924 y Fr(\)\))2486 3884 y Fs(T)2552 3847 y Fq(l)2676 3823 y Fs(X)2758 3790 y Fq(h)p Fn(\000)p Fr(\()p Fq(s)p Fr(+)p Fq(h)3010 3799 y Fl(2)3044 3790 y Fr(\))p 2588 3863 576 4 v 2588 3946 a Ft(\()p Fs(h)21 b Fp(\000)e Ft(\()p Fs(s)i Ft(+)f Fs(h)3028 3960 y Fr(2)3067 3946 y Ft(\)\)!)3173 3884 y Fp(\002)1525 4054 y Fo(\032)2203 4121 y Ft(1)p 1603 4162 1246 4 v 1603 4249 a Fs(q)1647 4222 y Fq(P)10 b Fr(\()p Fq(l)q(;h)p Fr(\))p Fn(\000)p Fq(P)g Fr(\()p Fq(l)q(;h)p Fn(\000)p Fr(\()p Fq(s)p Fr(+)p Fq(h)2270 4231 y Fl(2)2302 4222 y Fr(\)\))p Fn(\000)p Fq(h)2452 4231 y Fl(1)2487 4222 y Fq(l)q Fr(+)p Fq(m)2626 4231 y Fl(1)2661 4222 y Fr(\()p Fq(h)p Fn(\000)p Fq(s)p Fr(\))2868 4121 y Ft(\()p Fs(h)21 b Fp(\000)f Ft(\()p Fs(s)g Ft(+)g Fs(h)3308 4135 y Fr(2)3348 4121 y Ft(\)\)!)p 2868 4162 576 4 v 3005 4245 a(\()p Fs(h)h Fp(\000)f Fs(s)p Ft(\)!)3454 4182 y Fs(X)3536 4145 y Fq(s)p Fr(+)p Fq(h)3665 4154 y Fl(2)3703 4054 y Fo(\033)75 4397 y Ft(In)40 b(the)h(rest)h(of)f (the)g(pro)s(of,)i(w)m(e)f(will)f(sho)m(w)g(that)h(there)f(exists)g(a)h (constan)m(t)g Fs(C)50 b(>)42 b Ft(0)g(\(dep)s(ending)e(on)g Fs(q)s Ft(,)75 4509 y Fs(s;)15 b(h)210 4523 y Fr(1)250 4509 y Fs(;)g(h)342 4523 y Fr(2)382 4509 y Fs(;)g(m)502 4523 y Fr(1)542 4509 y Ft(\))30 b(suc)m(h)g(that)75 4712 y(\(7\))1846 4650 y(1)p 1246 4691 1246 4 v 1246 4778 a Fs(q)1290 4752 y Fq(P)10 b Fr(\()p Fq(l)q(;h)p Fr(\))p Fn(\000)p Fq(P)g Fr(\()p Fq(l)q(;h)p Fn(\000)p Fr(\()p Fq(s)p Fr(+)p Fq(h)1913 4761 y Fl(2)1945 4752 y Fr(\)\))p Fn(\000)p Fq(h)2095 4761 y Fl(1)2130 4752 y Fq(l)q Fr(+)p Fq(m)2269 4761 y Fl(1)2304 4752 y Fr(\()p Fq(h)p Fn(\000)p Fq(s)p Fr(\))2526 4712 y Fp(\024)25 b Fs(C)75 4918 y Ft(for)30 b(all)h Fs(l)d Fp(\025)d Ft(0,)31 b Fs(h)25 b Fp(\025)g Ft(0.)41 b(Indeed,)158 5145 y Fs(P)13 b Ft(\()p Fs(l)r(;)i(h)p Ft(\))22 b Fp(\000)e Fs(P)13 b Ft(\()p Fs(l)r(;)i(h)21 b Fp(\000)f Ft(\()p Fs(s)g Ft(+)g Fs(h)1113 5159 y Fr(2)1153 5145 y Ft(\)\))g Fp(\000)g Fs(h)1386 5159 y Fr(1)1426 5145 y Fs(l)i Ft(+)e Fs(m)1646 5159 y Fr(1)1685 5145 y Ft(\()p Fs(h)h Fp(\000)f Fs(s)p Ft(\))26 b(=)f Fs(l)r Ft(\()2158 5084 y Fs(s)20 b Ft(+)f Fs(h)2363 5098 y Fr(2)p 2158 5124 246 4 v 2258 5208 a Ft(2)2433 5145 y Fp(\000)h Fs(h)2576 5159 y Fr(1)2616 5145 y Ft(\))h(+)f Fs(h)p Ft(\()p Fs(m)2930 5159 y Fr(1)2990 5145 y Fp(\000)g Ft(\()p Fs(s)g Ft(+)g Fs(h)3322 5159 y Fr(2)3362 5145 y Ft(\)\))3032 5378 y Fp(\000)g Fs(m)3203 5392 y Fr(1)3242 5378 y Fs(s)g Ft(+)3406 5317 y(\()p Fs(s)g Ft(+)g Fs(h)3647 5331 y Fr(2)3687 5317 y Ft(\))3722 5284 y Fr(2)p 3406 5357 356 4 v 3561 5440 a Ft(2)75 5568 y(for)31 b(all)g Fs(l)r(;)15 b(h)27 b Fp(\025)e Ft(0.)43 b(F)-8 b(rom)31 b(the)g(assumption)f(\(4\),)i(w)m(e)f(deduce)g(that)g(the)g (inequalities)h(\(7\))g(hold)e(for)h(all)g Fs(l)d Fp(\025)e Ft(0,)75 5681 y Fs(h)g Fp(\025)f Ft(0.)41 b(Finally)-8 b(,)32 b(the)e(inequalit)m(y)i(\(5\))f(follo)m(ws)h(from)e(the)g (expression)h(\(6\))g(and)f(the)g(estimates)i(\(7\).)217 b Ff(2)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 3809 100 a Ft(5)75 399 y Fi(Lemma)35 b(2)45 b Fh(L)-5 b(et)33 b Fs(s;)15 b(h)835 413 y Fr(1)900 399 y Fp(\025)25 b Ft(0)33 b Fh(and)h Fs(T)1304 413 y Fr(0)1343 399 y Fs(;)15 b(X)1458 413 y Fr(0)1524 399 y Fs(>)25 b Ft(0)p Fh(.)42 b(Then,)33 b(for)g(al)5 b(l)33 b Fs(T)2328 413 y Fr(1)2368 399 y Fs(;)15 b(X)2483 413 y Fr(1)2548 399 y Fs(>)25 b Ft(0)33 b Fh(satisfying)75 603 y Ft(\(8\))1000 b(0)25 b Fs(<)g(X)1431 617 y Fr(1)1496 603 y Fp(\024)g Fs(X)1667 617 y Fr(0)1707 603 y Fs(q)1751 565 y Fn(\000)p Fq(s)1908 603 y Fs(;)80 b Ft(0)26 b Fs(<)f(T)2233 617 y Fr(1)2298 603 y Fp(\024)g Fs(T)2447 617 y Fr(0)2486 603 y Fs(q)2541 538 y Fk(s)p 2540 550 31 3 v 2540 591 a Fl(2)2580 565 y Fn(\000)p Fq(h)2676 574 y Fl(1)2715 603 y Fs(;)75 807 y Fh(ther)-5 b(e)34 b(exists)e(a)h(c)-5 b(onstant)35 b Fs(C)1057 821 y Fr(1)1122 807 y Fs(>)24 b Ft(0)33 b Fh(\(dep)-5 b(ending)34 b(on)g Fs(q)s(;)15 b(s;)g(h)2105 821 y Fr(1)2145 807 y Fs(;)g(T)2238 821 y Fr(0)2278 807 y Fs(;)g(X)2393 821 y Fr(0)2433 807 y Fh(\))32 b(such)h(that)75 1011 y Ft(\(9\))928 b Fp(jj)p Fs(x)1220 974 y Fq(s)1257 1011 y Fs(V)20 b Ft(\()p Fs(q)1409 974 y Fq(h)1450 983 y Fl(1)1489 1011 y Fs(\030)t(;)15 b(x)p Ft(\))p Fp(jj)1710 1030 y Fr(\()p Fq(T)1778 1039 y Fl(1)1814 1030 y Fq(;X)1892 1039 y Fl(1)1926 1030 y Fr(\))1983 1011 y Fp(\024)25 b Fs(C)2144 1025 y Fr(1)2184 1011 y Fp(jj)p Fs(V)20 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))p Fp(jj)2563 1030 y Fr(\()p Fq(T)2631 1039 y Fl(0)2668 1030 y Fq(;X)2746 1039 y Fl(0)2780 1030 y Fr(\))75 1215 y Fh(for)33 b(al)5 b(l)33 b Fs(V)21 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))26 b Fp(2)f Fm(E)803 1234 y Fq(q)r(;)p Fr(\()p Fq(T)925 1243 y Fl(0)959 1234 y Fq(;X)1037 1243 y Fl(0)1071 1234 y Fr(\))1103 1215 y Fh(.)75 1478 y Fi(Pro)s(of)31 b Ft(Let)g Fs(V)20 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))27 b Fp(2)d Fm(C)p Ft([[)p Fs(\030)t(;)15 b(x)p Ft(]].)42 b(F)-8 b(rom)31 b(the)g(de\014nition)f(of)g(the)h(norm)e Fp(jj)p Fs(:)p Fp(jj)2704 1496 y Fr(\()p Fq(T)6 b(;X)f Fr(\))2894 1478 y Ft(,)31 b(one)g(can)f(write)75 1797 y(\(10\))85 b Fp(jj)p Fs(x)422 1759 y Fq(s)459 1797 y Fs(V)20 b Ft(\()p Fs(q)611 1759 y Fq(h)652 1768 y Fl(1)691 1797 y Fs(\030)t(;)15 b(x)p Ft(\))p Fp(jj)912 1815 y Fr(\()p Fq(T)980 1824 y Fl(1)1016 1815 y Fq(;X)1094 1824 y Fl(1)1128 1815 y Fr(\))1185 1797 y Ft(=)1345 1710 y Fo(X)1281 1908 y Fq(l)q Fn(\025)p Fr(0)p Fq(;h)p Fn(\025)p Fq(s)1595 1733 y Fp(j)p Fs(v)1664 1748 y Fq(l)q(;h)p Fn(\000)p Fq(s)1838 1733 y Fp(j)p 1566 1776 328 4 v 1566 1863 a Fs(q)1610 1836 y Fq(P)10 b Fr(\()p Fq(l)q(;h)p Fn(\000)p Fq(s)p Fr(\))1903 1797 y Fs(T)1969 1759 y Fq(l)1956 1819 y Fr(0)2022 1735 y Fs(X)2104 1687 y Fr(\()p Fq(h)p Fn(\000)p Fq(s)p Fr(\))2097 1761 y(0)p 2006 1776 303 4 v 2006 1859 a Ft(\()p Fs(h)20 b Fp(\000)g Fs(s)p Ft(\)!)2318 1797 y Fp(\002)2311 1967 y Fo(\032)2729 2033 y Ft(1)p 2389 2074 726 4 v 2389 2161 a Fs(q)2433 2135 y Fq(P)10 b Fr(\()p Fq(l)q(;h)p Fr(\))p Fn(\000)p Fq(P)g Fr(\()p Fq(l)q(;h)p Fn(\000)p Fq(s)p Fr(\))p Fn(\000)p Fq(h)3056 2144 y Fl(1)3089 2135 y Fq(l)3125 2095 y Ft(\()3170 2033 y Fs(T)3223 2047 y Fr(1)p 3170 2074 93 4 v 3170 2157 a Fs(T)3223 2171 y Fr(0)3273 2095 y Ft(\))3308 2057 y Fq(l)3334 2095 y Ft(\()3379 2033 y Fs(X)3454 2047 y Fr(1)p 3379 2074 115 4 v 3379 2157 a Fs(X)3454 2171 y Fr(0)3504 2095 y Ft(\))3539 2057 y Fq(h)3584 2095 y Fs(X)3666 2057 y Fq(s)3659 2117 y Fr(0)3703 1967 y Fo(\033)75 2349 y Ft(In)27 b(the)i(rest)f(of)g(the)g(pro)s(of,)g(w)m(e)h(will)f(sho)m(w)g (that)h(there)f(exists)g(a)h(constan)m(t)g Fs(C)2732 2363 y Fr(1)2797 2349 y Fs(>)c Ft(0)j(\(dep)s(ending)f(on)h Fs(q)s(;)15 b(s;)g(h)3779 2363 y Fr(1)3819 2349 y Ft(\))75 2462 y(suc)m(h)30 b(that)75 2685 y(\(11\))1600 2624 y(1)p 1260 2664 726 4 v 1260 2752 a Fs(q)1304 2725 y Fq(P)10 b Fr(\()p Fq(l)q(;h)p Fr(\))p Fn(\000)p Fq(P)g Fr(\()p Fq(l)q(;h)p Fn(\000)p Fq(s)p Fr(\))p Fn(\000)p Fq(h)1927 2734 y Fl(1)1959 2725 y Fq(l)1995 2685 y Ft(\()2040 2624 y Fs(T)2093 2638 y Fr(1)p 2040 2664 93 4 v 2040 2748 a Fs(T)2093 2762 y Fr(0)2143 2685 y Ft(\))2178 2648 y Fq(l)2204 2685 y Ft(\()2249 2624 y Fs(X)2324 2638 y Fr(1)p 2249 2664 115 4 v 2249 2748 a Fs(X)2324 2762 y Fr(0)2374 2685 y Ft(\))2409 2648 y Fq(h)2480 2685 y Fp(\024)25 b Fs(C)2641 2699 y Fr(1)75 2932 y Ft(for)30 b(all)h Fs(l)d Fp(\025)d Ft(0,)31 b(all)g Fs(h)25 b Fp(\025)g Ft(0.)75 3095 y(W)-8 b(e)32 b(ha)m(v)m(e)928 3240 y Fs(P)13 b Ft(\()p Fs(l)r(;)i(h)p Ft(\))22 b Fp(\000)e Fs(P)13 b Ft(\()p Fs(l)r(;)i(h)21 b Fp(\000)f Fs(s)p Ft(\))g Fp(\000)g Fs(h)1883 3254 y Fr(1)1923 3240 y Fs(l)27 b Ft(=)e Fs(l)r Ft(\()2148 3179 y Fs(s)p 2147 3219 46 4 v 2147 3303 a Ft(2)2223 3240 y Fp(\000)20 b Fs(h)2366 3254 y Fr(1)2405 3240 y Ft(\))h(+)f Fs(h)p Ft(\()p Fp(\000)p Fs(s)p Ft(\))g(+)2909 3179 y Fs(s)2952 3146 y Fr(2)p 2909 3219 83 4 v 2928 3303 a Ft(2)75 3433 y(for)27 b(all)h Fs(h)e Fp(\025)f Ft(0.)40 b(F)-8 b(or)28 b(all)g Fs(T)954 3447 y Fr(1)993 3433 y Fs(;)15 b(X)1108 3447 y Fr(1)1176 3433 y Ft(satisfying)28 b(\(8\),)h(w)m(e)f(get)g(that)g(\(11\))h(holds)d(for)h(all)h Fs(l)g Fp(\025)d Ft(0,)j(all)g Fs(h)e Fp(\025)f Ft(0.)40 b(Finally)-8 b(,)75 3546 y(the)31 b(inequalit)m(y)g(\(9\))h(follo)m(ws) f(from)f(the)h(expression)f(\(10\))i(and)e(the)g(estimates)i(\(11\).) 738 b Ff(2)75 3808 y Fi(Lemma)35 b(3)45 b Fh(L)-5 b(et)33 b Fs(h)752 3822 y Fr(2)817 3808 y Fp(\025)25 b Ft(0)33 b Fh(and)h Fs(T)1221 3822 y Fr(0)1260 3808 y Fs(;)15 b(X)1375 3822 y Fr(0)1441 3808 y Fs(>)25 b Ft(0)p Fh(.)42 b(Then,)33 b(for)g(al)5 b(l)33 b Fs(T)2245 3822 y Fr(1)2285 3808 y Fs(;)15 b(X)2400 3822 y Fr(1)2465 3808 y Fs(>)25 b Ft(0)33 b Fh(satisfying)75 4013 y Ft(\(12\))951 b(0)25 b Fs(<)g(X)1427 4027 y Fr(1)1492 4013 y Fp(\024)g Fs(X)1663 4027 y Fr(0)1703 4013 y Fs(q)1747 3975 y Fn(\000)p Fq(h)1843 3984 y Fl(2)1946 4013 y Fs(;)81 b Ft(0)25 b Fs(<)g(T)2271 4027 y Fr(1)2336 4013 y Fp(\024)g Fs(T)2485 4027 y Fr(0)2525 4013 y Fs(q)2569 3975 y Fq(h)2610 3984 y Fl(2)2644 3975 y Fq(=)p Fr(2)2719 4013 y Fs(;)75 4217 y Fh(ther)-5 b(e)34 b(exists)e(a)h(c)-5 b(onstant)35 b Fs(C)1057 4231 y Fr(2)1122 4217 y Fs(>)24 b Ft(0)33 b Fh(\(dep)-5 b(ending)34 b(on)g Fs(q)s(;)15 b(h)2022 4231 y Fr(2)2062 4217 y Fs(;)g(T)2155 4231 y Fr(0)2195 4217 y Fs(;)g(X)2310 4231 y Fr(0)2350 4217 y Fh(\))32 b(such)h(that)75 4421 y Ft(\(13\))895 b Fp(jj)p Fs(@)1233 4383 y Fn(\000)p Fq(h)1329 4392 y Fl(2)1228 4443 y Fq(x)1368 4421 y Fs(V)20 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))p Fp(jj)1697 4440 y Fr(\()p Fq(T)1765 4449 y Fl(1)1802 4440 y Fq(;X)1880 4449 y Fl(1)1914 4440 y Fr(\))1971 4421 y Fp(\024)25 b Fs(C)2132 4435 y Fr(2)2171 4421 y Fp(jj)p Fs(V)c Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))p Fp(jj)2551 4440 y Fr(\()p Fq(T)2619 4449 y Fl(0)2656 4440 y Fq(;X)2734 4449 y Fl(0)2768 4440 y Fr(\))75 4625 y Fh(for)33 b(al)5 b(l)33 b Fs(V)21 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))26 b Fp(2)f Fm(E)803 4644 y Fq(q)r(;)p Fr(\()p Fq(T)925 4653 y Fl(0)959 4644 y Fq(;X)1037 4653 y Fl(0)1071 4644 y Fr(\))1103 4625 y Fh(.)75 4888 y Fi(Pro)s(of)31 b Ft(Let)g Fs(V)20 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))27 b Fp(2)d Fm(C)p Ft([[)p Fs(\030)t(;)15 b(x)p Ft(]].)42 b(F)-8 b(rom)31 b(the)g(de\014nition)f(of)g(the)h(norm)e Fp(jj)p Fs(:)p Fp(jj)2704 4906 y Fr(\()p Fq(T)6 b(;X)f Fr(\))2894 4888 y Ft(,)31 b(one)g(can)f(write)75 5206 y(\(14\))85 b Fp(jj)p Fs(@)423 5169 y Fn(\000)p Fq(h)519 5178 y Fl(2)418 5229 y Fq(x)558 5206 y Fs(V)20 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))p Fp(jj)887 5225 y Fr(\()p Fq(T)955 5234 y Fl(1)992 5225 y Fq(;X)1070 5234 y Fl(1)1104 5225 y Fr(\))1161 5206 y Ft(=)1342 5120 y Fo(X)1257 5318 y Fq(l)q Fn(\025)p Fr(0)p Fq(;h)p Fn(\025)p Fq(h)1526 5327 y Fl(2)1609 5143 y Fp(j)p Fs(v)1678 5158 y Fq(l)q(;h)p Fn(\000)p Fq(h)1857 5167 y Fl(2)1894 5143 y Fp(j)p 1584 5186 360 4 v 1584 5273 a(j)p Fs(q)s Fp(j)1678 5246 y Fq(P)10 b Fr(\()p Fq(l)q Fn(\000)p Fq(h)1878 5255 y Fl(2)1913 5246 y Fr(\))1954 5206 y Fs(T)2020 5169 y Fq(l)2007 5229 y Fr(0)2076 5145 y Fs(X)2158 5097 y Fr(\()p Fq(h)p Fn(\000)p Fq(h)2322 5106 y Fl(2)2357 5097 y Fr(\))2151 5171 y(0)p 2057 5186 352 4 v 2057 5269 a Ft(\()p Fs(h)21 b Fp(\000)e Fs(h)2307 5283 y Fr(2)2347 5269 y Ft(\)!)2418 5206 y Fp(\002)2007 5377 y Fo(\032)2370 5443 y Ft(1)p 2085 5484 617 4 v 2085 5571 a Fs(q)2129 5545 y Fq(P)10 b Fr(\()p Fq(l)q(;h)p Fr(\))p Fn(\000)p Fq(P)g Fr(\()p Fq(l)q(;h)p Fn(\000)p Fq(h)2637 5554 y Fl(2)2669 5545 y Fr(\))2711 5505 y Ft(\()2756 5443 y Fs(T)2809 5457 y Fr(1)p 2756 5484 93 4 v 2756 5567 a Fs(T)2809 5581 y Fr(0)2859 5505 y Ft(\))2894 5467 y Fq(l)2920 5505 y Ft(\()2965 5443 y Fs(X)3040 5457 y Fr(1)p 2965 5484 115 4 v 2965 5567 a Fs(X)3040 5581 y Fr(0)3090 5505 y Ft(\))3125 5467 y Fq(h)3180 5443 y Ft(\()p Fs(h)21 b Fp(\000)f Fs(h)3431 5457 y Fr(2)3471 5443 y Ft(\)!)p 3180 5484 352 4 v 3317 5567 a Fs(h)p Ft(!)3541 5505 y Fs(X)3623 5464 y Fq(h)3664 5473 y Fl(2)3616 5531 y Fr(0)3703 5377 y Fo(\033)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 3809 100 a Ft(6)75 399 y(In)32 b(the)g(rest)h(of)g(the)f(pro)s(of,)h(w)m(e)g(will)g(sho)m(w)f(that)h (there)g(exists)g(a)g(constan)m(t)h Fs(C)2795 413 y Fr(2)2863 399 y Fs(>)28 b Ft(0)33 b(\(dep)s(ending)f(on)g Fs(q)s(;)15 b(h)3779 413 y Fr(2)3819 399 y Ft(\))75 511 y(suc)m(h)30 b(that)75 733 y(\(15\))1600 672 y(1)p 1314 712 617 4 v 1314 799 a Fs(q)1358 773 y Fq(P)10 b Fr(\()p Fq(l)q(;h)p Fr(\))p Fn(\000)p Fq(P)g Fr(\()p Fq(l)q(;h)p Fn(\000)p Fq(h)1866 782 y Fl(2)1899 773 y Fr(\))1940 733 y Ft(\()1985 672 y Fs(T)2038 686 y Fr(1)p 1985 712 93 4 v 1985 796 a Fs(T)2038 810 y Fr(0)2088 733 y Ft(\))2123 696 y Fq(l)2149 733 y Ft(\()2194 672 y Fs(X)2269 686 y Fr(1)p 2194 712 115 4 v 2194 796 a Fs(X)2269 810 y Fr(0)2320 733 y Ft(\))2355 696 y Fq(h)2425 733 y Fp(\024)25 b Fs(C)2586 747 y Fr(2)75 979 y Ft(for)30 b(all)h Fs(l)d Fp(\025)d Ft(0,)31 b(all)g Fs(h)25 b Fp(\025)g Ft(0.)75 1141 y(W)-8 b(e)32 b(ha)m(v)m(e)1056 1286 y Fs(P)13 b Ft(\()p Fs(l)r(;)i(h)p Ft(\))21 b Fp(\000)f Fs(P)13 b Ft(\()p Fs(l)r(;)i(h)21 b Fp(\000)f Fs(h)1821 1300 y Fr(2)1861 1286 y Ft(\))26 b(=)f Fs(l)r Ft(\()2092 1225 y Fs(h)2144 1239 y Fr(2)p 2092 1265 92 4 v 2115 1349 a Ft(2)2194 1286 y(\))20 b(+)g Fs(h)p Ft(\()p Fp(\000)p Fs(h)2550 1300 y Fr(2)2590 1286 y Ft(\))h(+)2747 1225 y Fs(h)2799 1192 y Fr(2)2799 1249 y(2)p 2747 1265 V 2770 1349 a Ft(2)2849 1286 y Fs(;)75 1478 y Ft(for)38 b(all)i Fs(h)f Fp(\025)f Ft(0.)66 b(F)-8 b(or)39 b(all)h Fs(T)1052 1492 y Fr(1)1091 1478 y Fs(;)15 b(X)1206 1492 y Fr(1)1285 1478 y Ft(satisfying)39 b(\(12\),)k(w)m(e)c(get)g(that)g(\(15\))h (holds)e(for)h(all)g Fs(l)i Fp(\025)d Ft(0,)k(all)d Fs(h)g Fp(\025)g Ft(0.)75 1591 y(Finally)-8 b(,)32 b(the)f(inequalit)m(y)g (\(13\))h(follo)m(ws)g(from)e(the)g(expression)g(\(14\))i(and)e(the)h (estimates)h(\(15\).)365 b Ff(2)75 1754 y Ft(Let)35 b Fs(S;)15 b(m)418 1769 y Fr(0)p Fq(;k)516 1754 y Fs(;)g(m)636 1769 y Fr(1)p Fq(;k)733 1754 y Ft(,)36 b(0)d Fp(\024)f Fs(k)j Fp(\024)d Fs(S)c Fp(\000)22 b Ft(1)35 b(b)s(e)f(p)s(ositiv)m(e)i (in)m(tegers.)54 b(Let)35 b Fp(D)i Ft(the)e(linear)g(op)s(erator)g (from)f Fm(C)p Ft([[)p Fs(\030)t(;)15 b(x)p Ft(]])75 1867 y(in)m(to)31 b Fm(C)p Ft([[)p Fs(\030)t(;)15 b(x)p Ft(]])32 b(de\014ned)d(b)m(y)793 2156 y Fp(D)s Ft(\()p Fs(V)20 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\)\))26 b(:=)g Fs(@)1416 2118 y Fq(S)1411 2178 y(x)1467 2156 y Fs(V)20 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))21 b Fp(\000)1858 2042 y Fq(S)t Fn(\000)p Fr(1)1861 2069 y Fo(X)1862 2267 y Fq(k)r Fr(=0)2010 2156 y Fs(a)2058 2171 y Fq(k)2101 2156 y Ft(\()p Fs(x)p Ft(\)\()p Fs(@)2311 2118 y Fq(k)2306 2178 y(x)2355 2156 y Fs(V)f Ft(\)\()p Fs(q)2542 2118 y Fq(m)2604 2130 y Fl(0)p Fk(;k)2697 2156 y Fs(\030)t(;)15 b(x=q)2922 2118 y Fq(m)2984 2130 y Fl(1)p Fk(;k)3076 2156 y Ft(\))p Fs(;)75 2442 y Ft(for)39 b(all)g Fs(V)60 b Fp(2)38 b Fm(C)p Ft([[)p Fs(\030)t(;)15 b(x)p Ft(]],)42 b(where)d Fs(a)1258 2457 y Fq(k)1300 2442 y Ft(\()p Fs(x)p Ft(\))h(=)1572 2374 y Fo(P)1668 2469 y Fq(s)p Fn(2)p Fq(I)1779 2481 y Fk(k)1836 2442 y Fs(a)1884 2457 y Fq(k)r(s)1959 2442 y Fs(x)2011 2409 y Fq(s)2087 2442 y Fp(2)f Fm(C)p Ft([)p Fs(x)p Ft(],)i(with)e Fs(I)2677 2457 y Fq(k)2758 2442 y Ft(b)s(e)f(a)h(\014nite)g(subset)f(of)h Fm(N)p Ft(,)i(for)75 2555 y(0)26 b Fp(\024)f Fs(k)j Fp(\024)d Fs(S)g Fp(\000)20 b Ft(1.)75 2718 y(W)-8 b(e)32 b(mak)m(e)f(the)g (follo)m(wing)g(h)m(yp)s(othesis.)75 2880 y Fi(Assumption)36 b(\(A\))29 b Ft(F)-8 b(or)31 b(all)h(0)25 b Fp(\024)g Fs(k)k Fp(\024)c Fs(S)g Fp(\000)20 b Ft(1,)31 b(for)f(all)h Fs(s)25 b Fp(2)g Fs(I)2205 2895 y Fq(k)2248 2880 y Ft(,)30 b(w)m(e)h(ha)m(v)m(e)1174 3083 y Fs(s)20 b Ft(+)g Fs(S)25 b Fp(\000)20 b Fs(k)28 b Fp(\025)d Ft(2)p Fs(m)1796 3098 y Fr(0)p Fq(;k)1955 3083 y Fs(;)76 b(m)2136 3098 y Fr(1)p Fq(;k)2258 3083 y Fp(\025)25 b Fs(s)20 b Ft(+)g Fs(S)25 b Fp(\000)20 b Fs(k)s(:)75 3286 y Ft(W)-8 b(e)32 b(consider)e(the)g(op) s(erator)h Fp(A)f Ft(from)g Fm(C)p Ft([[)p Fs(\030)t(;)15 b(x)p Ft(]])31 b(in)m(to)h Fm(C)p Ft([[)p Fs(\030)t(;)15 b(x)p Ft(]])31 b(de\014ned)e(b)m(y)467 3575 y Fp(A)p Ft(\()p Fs(V)20 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\)\))27 b(=)e Fs(V)20 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))22 b Fp(\000)d(D)s Ft(\()p Fs(@)1564 3537 y Fn(\000)p Fq(S)1559 3597 y(x)1670 3575 y Fs(V)i Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\)\))26 b(=)2107 3461 y Fq(S)t Fn(\000)p Fr(1)2110 3488 y Fo(X)2111 3686 y Fq(k)r Fr(=0)2259 3575 y Fs(a)2307 3590 y Fq(k)2350 3575 y Ft(\()p Fs(x)p Ft(\)\()p Fs(@)2560 3537 y Fq(k)r Fn(\000)p Fq(S)2555 3597 y(x)2705 3575 y Fs(V)21 b Ft(\)\()p Fs(q)2893 3537 y Fq(m)2955 3549 y Fl(0)p Fk(;k)3047 3575 y Fs(\030)t(;)15 b(x=q)3272 3537 y Fq(m)3334 3549 y Fl(1)p Fk(;k)3427 3575 y Ft(\))75 3861 y(for)30 b(all)h Fs(V)46 b Fp(2)25 b Fm(C)p Ft([[)p Fs(\030)t(;)15 b(x)p Ft(]].)75 4024 y(F)-8 b(rom)31 b(the)f(lemma)h(1,) g(w)m(e)g(deduce)f(the)h(follo)m(wing)75 4210 y Fi(Lemma)k(4)45 b Fh(L)-5 b(et)34 b Fs(T)39 b(>)26 b Ft(0)p Fh(.)44 b(Then,)33 b(ther)-5 b(e)34 b(exists)g Fs(X)f(>)26 b Ft(0)34 b Fh(such)f(that)i Fp(A)d Fh(is)h(a)h(line)-5 b(ar)34 b(b)-5 b(ounde)g(d)35 b(op)-5 b(er)g(ator)36 b(fr)-5 b(om)75 4323 y Ft(\()p Fm(E)171 4342 y Fq(q)r(;)p Fr(\()p Fq(T)6 b(;X)f Fr(\))413 4323 y Fs(;)15 b Fp(jj)p Fs(:)p Fp(jj)578 4342 y Fr(\()p Fq(T)6 b(;X)f Fr(\))769 4323 y Ft(\))33 b Fh(into)g(itself,)f(with)i (the)f(fol)5 b(lowing)34 b(estimates)1217 4576 y Fp(jjA)p Ft(\()p Fs(V)21 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\)\))p Fp(jj)1740 4595 y Fr(\()p Fq(T)6 b(;X)f Fr(\))1956 4576 y Fp(\024)2062 4515 y Ft(1)p 2062 4556 46 4 v 2062 4639 a(2)2117 4576 y Fp(jj)p Fs(V)21 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))p Fp(jj)2497 4595 y Fr(\()p Fq(T)6 b(;X)f Fr(\))2687 4576 y Fs(;)75 4800 y Fh(for)33 b(al)5 b(l)33 b Fs(V)46 b Fp(2)25 b Fm(E)596 4819 y Fq(q)r(;)p Fr(\()p Fq(T)6 b(;X)f Fr(\))838 4800 y Fh(.)75 4987 y Ft(F)-8 b(rom)31 b(the)f(lemma)h(4,)g(w)m(e)g(deduce)f(the)h(next)75 5173 y Fi(Corollary)k(1)46 b Fh(L)-5 b(et)39 b Fs(T)49 b(>)36 b Ft(0)p Fh(.)61 b(Then,)41 b(ther)-5 b(e)40 b(exists)f Fs(X)44 b(>)36 b Ft(0)j Fh(such)g(that)h Fp(D)28 b(\016)d Fs(@)2841 5140 y Fn(\000)p Fq(S)2836 5196 y(x)2986 5173 y Fh(is)38 b(an)i(invertible)e(line)-5 b(ar)75 5286 y(op)g(er)g(ator)36 b(fr)-5 b(om)34 b Ft(\()p Fm(E)744 5305 y Fq(q)r(;)p Fr(\()p Fq(T)6 b(;X)f Fr(\))986 5286 y Fs(;)15 b Fp(jj)p Fs(:)p Fp(jj)1151 5305 y Fr(\()p Fq(T)6 b(;X)f Fr(\))1341 5286 y Ft(\))33 b Fh(into)h(itself.)41 b(In)33 b(p)-5 b(articular,)35 b(ther)-5 b(e)33 b(exists)g Fs(C)f(>)25 b Ft(0)33 b Fh(such)g(that)75 5502 y Ft(\(16\))946 b Fp(jjD)s Ft(\()p Fs(@)1392 5465 y Fn(\000)p Fq(S)1387 5525 y(x)1499 5502 y Fs(b)p Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\)\))p Fp(jj)1829 5521 y Fr(\()p Fq(T)6 b(;X)f Fr(\))2045 5502 y Fp(\024)25 b Fs(C)7 b Fp(jj)p Fs(b)p Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))p Fp(jj)2558 5521 y Fr(\()p Fq(T)6 b(;X)f Fr(\))75 5705 y Fh(for)33 b(al)5 b(l)33 b Fs(b)p Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))27 b Fp(2)d Fm(E)768 5724 y Fq(q)r(;)p Fr(\()p Fq(T)6 b(;X)f Fr(\))1010 5705 y Fh(.)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 3809 100 a Ft(7)75 399 y Fi(Theorem)35 b(1)46 b Fh(L)-5 b(et)34 b Fs(S)d Fp(\025)c Ft(1)34 b Fh(b)-5 b(e)34 b(an)g(inte)-5 b(ger.)45 b(F)-7 b(or)34 b(al)5 b(l)34 b Ft(0)28 b Fp(\024)e Fs(k)k Fp(\024)d Fs(S)f Fp(\000)21 b Ft(1)p Fh(,)34 b(let)f Fs(m)2759 414 y Fr(0)p Fq(;k)2857 399 y Fs(;)15 b(m)2977 414 y Fr(1)p Fq(;k)3108 399 y Fh(b)-5 b(e)34 b(p)-5 b(ositive)34 b(inte)-5 b(gers)75 511 y(and)34 b Fs(a)300 526 y Fq(k)342 511 y Ft(\()p Fs(x)p Ft(\))26 b(=)586 443 y Fo(P)682 538 y Fq(s)p Fn(2)p Fq(I)793 550 y Fk(k)850 511 y Fs(a)898 526 y Fq(k)r(s)973 511 y Fs(x)1025 478 y Fq(s)1087 511 y Fp(2)f Fm(C)p Ft([)p Fs(x)p Ft(])p Fh(.)42 b(We)32 b(make)h(the)g(hyp)-5 b(othesis)35 b(that)f(the)f(assumption)i Fi(\(A\))c Fh(holds.)75 672 y(We)h(c)-5 b(onsider)34 b(the)f(fol)5 b(lowing)34 b(functional)f(e)-5 b(quation)75 952 y Ft(\(17\))850 b Fs(@)1138 914 y Fq(S)1133 974 y(x)1189 952 y Fs(V)21 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))26 b(=)1591 838 y Fq(S)t Fn(\000)p Fr(1)1594 865 y Fo(X)1595 1063 y Fq(k)r Fr(=0)1743 952 y Fs(a)1791 967 y Fq(k)1833 952 y Ft(\()p Fs(x)p Ft(\)\()p Fs(@)2043 914 y Fq(k)2038 974 y(x)2087 952 y Fs(V)21 b Ft(\)\()p Fs(q)2275 914 y Fq(m)2337 926 y Fl(0)p Fk(;k)2429 952 y Fs(\030)t(;)15 b(x=q)2654 914 y Fq(m)2716 926 y Fl(1)p Fk(;k)2809 952 y Ft(\))75 1229 y Fh(with)33 b(initial)h(c)-5 b(onditions)75 1428 y Ft(\(18\))976 b(\()p Fs(@)1299 1390 y Fq(j)1294 1450 y(x)1339 1428 y Fs(V)20 b Ft(\)\()p Fs(\030)t(;)15 b Ft(0\))27 b(=)e Fs(\036)1823 1442 y Fq(j)1860 1428 y Ft(\()p Fs(\030)t Ft(\))65 b Fs(;)81 b Ft(0)26 b Fp(\024)f Fs(j)30 b Fp(\024)25 b Fs(S)h Fp(\000)20 b Ft(1)p Fs(:)75 1626 y Fh(We)34 b(assume)h(that)h Fs(\036)804 1640 y Fq(j)841 1626 y Ft(\()p Fs(\030)t Ft(\))30 b Fp(2)e Fm(E)1135 1645 y Fq(q)r(;)p Fr(\()p Fq(T)1257 1654 y Fl(0)1291 1645 y Fq(;X)1369 1654 y Fl(0)1404 1645 y Fr(\))1435 1626 y Fh(,)35 b(for)g Ft(0)30 b Fp(\024)e Fs(j)35 b Fp(\024)28 b Fs(S)f Fp(\000)22 b Ft(1)p Fh(,)35 b(wher)-5 b(e)36 b Fs(T)2586 1640 y Fr(0)2625 1626 y Fs(;)15 b(X)2740 1640 y Fr(0)2809 1626 y Fs(>)29 b Ft(0)35 b Fh(ar)-5 b(e)35 b(given)f(p)-5 b(ositive)36 b(r)-5 b(e)g(al)75 1739 y(numb)g(ers.)42 b(Then,)33 b(ther)-5 b(e)33 b(exist)f Fs(T)8 b(;)15 b(X)33 b(>)25 b Ft(0)32 b Fh(with)h Ft(0)26 b Fs(<)f(T)38 b Fp(\024)25 b Fs(T)2161 1753 y Fr(0)2200 1739 y Fh(,)32 b Ft(0)26 b Fs(<)f(X)33 b Fp(\024)25 b Fs(X)2706 1753 y Fr(0)2777 1739 y Fh(such)33 b(that)g(the)g(pr)-5 b(oblem)33 b(\(17\),)75 1852 y(\(18\))h(has)f(a)g(unique)f(solution)i Fs(V)20 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))26 b Fp(2)f Fm(E)1605 1870 y Fq(q)r(;)p Fr(\()p Fq(T)6 b(;X)f Fr(\))1847 1852 y Fh(.)75 2107 y Fi(Pro)s(of)26 b Ft(A)g(formal)g(series)f Fs(V)c Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))26 b Fp(2)f Fm(C)p Ft([[)p Fs(\030)t(;)15 b(x)p Ft(]])26 b(whic)m(h)f(satis\014es)h (\(18\))h(can)f(b)s(e)f(uniquely)g(written)g(in)g(the)h(form)75 2219 y Fs(V)20 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))27 b(=)d Fs(@)529 2186 y Fn(\000)p Fq(S)524 2242 y(x)635 2219 y Fs(U)10 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))22 b(+)e Fs(I)7 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))31 b(where)1544 2504 y Fs(I)7 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))27 b(=)1920 2390 y Fq(S)t Fn(\000)p Fr(1)1922 2418 y Fo(X)1927 2613 y Fq(j)t Fr(=0)2072 2504 y Fs(\036)2126 2518 y Fq(j)2162 2504 y Ft(\()p Fs(\030)t Ft(\))2286 2443 y Fs(x)2338 2410 y Fq(j)p 2286 2483 89 4 v 2296 2566 a Fs(j)5 b Ft(!)75 2797 y(and)29 b Fs(U)10 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))26 b Fp(2)f Fm(C)p Ft([[)p Fs(\030)t(;)15 b(x)p Ft(]].)41 b(A)30 b(formal)f(series)h Fs(V)20 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))26 b Fp(2)f Fm(C)p Ft([[)p Fs(\030)t(;)15 b(x)p Ft(]])31 b(is)e(a)h(solution)g(of)f (the)h(problem)f(\(17\),)i(\(18\))75 2910 y(if)f(and)g(only)h(if)f Fs(U)10 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))31 b(satis\014es)g(the)g (equation)75 3109 y(\(19\))1132 b Fp(D)s Ft(\()p Fs(@)1528 3071 y Fn(\000)p Fq(S)1523 3131 y(x)1634 3109 y Fs(U)10 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\)\))26 b(=)f Fp(\000D)s Ft(\()p Fs(I)7 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\)\))p Fs(:)75 3307 y Ft(By)31 b(construction,)g(w)m(e)g(ha)m(v)m(e)g(that)751 3587 y Fp(\000D)s Ft(\()p Fs(I)7 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\)\))26 b(=)1340 3473 y Fq(S)t Fn(\000)p Fr(1)1343 3500 y Fo(X)1344 3698 y Fq(k)r Fr(=0)1492 3473 y Fq(S)t Fn(\000)p Fr(1)1495 3500 y Fo(X)1498 3698 y Fq(j)t Fr(=)p Fq(k)1653 3500 y Fo(X)1644 3697 y Fq(s)p Fn(2)p Fq(I)1755 3709 y Fk(k)2095 3525 y Fs(a)2143 3540 y Fq(k)r(s)p 1818 3566 679 4 v 1818 3658 a Fs(q)1862 3626 y Fq(m)1924 3638 y Fl(1)p Fk(;k)2011 3626 y Fr(\()p Fq(j)t Fn(\000)p Fq(k)r Fr(\))2196 3658 y Ft(\()p Fs(j)g Fp(\000)20 b Fs(k)s Ft(\)!)2506 3587 y Fs(x)2558 3549 y Fq(s)p Fr(+)p Fq(j)t Fn(\000)p Fq(k)2775 3587 y Fs(\036)2829 3601 y Fq(j)2866 3587 y Ft(\()p Fs(q)2945 3549 y Fq(m)3007 3561 y Fl(0)p Fk(;k)3099 3587 y Fs(\030)t Ft(\))75 3877 y(F)-8 b(rom)31 b(the)f(lemma)h(2,)g(there)g(exist)g Fs(T)1365 3891 y Fr(1)1405 3877 y Fs(;)15 b(X)1520 3891 y Fr(1)1585 3877 y Fs(>)25 b Ft(0)31 b(suc)m(h)f(that)1392 4075 y Fs(x)1444 4038 y Fq(s)p Fr(+)p Fq(j)t Fn(\000)p Fq(k)1662 4075 y Fs(\036)1716 4089 y Fq(j)1752 4075 y Ft(\()p Fs(q)1831 4038 y Fq(m)1893 4050 y Fl(0)p Fk(;k)1986 4075 y Fs(\030)t Ft(\))25 b Fp(2)g Fm(E)2237 4094 y Fq(q)r(;)p Fr(\()p Fq(T)2359 4103 y Fl(1)2393 4094 y Fq(;X)2471 4103 y Fl(1)2506 4094 y Fr(\))75 4274 y Ft(for)i(all)g(0)f Fp(\024)f Fs(k)k Fp(\024)24 b Fs(S)19 b Fp(\000)13 b Ft(1,)28 b(all)g Fs(k)g Fp(\024)d Fs(j)31 b Fp(\024)25 b Fs(S)18 b Fp(\000)13 b Ft(1,)28 b(all)g Fs(s)d Fp(2)g Fs(I)1959 4289 y Fq(k)2001 4274 y Ft(.)40 b(W)-8 b(e)28 b(deduce)e(that)i Fp(D)s Ft(\()p Fs(I)7 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\)\))27 b Fp(2)d Fm(E)3281 4292 y Fq(q)r(;)p Fr(\()p Fq(T)3403 4301 y Fl(1)3438 4292 y Fq(;X)3516 4301 y Fl(1)3550 4292 y Fr(\))3581 4274 y Ft(.)40 b(F)-8 b(rom)75 4387 y(the)39 b(corollary)h(1,)h(w)m(e)f(deduce)e(that)h(the)g(equation)h(\(19\))g (has)e(a)h(unique)f(solution)h Fs(U)10 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))41 b Fp(2)d Fm(E)3529 4405 y Fq(q)r(;)p Fr(\()p Fq(T)3651 4414 y Fl(1)3685 4405 y Fq(;X)3763 4414 y Fl(1)3798 4405 y Fr(\))3829 4387 y Ft(.)75 4508 y(F)-8 b(rom)29 b(the)f(lemma)h(3,)g(there)g(exist)g(0)d Fs(<)f(T)1520 4522 y Fr(2)1584 4508 y Fs(<)g(T)1733 4522 y Fr(1)1801 4508 y Ft(and)j(0)d Fs(<)g(X)2217 4522 y Fr(2)2282 4508 y Fs(<)g(X)2453 4522 y Fr(1)2521 4508 y Ft(suc)m(h)j(that)h Fs(@)2972 4475 y Fn(\000)p Fq(S)2967 4531 y(x)3078 4508 y Fs(U)10 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))26 b Fp(2)f Fm(E)3529 4527 y Fq(q)r(;)p Fr(\()p Fq(T)3651 4536 y Fl(2)3685 4527 y Fq(;X)3763 4536 y Fl(2)3798 4527 y Fr(\))3829 4508 y Ft(.)75 4621 y(F)-8 b(rom)42 b(the)g(lemma)h(2,)i(there)d(exist) h(0)i Fs(<)f(T)1642 4635 y Fr(3)1726 4621 y Fs(<)g(T)1894 4635 y Fr(2)1934 4621 y Ft(,)h(0)g Fs(<)f(X)2284 4635 y Fr(3)2368 4621 y Fs(<)g(X)2558 4635 y Fr(2)2640 4621 y Ft(suc)m(h)d(that)i Fs(I)7 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))45 b Fp(2)f Fm(E)3529 4640 y Fq(q)r(;)p Fr(\()p Fq(T)3651 4649 y Fl(3)3685 4640 y Fq(;X)3763 4649 y Fl(3)3798 4640 y Fr(\))3829 4621 y Ft(.)75 4743 y(Finally)-8 b(,)37 b(the)e(formal)g(series)g Fs(V)21 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))33 b(=)f Fs(@)1575 4710 y Fn(\000)p Fq(S)1570 4766 y(x)1681 4743 y Fs(U)10 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\))25 b(+)d Fs(I)7 b Ft(\()p Fs(\030)t(;)15 b(x)p Ft(\),)38 b(unique)33 b(solution)j(of)f(the)g(problem)f(\(17\),) 75 4856 y(\(18\),)e(b)s(elongs)e(to)i Fm(E)791 4875 y Fq(q)r(;)p Fr(\()p Fq(T)913 4884 y Fl(3)947 4875 y Fq(;X)1025 4884 y Fl(3)1059 4875 y Fr(\))1091 4856 y Ft(.)2640 b Ff(2)75 5189 y Fu(3)135 b(A)37 b(parameter)i(dep)t(ending)e(Cauc)l(h)l (y)h(problem)g(in)f(analytic)i(spaces)f(of)277 5339 y Fe(q)t Fu(-exp)t(onen)l(tial)46 b(gro)l(wth)75 5542 y Ft(Let)31 b Fs(U;)15 b(V)51 b Ft(b)s(e)29 b(op)s(en)h(b)s(ounded)e (sets)j(in)f Fm(C)1508 5509 y Fn(\003)1578 5542 y Ft(and)f(let)j Fs(q)c Fp(2)d Fm(C)2107 5509 y Fn(\003)2176 5542 y Ft(suc)m(h)30 b(that)h Fp(j)p Fs(q)s Fp(j)26 b Fs(>)f Ft(1.)41 b(W)-8 b(e)31 b(de\014ne)541 5740 y Fs(U)10 b(q)657 5703 y Fn(\000)p Fj(N)789 5740 y Ft(=)25 b Fp(f)p Fs(\017)967 5754 y Fq(u)1012 5740 y Fs(q)1056 5703 y Fn(\000)p Fq(n)1183 5740 y Fp(2)g Fm(C)p Fs(=\017)1417 5754 y Fq(u)1487 5740 y Fp(2)g Fs(U;)15 b(n)25 b Fp(2)g Fm(N)p Fp(g)61 b Fs(;)76 b(V)20 b(q)2231 5703 y Fj(N)2308 5740 y Ft(=)25 b Fp(f)p Fs(\034)2489 5754 y Fq(v)2530 5740 y Fs(q)2574 5703 y Fq(l)2625 5740 y Fp(2)g Fm(C)p Fs(=\034)2862 5754 y Fq(v)2928 5740 y Fp(2)g Fs(V)5 b(;)15 b(l)28 b Fp(2)d Fm(N)p Fp(g)p Fs(:)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 3809 100 a Ft(8)75 399 y(W)-8 b(e)31 b(assume)e(that)h(the)f(follo)m(wing)i(condition)f(is)g(satis\014ed:)40 b(there)30 b(exists)g Fs(\016)f(>)c Ft(0)30 b(suc)m(h)f(that)h Fp(j)p Fs(\034)e Ft(+)18 b(1)p Fp(j)26 b Fs(>)f(\016)s Ft(,)31 b(for)75 511 y(all)g Fs(\034)36 b Fp(2)24 b Fs(V)d(q)480 478 y Fj(N)532 511 y Ft(.)75 674 y(Let)30 b Fs(S)g Fp(\025)25 b Ft(1)30 b(b)s(e)f(an)g(in)m(teger.)42 b(F)-8 b(or)30 b(all)h(0)25 b Fp(\024)g Fs(k)k Fp(\024)c Fs(S)e Fp(\000)18 b Ft(1,)31 b(let)f Fs(m)2183 689 y Fr(0)p Fq(;k)2281 674 y Ft(,)p Fs(m)2386 689 y Fr(1)p Fq(;k)2513 674 y Ft(b)s(e)e(p)s(ositiv)m(e)j(in)m(tegers)g(and)e Fs(b)3522 689 y Fq(k)3564 674 y Ft(\()p Fs(\017;)15 b(z)t Ft(\))27 b(=)75 719 y Fo(P)171 814 y Fq(s)p Fn(2)p Fq(I)282 826 y Fk(k)339 787 y Fs(b)378 802 y Fq(k)r(s)453 787 y Ft(\()p Fs(\017)p Ft(\))p Fs(z)606 754 y Fq(s)687 787 y Ft(b)s(e)43 b(a)g(p)s(olynomial)h(in)f Fs(z)t Ft(,)k(where)c Fs(I)1948 802 y Fq(k)2034 787 y Ft(is)h(a)f(subset)g(of)h Fm(N)f Ft(and)f Fs(b)2973 802 y Fq(k)r(s)3049 787 y Ft(\()p Fs(\017)p Ft(\))i(are)f(holomorphic)75 913 y(b)s(ounded)28 b(functions)i(on)g(some)h(op)s(en)f(disc)g Fs(D)s Ft(\(0)p Fs(;)15 b(r)1833 927 y Fr(0)1874 913 y Ft(\))30 b(with)g Fs(r)2187 927 y Fr(0)2252 913 y Fs(>)25 b Ft(0)31 b(suc)m(h)f(that)h Fs(U)10 b(q)2942 880 y Fn(\000)p Fj(N)3074 913 y Fp(\032)25 b Fs(D)s Ft(\(0)p Fs(;)15 b(r)3409 927 y Fr(0)3449 913 y Ft(\).)75 1026 y(W)-8 b(e)22 b(consider)f(functions)f(\()p Fs(\017;)15 b(\034)10 b Ft(\))26 b Fp(7!)g Fs(W)1375 1040 y Fq(j)1411 1026 y Ft(\()p Fs(\017;)15 b(\034)10 b Ft(\),)24 b(for)c(0)26 b Fp(\024)f Fs(j)31 b Fp(\024)25 b Fs(S)6 b Fp(\000)q Ft(1,)23 b(whic)m(h)e(are)g(holomorphic)g(on)f Fs(U)10 b(q)3480 993 y Fn(\000)p Fj(N)3588 1026 y Fp(\002)q Fs(V)20 b(q)3777 993 y Fj(N)3829 1026 y Ft(.)75 1139 y(W)-8 b(e)32 b(consider)e(the)g(follo)m(wing)i(Cauc)m(h)m(y)f(problem) 75 1424 y(\(20\))703 b Fs(@)991 1386 y Fq(S)986 1446 y(z)1042 1424 y Fs(W)13 b Ft(\()p Fs(\017;)i(\034)5 b(;)15 b(z)t Ft(\))27 b(=)1542 1310 y Fq(S)t Fn(\000)p Fr(1)1544 1338 y Fo(X)1546 1535 y Fq(k)r Fr(=0)1704 1362 y Fs(b)1743 1377 y Fq(k)1785 1362 y Ft(\()p Fs(\017;)15 b(z)t Ft(\))p 1704 1403 276 4 v 1738 1486 a Fs(\034)31 b Ft(+)19 b(1)1990 1424 y Fs(\034)2040 1386 y Fq(m)2102 1398 y Fl(0)p Fk(;k)2194 1424 y Ft(\()p Fs(@)2282 1386 y Fq(k)2277 1446 y(z)2325 1424 y Fs(W)13 b Ft(\)\()p Fs(\017;)i(\034)5 b(;)15 b(z)t(q)2746 1386 y Fn(\000)p Fq(m)2863 1398 y Fl(1)p Fk(;k)2956 1424 y Ft(\))75 1707 y(with)30 b(initial)i(conditions)75 1911 y(\(21\))873 b(\()p Fs(@)1196 1874 y Fq(j)1191 1934 y(z)1233 1911 y Fs(W)13 b Ft(\)\()p Fs(\017;)i(\034)5 b(;)15 b Ft(0\))27 b(=)e Fs(W)1853 1925 y Fq(j)1889 1911 y Ft(\()p Fs(\017;)15 b(\034)10 b Ft(\))62 b Fs(;)76 b Ft(0)25 b Fp(\024)g Fs(j)31 b Fp(\024)25 b Fs(S)g Fp(\000)20 b Ft(1)p Fs(:)75 2124 y Fi(Theorem)35 b(2)46 b Fh(L)-5 b(et)37 b(the)h(assumption)h Fi(\(A\))d Fh(\(of)i(se)-5 b(ction)37 b(2\))h(b)-5 b(e)37 b(ful\014l)5 b(le)-5 b(d)38 b(by)f(the)g(sets)h Fs(I)3133 2139 y Fq(k)3212 2124 y Fh(and)h(the)e(inte)-5 b(gers)75 2237 y Fs(m)155 2252 y Fr(0)p Fq(;k)252 2237 y Fh(,)p Fs(m)360 2252 y Fr(1)p Fq(;k)458 2237 y Fh(,)40 b(for)f Ft(0)e Fp(\024)f Fs(k)j Fp(\024)d Fs(S)30 b Fp(\000)24 b Ft(1)p Fh(.)60 b(We)39 b(make)g(the)g(fol)5 b(lowing)39 b(assumptions)i(on)e(the)g(initial)h (c)-5 b(onditions.)75 2350 y(Ther)g(e)33 b(exist)g(c)-5 b(onstants)35 b Fs(K)q(;)15 b(T)1119 2364 y Fr(0)1185 2350 y Fs(>)25 b Ft(0)33 b Fh(such)f(that)75 2587 y Ft(\(22\))1000 b(sup)1232 2665 y Fq(x)p Fn(2)p Fq(V)1391 2587 y Fp(j)p Fs(W)1502 2601 y Fq(j)1538 2587 y Ft(\()p Fs(\017;)15 b(xq)1746 2549 y Fq(l)1773 2587 y Ft(\))p Fp(j)26 b(\024)f Fs(K)7 b Fp(j)p Fs(q)s Fp(j)2143 2522 y Fl(1)p 2143 2534 31 3 v 2143 2576 a(4)2183 2549 y Fq(l)2205 2526 y Fl(2)2244 2587 y Ft(\()2312 2526 y(1)p 2289 2566 93 4 v 2289 2649 a Fs(T)2342 2663 y Fr(0)2391 2587 y Ft(\))2426 2549 y Fq(l)2552 2526 y Ft(1)p 2463 2566 225 4 v 2463 2649 a(1)20 b(+)g Fs(l)2648 2623 y Fr(2)75 2857 y Fh(for)32 b(al)5 b(l)32 b Ft(0)26 b Fp(\024)f Fs(j)31 b Fp(\024)25 b Fs(S)e Fp(\000)17 b Ft(1)p Fh(,)32 b(al)5 b(l)32 b Fs(l)c Fp(\025)d Ft(0)p Fh(,)32 b(al)5 b(l)32 b Fs(\017)25 b Fp(2)g Fs(U)10 b(q)1729 2824 y Fn(\000)p Fj(N)1835 2857 y Fh(.)42 b(Then,)32 b(ther)-5 b(e)32 b(exists)g(a)g(unique)f(solution)i(of)e(\(20\),)i (\(21\))1211 3120 y Ft(\()p Fs(\017;)15 b(\034)5 b(;)15 b(z)t Ft(\))27 b Fp(7!)f Fs(W)13 b Ft(\()p Fs(\017;)i(\034)5 b(;)15 b(z)t Ft(\))26 b(=)2132 3034 y Fo(X)2132 3231 y Fq(h)p Fn(\025)p Fr(0)2278 3120 y Fs(W)2364 3135 y Fq(h)2409 3120 y Ft(\()p Fs(\017;)15 b(\034)10 b Ft(\))2616 3058 y Fs(z)2662 3025 y Fq(h)p 2616 3099 92 4 v 2623 3182 a Fs(h)p Ft(!)75 3430 y Fh(which)29 b(is)e(holomorphic)k(on)d Fs(U)10 b(q)1171 3397 y Fn(\000)p Fj(N)1288 3430 y Fp(\002)f Fs(V)20 b(q)1485 3397 y Fj(N)1547 3430 y Fp(\002)9 b Fs(D)s Ft(\(0)p Fs(;)15 b(\032)p Ft(\))p Fh(,)30 b(for)e(every)f(discs) h Fs(D)s Ft(\(0)p Fs(;)15 b(\032)p Ft(\))29 b Fh(c)-5 b(enter)g(e)g(d)29 b(at)g(0,)g(with)f(r)-5 b(adius)75 3543 y Fs(\032)35 b(>)g Ft(0)p Fh(.)58 b(Mor)-5 b(e)g(over,)40 b(ther)-5 b(e)39 b(exist)f(two)h(c)-5 b(onstants)40 b Fs(C)q(;)15 b(T)48 b(>)35 b Ft(0)j Fh(\(dep)-5 b(ending)40 b(on)e Fs(\032)p Fh(,)p Fs(S)5 b Fh(,)p Fp(j)p Fs(q)s Fp(j)p Fh(,)p Fs(b)3179 3558 y Fq(k)3222 3543 y Ft(\()p Fs(\017;)15 b(z)t Ft(\))p Fh(,)p Fs(m)3523 3558 y Fr(0)p Fq(;k)3621 3543 y Fh(,)p Fs(m)3729 3558 y Fr(1)p Fq(;k)3827 3543 y Fh(,)75 3656 y(for)33 b Ft(0)26 b Fp(\024)f Fs(k)j Fp(\024)d Fs(S)g Fp(\000)20 b Ft(1)33 b Fh(and)h Fs(T)1038 3670 y Fr(0)1077 3656 y Fh(\))f(such)g(that)75 3898 y Ft(\(23\))1111 b(sup)1194 3981 y Fq(x)p Fn(2)p Fq(V)t(;z)s Fn(2)p Fq(D)r Fr(\(0)p Fq(;\032)p Fr(\))1649 3898 y Fp(j)p Fs(W)13 b Ft(\()p Fs(\017;)i(xq)1981 3861 y Fq(l)2008 3898 y Fs(;)g(z)t Ft(\))p Fp(j)26 b(\024)f Fs(C)7 b Fp(j)p Fs(q)s Fp(j)2452 3833 y Fl(1)p 2452 3845 31 3 v 2452 3887 a(2)2492 3861 y Fq(l)2514 3837 y Fl(2)2553 3898 y Ft(\()2608 3837 y(1)p 2598 3877 66 4 v 2598 3960 a Fs(T)2674 3898 y Ft(\))2709 3861 y Fq(l)75 4185 y Fh(for)33 b(al)5 b(l)33 b Fs(l)27 b Fp(\025)e Ft(0)p Fh(,)33 b(al)5 b(l)33 b Fs(\017)26 b Fp(2)e Fs(U)10 b(q)1000 4152 y Fn(\000)p Fj(N)1107 4185 y Fh(.)75 4447 y Fi(Pro)s(of)31 b Ft(W)-8 b(e)32 b(consider)e(the)h(sequence)f(of)h(functions)f Fs(W)1985 4462 y Fq(h)2030 4447 y Ft(\()p Fs(\017;)15 b(\034)10 b Ft(\),)31 b Fs(h)26 b Fp(\025)f Fs(S)5 b Ft(,)30 b(de\014ned)f(as)i(follo)m(ws)75 4738 y(\(24\))685 4676 y Fs(W)771 4691 y Fq(h)p Fr(+)p Fq(S)917 4676 y Ft(\()p Fs(\017;)15 b(xq)1125 4643 y Fq(l)1151 4676 y Ft(\))p 685 4717 503 4 v 897 4800 a Fs(h)p Ft(!)1222 4738 y(=)1318 4624 y Fq(S)t Fn(\000)p Fr(1)1320 4651 y Fo(X)1322 4849 y Fq(k)r Fr(=0)1660 4651 y Fo(X)1470 4849 y Fq(h)1511 4858 y Fl(1)1545 4849 y Fr(+)p Fq(h)1641 4858 y Fl(2)1675 4849 y Fr(=)p Fq(h;h)1832 4858 y Fl(1)1865 4849 y Fn(2)p Fq(I)1943 4861 y Fk(k)2006 4676 y Fs(b)2045 4691 y Fq(k)r(h)2125 4700 y Fl(1)2163 4676 y Ft(\()p Fs(\017)p Ft(\))p Fs(x)2322 4643 y Fq(m)2384 4655 y Fl(0)p Fk(;k)2477 4676 y Fs(q)2521 4643 y Fq(m)2583 4655 y Fl(0)p Fk(;k)2671 4643 y Fq(l)p 2006 4717 691 4 v 2177 4800 a Ft(\()p Fs(xq)2308 4774 y Fq(l)2354 4800 y Ft(+)20 b(1\))2717 4676 y Fs(W)2803 4691 y Fq(h)2844 4700 y Fl(2)2878 4691 y Fr(+)p Fq(k)2975 4676 y Ft(\()p Fs(\017;)15 b(xq)3183 4643 y Fq(l)3210 4676 y Ft(\))p 2717 4717 529 4 v 2786 4805 a Fs(h)2838 4819 y Fr(2)2877 4805 y Ft(!)p Fs(q)2946 4774 y Fq(m)3008 4786 y Fl(1)p Fk(;k)3096 4774 y Fq(h)3137 4783 y Fl(2)75 5049 y Ft(for)35 b(all)i Fs(h)d Fp(\025)g Ft(0,)k(all)e Fs(l)g Fp(2)e Fm(N)p Ft(,)j(all)f Fs(x)e Fp(2)g Fs(V)56 b Ft(and)35 b(all)h Fs(\017)e Fp(2)g Fs(U)10 b(q)2083 5016 y Fn(\000)p Fj(N)2190 5049 y Ft(.)56 b(One)35 b(c)m(hec)m(ks)i(that)g(the)f(sequence)g Fs(W)3587 5064 y Fq(h)3631 5049 y Ft(\()p Fs(\017;)15 b(\034)10 b Ft(\),)75 5162 y Fs(h)31 b Fp(\025)f Ft(0,)35 b(of)f(holomorphic)g(functions)f (on)g Fs(U)10 b(q)1635 5129 y Fn(\000)p Fj(N)1765 5162 y Fp(\002)22 b Fs(V)e(q)1975 5129 y Fj(N)2027 5162 y Ft(,)34 b(satis\014es)g(the)g(recursion)f(\(24\))i(if)f(and)f(only)h (if)f(the)75 5275 y(formal)e(series)1422 5424 y Fs(W)13 b Ft(\()p Fs(\017;)i(\034)5 b(;)15 b(z)t Ft(\))26 b(=)1921 5338 y Fo(X)1922 5535 y Fq(h)p Fn(\025)p Fr(0)2068 5424 y Fs(W)2154 5439 y Fq(h)2198 5424 y Ft(\()p Fs(\017;)15 b(\034)10 b Ft(\))2405 5362 y Fs(z)2451 5329 y Fq(h)p 2405 5403 92 4 v 2412 5486 a Fs(h)p Ft(!)75 5686 y(in)30 b(the)h Fs(z)j Ft(v)-5 b(ariable,)32 b(satis\014es)e(the)h(problem)f (\(20\),)i(\(21\).)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 3809 100 a Ft(9)75 399 y(W)-8 b(e)32 b(consider)e(the)g(sequence)h(of)g(p)s(ositiv)m(e)g(real)g(n)m(um)m(b)s (ers)e(de\014ned)g(b)m(y)1457 570 y Fs(w)1522 585 y Fq(l)q(;h)1609 570 y Ft(\()p Fs(\017)p Ft(\))c(=)k(sup)1837 648 y Fq(x)p Fn(2)p Fq(V)1996 570 y Fp(j)p Fs(W)2107 585 y Fq(h)2152 570 y Ft(\()p Fs(\017;)15 b(xq)2360 533 y Fq(l)2386 570 y Ft(\))p Fp(j)p Fs(;)75 807 y Ft(for)41 b(all)h Fs(l)j Fp(\025)e Ft(0,)i(all)d Fs(h)h Fp(\025)g Ft(0,)i(all)d Fs(\017)h Fp(2)g Fs(U)10 b(q)1561 774 y Fn(\000)p Fj(N)1667 807 y Ft(.)73 b(The)41 b(sequence)h(\()p Fs(w)2451 822 y Fq(l)q(;h)2537 807 y Ft(\()p Fs(\017)p Ft(\)\))2679 822 y Fq(l)q Fn(\025)p Fr(0)p Fq(;h)p Fn(\025)p Fr(0)2988 807 y Ft(satis\014es)g(the)f(follo)m(wing)75 920 y(inequalities:)h (there)31 b(exists)g Fs(r)d(>)c Ft(0)31 b(\(dep)s(ending)e(on)i Fs(m)1991 935 y Fr(0)p Fq(;k)2088 920 y Ft(,)p Fs(V)21 b Ft(\))30 b(suc)m(h)g(that)75 1183 y(\(25\))802 1120 y Fs(w)867 1135 y Fq(l)q(;h)p Fr(+)p Fq(S)1055 1120 y Ft(\()p Fs(\017)p Ft(\))p 802 1162 361 4 v 944 1246 a Fs(h)p Ft(!)1198 1183 y Fp(\024)1294 1070 y Fq(S)t Fn(\000)p Fr(1)1297 1097 y Fo(X)1298 1295 y Fq(k)r Fr(=0)1636 1097 y Fo(X)1446 1295 y Fq(h)1487 1304 y Fl(1)1521 1295 y Fr(+)p Fq(h)1617 1304 y Fl(2)1651 1295 y Fr(=)p Fq(h;h)1808 1304 y Fl(1)1842 1295 y Fn(2)p Fq(I)1920 1307 y Fk(k)1983 1122 y Fp(j)p Fs(b)2047 1137 y Fq(k)r(h)2127 1146 y Fl(1)2165 1122 y Ft(\()p Fs(\017)p Ft(\))p Fp(j)p Fs(r)s Fp(j)p Fs(q)s Fp(j)2435 1089 y Fq(m)2497 1101 y Fl(0)p Fk(;k)2586 1089 y Fq(l)p 1983 1162 629 4 v 2275 1246 a Fs(\016)2686 1120 y(w)2751 1135 y Fq(l)q(;h)2834 1144 y Fl(2)2868 1135 y Fr(+)p Fq(k)2965 1120 y Ft(\()p Fs(\017)p Ft(\))p 2631 1162 496 4 v 2631 1254 a Fs(h)2683 1268 y Fr(2)2723 1254 y Ft(!)p Fp(j)p Fs(q)s Fp(j)2842 1223 y Fr(\()p Fq(m)2931 1235 y Fl(1)p Fk(;k)3020 1223 y Fq(h)3061 1232 y Fl(2)3095 1223 y Fr(\))75 1462 y Ft(for)g(all)h Fs(l)d Fp(\025)d Ft(0,)31 b(all)g Fs(h)25 b Fp(\025)g Ft(0,)31 b(all)h Fs(\017)25 b Fp(2)g Fs(U)10 b(q)1383 1429 y Fn(\000)p Fj(N)1490 1462 y Ft(.)75 1575 y(W)-8 b(e)32 b(consider)e(the)g (sequence)h(of)g(real)g(n)m(um)m(b)s(ers)e(\()p Fs(v)1846 1590 y Fq(l)q(;h)1932 1575 y Ft(\()p Fs(\017)p Ft(\)\))2074 1590 y Fq(l)q Fn(\025)p Fr(0)p Fq(;h)p Fn(\025)p Fr(0)2372 1575 y Ft(de\014ned)g(b)m(y)i(the)f(follo)m(wing)i(recursion)75 1838 y(\(26\))813 1775 y Fs(v)857 1790 y Fq(l)q(;h)p Fr(+)p Fq(S)1045 1775 y Ft(\()p Fs(\017)p Ft(\))p 813 1817 340 4 v 944 1900 a Fs(h)p Ft(!)1188 1838 y(=)1284 1724 y Fq(S)t Fn(\000)p Fr(1)1286 1752 y Fo(X)1288 1949 y Fq(k)r Fr(=0)1626 1752 y Fo(X)1436 1949 y Fq(h)1477 1958 y Fl(1)1511 1949 y Fr(+)p Fq(h)1607 1958 y Fl(2)1641 1949 y Fr(=)p Fq(h;h)1798 1958 y Fl(1)1831 1949 y Fn(2)p Fq(I)1909 1961 y Fk(k)1972 1777 y Fp(j)p Fs(b)2036 1792 y Fq(k)r(h)2116 1801 y Fl(1)2154 1777 y Ft(\()p Fs(\017)p Ft(\))p Fp(j)p Fs(r)s Fp(j)p Fs(q)s Fp(j)2424 1744 y Fq(m)2486 1756 y Fl(0)p Fk(;k)2575 1744 y Fq(l)p 1972 1817 629 4 v 2265 1900 a Fs(\016)2686 1775 y(v)2730 1790 y Fq(l)q(;h)2813 1799 y Fl(2)2847 1790 y Fr(+)p Fq(k)2944 1775 y Ft(\()p Fs(\017)p Ft(\))p 2621 1817 496 4 v 2621 1909 a Fs(h)2673 1923 y Fr(2)2713 1909 y Ft(!)p Fp(j)p Fs(q)s Fp(j)2832 1878 y Fr(\()p Fq(m)2921 1890 y Fl(1)p Fk(;k)3010 1878 y Fq(h)3051 1887 y Fl(2)3085 1878 y Fr(\))75 2117 y Ft(with)39 b(initial)h(conditions)f Fs(v)1049 2132 y Fq(l)q(;j)1127 2117 y Ft(\()p Fs(\017)p Ft(\))i(=)e Fs(w)1450 2132 y Fq(l)q(;j)1528 2117 y Ft(\()p Fs(\017)p Ft(\),)j(for)d(0)h Fp(\024)f Fs(j)45 b Fp(\024)40 b Fs(S)30 b Fp(\000)c Ft(1,)42 b(all)e Fs(l)h Fp(\025)f Ft(0)f(and)f(all)i Fs(\017)g Fp(2)f Fs(U)10 b(q)3544 2084 y Fn(\000)p Fj(N)3651 2117 y Ft(.)66 b(By)75 2230 y(construction,)31 b(w)m(e)g(ha)m(v)m(e)h (that)75 2402 y(\(27\))1421 b Fs(w)1721 2417 y Fq(l)q(;h)1807 2402 y Ft(\()p Fs(\017)p Ft(\))26 b Fp(\024)f Fs(v)2080 2417 y Fq(l)q(;h)2166 2402 y Ft(\()p Fs(\017)p Ft(\))75 2573 y(for)30 b(all)h Fs(l)d Fp(\025)d Ft(0,)31 b(all)g Fs(h)25 b Fp(\025)g Ft(0,)31 b(all)h Fs(\017)25 b Fp(2)g Fs(U)10 b(q)1383 2540 y Fn(\000)p Fj(N)1490 2573 y Ft(.)75 2725 y(In)31 b(the)h(follo)m(wing,)h(w)m(e)f(put)f Fs(a)1109 2740 y Fq(k)1152 2725 y Ft(\()p Fs(\017;)15 b(x)p Ft(\))28 b(=)1477 2657 y Fo(P)1573 2752 y Fq(s)p Fn(2)p Fq(I)1684 2764 y Fk(k)1726 2725 y Ft(\()p Fp(j)p Fs(b)1825 2740 y Fq(k)r(s)1901 2725 y Ft(\()p Fs(\017)p Ft(\))p Fp(j)p Fs(r)s(=\016)s Ft(\))p Fs(x)2252 2692 y Fq(s)2322 2725 y Ft(for)k(0)27 b Fp(\024)g Fs(k)k Fp(\024)c Fs(S)f Fp(\000)21 b Ft(1)32 b(and)f(w)m(e)h(consider)f(the)75 2838 y(formal)g(p)s(o)m(w)m (er)f(series)1362 3005 y Fs(V)20 b Ft(\()p Fs(\017;)15 b(\030)t(;)g(x)p Ft(\))26 b(=)1906 2918 y Fo(X)1840 3116 y Fq(l)q Fn(\025)p Fr(0)p Fq(;h)p Fn(\025)p Fr(0)2118 3005 y Fs(v)2162 3020 y Fq(l)q(;h)2248 3005 y Ft(\()p Fs(\017)p Ft(\))p Fs(\030)2399 2967 y Fq(l)2436 2943 y Fs(x)2488 2910 y Fq(h)p 2436 2984 97 4 v 2446 3067 a Fs(h)p Ft(!)2543 3005 y Fs(:)75 3252 y Ft(F)-8 b(rom)31 b(the)f(recursion)g(\(26\),)j(w)m(e)d(get)i(that)f Fs(V)20 b Ft(\()p Fs(\017;)15 b(\030)t(;)g(x)p Ft(\))32 b(satis\014es)f(the)f (follo)m(wing)i(Cauc)m(h)m(y)f(problem)75 3510 y(\(28\))709 b Fs(@)997 3473 y Fq(S)992 3533 y(x)1048 3510 y Fs(V)20 b Ft(\()p Fs(\017;)15 b(\030)t(;)g(x)p Ft(\))27 b(=)1527 3396 y Fq(S)t Fn(\000)p Fr(1)1530 3424 y Fo(X)1531 3621 y Fq(k)r Fr(=0)1679 3510 y Fs(a)1727 3525 y Fq(k)1770 3510 y Ft(\()p Fs(\017;)15 b(x)p Ft(\)\()p Fs(@)2057 3473 y Fq(k)2052 3533 y(x)2101 3510 y Fs(V)20 b Ft(\)\()p Fs(\017;)15 b(q)2365 3473 y Fq(m)2427 3485 y Fl(0)p Fk(;k)2520 3510 y Fs(\030)t(;)g(x=)p Fp(j)p Fs(q)s Fp(j)2795 3473 y Fq(m)2857 3485 y Fl(1)p Fk(;k)2950 3510 y Ft(\))75 3761 y(with)30 b(initial)i(conditions)75 3932 y(\(29\))597 b(\()p Fs(@)920 3895 y Fq(j)915 3955 y(x)959 3932 y Fs(V)20 b Ft(\)\()p Fs(\017;)15 b(\030)t(;)g Ft(0\))28 b(=)d Fs(\036)1521 3946 y Fq(j)1557 3932 y Ft(\()p Fs(\017;)15 b(\030)t Ft(\))27 b(:=)1896 3846 y Fo(X)1906 4044 y Fq(l)q Fn(\025)p Fr(0)2042 3932 y Fs(w)2107 3947 y Fq(l)q(;j)2185 3932 y Ft(\()p Fs(\017)p Ft(\))p Fs(\030)2336 3895 y Fq(l)2424 3932 y Fs(;)76 b Ft(0)25 b Fp(\024)g Fs(j)31 b Fp(\024)25 b Fs(S)g Fp(\000)20 b Ft(1)p Fs(:)75 4210 y Ft(F)-8 b(rom)24 b(the)g(h)m(yp)s(othesis)g(\(22\),)j(w)m(e)d(get)h (that)f Fs(\036)1627 4224 y Fq(j)1664 4210 y Ft(\()p Fs(\017;)15 b(\030)t Ft(\))25 b(b)s(elongs)f(to)g Fm(E)2365 4228 y Fn(j)p Fq(q)r Fn(j)p Fq(;)p Fr(\()p Fq(T)2527 4237 y Fl(0)2561 4228 y Fq(;X)2639 4237 y Fl(0)2673 4228 y Fr(\))2704 4210 y Ft(,)i(for)d(all)i Fs(X)3082 4224 y Fr(0)3147 4210 y Fs(>)g Ft(0,)h(all)e Fs(\017)i Fp(2)e Fs(U)10 b(q)3722 4177 y Fn(\000)p Fj(N)3829 4210 y Ft(.)75 4323 y(By)40 b(h)m(yp)s(othesis,)j(the)d(assumption)f Fi(\(A\))h Ft(holds)f(for)h(the)g(sets)h Fs(I)2360 4338 y Fq(k)2442 4323 y Ft(and)f(the)g(n)m(um)m(b)s(ers)e Fs(m)3251 4338 y Fr(0)p Fq(;k)3349 4323 y Ft(,)p Fs(m)3454 4338 y Fr(1)p Fq(;k)3551 4323 y Ft(.)70 b(F)-8 b(rom)75 4435 y(Theorem)40 b(1,)k(w)m(e)d(deduce)g(that)g(the)g(unique)f (solution)h Fs(V)20 b Ft(\()p Fs(\017;)15 b(\030)t(;)g(x)p Ft(\))42 b(of)f(the)g(problem)f(\(28\),)45 b(\(29\))d(satis\014es)75 4548 y Fs(V)20 b Ft(\()p Fs(\017;)15 b(\030)t(;)g(x)p Ft(\))40 b Fp(2)e Fs(E)637 4567 y Fn(j)p Fq(q)r Fn(j)p Fq(;)p Fr(\()p Fq(T)6 b(;X)f Fr(\))956 4548 y Ft(for)38 b(real)h(n)m(um)m(b)s(ers)e Fs(T)8 b(;)15 b(X)46 b(>)38 b Ft(0)g(with)g(0)h Fs(<)f(T)51 b(<)38 b(T)2751 4562 y Fr(0)2790 4548 y Ft(,)i(0)f Fs(<)f(X)46 b(<)38 b(X)3353 4562 y Fr(0)3393 4548 y Ft(.)64 b(F)-8 b(rom)38 b(the)75 4683 y(inequalit)m(y)31 b Fs(P)13 b Ft(\()p Fs(l)r(;)i(h)p Ft(\))27 b Fp(\024)890 4647 y Fq(l)912 4624 y Fl(2)p 890 4662 57 4 v 901 4715 a Fr(2)975 4683 y Fp(\000)1075 4647 y Fq(h)1116 4624 y Fl(2)p 1075 4662 76 4 v 1095 4715 a Fr(4)1160 4683 y Ft(,)j(for)f(all)i Fs(l)c Fp(\025)e Ft(0,)30 b Fs(h)c Fp(\025)f Ft(0,)30 b(w)m(e)g(get)h(the)e(estimates)j (:)40 b(there)30 b(exists)g Fs(C)3461 4650 y Fn(0)3509 4683 y Fs(>)25 b Ft(0)30 b(suc)m(h)75 4796 y(that)75 4983 y(\(30\))1034 b Fp(j)p Fs(v)1338 4998 y Fq(l)q(;h)1424 4983 y Ft(\()p Fs(\017)p Ft(\))p Fp(j)26 b(\024)f Fs(C)1750 4946 y Fn(0)1773 4983 y Fp(j)p Fs(q)s Fp(j)1877 4919 y Fk(l)1897 4898 y Fl(2)p 1877 4931 55 3 v 1889 4972 a(2)1946 4983 y Fp(j)p Fs(q)s Fp(j)2050 4918 y Fg(\000)p Fk(h)2134 4897 y Fl(2)p 2050 4931 119 3 v 2094 4972 a(4)2183 4983 y Fs(h)p Ft(!\()2316 4922 y(1)p 2305 4962 66 4 v 2305 5046 a Fs(T)2382 4983 y Ft(\))2417 4946 y Fq(l)2443 4983 y Ft(\()2507 4922 y(1)p 2488 4962 83 4 v 2488 5046 a Fs(X)2581 4983 y Ft(\))2616 4946 y Fq(h)75 5191 y Ft(for)30 b(all)h Fs(l)d Fp(\025)d Ft(0,)31 b(all)g Fs(h)25 b Fp(\025)g Ft(0,)31 b(all)h Fs(\017)25 b Fp(2)g Fs(U)10 b(q)1383 5158 y Fn(\000)p Fj(N)1490 5191 y Ft(.)40 b(F)-8 b(rom)31 b(the)g(inequalities)g(\(27\))h(and)e(\(30\),)i(w)m(e)f(get)h(that)75 5401 y(\(31\))748 b(sup)832 5484 y Fq(x)p Fn(2)p Fq(V)t(;z)s Fn(2)p Fq(D)r Fr(\(0)p Fq(;\032)p Fr(\))1287 5401 y Fp(j)p Fs(W)13 b Ft(\()p Fs(\017;)i(xq)1619 5364 y Fq(l)1645 5401 y Fs(;)g(z)t Ft(\))p Fp(j)26 b(\024)f Fs(C)1985 5364 y Fn(0)2008 5401 y Fp(j)p Fs(q)s Fp(j)2112 5337 y Fk(l)2132 5316 y Fl(2)p 2112 5349 55 3 v 2125 5390 a(2)2181 5401 y Ft(\()2237 5340 y(1)p 2226 5380 66 4 v 2226 5464 a Fs(T)2302 5401 y Ft(\))2337 5364 y Fq(l)2364 5401 y Ft(\()2399 5315 y Fo(X)2399 5512 y Fq(h)p Fn(\025)p Fr(0)2545 5401 y Fp(j)p Fs(q)s Fp(j)2639 5364 y Fn(\000)p Fq(h)2735 5340 y Fl(2)2770 5364 y Fq(=)p Fr(4)2845 5401 y Ft(\()2908 5340 y Fs(\032)p 2890 5380 83 4 v 2890 5464 a(X)2982 5401 y Ft(\))3017 5364 y Fq(h)3062 5401 y Ft(\))75 5679 y(for)30 b(all)h Fs(l)d Fp(\025)d Ft(0,)31 b(all)g Fs(\032)25 b(>)g Ft(0,)31 b(all)g Fs(\017)25 b Fp(2)g Fs(U)10 b(q)1377 5646 y Fn(\000)p Fj(N)1484 5679 y Ft(.)41 b(So)30 b(that)h(the)g(estimates)h(\(23\))g(hold.)937 b Ff(2)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 3764 100 a Ft(10)75 399 y Fi(Theorem)35 b(3)46 b Fh(L)-5 b(et)37 b(the)h(assumption)h Fi(\(A\))d Fh(\(of)i(se)-5 b(ction)37 b(2\))h(b)-5 b(e)37 b(ful\014l)5 b(le)-5 b(d)38 b(by)f(the)g(sets)h Fs(I)3133 414 y Fq(k)3212 399 y Fh(and)h(the)e(inte)-5 b(gers)75 511 y Fs(m)155 526 y Fr(0)p Fq(;k)252 511 y Fh(,)p Fs(m)360 526 y Fr(1)p Fq(;k)458 511 y Fh(,)33 b(for)h Ft(0)26 b Fp(\024)g Fs(k)k Fp(\024)c Fs(S)f Fp(\000)c Ft(1)p Fh(.)44 b(Assume)33 b(that)h(the)g(initial)f(c)-5 b(onditions)35 b Ft(\()p Fs(\017;)15 b(\034)10 b Ft(\))28 b Fp(7!)e Fs(W)3105 525 y Fq(j)3141 511 y Ft(\()p Fs(\017;)15 b(\034)10 b Ft(\))p Fh(,)34 b(for)g Ft(0)27 b Fp(\024)f Fs(j)32 b Fp(\024)75 624 y Fs(S)27 b Fp(\000)21 b Ft(1)p Fh(,)35 b(ar)-5 b(e)35 b(holomorphic)j(and)e(b)-5 b(ounde)g(d)36 b(on)e Fs(D)s Ft(\(0)p Fs(;)15 b(r)1930 638 y Fr(0)1971 624 y Ft(\))22 b Fp(\002)f Fs(D)s Ft(\(0)p Fs(;)15 b(\032)2365 638 y Fr(0)2406 624 y Ft(\))p Fh(,)35 b(for)g(some)g Ft(0)29 b Fs(<)f(\032)3105 638 y Fr(0)3173 624 y Fs(<)h Ft(1)p Fh(.)47 b(Then,)36 b(ther)-5 b(e)75 737 y(exists)33 b(a)g(unique)f(solution)h(of)g(\(20\),)h(\(21\))1211 987 y Ft(\()p Fs(\017;)15 b(\034)5 b(;)15 b(z)t Ft(\))27 b Fp(7!)f Fs(W)13 b Ft(\()p Fs(\017;)i(\034)5 b(;)15 b(z)t Ft(\))26 b(=)2132 901 y Fo(X)2132 1098 y Fq(h)p Fn(\025)p Fr(0)2278 987 y Fs(W)2364 1002 y Fq(h)2409 987 y Ft(\()p Fs(\017;)15 b(\034)10 b Ft(\))2616 926 y Fs(z)2662 893 y Fq(h)p 2616 966 92 4 v 2623 1049 a Fs(h)p Ft(!)75 1273 y Fh(which)33 b(is)f(b)-5 b(ounde)g(d)34 b(holomorphic)h(on)e Fs(D)s Ft(\(0)p Fs(;)15 b(r)1658 1287 y Fr(0)1698 1273 y Ft(\))20 b Fp(\002)f Fs(D)s Ft(\(0)p Fs(;)c(\032)2088 1287 y Fr(0)2128 1273 y Ft(\))20 b Fp(\002)f Fs(D)s Ft(\(0)p Fs(;)c(\032)p Ft(\))p Fh(,)33 b(for)g(every)f(discs)h Fs(D)s Ft(\(0)p Fs(;)15 b(\032)p Ft(\))33 b Fh(c)-5 b(enter)g(e)g(d)75 1386 y(at)33 b(0,)g(with)h(r)-5 b(adius)34 b Fs(\032)25 b(>)g Ft(0)p Fh(.)75 1631 y Fi(Pro)s(of)31 b Ft(W)-8 b(e)32 b(consider)e(the)h(sequence)f(of)h(functions)f Fs(W)1985 1646 y Fq(h)2030 1631 y Ft(\()p Fs(\017;)15 b(\034)10 b Ft(\),)31 b Fs(h)26 b Fp(\025)f Fs(S)5 b Ft(,)30 b(de\014ned)f(as)i(follo)m(ws)75 1908 y(\(32\))867 1847 y Fs(W)953 1862 y Fq(h)p Fr(+)p Fq(S)1099 1847 y Ft(\()p Fs(\017;)15 b(\034)10 b Ft(\))p 867 1887 431 4 v 1043 1971 a Fs(h)p Ft(!)1333 1908 y(=)1429 1795 y Fq(S)t Fn(\000)p Fr(1)1431 1822 y Fo(X)1433 2020 y Fq(k)r Fr(=0)1771 1822 y Fo(X)1580 2020 y Fq(h)1621 2029 y Fl(1)1656 2020 y Fr(+)p Fq(h)1752 2029 y Fl(2)1786 2020 y Fr(=)p Fq(h;h)1943 2029 y Fl(1)1976 2020 y Fn(2)p Fq(I)2054 2032 y Fk(k)2117 1847 y Fs(b)2156 1862 y Fq(k)r(h)2236 1871 y Fl(1)2274 1847 y Ft(\()p Fs(\017)p Ft(\))p Fs(\034)2431 1814 y Fq(m)2493 1826 y Fl(0)p Fk(;k)p 2117 1887 469 4 v 2213 1971 a Ft(\()p Fs(\034)30 b Ft(+)20 b(1\))2606 1847 y Fs(W)2692 1862 y Fq(h)2733 1871 y Fl(2)2767 1862 y Fr(+)p Fq(k)2864 1847 y Ft(\()p Fs(\017;)15 b(\034)10 b Ft(\))p 2606 1887 457 4 v 2639 1976 a Fs(h)2691 1990 y Fr(2)2731 1976 y Ft(!)p Fs(q)2800 1944 y Fq(m)2862 1956 y Fl(1)p Fk(;k)2950 1944 y Fq(h)2991 1953 y Fl(2)75 2196 y Ft(for)25 b(all)i Fs(h)e Fp(\025)g Ft(0,)j(all)e Fs(\034)35 b Fp(2)25 b Fs(D)s Ft(\(0)p Fs(;)15 b(\032)1129 2210 y Fr(0)1170 2196 y Ft(\))25 b(and)g(all)i Fs(\017)e Fp(2)g Fs(D)s Ft(\(0)p Fs(;)15 b(r)1911 2210 y Fr(0)1951 2196 y Ft(\).)40 b(One)25 b(c)m(hec)m(ks)i(that)f(the)g(sequence)g Fs(W)3316 2211 y Fq(h)3360 2196 y Ft(\()p Fs(\017;)15 b(\034)10 b Ft(\),)28 b Fs(h)e Fp(\025)f Ft(0,)75 2309 y(of)37 b(holomorphic)g(functions)f(on)h Fs(D)s Ft(\(0)p Fs(;)15 b(r)1482 2323 y Fr(0)1522 2309 y Ft(\))25 b Fp(\002)f Fs(D)s Ft(\(0)p Fs(;)15 b(\032)1922 2323 y Fr(0)1962 2309 y Ft(\),)39 b(satis\014es)e(the)g(recursion)g(\(32\))h(if)f(and)f (only)h(if)f(the)75 2422 y(formal)31 b(series)1422 2571 y Fs(W)13 b Ft(\()p Fs(\017;)i(\034)5 b(;)15 b(z)t Ft(\))26 b(=)1921 2485 y Fo(X)1922 2682 y Fq(h)p Fn(\025)p Fr(0)2068 2571 y Fs(W)2154 2586 y Fq(h)2198 2571 y Ft(\()p Fs(\017;)15 b(\034)10 b Ft(\))2405 2509 y Fs(z)2451 2476 y Fq(h)p 2405 2550 92 4 v 2412 2633 a Fs(h)p Ft(!)75 2826 y(in)30 b(the)h Fs(z)j Ft(v)-5 b(ariable,)32 b(satis\014es)e(the)h(problem)f (\(20\),)i(\(21\).)75 2939 y(W)-8 b(e)32 b(consider)e(the)g(sequence)h (of)g(p)s(ositiv)m(e)g(real)g(n)m(um)m(b)s(ers)e(de\014ned)g(b)m(y)1422 3130 y Fs(w)1487 3145 y Fq(h)1532 3130 y Ft(\()p Fs(\017)p Ft(\))d(=)120 b(sup)1761 3213 y Fq(\034)8 b Fn(2)p Fq(D)r Fr(\(0)p Fq(;\032)2025 3222 y Fl(0)2060 3213 y Fr(\))2103 3130 y Fp(j)p Fs(W)2214 3145 y Fq(h)2259 3130 y Ft(\()p Fs(\017;)15 b(\034)10 b Ft(\))p Fp(j)p Fs(;)75 3393 y Ft(for)30 b(all)h Fs(h)26 b Fp(\025)f Ft(0,)31 b(all)g Fs(\017)25 b Fp(2)g Fs(D)s Ft(\(0)p Fs(;)15 b(r)1128 3407 y Fr(0)1169 3393 y Ft(\).)41 b(The)30 b(sequence)g(\()p Fs(w)1933 3408 y Fq(h)1979 3393 y Ft(\()p Fs(\017)p Ft(\)\))2121 3408 y Fq(h)p Fn(\025)p Fr(0)2287 3393 y Ft(satis\014es)h(the)f(follo)m (wing)i(inequalities)75 3674 y(\(33\))860 3612 y Fs(w)925 3627 y Fq(h)p Fr(+)p Fq(S)1071 3612 y Ft(\()p Fs(\017)p Ft(\))p 860 3653 320 4 v 981 3736 a Fs(h)p Ft(!)1214 3674 y Fp(\024)1310 3560 y Fq(S)t Fn(\000)p Fr(1)1313 3588 y Fo(X)1314 3785 y Fq(k)r Fr(=0)1652 3588 y Fo(X)1462 3785 y Fq(h)1503 3794 y Fl(1)1537 3785 y Fr(+)p Fq(h)1633 3794 y Fl(2)1667 3785 y Fr(=)p Fq(h;h)1824 3794 y Fl(1)1858 3785 y Fn(2)p Fq(I)1936 3797 y Fk(k)1999 3612 y Fp(j)p Fs(b)2063 3627 y Fq(k)r(h)2143 3636 y Fl(1)2181 3612 y Ft(\()p Fs(\017)p Ft(\))p Fp(j)p Fs(\032)2360 3626 y Fr(0)2400 3579 y Fq(m)2462 3591 y Fl(0)p Fk(;k)p 1999 3653 556 4 v 2155 3736 a Ft(1)21 b Fp(\000)e Fs(\032)2358 3750 y Fr(0)2649 3612 y Fs(w)2714 3627 y Fq(h)2755 3636 y Fl(2)2789 3627 y Fr(+)p Fq(k)2887 3612 y Ft(\()p Fs(\017)p Ft(\))p 2574 3653 496 4 v 2574 3745 a Fs(h)2626 3759 y Fr(2)2666 3745 y Ft(!)p Fp(j)p Fs(q)s Fp(j)2785 3714 y Fr(\()p Fq(m)2874 3726 y Fl(1)p Fk(;k)2963 3714 y Fq(h)3004 3723 y Fl(2)3038 3714 y Fr(\))75 3962 y Ft(for)30 b(all)h Fs(h)26 b Fp(\025)f Ft(0,)31 b(all)g Fs(\017)25 b Fp(2)g Fs(D)s Ft(\(0)p Fs(;)15 b(r)1128 3976 y Fr(0)1169 3962 y Ft(\).)75 4075 y(W)-8 b(e)32 b(consider)e(the)g(sequence)h(of)g(real) g(n)m(um)m(b)s(ers)e(\()p Fs(v)1846 4090 y Fq(h)1891 4075 y Ft(\()p Fs(\017)p Ft(\)\))2033 4090 y Fq(h)p Fn(\025)p Fr(0)2199 4075 y Ft(de\014ned)g(b)m(y)i(the)f(follo)m(wing)i(recursion) 75 4355 y(\(34\))870 4294 y Fs(v)914 4309 y Fq(h)p Fr(+)p Fq(S)1061 4294 y Ft(\()p Fs(\017)p Ft(\))p 870 4334 299 4 v 981 4418 a Fs(h)p Ft(!)1204 4355 y(=)1300 4242 y Fq(S)t Fn(\000)p Fr(1)1302 4269 y Fo(X)1304 4467 y Fq(k)r Fr(=0)1642 4269 y Fo(X)1451 4467 y Fq(h)1492 4476 y Fl(1)1527 4467 y Fr(+)p Fq(h)1623 4476 y Fl(2)1657 4467 y Fr(=)p Fq(h;h)1814 4476 y Fl(1)1847 4467 y Fn(2)p Fq(I)1925 4479 y Fk(k)1988 4294 y Fp(j)p Fs(b)2052 4309 y Fq(k)r(h)2132 4318 y Fl(1)2170 4294 y Ft(\()p Fs(\017)p Ft(\))p Fp(j)p Fs(\032)2349 4308 y Fr(0)2390 4261 y Fq(m)2452 4273 y Fl(0)p Fk(;k)p 1988 4334 556 4 v 2144 4418 a Ft(1)21 b Fp(\000)f Fs(\032)2348 4432 y Fr(0)2649 4294 y Fs(v)2693 4309 y Fq(h)2734 4318 y Fl(2)2768 4309 y Fr(+)p Fq(k)2866 4294 y Ft(\()p Fs(\017)p Ft(\))p 2564 4334 496 4 v 2564 4426 a Fs(h)2616 4440 y Fr(2)2655 4426 y Ft(!)p Fp(j)p Fs(q)s Fp(j)2774 4395 y Fr(\()p Fq(m)2863 4407 y Fl(1)p Fk(;k)2952 4395 y Fq(h)2993 4404 y Fl(2)3028 4395 y Fr(\))75 4643 y Ft(with)34 b(initial)h(conditions)f Fs(v)1034 4657 y Fq(j)1071 4643 y Ft(\()p Fs(\017)p Ft(\))d(=)g Fs(w)1376 4657 y Fq(j)1412 4643 y Ft(\()p Fs(\017)p Ft(\),)36 b(for)e(0)d Fp(\024)g Fs(j)36 b Fp(\024)31 b Fs(S)d Fp(\000)22 b Ft(1)34 b(and)f(all)i Fs(\017)c Fp(2)g Fs(D)s Ft(\(0)p Fs(;)15 b(r)3041 4657 y Fr(0)3081 4643 y Ft(\).)51 b(By)35 b(construction,)75 4756 y(w)m(e)c(ha)m(v)m(e)g(that)75 4947 y(\(35\))1462 b Fs(w)1762 4962 y Fq(h)1807 4947 y Ft(\()p Fs(\017)p Ft(\))26 b Fp(\024)f Fs(v)2080 4962 y Fq(h)2125 4947 y Ft(\()p Fs(\017)p Ft(\))75 5138 y(for)30 b(all)h Fs(h)26 b Fp(\025)f Ft(0,)31 b(all)g Fs(\017)25 b Fp(2)g Fs(D)s Ft(\(0)p Fs(;)15 b(r)1128 5152 y Fr(0)1169 5138 y Ft(\).)216 5297 y(In)33 b(the)h(follo)m(wing,)j(w)m(e)d(put)f Fs(c)1253 5312 y Fq(k)1296 5297 y Ft(\()p Fs(\017;)15 b(x)p Ft(\))32 b(=)1629 5228 y Fo(P)1725 5324 y Fq(s)p Fn(2)p Fq(I)1836 5336 y Fk(k)1878 5297 y Ft(\()p Fp(j)p Fs(b)1977 5312 y Fq(k)r(s)2053 5297 y Ft(\()p Fs(\017)p Ft(\))p Fp(j)p Fs(\032)2232 5311 y Fr(0)2272 5264 y Fq(m)2334 5276 y Fl(0)p Fk(;k)2426 5297 y Fs(=)p Ft(\(1)24 b Fp(\000)e Fs(\032)2715 5311 y Fr(0)2755 5297 y Ft(\)\))p Fs(x)2877 5264 y Fq(s)2948 5297 y Ft(for)33 b(0)f Fp(\024)f Fs(k)j Fp(\024)d Fs(S)d Fp(\000)22 b Ft(1)34 b(and)75 5410 y(w)m(e)d(consider) f(the)h(formal)f(p)s(o)m(w)m(er)h(series)1525 5654 y Fs(V)20 b Ft(\()p Fs(\017;)15 b(x)p Ft(\))27 b(=)1920 5568 y Fo(X)1920 5766 y Fq(h)p Fn(\025)p Fr(0)2066 5654 y Fs(v)2110 5669 y Fq(h)2155 5654 y Ft(\()p Fs(\017)p Ft(\))2272 5593 y Fs(x)2324 5560 y Fq(h)p 2272 5633 97 4 v 2282 5717 a Fs(h)p Ft(!)2379 5654 y Fs(:)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 3764 100 a Ft(11)75 399 y(F)-8 b(rom)31 b(the)f(recursion)g(\(34\),)j(w)m(e)d(get)i(that)f Fs(V)20 b Ft(\()p Fs(\017;)15 b(x)p Ft(\))31 b(satis\014es)g(the)g (follo)m(wing)h(Cauc)m(h)m(y)e(problem)75 689 y(\(36\))897 b Fs(@)1185 651 y Fq(S)1180 711 y(x)1236 689 y Fs(V)20 b Ft(\()p Fs(\017;)15 b(x)p Ft(\))26 b(=)1630 575 y Fq(S)t Fn(\000)p Fr(1)1633 603 y Fo(X)1634 800 y Fq(k)r Fr(=0)1782 689 y Fs(c)1821 704 y Fq(k)1864 689 y Ft(\()p Fs(\017;)15 b(x)p Ft(\)\()p Fs(@)2151 651 y Fq(k)2146 711 y(x)2195 689 y Fs(V)21 b Ft(\)\()p Fs(\017;)15 b(x=)p Fp(j)p Fs(q)s Fp(j)2607 651 y Fq(m)2669 663 y Fl(1)p Fk(;k)2762 689 y Ft(\))75 972 y(with)30 b(initial)i(conditions)75 1176 y(\(37\))810 b(\()p Fs(@)1133 1139 y Fq(j)1128 1199 y(x)1172 1176 y Fs(V)21 b Ft(\)\()p Fs(\017;)15 b Ft(0\))27 b(=)e Fs(\036)1650 1190 y Fq(j)1686 1176 y Ft(\()p Fs(\017)p Ft(\))h(:=)f Fs(w)2005 1190 y Fq(j)2042 1176 y Ft(\()p Fs(\017)p Ft(\))61 b Fs(;)76 b Ft(0)26 b Fp(\024)f Fs(j)31 b Fp(\024)25 b Fs(S)g Fp(\000)20 b Ft(1)p Fs(:)75 1381 y Ft(W)-8 b(e)27 b(see)f(that)g Fs(\036)615 1395 y Fq(j)651 1381 y Ft(\()p Fs(\017)p Ft(\))g Fp(2)f Fs(E)937 1399 y Fn(j)p Fq(q)r Fn(j)p Fq(;)p Fr(\()p Fq(T)1099 1408 y Fl(0)1133 1399 y Fq(;X)1211 1408 y Fl(0)1245 1399 y Fr(\))1277 1381 y Ft(,)i(for)e(all)h Fs(T)1637 1395 y Fr(0)1677 1381 y Fs(;)15 b(X)1792 1395 y Fr(0)1857 1381 y Fs(>)25 b Ft(0,)i(all)f Fs(\017)g Fp(2)e Fs(D)s Ft(\(0)p Fs(;)15 b(r)2558 1395 y Fr(0)2599 1381 y Ft(\).)39 b(F)-8 b(rom)26 b(our)f(h)m(yp)s(otheses,)i(w)m(e)f(also)75 1493 y(get)33 b(that)g(the)f(assumptions)g(of)g(Theorem)g(1)g(are)h (ful\014lled)e(for)h(the)h(problem)e(\(36\),)k(\(37\).)47 b(F)-8 b(rom)33 b(Theorem)75 1606 y(1,)d(w)m(e)f(get)h(that)f(there)g (exists)h(a)f(solution)g Fs(V)20 b Ft(\()p Fs(\017;)15 b(x)p Ft(\))30 b(of)f(\(36\),)i(\(37\))f(suc)m(h)f(that)g Fs(V)20 b Ft(\()p Fs(\017;)15 b(x)p Ft(\))26 b Fp(2)f Fs(E)3209 1625 y Fn(j)p Fq(q)r Fn(j)p Fq(;)p Fr(\()p Fq(T)6 b(;X)f Fr(\))3520 1606 y Ft(for)28 b(some)75 1719 y(0)f Fs(<)f(T)39 b(<)26 b(T)486 1733 y Fr(0)526 1719 y Ft(,)31 b(0)c Fs(<)f(X)34 b(<)26 b(X)1032 1733 y Fr(0)1072 1719 y Ft(,)32 b(for)f(all)g Fs(\017)c Fp(2)f Fs(D)s Ft(\(0)p Fs(;)15 b(r)1785 1733 y Fr(0)1825 1719 y Ft(\).)43 b(In)31 b(particular,)h(w)m(e)f(get)h(the)g(estimates)g(:)43 b(there)31 b(exists)75 1832 y Fs(C)h(>)25 b Ft(0)30 b(suc)m(h)g(that)75 2055 y(\(38\))1265 b Fs(v)1544 2070 y Fq(h)1589 2055 y Ft(\()p Fs(\017)p Ft(\))26 b Fp(\024)f Fs(C)7 b Ft(\()1953 1994 y(1)p 1935 2035 83 4 v 1935 2118 a Fs(X)2027 2055 y Ft(\))2062 2018 y Fq(h)2107 2055 y Fs(h)p Ft(!)p Fp(j)p Fs(q)s Fp(j)2278 2018 y Fn(\000)2343 1991 y Fk(h)2379 1970 y Fl(2)p 2344 2003 71 3 v 2364 2044 a(2)75 2286 y Ft(for)33 b(all)h Fs(h)c Fp(\025)f Ft(0,)34 b(all)g Fs(\017)c Fp(2)f Fs(D)s Ft(\(0)p Fs(;)15 b(r)1157 2300 y Fr(0)1197 2286 y Ft(\).)49 b(F)-8 b(rom)34 b(the)f(estimates)h (\(35\))h(and)d(\(38\),)k(w)m(e)d(get)h(that)g(the)f(formal)g(series)75 2399 y Fs(W)13 b Ft(\()p Fs(\017;)i(\034)5 b(;)15 b(z)t Ft(\))38 b(de\014nes)e(a)i(b)s(ounded)c(holomorphic)k(function)e(on)h Fs(D)s Ft(\(0)p Fs(;)15 b(r)2517 2413 y Fr(0)2558 2399 y Ft(\))25 b Fp(\002)f Fs(D)s Ft(\(0)p Fs(;)15 b(\032)2958 2413 y Fr(0)2998 2399 y Ft(\))25 b Fp(\002)g Fs(D)s Ft(\(0)p Fs(;)15 b(\032)p Ft(\),)40 b(for)d(ev)m(ery)75 2511 y(discs)30 b Fs(D)s Ft(\(0)p Fs(;)15 b(\032)p Ft(\))32 b(cen)m(tered)f(at)g(0,)g (with)f(radius)g Fs(\032)25 b(>)g Ft(0.)1859 b Ff(2)75 2848 y Fu(4)135 b(Analytic)55 b(solutions)g(in)g(a)f(small)i(parameter) g(of)f(a)f(singularly)i(p)t(er-)277 2997 y(turb)t(ed)44 b(Cauc)l(h)l(y)h(problem)75 3203 y Fd(4.1)112 b(A)37 b Fc(q)t Fd(-analog)i(of)f(the)f(Laplace)i(transform)f(and)h Fc(q)t Fd(-asymptotic)f(expansion)75 3375 y Ft(In)f(this)h(section,)k (w)m(e)c(recall)i(the)e(de\014nition)g(of)g(a)h Fs(q)s Ft(-analog)h(of)e(the)g(Laplace)i(transform)d(in)m(tro)s(duced)h(b)m(y) 75 3488 y(J-P)-8 b(.)29 b(Ramis)g(and)f(C.)h(Zhang)f(in)h(the)g(pap)s (ers)e([11)r(],)i([12)q(])g(and)f(some)i(prop)s(erties)e(that)h(will)g (b)s(e)f(usefull)h(in)f(the)75 3601 y(sequel.)216 3764 y(In)35 b(the)g(next)g(prop)s(osition,)i(whic)m(h)e(is)g(a)g(Banac)m (hic)i(v)m(ersion)f(of)f(Theorem)g(1.3.2)i(in)e([12)q(],)i(w)m(e)f(giv) m(e)g(the)75 3876 y(de\014nition)31 b(of)g(the)g Fs(q)s Ft(-Laplace)i(transform)d(and)h(w)m(e)h(recall)g(estimates)h(for)e(its) g(asymptotic)i(expansion.)43 b(F)-8 b(or)75 3989 y(the)31 b(pro)s(of,)e(w)m(e)i(refer)f(to)h([12)r(].)75 4202 y Fi(Prop)s(osition)36 b(1)46 b Fh(L)-5 b(et)41 b Fs(q)j Fp(2)c Fm(C)h Fh(such)g(that)i Fp(j)p Fs(q)s Fp(j)e Fs(>)f Ft(1)p Fh(.)68 b(L)-5 b(et)42 b Fs(V)61 b Fh(b)-5 b(e)41 b(an)h(op)-5 b(en)42 b(and)g(b)-5 b(ounde)g(d)43 b(set)e(in)g Fm(C)3630 4169 y Fn(\003)3710 4202 y Fh(and)75 4315 y Fs(D)s Ft(\(0)p Fs(;)15 b(\032)320 4329 y Fr(0)360 4315 y Ft(\))45 b Fh(a)f(disc)g(such)g(that)h Fs(V)k Fp(\\)28 b Fs(D)s Ft(\(0)p Fs(;)15 b(\032)1578 4329 y Fr(0)1618 4315 y Ft(\))46 b Fp(6)p Ft(=)f Fp(;)p Fh(.)76 b(L)-5 b(et)44 b Ft(\()p Fm(F)p Fs(;)15 b Fp(jj)p Fs(:)p Fp(jj)2388 4329 y Fj(F)2435 4315 y Ft(\))44 b Fh(b)-5 b(e)44 b(a)g(c)-5 b(omplex)45 b(Banach)g(sp)-5 b(ac)g(e.)77 b(L)-5 b(et)75 4428 y Fs(\036)25 b Ft(:)h Fs(V)20 b(q)322 4395 y Fj(N)389 4428 y Fp([)15 b Fs(D)s Ft(\(0)p Fs(;)g(\032)710 4442 y Fr(0)751 4428 y Ft(\))25 b Fp(!)h Fm(F)j Fh(b)-5 b(e)31 b(a)g(holomorphic)i(function)e(which)g(satis\014es)g(the)g(fol)5 b(lowing)32 b(estimates)f(:)41 b(ther)-5 b(e)75 4541 y(exist)33 b Fs(C)q(;)15 b(M)36 b(>)25 b Ft(0)33 b Fh(such)g(that)75 4745 y Ft(\(39\))1202 b Fp(jj)p Fs(\036)p Ft(\()p Fs(xq)1672 4707 y Fq(m)1739 4745 y Ft(\))p Fp(jj)1824 4759 y Fj(F)1895 4745 y Fp(\024)25 b Fs(M)10 b Fp(j)p Fs(q)s Fp(j)2183 4707 y Fq(m)2245 4684 y Fl(2)2280 4707 y Fq(=)p Fr(2)2355 4745 y Fs(C)2427 4707 y Fq(m)75 4949 y Fh(for)33 b(al)5 b(l)33 b Fs(m)25 b Fp(\025)g Ft(0)p Fh(,)33 b(al)5 b(l)33 b Fs(x)25 b Fp(2)g Fs(V)c Fh(.)41 b(L)-5 b(et)33 b Ft(\002)f Fh(b)-5 b(e)33 b(the)g(Theta)g(Jac)-5 b(obi)34 b(function)e(de\014ne)-5 b(d)34 b(in)e Fm(C)3000 4916 y Fn(\003)3072 4949 y Fh(by)1469 5162 y Ft(\002\()p Fs(x)p Ft(\))25 b(=)1785 5076 y Fo(X)1783 5273 y Fq(n)p Fn(2)p Fj(Z)1933 5162 y Fs(q)1977 5124 y Fn(\000)p Fq(n)p Fr(\()p Fq(n)p Fn(\000)p Fr(1\))p Fq(=)p Fr(2)2337 5162 y Fs(x)2389 5124 y Fq(n)2436 5162 y Fs(:)75 5454 y Fh(L)-5 b(et)33 b Fs(\016)c(>)c Ft(0)33 b Fh(and)h Fs(\025)25 b Fp(2)g Fs(V)40 b Fp(\\)20 b Fs(D)s Ft(\(0)p Fs(;)15 b(\032)1235 5468 y Fr(0)1275 5454 y Ft(\))p Fh(.)42 b(We)33 b(denote)g(by)985 5701 y Fp(T)1035 5716 y Fq(\025;q)r(;\016)o(;r)1232 5725 y Fl(1)1294 5701 y Ft(=)25 b Fp(f)p Fs(t)g Fp(2)g Fm(C)1645 5663 y Fn(\003)1684 5701 y Fs(=)p Fp(j)p Ft(1)c(+)1955 5639 y Fs(\025)p 1921 5680 120 4 v 1921 5763 a(tq)1998 5737 y Fq(k)2051 5701 y Fp(j)k Fs(>)g(\016)n(;)15 b Fp(8)p Fs(k)29 b Fp(2)c Fm(Z)p Fs(;)15 b Fp(j)p Fs(t)p Fp(j)26 b Fs(<)f(r)2835 5715 y Fr(1)2874 5701 y Fp(g)p Fs(:)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 3764 100 a Ft(12)75 399 y Fh(The)33 b Fs(q)s Fh(-L)-5 b(aplac)g(e)34 b(tr)-5 b(ansform)35 b(of)e Fs(\036)f Fh(in)h(the)g(dir)-5 b(e)g(ction)34 b Fs(\025q)2010 366 y Fj(Z)2091 399 y Fh(is)f(de\014ne)-5 b(d)33 b(by)1310 640 y Fp(L)1373 603 y Fq(\025)1373 663 y(q)1418 640 y Ft(\()p Fs(\036)p Ft(\)\()p Fs(t)p Ft(\))26 b(:=)1804 554 y Fo(X)1792 751 y Fq(m)p Fn(2)p Fj(Z)1961 640 y Fs(\036)p Ft(\()p Fs(q)2094 603 y Fq(m)2161 640 y Fs(\025)p Ft(\))p Fs(=)p Ft(\002\()2410 579 y Fs(q)2454 546 y Fq(m)2522 579 y Fs(\025)p 2410 620 164 4 v 2476 703 a(t)2585 640 y Ft(\))75 947 y Fh(for)f(al)5 b(l)24 b Fs(t)i Fp(2)e(T)527 962 y Fq(\025;q)r(;\016)o(;r)724 971 y Fl(1)761 947 y Fh(,)i(if)e Fs(r)936 961 y Fr(1)1000 947 y Fs(<)h Fp(j)p Fs(\025q)1218 914 y Fr(1)p Fq(=)p Fr(2)1328 947 y Fp(j)p Fs(=C)7 b Fh(.)39 b(Mor)-5 b(e)g(over,)27 b Fp(L)2021 914 y Fq(\025)2021 970 y(q)2066 947 y Ft(\()p Fs(\036)p Ft(\)\()p Fs(t)p Ft(\))f Fh(de\014nes)f(a)f(b)-5 b(ounde)g(d)26 b(holomorphic)h(function)75 1074 y(on)33 b Fp(T)255 1089 y Fq(\025;q)r(;\016)o(;r)452 1098 y Fl(1)521 1074 y Fh(with)h(values)e(in)h Fm(F)f Fh(when)h Fs(r)1463 1088 y Fr(1)1528 1074 y Fs(<)25 b Fp(j)p Fs(\025q)1746 1041 y Fr(1)p Fq(=)p Fr(2)1856 1074 y Fp(j)p Fs(=C)7 b Fh(.)42 b(Assume)32 b(that)i(the)f(function)f Fs(\036)h Fh(has)g(the)g(fol)5 b(lowing)75 1186 y(T)-7 b(aylor)34 b(exp)-5 b(ansion)75 1428 y Ft(\(40\))1275 b Fs(\036)p Ft(\()p Fs(\034)10 b Ft(\))26 b(=)1806 1342 y Fo(X)1806 1538 y Fq(n)p Fn(\025)p Fr(0)2093 1367 y Fs(f)2138 1381 y Fq(n)p 1964 1407 350 4 v 1964 1495 a Fs(q)2008 1468 y Fq(n)p Fr(\()p Fq(n)p Fn(\000)p Fr(1\))p Fq(=)p Fr(2)2323 1428 y Fs(\034)2373 1391 y Fq(n)75 1726 y Fh(on)33 b Fs(D)s Ft(\(0)p Fs(;)15 b(\032)450 1740 y Fr(0)491 1726 y Ft(\))p Fh(,)32 b(wher)-5 b(e)34 b Fs(f)888 1740 y Fq(n)960 1726 y Fp(2)25 b Fm(F)p Fh(,)32 b Fs(n)24 b Fp(\025)h Ft(0)p Fh(.)42 b(Then,)33 b(ther)-5 b(e)34 b(exist)f(two)g(c)-5 b(onstants)35 b Fs(D)s(;)15 b(B)30 b(>)25 b Ft(0)33 b Fh(such)f(that)75 2014 y Ft(\(41\))819 b Fp(jjL)1167 1976 y Fq(\025)1167 2036 y(q)1213 2014 y Ft(\()p Fs(\036)p Ft(\)\()p Fs(t)p Ft(\))21 b Fp(\000)1562 1900 y Fq(n)p Fn(\000)p Fr(1)1563 1928 y Fo(X)1552 2122 y Fq(m)p Fr(=0)1720 2014 y Fs(f)1765 2028 y Fq(m)1831 2014 y Fs(t)1864 1976 y Fq(m)1930 2014 y Fp(jj)1980 2028 y Fj(F)2051 2014 y Fp(\024)k Fs(D)s(B)2299 1976 y Fq(n)2345 2014 y Fp(j)p Fs(q)s Fp(j)2439 1976 y Fq(n)p Fr(\()p Fq(n)p Fn(\000)p Fr(1\))p Fq(=)p Fr(2)2745 2014 y Fp(j)p Fs(t)p Fp(j)2828 1976 y Fq(n)75 2294 y Fh(for)33 b(al)5 b(l)33 b Fs(n)25 b Fp(\025)g Ft(1)p Fh(,)33 b(for)g(al)5 b(l)33 b Fs(t)25 b Fp(2)g(T)1101 2309 y Fq(\025;q)r(;\016)o(;r)1298 2318 y Fl(1)1335 2294 y Fh(.)216 2506 y Ft(In)34 b(the)h(follo)m(wing)h (prop)s(osition,)f(w)m(e)g(recall)h(a)f(comm)m(utation)h(form)m(ula)e (for)h(the)f Fs(q)s Ft(-Laplace)i(transform)75 2619 y(and)30 b(the)g(m)m(ultiplication)j(b)m(y)d(a)h(monomial.)75 2832 y Fi(Prop)s(osition)36 b(2)46 b Fh(L)-5 b(et)47 b Fs(V)66 b Fh(b)-5 b(e)46 b(an)h(op)-5 b(en)47 b(and)h(b)-5 b(ounde)g(d)47 b(set)g(in)f Fm(C)2421 2799 y Fn(\003)2506 2832 y Fh(and)h Fs(D)s Ft(\(0)p Fs(;)15 b(\032)2941 2846 y Fr(0)2982 2832 y Ft(\))46 b Fh(b)-5 b(e)47 b(a)f(disc)h(such)f(that) 75 2945 y Fs(V)d Fp(\\)22 b Fs(D)s Ft(\(0)p Fs(;)15 b(\032)499 2959 y Fr(0)539 2945 y Ft(\))32 b Fp(6)p Ft(=)f Fp(;)p Fh(.)52 b(L)-5 b(et)36 b Fs(\036)g Fh(b)-5 b(e)36 b(a)g(holomorphic)j (function)d(on)h Fs(V)20 b(q)2412 2912 y Fj(N)2486 2945 y Fp([)j Fs(D)s Ft(\(0)p Fs(;)15 b(\032)2815 2959 y Fr(0)2855 2945 y Ft(\))36 b Fh(with)h(values)f(in)g Ft(\()p Fm(F)p Fs(;)15 b Fp(jj)p Fs(:)p Fp(jj)3773 2959 y Fj(F)3819 2945 y Ft(\))75 3058 y Fh(which)34 b(satis\014es)f(the)g(estimates)h(:) 41 b(ther)-5 b(e)34 b(exist)f Fs(C)q(;)15 b(K)32 b(>)25 b Ft(0)p Fh(,)33 b(such)g(that)75 3262 y Ft(\(42\))1209 b Fp(jj)p Fs(\036)p Ft(\()p Fs(xq)1679 3224 y Fq(m)1746 3262 y Ft(\))p Fp(jj)1831 3276 y Fj(F)1902 3262 y Fp(\024)25 b Fs(K)7 b Fp(j)p Fs(q)s Fp(j)2176 3224 y Fq(m)2238 3201 y Fl(2)2273 3224 y Fq(=)p Fr(2)2348 3262 y Fs(C)2420 3224 y Fq(m)75 3466 y Fh(for)34 b(al)5 b(l)34 b Fs(m)26 b Fp(\025)g Ft(0)p Fh(,)34 b(al)5 b(l)34 b Fs(x)26 b Fp(2)g Fs(V)21 b Fh(.)43 b(Then,)34 b(the)g(function)f Fs(m\036)p Ft(\()p Fs(\034)10 b Ft(\))27 b(=)g Fs(\034)10 b(\036)p Ft(\()p Fs(\034)g Ft(\))34 b Fh(is)f(holomorphic)k(on)d Fs(V)20 b(q)3380 3433 y Fj(N)3452 3466 y Fp([)h Fs(D)s Ft(\(0)p Fs(;)15 b(\032)3779 3480 y Fr(0)3819 3466 y Ft(\))75 3579 y Fh(and)34 b(satis\014es)f(estimates)h(of)f(the)g(form)g (\(39\).)43 b(L)-5 b(et)33 b Fs(\025)25 b Fp(2)g Fs(V)41 b Fp(\\)19 b Fs(D)s Ft(\(0)p Fs(;)c(\032)2447 3593 y Fr(0)2488 3579 y Ft(\))p Fh(.)42 b(We)32 b(have)h(the)g(fol)5 b(lowing)34 b(e)-5 b(quality)75 3783 y Ft(\(43\))1254 b Fp(L)1552 3746 y Fq(\025)1552 3806 y(q)1598 3783 y Ft(\()p Fs(m\036)p Ft(\)\()p Fs(t)p Ft(\))26 b(=)f Fs(t)p Fp(L)2123 3746 y Fq(\025)2123 3806 y(q)2168 3783 y Ft(\()p Fs(\036)p Ft(\)\()p Fs(q)s(t)p Ft(\))75 4001 y Fh(for)33 b(al)5 b(l)33 b Fs(t)25 b Fp(2)g(T)544 4016 y Fq(\025;q)r(;\016)o(;r) 741 4025 y Fl(1)778 4001 y Fh(,)32 b(if)h Fs(r)968 4015 y Fr(1)1032 4001 y Fs(<)25 b Fp(j)p Fs(\025q)1250 3968 y Fr(1)p Fq(=)p Fr(2)1360 4001 y Fp(j)p Fs(=)p Ft(\()p Fs(C)7 b Fp(j)p Fs(q)s Fp(j)p Ft(\))p Fh(.)75 4263 y Fi(Pro)s(of)31 b Ft(F)-8 b(rom)31 b(the)g(estimates)h(\(42\),)g(w)m(e)f (get)g(a)g(constan)m(t)h Fs(r)27 b(>)e Ft(0)31 b(suc)m(h)f(that)1264 4479 y Fp(jj)p Ft(\()p Fs(m\036)p Ft(\)\()p Fs(xq)1649 4441 y Fq(m)1717 4479 y Ft(\))p Fp(jj)1802 4493 y Fj(F)1873 4479 y Fp(\024)25 b Fs(r)s(K)7 b Fp(j)p Fs(q)s Fp(j)2191 4441 y Fq(m)2253 4418 y Fl(2)2288 4441 y Fq(=)p Fr(2)2362 4479 y Ft(\()p Fp(j)p Fs(q)s Fp(j)p Fs(C)g Ft(\))2598 4441 y Fq(m)75 4683 y Ft(for)31 b(all)g Fs(m)26 b Fp(\025)g Ft(0,)31 b(all)h Fs(x)26 b Fp(2)f Fs(V)c Ft(.)42 b(F)-8 b(rom)31 b(Prop)s(osition)g(1,)g(w)m(e)g(deduce)g(that)g Fp(L)2605 4650 y Fq(\025)2605 4705 y(q)2650 4683 y Ft(\()p Fs(m\036)p Ft(\)\()p Fs(t)p Ft(\))h(de\014nes)e(a)h(holomorphic)75 4809 y(function)f(on)g Fp(T)607 4824 y Fq(\025;q)r(;\016)o(;r)804 4833 y Fl(1)841 4809 y Ft(,)h(if)f Fs(r)1021 4823 y Fr(1)1086 4809 y Fs(<)25 b Fp(j)p Fs(\025q)1304 4776 y Fr(1)p Fq(=)p Fr(2)1414 4809 y Fp(j)p Fs(=)p Ft(\()p Fs(C)7 b Fp(j)p Fs(q)s Fp(j)p Ft(\).)42 b(On)29 b(the)i(other)f(hand,)75 5067 y(\(44\))1119 b Fs(t)p Fp(L)1450 5029 y Fq(\025)1450 5089 y(q)1495 5067 y Ft(\()p Fs(\036)p Ft(\)\()p Fs(q)s(t)p Ft(\))27 b(=)1900 4980 y Fo(X)1889 5177 y Fq(m)p Fn(2)p Fj(Z)2156 5005 y Fs(t\036)p Ft(\()p Fs(q)2322 4972 y Fq(m)2389 5005 y Fs(\025)p Ft(\))p 2068 5046 498 4 v 2068 5129 a(\002\()p Fs(q)2218 5103 y Fq(m)2284 5129 y Fs(\025=)p Ft(\()p Fs(q)s(t)p Ft(\)\))75 5353 y(But)k(w)m(e)f(ha)m(v) m(e)i(that)1686 5425 y Fs(t)p 1454 5465 V 1454 5549 a Ft(\002\()p Fs(q)1604 5522 y Fq(m)1670 5549 y Fs(\025=)p Ft(\()p Fs(q)s(t)p Ft(\)\))1987 5486 y(=)2202 5425 y Fs(q)2246 5392 y Fq(m)2313 5425 y Fs(\025)p 2093 5465 384 4 v 2093 5549 a Ft(\002\()p Fs(q)2243 5522 y Fq(m)2309 5549 y Fs(\025=t)p Ft(\))p eop end %%Page: 13 13 TeXDict begin 13 12 bop 3764 100 a Ft(13)75 399 y(for)35 b(all)h Fs(m)d Fp(2)g Fm(Z)p Ft(.)55 b(Indeed,)35 b(put)g Fs(y)h Ft(=)d Fs(q)1425 366 y Fq(m)1491 399 y Fs(\025=)p Ft(\()p Fs(q)s(t)p Ft(\))j(in)f(the)h(equation)g(\002\()p Fs(q)s(y)s Ft(\))d(=)g Fs(q)s(y)s Ft(\002\()p Fs(y)s Ft(\))i(whic)m(h)g(holds)g(for)g(all)75 511 y Fs(y)28 b Fp(2)d Fm(C)300 478 y Fn(\003)339 511 y Ft(.)41 b(F)-8 b(rom)31 b(\(44\),)h(w)m(e)f(get)g(that)1109 754 y Fs(t)p Fp(L)1205 716 y Fq(\025)1205 776 y(q)1249 754 y Ft(\()p Fs(\036)p Ft(\)\()p Fs(q)s(t)p Ft(\))c(=)1654 667 y Fo(X)1643 865 y Fq(m)p Fn(2)p Fj(Z)1822 692 y Fs(q)1866 659 y Fq(m)1932 692 y Fs(\025\036)p Ft(\()p Fs(q)2118 659 y Fq(m)2185 692 y Fs(\025)p Ft(\))p 1822 733 452 4 v 1856 816 a(\002\()p Fs(q)2006 790 y Fq(m)2073 816 y Fs(\025=t)p Ft(\))2309 754 y(=)e Fp(L)2468 716 y Fq(\025)2468 776 y(q)2513 754 y Ft(\()p Fs(m\036)p Ft(\)\()p Fs(t)p Ft(\))75 1031 y(for)30 b(all)h Fs(t)25 b Fp(2)g(T)534 1046 y Fq(\025;q)r(;\016)o(;r)731 1055 y Fl(1)768 1031 y Ft(.)2963 b Ff(2)216 1191 y Ft(In)23 b(the)g(follo)m(wing,)j(w)m(e)e(recall)g(the)g(de\014nition)e(of)i Fs(q)s Ft(-asymptotic)g(expansion)f(as)h(in)m(tro)s(duced)e(in)h(the)h (pap)s(er)75 1304 y([10)q(].)75 1506 y Fi(De\014nition)35 b(2)46 b Fh(L)-5 b(et)45 b Fs(U)55 b Fh(b)-5 b(e)45 b(an)g(op)-5 b(en)46 b(set)f(in)g Fm(C)1783 1473 y Fn(\003)1822 1506 y Fh(.)79 b(L)-5 b(et)45 b Ft(\()p Fm(H)p Fs(;)15 b Fp(jj)p Fs(:)p Fp(jj)2369 1520 y Fj(H)2426 1506 y Ft(\))45 b Fh(b)-5 b(e)45 b(a)h(c)-5 b(omplex)46 b(Banach)g(sp)-5 b(ac)g(e.)80 b(L)-5 b(et)75 1619 y Fs(f)37 b Ft(:)29 b Fs(U)10 b(q)327 1586 y Fn(\000)p Fj(N)462 1619 y Fp(!)28 b Fm(H)34 b Fh(b)-5 b(e)34 b(a)h(holomorphic)i(function)e(and)g(let) 2085 1595 y Ft(^)2065 1619 y Fs(f)9 b Ft(\()p Fs(\017)p Ft(\))29 b(=)2354 1551 y Fo(P)2450 1646 y Fq(n)p Fn(\025)p Fr(0)2602 1619 y Fs(a)2650 1633 y Fq(n)2697 1619 y Fs(\017)2734 1586 y Fq(n)2810 1619 y Fp(2)e Fm(H)p Ft([[)p Fs(\017)p Ft(]])35 b Fh(b)-5 b(e)34 b(a)h(formal)h(p)-5 b(ower)75 1753 y(series)35 b(with)h(c)-5 b(o)g(e\016cients)36 b(in)f Fm(H)p Fh(.)49 b(The)35 b(formal)h(series)2020 1729 y Ft(^)2001 1753 y Fs(f)44 b Fh(is)35 b(the)g Fs(q)s Fh(-asymptotic)i (exp)-5 b(ansion)37 b(of)e Fs(f)44 b Fh(on)36 b Fs(U)10 b(q)3748 1720 y Fn(\000)p Fj(N)75 1866 y Fh(if)32 b(for)h(al)5 b(l)34 b Fs(n)24 b Fp(\025)h Ft(1)p Fh(,)33 b(al)5 b(l)33 b(c)-5 b(omp)g(act)35 b(set)e Fs(K)f Fp(\032)25 b Fs(U)10 b Fh(,)32 b(ther)-5 b(e)34 b(exists)f(a)g(c)-5 b(onstant)34 b Fs(C)2662 1880 y Fq(K)q(;n)2814 1866 y Fs(>)25 b Ft(0)33 b Fh(such)g(that)75 2148 y Ft(\(45\))1119 b Fp(jj)p Fs(f)10 b Ft(\()p Fs(\017)p Ft(\))21 b Fp(\000)1678 2035 y Fq(n)p Fn(\000)p Fr(1)1679 2062 y Fo(X)1680 2260 y Fq(k)r Fr(=0)1826 2148 y Fs(a)1874 2163 y Fq(k)1917 2148 y Fs(\017)1954 2111 y Fq(k)1996 2148 y Fp(jj)2046 2162 y Fj(H)2128 2148 y Fp(\024)k Fs(C)2289 2162 y Fq(K)q(;n)2416 2148 y Fp(j)p Fs(\017)p Fp(j)2503 2111 y Fq(n)2550 2148 y Fs(;)75 2437 y Fh(for)33 b(al)5 b(l)33 b Fs(\017)25 b Fp(2)g Fs(K)7 b(q)626 2404 y Fn(\000)p Fj(N)733 2437 y Fh(.)75 2640 y Fi(Remark:)40 b Ft(If)28 b Fs(f)38 b Ft(admits)28 b(a)h Fs(q)s Ft(-asymptotic)h(expansion)2028 2616 y(^)2009 2640 y Fs(f)9 b Ft(,)29 b(then)f(it)h(is)g(unique.)39 b(Moreo)m(v)m(er,)31 b(if)e(w)m(e)g(denote)g(b)m(y)75 2753 y Fs(f)130 2720 y Fn(0)180 2753 y Ft(the)g(complex)f(deriv)-5 b(ativ)m(e)30 b(of)e Fs(f)37 b Ft(on)28 b Fs(U)10 b(q)1526 2720 y Fn(\000)p Fj(N)1660 2753 y Ft(and)1854 2729 y(^)1835 2753 y Fs(f)1890 2720 y Fn(0)1940 2753 y Ft(the)28 b(formal)g(deriv)-5 b(ativ)m(e)30 b(of)2912 2729 y(^)2893 2753 y Fs(f)9 b Ft(,)29 b(w)m(e)f(ha)m(v)m(e)h(that)3553 2729 y(^)3534 2753 y Fs(f)3589 2720 y Fn(0)3639 2753 y Ft(is)f(the)75 2866 y Fs(q)s Ft(-asymptotic)k(expansion)e(of)h Fs(f)1198 2833 y Fn(0)1250 2866 y Ft(on)g Fs(U)10 b(q)1493 2833 y Fn(\000)p Fj(N)1600 2866 y Ft(,)30 b(see)h([10)q(])g(p.31.)75 3155 y Fd(4.2)112 b(Analytic)38 b(solutions)g(in)g(a)g(parameter)g(of)g (a)g(singular)g(Cauc)m(h)m(y)g(problem)75 3326 y Ft(Let)45 b Fs(S)55 b Fp(\025)49 b Ft(1)d(b)s(e)e(an)h(in)m(teger.)85 b(F)-8 b(or)46 b(all)g(0)k Fp(\024)f Fs(k)k Fp(\024)c Fs(S)35 b Fp(\000)30 b Ft(1,)49 b(let)d Fs(m)2523 3341 y Fr(0)p Fq(;k)2620 3326 y Ft(,)p Fs(m)2725 3341 y Fr(1)p Fq(;k)2868 3326 y Ft(b)s(e)e(p)s(ositiv)m(e)i(in)m(tegers)g(and)75 3439 y Fs(b)114 3454 y Fq(k)157 3439 y Ft(\()p Fs(\017;)15 b(z)t Ft(\))26 b(=)472 3371 y Fo(P)568 3466 y Fq(s)p Fn(2)p Fq(I)679 3478 y Fk(k)736 3439 y Fs(b)775 3454 y Fq(k)r(s)850 3439 y Ft(\()p Fs(\017)p Ft(\))p Fs(z)1003 3406 y Fq(s)1069 3439 y Ft(b)s(e)h(a)i(p)s(olynomial)f(in)g Fs(z)t Ft(,)h(where)e Fs(I)2235 3454 y Fq(k)2306 3439 y Ft(is)h(a)g(subset)g(of)g Fm(N)g Ft(and)f Fs(b)3153 3454 y Fq(k)r(s)3229 3439 y Ft(\()p Fs(\017)p Ft(\))h(are)h(b)s(ounded) 75 3552 y(holomorphic)k(functions)g(on)g(some)g(disc)g Fs(D)s Ft(\(0)p Fs(;)15 b(r)1776 3566 y Fr(0)1816 3552 y Ft(\))34 b(with)f Fs(r)2136 3566 y Fr(0)2205 3552 y Fs(>)c Ft(0.)49 b(Let)34 b Fs(U;)15 b(V)53 b Ft(b)s(e)33 b(op)s(en)f(sets)h(in)g Fm(C)3502 3519 y Fn(\003)3574 3552 y Ft(and)g(let)75 3665 y Fs(q)28 b Fp(2)d Fm(C)j Ft(suc)m(h)g(that)h Fp(j)p Fs(q)s Fp(j)c Fs(>)g Ft(1.)41 b(W)-8 b(e)29 b(assume)f(that)h(there)f(exists)h Fs(\016)g(>)c Ft(0)k(suc)m(h)f(that)h Fp(j)p Fs(\034)d Ft(+)16 b(1)p Fp(j)26 b Fs(>)f(\016)32 b Ft(for)c(all)h Fs(\034)35 b Fp(2)25 b Fs(V)20 b(q)3806 3632 y Fj(Z)75 3778 y Ft(and)30 b(that)h Fs(U)10 b(q)565 3745 y Fn(\000)p Fj(N)697 3778 y Fp(\032)25 b Fs(D)s Ft(\(0)p Fs(;)15 b(r)1032 3792 y Fr(0)1072 3778 y Ft(\).)75 3981 y Fi(De\014nition)35 b(3)46 b Fh(L)-5 b(et)46 b Fs(T)884 3995 y Fr(0)974 3981 y Fs(>)k Ft(0)p Fh(,)f Ft(0)i Fs(<)f(\032)1481 3995 y Fr(0)1570 3981 y Fs(<)g Ft(1)c Fh(b)-5 b(e)46 b(r)-5 b(e)g(al)48 b(numb)-5 b(ers)47 b(such)f(that)i Fs(V)i Fp(\\)30 b Fs(D)s Ft(\(0)p Fs(;)15 b(\032)3328 3995 y Fr(0)3368 3981 y Ft(\))50 b Fp(6)p Ft(=)g Fp(;)p Fh(.)83 b(L)-5 b(et)75 4094 y Ft(\()p Fs(\017;)15 b(\034)10 b Ft(\))26 b Fp(7!)f Fs(W)13 b Ft(\()p Fs(\017;)i(\034)10 b Ft(\))28 b Fh(b)-5 b(e)28 b(a)g(b)-5 b(ounde)g(d)29 b(holomorphic)i(function)d(on)g Fs(D)s Ft(\(0)p Fs(;)15 b(r)2489 4108 y Fr(0)2529 4094 y Ft(\))9 b Fp(\002)g Fs(D)s Ft(\(0)p Fs(;)15 b(\032)2898 4108 y Fr(0)2939 4094 y Ft(\))p Fh(.)41 b(Assume)27 b(that)i Fs(W)13 b Ft(\()p Fs(\017;)i(\034)10 b Ft(\))75 4207 y Fh(c)-5 b(an)29 b(b)-5 b(e)27 b(extende)-5 b(d)30 b(to)e(an)h(analytic)g (function)f Ft(\()p Fs(\017;)15 b(\034)10 b Ft(\))26 b Fp(7!)f Fs(W)2057 4221 y Fq(U)7 b(V)2173 4207 y Ft(\()p Fs(\017;)15 b(\034)10 b Ft(\))29 b Fh(on)f Fs(U)10 b(q)2640 4174 y Fn(\000)p Fj(N)2757 4207 y Fp(\002)g Ft(\()p Fs(V)21 b(q)2991 4174 y Fj(N)3053 4207 y Fp([)10 b Fs(D)s Ft(\(0)p Fs(;)15 b(\032)3369 4221 y Fr(0)3409 4207 y Ft(\)\))p Fh(.)41 b(Assume)75 4320 y(mor)-5 b(e)g(over)35 b(that)e(ther)-5 b(e)34 b(exists)f Fs(K)f(>)25 b Ft(0)33 b Fh(such)g(that)75 4533 y Ft(\(46\))961 b(sup)1192 4611 y Fq(x)p Fn(2)p Fq(V)1351 4533 y Fp(j)p Fs(W)1462 4547 y Fq(U)7 b(V)1578 4533 y Ft(\()p Fs(\017;)15 b(xq)1786 4496 y Fq(l)1812 4533 y Ft(\))p Fp(j)26 b(\024)f Fs(K)7 b Fp(j)p Fs(q)s Fp(j)2182 4469 y Fl(1)p 2182 4481 31 3 v 2182 4522 a(4)2223 4496 y Fq(l)2245 4472 y Fl(2)2283 4533 y Ft(\()2352 4472 y(1)p 2328 4512 93 4 v 2328 4596 a Fs(T)2381 4610 y Fr(0)2431 4533 y Ft(\))2466 4496 y Fq(l)2592 4472 y Ft(1)p 2502 4512 225 4 v 2502 4596 a(1)21 b(+)f Fs(l)2688 4569 y Fr(2)75 4793 y Fh(for)33 b(al)5 b(l)33 b Fs(l)27 b Fp(\025)e Ft(0)p Fh(,)33 b(al)5 b(l)33 b Fs(\017)26 b Fp(2)e Fs(U)10 b(q)1000 4760 y Fn(\000)p Fj(N)1107 4793 y Fh(.)42 b(We)32 b(say)h(that)h(the)f(set)g Fp(f)p Fs(W)m(;)15 b(W)2235 4807 y Fq(U)7 b(V)2352 4793 y Fs(;)15 b(T)2445 4807 y Fr(0)2485 4793 y Fs(;)g(\032)2572 4807 y Fr(0)2612 4793 y Fp(g)33 b Fh(is)f(admissible.)216 4996 y Ft(W)-8 b(e)32 b(consider)e(the)h(follo)m(wing)g(singular)g(Cauc)m(h)m(y)f(problem)75 5272 y(\(47\))220 b Fs(t@)541 5235 y Fq(S)536 5295 y(z)592 5272 y Fs(Y)645 5286 y Fq(U)7 b(V)760 5272 y Ft(\()p Fs(\017;)15 b(q)s(t;)g(z)t Ft(\))22 b(+)e Fs(@)1236 5235 y Fq(S)1231 5295 y(z)1287 5272 y Fs(Y)1340 5286 y Fq(U)7 b(V)1455 5272 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))27 b(=)1844 5159 y Fq(S)t Fn(\000)p Fr(1)1847 5186 y Fo(X)1848 5383 y Fq(k)r Fr(=0)1996 5272 y Fs(b)2035 5287 y Fq(k)2078 5272 y Ft(\()p Fs(\017;)15 b(z)t Ft(\)\()p Fs(t\033)2391 5286 y Fq(q)2430 5272 y Ft(\))2465 5235 y Fq(m)2527 5247 y Fl(0)p Fk(;k)2619 5272 y Ft(\()p Fs(@)2707 5235 y Fq(k)2702 5295 y(z)2751 5272 y Fs(Y)2804 5286 y Fq(U)7 b(V)2919 5272 y Ft(\)\()p Fs(\017;)15 b(t;)g(z)t(q)3229 5235 y Fn(\000)p Fq(m)3346 5247 y Fl(1)p Fk(;k)3440 5272 y Ft(\))75 5546 y(for)30 b(giv)m(en)i(initial)f(conditions)75 5740 y(\(48\))806 b(\()p Fs(@)1129 5703 y Fq(j)1124 5763 y(z)1166 5740 y Fs(Y)1219 5754 y Fq(U)7 b(V)1334 5740 y Ft(\)\()p Fs(\017;)15 b(t;)g Ft(0\))28 b(=)d Fs(')1817 5754 y Fq(U)7 b(V)t(;j)1973 5740 y Ft(\()p Fs(\017;)15 b(t)p Ft(\))62 b Fs(;)76 b Ft(0)25 b Fp(\024)g Fs(j)31 b Fp(\024)25 b Fs(S)g Fp(\000)20 b Ft(1)p Fs(;)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 3764 100 a Ft(14)75 399 y(where)35 b Fs(\033)395 413 y Fq(q)468 399 y Ft(is)h(the)f(dilation)i(op)s (erator)e(de\014ned)g(b)m(y)g(\()p Fs(\033)1973 413 y Fq(q)2011 399 y Fs(Y)20 b Ft(\)\()p Fs(\017;)15 b(t;)g(z)t Ft(\))36 b(=)d Fs(Y)20 b Ft(\()p Fs(\017;)15 b(q)s(t;)g(z)t Ft(\).)57 b(The)35 b(initial)i(conditions)75 511 y Fs(')134 525 y Fq(U)7 b(V)t(;j)290 511 y Ft(\()p Fs(\017;)15 b(t)p Ft(\))22 b(are)f(constructed)g(as)g(follo)m(ws.)39 b(Let)21 b Fs(T)1754 525 y Fr(0)1794 511 y Fs(;)15 b(\032)1881 525 y Fr(0)1946 511 y Fs(>)25 b Ft(0.)38 b(F)-8 b(or)21 b(all)h(0)k Fp(\024)f Fs(j)30 b Fp(\024)25 b Fs(S)6 b Fp(\000)q Ft(1,)23 b(let)f Fp(f)p Fs(W)3230 525 y Fq(j)3267 511 y Fs(;)15 b(W)3393 525 y Fq(U)7 b(V)t(;j)3549 511 y Fs(;)15 b(T)3642 525 y Fr(0)3682 511 y Fs(;)g(\032)3769 525 y Fr(0)3809 511 y Fp(g)75 624 y Ft(b)s(e)30 b(admissible)g(sets.) 216 784 y(Let)h Fs(\025)432 798 y Fq(V)518 784 y Fp(2)25 b Fs(V)41 b Fp(\\)19 b Fs(D)s Ft(\(0)p Fs(;)c(\032)1023 798 y Fr(0)1064 784 y Ft(\))30 b(b)s(e)g(a)h(complex)g(n)m(um)m(b)s (er.)39 b(W)-8 b(e)32 b(put)1225 981 y Fs(')1284 995 y Fq(U)7 b(V)t(;j)1440 981 y Ft(\()p Fs(\017;)15 b(t)p Ft(\))27 b(=)e Fp(L)1806 943 y Fq(\025)1847 954 y Fk(V)1806 1003 y Fq(q)1904 981 y Ft(\()p Fs(\034)36 b Fp(7!)25 b Fs(W)2217 995 y Fq(U)7 b(V)t(;j)2373 981 y Ft(\()p Fs(\034)e(;)15 b(\017)p Ft(\)\)\()p Fs(t)p Ft(\))75 1177 y(for)34 b(all)i(0)c Fp(\024)g Fs(j)38 b Fp(\024)32 b Fs(S)c Fp(\000)23 b Ft(1.)53 b(F)-8 b(rom)35 b(the)g(prop)s(osition)f (1)h(and)f(the)h(estimates)h(\(46\),)i(w)m(e)d(deduce)f(that,)i(for)f (all)75 1290 y(0)h Fp(\024)f Fs(j)41 b Fp(\024)35 b Fs(S)29 b Fp(\000)24 b Ft(1,)38 b Fs(')793 1304 y Fq(U)7 b(V)t(;j)950 1290 y Ft(\()p Fs(\017;)15 b(t)p Ft(\))37 b(is)f(holomorphic)h(and)e(b) s(ounded)g(on)h Fs(U)10 b(q)2599 1257 y Fn(\000)p Fj(N)2730 1290 y Fp(\002)24 b(T)2875 1305 y Fq(\025)2916 1316 y Fk(V)2969 1305 y Fq(;q)r(;\016)o(;r)3125 1314 y Fl(1)3198 1290 y Ft(for)36 b Fs(r)3384 1304 y Fr(1)3459 1290 y Fs(>)f Ft(0)i(small)75 1403 y(enough.)j(W)-8 b(e)32 b(ha)m(v)m(e)g(the) e(follo)m(wing)i(result.)75 1608 y Fi(Theorem)j(4)46 b Fh(L)-5 b(et)37 b(the)h(assumption)h Fi(\(A\))d Fh(\(of)i(se)-5 b(ction)37 b(2\))h(b)-5 b(e)37 b(ful\014l)5 b(le)-5 b(d)38 b(by)f(the)g(sets)h Fs(I)3133 1623 y Fq(k)3212 1608 y Fh(and)h(the)e(inte)-5 b(gers)75 1721 y Fs(m)155 1736 y Fr(0)p Fq(;k)252 1721 y Fh(,)p Fs(m)360 1736 y Fr(1)p Fq(;k)458 1721 y Fh(,)32 b(for)h Ft(0)26 b Fp(\024)f Fs(k)j Fp(\024)d Fs(S)g Fp(\000)20 b Ft(1)p Fh(.)43 b(L)-5 b(et)32 b(the)h(initial)h(data)g(\(48\))g(c)-5 b(onstructe)g(d)34 b(as)f(ab)-5 b(ove.)75 1834 y(Then,)41 b(the)e(pr)-5 b(oblem)41 b(\(47\),\(48\))g(has)f(a)f(solution)i Fs(Y)1911 1848 y Fq(U)7 b(V)2026 1834 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))40 b Fh(which)g(is)f(holomorphic)k(and)c(b)-5 b(ounde)g(d)41 b(on)75 1947 y Fs(U)10 b(q)191 1914 y Fn(\000)p Fj(N)318 1947 y Fp(\002)20 b(T)459 1962 y Fq(\025)500 1973 y Fk(V)553 1962 y Fq(;q)r(;\016)o(;r)709 1971 y Fl(2)766 1947 y Fp(\002)g Fs(D)s Ft(\(0)p Fs(;)15 b(\032)p Ft(\))p Fh(,)34 b(for)f Fs(r)1385 1961 y Fr(2)1450 1947 y Fp(\024)25 b Fs(r)1587 1961 y Fr(1)1659 1947 y Fh(and)33 b(al)5 b(l)33 b Fs(\032)26 b(>)f Ft(0)p Fh(.)75 2060 y(Mor)-5 b(e)g(over,)30 b(ther)-5 b(e)30 b(exists)f(a)g(se)-5 b(quenc)g(e)28 b Ft(\()p Fs(W)1528 2074 y Fq(n)1575 2060 y Ft(\()p Fs(\017;)15 b(z)t Ft(\)\))1803 2074 y Fq(n)p Fn(\025)p Fr(0)1970 2060 y Fh(of)29 b(functions)g(which)g(ar)-5 b(e)30 b(indep)-5 b(endent)30 b(of)f Fs(U;)15 b(V)20 b Fh(,)29 b(that)75 2173 y(b)-5 b(elongs)29 b(to)f(the)g(Banach)h(sp)-5 b(ac)g(e)29 b(of)g(b)-5 b(ounde)g(d)29 b(holomorphic)i(functions)d(on)h Fs(D)s Ft(\(0)p Fs(;)15 b(r)2899 2187 y Fr(0)2939 2173 y Ft(\))10 b Fp(\002)g Fs(D)s Ft(\(0)p Fs(;)15 b(\032)p Ft(\))29 b Fh(and)g(satis\014es)75 2285 y(the)k(fol)5 b(lowing)34 b(pr)-5 b(op)g(erty:)44 b(ther)-5 b(e)34 b(exist)f Fs(C)q(;)15 b(A)26 b(>)f Ft(0)33 b Fh(such)g(that)75 2561 y Ft(\(49\))629 b(sup)788 2643 y Fq(z)s Fn(2)p Fq(D)r Fr(\(0)p Fq(;\032)p Fr(\))1092 2561 y Fp(j)p Fs(Y)1170 2575 y Fq(U)7 b(V)1286 2561 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))21 b Fp(\000)1674 2447 y Fq(n)p Fn(\000)p Fr(1)1675 2474 y Fo(X)1664 2668 y Fq(m)p Fr(=0)1832 2561 y Fs(W)1918 2575 y Fq(m)1984 2561 y Ft(\()p Fs(\017;)15 b(z)t Ft(\))p Fs(t)2210 2523 y Fq(m)2278 2561 y Fp(j)26 b(\024)f Fs(C)7 b(A)2565 2523 y Fq(n)2611 2561 y Fp(j)p Fs(q)s Fp(j)2705 2523 y Fq(n)p Fr(\()p Fq(n)p Fn(\000)p Fr(1\))p Fq(=)p Fr(2)3011 2561 y Fp(j)p Fs(t)p Fp(j)3094 2523 y Fq(n)75 2848 y Fh(for)33 b(al)5 b(l)33 b Fs(n)25 b Fp(\025)g Ft(1)p Fh(,)33 b(for)g(al)5 b(l)33 b Fs(\017)25 b Fp(2)g Fs(U)10 b(q)1171 2815 y Fn(\000)p Fj(N)1278 2848 y Fh(,)32 b(al)5 b(l)34 b Fs(t)25 b Fp(2)g(T)1663 2863 y Fq(\025)1704 2874 y Fk(V)1757 2863 y Fq(;q)r(;\016)o(;r)1913 2872 y Fl(2)1950 2848 y Fh(.)75 3100 y Fi(Pro)s(of)35 b Ft(W)-8 b(e)35 b(consider)e(the)h(Cauc)m(h)m(y)h(problem)e(\(20\),\(21\))k(for) d(the)g(initial)h(conditions)f Fs(W)3202 3114 y Fq(U)7 b(V)t(;j)3358 3100 y Ft(\()p Fs(\017;)15 b(\034)10 b Ft(\),)36 b(for)e(all)75 3213 y(0)26 b Fp(\024)f Fs(j)31 b Fp(\024)25 b Fs(S)18 b Fp(\000)c Ft(1.)40 b(F)-8 b(rom)27 b(the)h(fact)g(that)f(the)h(initial)g(conditions)g(are)f(holomorphic)h (functions)e(on)h Fs(D)s Ft(\(0)p Fs(;)15 b(r)3694 3227 y Fr(0)3735 3213 y Ft(\))f Fp(\002)75 3326 y Fs(D)s Ft(\(0)p Fs(;)h(\032)320 3340 y Fr(0)360 3326 y Ft(\),)31 b(w)m(e)e(deduce)g (from)g(Theorem)g(3)h(that)g(the)f(problem)g(\(20\),\(21\))k(has)c(a)g (solution)h Fs(W)13 b Ft(\()p Fs(\017;)i(\034)5 b(;)15 b(z)t Ft(\))31 b(whic)m(h)75 3439 y(is)f(holomorphic)h(and)f(b)s (ounded)e(on)i Fs(D)s Ft(\(0)p Fs(;)15 b(r)1600 3453 y Fr(0)1641 3439 y Ft(\))20 b Fp(\002)g Fs(D)s Ft(\(0)p Fs(;)15 b(\032)2032 3453 y Fr(0)2072 3439 y Ft(\))21 b Fp(\002)f Fs(D)s Ft(\(0)p Fs(;)15 b(\032)p Ft(\),)32 b(for)e(all)h Fs(\032)26 b(>)e Ft(0.)42 b(Let)1408 3683 y Fs(W)13 b Ft(\()p Fs(\017;)i(\034)5 b(;)15 b(z)t Ft(\))26 b(=)1908 3597 y Fo(X)1907 3793 y Fq(n)p Fn(\025)p Fr(0)2076 3622 y Fs(W)2162 3636 y Fq(n)2209 3622 y Ft(\()p Fs(\017;)15 b(z)t Ft(\))p 2065 3662 350 4 v 2065 3749 a Fs(q)2109 3723 y Fq(n)p Fr(\()p Fq(n)p Fn(\000)p Fr(1\))p Fq(=)p Fr(2)2425 3683 y Fs(\034)2475 3646 y Fq(n)75 3973 y Ft(it's)31 b(T)-8 b(a)m(ylor)32 b(expansion)e(with)g(resp)s(ect)g(to)h Fs(\034)41 b Ft(on)30 b Fs(D)s Ft(\(0)p Fs(;)15 b(\032)2016 3987 y Fr(0)2056 3973 y Ft(\).)216 4086 y(F)-8 b(rom)45 b(the)f(estimates)i(\(46\))g(for)e(the)g(initial)i(conditions,)i(w)m(e) d(deduce)f(from)g(Theorem)g(2)g(that)h(the)75 4199 y(function)29 b Fs(W)13 b Ft(\()p Fs(\017;)i(\034)5 b(;)15 b(z)t Ft(\))30 b(can)f(b)s(e)g(extended)g(to)h(a)f(solution)h Fs(W)2123 4213 y Fq(U)7 b(V)2238 4199 y Ft(\()p Fs(\017;)15 b(\034)5 b(;)15 b(z)t Ft(\))31 b(of)e(the)h(problem)e(\(20\),\(21\))33 b(whic)m(h)28 b(is)75 4312 y(a)34 b(holomorphic)g(function)f(on)h Fs(U)10 b(q)1283 4279 y Fn(\000)p Fj(N)1412 4312 y Fp(\002)23 b Fs(V)d(q)1623 4279 y Fj(N)1697 4312 y Fp(\002)i Fs(D)s Ft(\(0)p Fs(;)15 b(\032)p Ft(\),)37 b(for)c(all)i Fs(\032)c(>)f Ft(0.)52 b(Moreo)m(v)m(er,)37 b(the)c(function)h Fs(W)3739 4326 y Fq(U)7 b(V)75 4425 y Ft(satis\014es)31 b(the)f(follo)m(wing)i (estimates:)43 b(there)30 b(exist)h(t)m(w)m(o)h(constan)m(ts)g Fs(C)q(;)15 b(T)38 b(>)25 b Ft(0,)31 b(suc)m(h)f(that)75 4654 y(\(50\))1059 b(sup)1143 4737 y Fq(x)p Fn(2)p Fq(V)t(;z)s Fn(2)p Fq(D)r Fr(\(0)p Fq(;\032)p Fr(\))1598 4654 y Fp(j)p Fs(W)1709 4668 y Fq(U)7 b(V)1825 4654 y Ft(\()p Fs(\017;)15 b(xq)2033 4617 y Fq(l)2059 4654 y Fs(;)g(z)t Ft(\))p Fp(j)27 b(\024)e Fs(C)7 b Fp(j)p Fs(q)s Fp(j)2504 4590 y Fl(1)p 2503 4602 31 3 v 2503 4643 a(2)2544 4617 y Fq(l)2566 4593 y Fl(2)2604 4654 y Ft(\()2660 4593 y(1)p 2649 4633 66 4 v 2649 4717 a Fs(T)2725 4654 y Ft(\))2760 4617 y Fq(l)75 4933 y Ft(for)31 b(all)g Fs(l)d Fp(\025)e Ft(0,)31 b(all)h Fs(\017)26 b Fp(2)f Fs(U)932 4947 y Fq(I)972 4933 y Fs(q)1016 4900 y Fn(\000)p Fj(N)1123 4933 y Ft(.)42 b(F)-8 b(rom)31 b(the)g(prop)s(osition)g(1)g(and)f(the)h(estimates)i (\(50\),)f(w)m(e)f(deduce)g(that)g(the)75 5046 y(function)1169 5159 y Fs(Y)1222 5173 y Fq(U)7 b(V)1338 5159 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))27 b(:=)e Fp(L)1815 5122 y Fq(\025)1856 5133 y Fk(V)1815 5182 y Fq(q)1913 5159 y Ft(\()p Fs(\034)36 b Fp(7!)25 b Fs(W)2226 5173 y Fq(U)7 b(V)2342 5159 y Ft(\()p Fs(\017;)15 b(\034)5 b(;)15 b(z)t Ft(\)\)\()p Fs(t)p Ft(\))75 5332 y(is)30 b(a)g(holomorphic)g(and)f(b)s (ounded)f(function)i(on)g Fs(U)10 b(q)1906 5299 y Fn(\000)p Fj(N)2032 5332 y Fp(\002)19 b(T)2172 5347 y Fq(\025)2213 5358 y Fk(V)2266 5347 y Fq(;q)r(;\016)o(;r)2422 5356 y Fl(2)2478 5332 y Fp(\002)g Fs(D)s Ft(\(0)p Fs(;)c(\032)p Ft(\),)31 b(for)f(some)g Fs(r)3311 5346 y Fr(2)3376 5332 y Fp(\024)25 b Fs(r)3513 5346 y Fr(1)3582 5332 y Ft(and)30 b(all)75 5444 y Fs(\032)25 b(>)g Ft(0.)38 b(Moreo)m(v)m(er,)26 b(the)c(estimates)h(\(49\))g(hold.)38 b(Finally)-8 b(,)25 b(from)c(the)g(prop)s(osition)h(2,)i(w)m(e)e(get)g(that)g Fs(Y)3471 5458 y Fq(U)7 b(V)3587 5444 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))75 5557 y(is)38 b(a)h(solution)g(of)f(\(47\))i(and)d (b)m(y)h(construction)h(of)g(the)f(initial)h(conditions)g Fs(')2816 5571 y Fq(U)7 b(V)t(;j)2972 5557 y Ft(\()p Fs(\017;)15 b(t)p Ft(\),)42 b(0)c Fp(\024)g Fs(j)44 b Fp(\024)38 b Fs(S)31 b Fp(\000)25 b Ft(1,)75 5670 y Fs(Y)128 5684 y Fq(U)7 b(V)244 5670 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))31 b(satis\014es)g(also)h(\(48\).)2512 b Ff(2)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 3764 100 a Ft(15)75 399 y Fd(4.3)112 b(Analytic)29 b(solutions)h(in)e(a)i(parameter)f(of)g(a)g(singularly)h (p)s(erturb)s(ed)g(Cauc)m(h)m(y)f(prob-)330 515 y(lem)75 686 y Ft(W)-8 b(e)32 b(recall)f(the)g(de\014nition)f(of)g(a)h(go)s(o)s (d)f(co)m(v)m(ering)j(as)d(de\014ned)f(in)h([10)r(],)g(p.36.)75 899 y Fi(De\014nition)35 b(4)46 b Fh(F)-7 b(or)30 b(any)g(tuple)g Fs(I)j Ft(=)24 b(\()p Fs(A;)15 b(B)5 b Ft(\))p Fh(,)31 b(wher)-5 b(e)31 b Fs(A)p Fh(,)f Fs(B)j Fh(ar)-5 b(e)31 b(op)-5 b(en)30 b(intervals)h(of)e Fm(R)g Fh(with)h(length)g(smal)5 b(ler)75 1012 y(than)34 b Ft(1)p Fs(=)p Ft(4)p Fh(,)g(let)e Fs(U)668 1026 y Fq(I)741 1012 y Fh(b)-5 b(e)32 b(the)h(b)-5 b(ounde)g(d)35 b(op)-5 b(en)33 b(sets)g(in)g Fm(C)1920 979 y Fn(\003)1991 1012 y Fh(de\014ne)-5 b(d)34 b(by)1040 1216 y Fs(U)1102 1230 y Fq(I)1167 1216 y Ft(=)25 b Fp(f)p Ft(exp\(2)1527 1143 y Fp(p)p 1604 1143 71 4 v 1604 1216 a(\000)o Ft(1)p Fs(\031)s(u)p Ft(\))p Fs(q)1905 1179 y Fq(\027)1975 1216 y Fp(2)g Fm(C)2127 1179 y Fn(\003)2166 1216 y Fs(=)p Fp(8)p Fs(u)g Fp(2)g Fs(A;)15 b Fp(8)p Fs(\027)31 b Fp(2)25 b Fs(B)5 b Fp(g)p Fs(:)75 1420 y Fh(We)32 b(cho)-5 b(ose)34 b(a)f(\014nite)g(family)g Fp(I)39 b Fh(of)33 b(tuple)g Fs(I)40 b Fh(as)33 b(ab)-5 b(ove,)33 b(such)g(that)75 1583 y(1\))g Fp([)252 1597 y Fq(I)5 b Fn(2I)382 1583 y Ft(\()p Fs(U)479 1597 y Fq(I)520 1583 y Fs(q)564 1550 y Fn(\000)p Fj(N)670 1583 y Ft(\))26 b(=)f Fp(V)i(n)21 b(f)p Ft(0)p Fp(g)p Fh(,)34 b(wher)-5 b(e)33 b Fp(V)40 b Fh(is)33 b(a)g(neighb)-5 b(orho)g(o)g(d)36 b(of)d Ft(0)g Fh(in)f Fm(C)p Fh(.)75 1696 y(2\))h(The)g(op)-5 b(en)34 b(sets)f Fs(U)832 1710 y Fq(I)872 1696 y Fs(q)916 1663 y Fn(\000)p Fj(N)1022 1696 y Fh(,)g Fs(I)f Fp(2)25 b(I)7 b Fh(,)32 b(ar)-5 b(e)33 b(four)g(by)g(four)g(disjoint.)75 1859 y(Then,)g(the)g(family)g Ft(\()p Fs(U)864 1873 y Fq(I)905 1859 y Fs(q)949 1826 y Fn(\000)p Fj(N)1056 1859 y Ft(\))1091 1873 y Fq(I)5 b Fn(2I)1254 1859 y Fh(is)32 b(c)-5 b(al)5 b(le)-5 b(d)34 b(a)f(go)-5 b(o)g(d)35 b(c)-5 b(overing.)75 2071 y Fi(De\014nition)35 b(5)46 b Fh(L)-5 b(et)35 b Ft(\()p Fs(U)917 2085 y Fq(I)957 2071 y Fs(q)1001 2038 y Fn(\000)p Fj(N)1108 2071 y Ft(\))1143 2085 y Fq(I)5 b Fn(2I)1308 2071 y Fh(b)-5 b(e)34 b(a)h(go)-5 b(o)g(d)37 b(c)-5 b(overing.)47 b(L)-5 b(et)35 b Fs(\016)n(;)15 b(r)2388 2085 y Fr(2)2457 2071 y Fs(>)29 b Ft(0)p Fh(.)48 b(We)34 b(c)-5 b(onsider)35 b(a)g(family)h(of)e(op)-5 b(en)75 2184 y(b)g(ounde)g(d)34 b(sets)f Ft(\()p Fs(V)688 2198 y Fq(I)728 2184 y Ft(\))763 2198 y Fq(I)5 b Fn(2I)894 2184 y Fh(,)33 b Fp(T)55 b Fh(in)32 b Fm(C)1237 2151 y Fn(\003)1309 2184 y Fh(satisfying)h(the)g(fol)5 b(lowing)34 b(pr)-5 b(op)g(erties:)75 2297 y(1\))33 b(Ther)-5 b(e)34 b(exists)e Ft(0)26 b Fs(<)f(\032)915 2311 y Fr(0)980 2297 y Fs(<)g Ft(1)33 b Fh(such)g(that)g Fs(V)1599 2311 y Fq(I)1660 2297 y Fp(\\)19 b Fs(D)s Ft(\(0)p Fs(;)c(\032)1985 2311 y Fr(0)2026 2297 y Ft(\))25 b Fp(6)p Ft(=)g Fp(;)p Fh(,)33 b(for)g(al)5 b(l)34 b Fs(I)e Fp(2)25 b(I)7 b Fh(.)75 2410 y(2\))33 b(F)-7 b(or)34 b(al)5 b(l)33 b Fs(I)f Fp(2)25 b(I)7 b Fh(,)32 b(for)h(al)5 b(l)33 b Fs(\034)i Fp(2)25 b Fs(V)1255 2424 y Fq(I)1295 2410 y Fs(q)1339 2377 y Fj(Z)1387 2410 y Fh(,)33 b Fp(j)p Fs(\034)d Ft(+)20 b(1)p Fp(j)26 b Fs(>)f(\016)s Fh(.)75 2523 y(3\))33 b(F)-7 b(or)34 b(al)5 b(l)33 b Fs(I)f Fp(2)25 b(I)7 b Fh(,)32 b(for)h(al)5 b(l)33 b Fs(t)25 b Fp(2)g(T)e Fh(,)32 b(al)5 b(l)33 b Fs(\017)1485 2537 y Fq(u)1555 2523 y Fp(2)25 b Fs(U)1703 2537 y Fq(I)1743 2523 y Fh(,)33 b(al)5 b(l)33 b Fs(\025)1987 2537 y Fq(v)2053 2523 y Fp(2)25 b Fs(V)2192 2537 y Fq(I)2252 2523 y Fp(\\)20 b Fs(D)s Ft(\(0)p Fs(;)15 b(\032)2578 2537 y Fr(0)2618 2523 y Ft(\))p Fh(,)33 b(we)g(have)1387 2770 y Fp(j)p Ft(1)21 b(+)1632 2708 y Fs(\025)1685 2722 y Fq(v)p 1579 2749 202 4 v 1579 2832 a Fs(\017)1616 2846 y Fq(u)1660 2832 y Fs(tq)1737 2806 y Fq(k)1790 2770 y Fp(j)k Fs(>)g(\016)69 b(;)81 b Fp(j)p Fs(\017)2213 2784 y Fq(u)2258 2770 y Fs(t)p Fp(j)25 b Fs(<)g(r)2478 2784 y Fr(2)2517 2770 y Fs(;)75 3028 y Fh(for)i(al)5 b(l)27 b Fs(k)i Fp(2)c Fm(Z)p Fh(.)39 b(We)26 b(say)i(that)g(the)f(family)g Fp(f)p Ft(\()p Fs(V)1670 3042 y Fq(I)1711 3028 y Ft(\))1746 3042 y Fq(I)5 b Fn(2I)1877 3028 y Fs(;)15 b Fp(T)23 b(g)k Fh(is)g(asso)-5 b(ciate)g(d)29 b(to)e(the)g(go)-5 b(o)g(d)29 b(c)-5 b(overing)27 b Ft(\()p Fs(U)3470 3042 y Fq(I)3510 3028 y Fs(q)3554 2995 y Fn(\000)p Fj(N)3661 3028 y Ft(\))3696 3042 y Fq(I)5 b Fn(2I)3827 3028 y Fh(.)216 3241 y Ft(Let)40 b Fs(S)46 b Fp(\025)40 b Ft(1)g(b)s(e)f(an)h(in)m(teger.)70 b(F)-8 b(or)40 b(all)h(0)g Fp(\024)f Fs(k)k Fp(\024)c Fs(S)32 b Fp(\000)26 b Ft(1,)42 b(let)f Fs(m)2544 3256 y Fr(0)p Fq(;k)2641 3241 y Ft(,)p Fs(m)2746 3256 y Fr(1)p Fq(;k)2884 3241 y Ft(b)s(e)e(p)s(ositiv)m(e)h(in)m(tegers)h(and)75 3354 y Fs(b)114 3369 y Fq(k)157 3354 y Ft(\()p Fs(\017;)15 b(z)t Ft(\))26 b(=)472 3285 y Fo(P)568 3381 y Fq(s)p Fn(2)p Fq(I)679 3393 y Fk(k)736 3354 y Fs(b)775 3369 y Fq(k)r(s)850 3354 y Ft(\()p Fs(\017)p Ft(\))p Fs(x)1009 3321 y Fq(s)1074 3354 y Ft(b)s(e)g(a)i(p)s(olynomial)g(in)f Fs(x)p Ft(,)h(where)f Fs(I)2242 3369 y Fq(k)2312 3354 y Ft(is)g(a)h(subset)f(of)g Fm(N)g Ft(and)g Fs(b)3155 3369 y Fq(k)r(s)3230 3354 y Ft(\()p Fs(\017)p Ft(\))h(are)g(b)s(ounded) 75 3479 y(holomorphic)f(functions)g(on)f(some)i(disc)f Fs(D)s Ft(\(0)p Fs(;)15 b(r)1746 3493 y Fr(0)1786 3479 y Ft(\))28 b(in)e Fm(C)p Ft(.)39 b(Let)28 b(\()p Fs(U)2338 3493 y Fq(I)2378 3479 y Fs(q)2422 3446 y Fn(\000)p Fj(N)2529 3479 y Ft(\))2564 3493 y Fq(I)5 b Fn(2I)2722 3479 y Ft(b)s(e)26 b(a)h(go)s(o)s(d)g(co)m(v)m(ering)i(suc)m(h)e(that)75 3592 y Fs(U)137 3606 y Fq(I)177 3592 y Fs(q)221 3559 y Fn(\000)p Fj(N)353 3592 y Fp(\032)e Fs(D)s Ft(\(0)p Fs(;)15 b(r)688 3606 y Fr(0)729 3592 y Ft(\))30 b(for)g(all)i Fs(I)g Fp(2)25 b(I)7 b Ft(.)75 3755 y(F)-8 b(or)31 b(all)g Fs(I)i Fp(2)25 b(I)7 b Ft(,)29 b(w)m(e)i(consider)f(the)h(follo)m(wing) h(singularly)e(p)s(erturb)s(ed)e(Cauc)m(h)m(y)j(problem)75 4040 y(\(51\))263 b Fs(\017t@)621 4003 y Fq(S)616 4063 y(z)671 4040 y Fs(X)746 4054 y Fq(I)787 4040 y Ft(\()p Fs(\017;)15 b(q)s(t;)g(z)t Ft(\))21 b(+)f Fs(@)1262 4003 y Fq(S)1257 4063 y(z)1313 4040 y Fs(X)1388 4054 y Fq(I)1429 4040 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))26 b(=)1817 3927 y Fq(S)t Fn(\000)p Fr(1)1820 3954 y Fo(X)1821 4152 y Fq(k)r Fr(=0)1969 4040 y Fs(b)2008 4055 y Fq(k)2051 4040 y Ft(\()p Fs(\017;)15 b(z)t Ft(\)\()p Fs(\017t\033)2401 4054 y Fq(q)2440 4040 y Ft(\))2475 4003 y Fq(m)2537 4015 y Fl(0)p Fk(;k)2630 4040 y Ft(\()p Fs(@)2718 4003 y Fq(k)2713 4063 y(z)2761 4040 y Fs(X)2836 4054 y Fq(I)2876 4040 y Ft(\)\()p Fs(\017;)g(t;)g(z)t(q)3186 4003 y Fn(\000)p Fq(m)3303 4015 y Fl(1)p Fk(;k)3397 4040 y Ft(\))75 4323 y(for)30 b(giv)m(en)i(initial)f(conditions)75 4528 y(\(52\))876 b(\()p Fs(@)1199 4490 y Fq(j)1194 4550 y(z)1236 4528 y Fs(Y)1289 4542 y Fq(I)1329 4528 y Ft(\)\()p Fs(\017;)15 b(t;)g Ft(0\))27 b(=)e Fs(')1811 4542 y Fq(I)5 b(;j)1903 4528 y Ft(\()p Fs(\017;)15 b(t)p Ft(\))62 b Fs(;)76 b Ft(0)25 b Fp(\024)g Fs(j)31 b Fp(\024)25 b Fs(S)g Fp(\000)20 b Ft(1)p Fs(;)75 4732 y Ft(where)33 b(the)h(functions)g Fs(')956 4746 y Fq(I)5 b(;j)1048 4732 y Ft(\()p Fs(\017;)15 b(t)p Ft(\))35 b(are)f(constructed)g(as)h(follo)m(ws.)52 b(Let)34 b Fp(f)p Ft(\()p Fs(V)2671 4746 y Fq(I)2712 4732 y Ft(\))2747 4746 y Fq(I)5 b Fn(2I)2878 4732 y Fs(;)15 b Fp(T)23 b(g)34 b Ft(b)s(e)f(a)i(family)f(of)g(op)s(en)75 4845 y(sets)41 b(asso)s(ciated)h(to)f(the)g(go)s(o)s(d)g(co)m(v)m (ering)h(\()p Fs(U)1690 4859 y Fq(I)1730 4845 y Fs(q)1774 4812 y Fn(\000)p Fj(N)1881 4845 y Ft(\))1916 4859 y Fq(I)5 b Fn(2I)2047 4845 y Ft(.)71 b(F)-8 b(or)42 b(all)f(0)i Fp(\024)f Fs(j)48 b Fp(\024)42 b Fs(S)32 b Fp(\000)26 b Ft(1,)44 b(for)c(all)i Fs(I)49 b Fp(2)42 b(I)7 b Ft(,)43 b(let)75 4958 y Fp(f)p Fs(W)206 4972 y Fq(j)243 4958 y Fs(;)15 b(W)369 4972 y Fq(U)417 4983 y Fk(I)452 4972 y Fq(V)493 4983 y Fk(I)528 4972 y Fq(;j)585 4958 y Fs(;)g(T)678 4972 y Fr(0)718 4958 y Fs(;)g(\032)805 4972 y Fr(0)844 4958 y Fp(g)36 b Ft(b)s(e)e(an)g(admissible)h(set.)55 b(Let)35 b Fs(\025)2039 4972 y Fq(I)2114 4958 y Ft(b)s(e)f(a)h(complex) h(n)m(um)m(b)s(er)d(in)i Fs(V)3182 4972 y Fq(I)3245 4958 y Fp(\\)23 b Fs(D)s Ft(\(0)p Fs(;)15 b(\032)3574 4972 y Fr(0)3614 4958 y Ft(\).)54 b(W)-8 b(e)75 5071 y(put)1205 5297 y Fs(')1264 5311 y Fq(I)5 b(;j)1357 5297 y Ft(\()p Fs(\017;)15 b(t)p Ft(\))26 b(:=)f Fp(L)1747 5259 y Fq(\025)1788 5270 y Fk(I)1747 5319 y Fq(q)1827 5297 y Ft(\()p Fs(\034)36 b Fp(7!)25 b Fs(W)2140 5311 y Fq(U)2188 5322 y Fk(I)2223 5311 y Fq(V)2264 5322 y Fk(I)2299 5311 y Fq(;j)2355 5297 y Ft(\()p Fs(\034)5 b(;)15 b(\017)p Ft(\)\)\()p Fs(\017t)p Ft(\))75 5463 y(F)-8 b(rom)40 b(the)f(prop)s(osition)g(1)g(and)g(the)g (estimates)i(\(46\),)i(w)m(e)d(deduce)f(that)g Fs(')2766 5477 y Fq(I)5 b(;j)2859 5463 y Ft(\()p Fs(\017;)15 b(t)p Ft(\))40 b(is)f(holomorphic)g(and)75 5576 y(b)s(ounded)28 b(on)i Fs(U)634 5590 y Fq(I)674 5576 y Fs(q)718 5543 y Fn(\000)p Fj(N)846 5576 y Fp(\002)19 b(T)k Ft(,)31 b(for)f(all)h Fs(I)h Fp(2)25 b(I)7 b Ft(,)30 b(0)c Fp(\024)f Fs(j)31 b Fp(\024)25 b Fs(S)g Fp(\000)20 b Ft(1,)31 b(if)f Fs(r)2328 5590 y Fr(2)2398 5576 y Ft(is)g(small)h(enough.)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 3764 100 a Ft(16)75 399 y Fi(Prop)s(osition)36 b(3)46 b Fh(L)-5 b(et)29 b(the)h(assumption)h Fi(\(A\))d Fh(\(of)h(se)-5 b(ction)30 b(2\))g(b)-5 b(e)29 b(ful\014l)5 b(le)-5 b(d)29 b(by)g(the)h(sets)f Fs(I)3157 414 y Fq(k)3229 399 y Fh(and)h(the)f(inte)-5 b(gers)75 511 y Fs(m)155 526 y Fr(0)p Fq(;k)252 511 y Fh(,)p Fs(m)360 526 y Fr(1)p Fq(;k)458 511 y Fh(,)32 b(for)h Ft(0)26 b Fp(\024)f Fs(k)j Fp(\024)d Fs(S)g Fp(\000)20 b Ft(1)p Fh(.)43 b(L)-5 b(et)32 b(the)h(initial)h(data)g(\(52\))g(c)-5 b(onstructe)g(d)34 b(as)f(ab)-5 b(ove.)75 624 y(Then,)35 b(for)g(al)5 b(l)34 b Fs(I)i Fp(2)27 b(I)7 b Fh(,)34 b(the)h(pr)-5 b(oblem)35 b(\(51\),)h(\(52\))g(has)f(a)f(solution)i Fs(X)2494 638 y Fq(I)2534 624 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))36 b Fh(which)f(is)f(holomorphic)j(and)75 737 y(b)-5 b(ounde)g(d)34 b(on)f Fs(U)613 751 y Fq(I)653 737 y Fs(q)697 704 y Fn(\000)p Fj(N)824 737 y Fp(\002)20 b(T)43 b(\002)20 b Fs(D)s Ft(\(0)p Fs(;)15 b(\032)p Ft(\))p Fh(,)34 b(for)f(al)5 b(l)33 b Fs(\032)25 b(>)g Ft(0)p Fh(.)75 850 y(Mor)-5 b(e)g(over,)34 b(ther)-5 b(e)33 b(exist)g(two)h(c)-5 b(onstants)34 b Fs(C)q(;)15 b(A)26 b(>)f Ft(0)33 b Fh(such)g(that)75 1052 y Ft(\(53\))606 b(sup)845 1130 y Fq(t)p Fn(2T)1069 1052 y Ft(sup)993 1135 y Fq(z)s Fn(2)p Fq(D)r Fr(\(0)p Fq(;\032)p Fr(\))1297 1052 y Fp(j)p Fs(X)1397 1066 y Fq(I)1437 1052 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))22 b Fp(\000)e Fs(X)1891 1067 y Fq(I)1927 1048 y Fg(0)1954 1052 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))p Fp(j)27 b(\024)e Fs(C)7 b(A)2508 1014 y Fq(n)2554 1052 y Fp(j)p Fs(q)s Fp(j)2648 1014 y Fq(n)p Fr(\()p Fq(n)p Fn(\000)p Fr(1\))p Fq(=)p Fr(2)2954 1052 y Fp(j)p Fs(\017)p Fp(j)3041 1014 y Fq(n)75 1336 y Fh(for)31 b(al)5 b(l)31 b Fs(I)7 b(;)15 b(I)480 1303 y Fn(0)530 1336 y Fp(2)24 b(I)7 b Fh(,)31 b Fs(I)h Fp(6)p Ft(=)25 b Fs(I)946 1303 y Fn(0)1000 1336 y Fh(such)31 b(that)h Fs(U)1451 1350 y Fq(I)1491 1336 y Fs(q)1535 1303 y Fn(\000)p Fj(N)1657 1336 y Fp(\\)16 b Fs(U)1796 1352 y Fq(I)1832 1333 y Fg(0)1859 1336 y Fs(q)1903 1303 y Fn(\000)p Fj(N)2035 1336 y Fp(6)p Ft(=)25 b Fp(;)p Fh(,)31 b(for)g(al)5 b(l)31 b Fs(\017)25 b Fp(2)g Fs(U)2716 1350 y Fq(I)2756 1336 y Fs(q)2800 1303 y Fn(\000)p Fj(N)2923 1336 y Fp(\\)16 b Fs(U)3062 1352 y Fq(I)3098 1333 y Fg(0)3124 1336 y Fs(q)3168 1303 y Fn(\000)p Fj(N)3275 1336 y Fh(,)31 b(for)g(al)5 b(l)31 b Fs(n)25 b Fp(\025)g Ft(1)p Fh(.)75 1449 y(In)37 b(other)h(wor)-5 b(ds,)40 b(if)c(we)h(denote)h(by)e Fm(H)1447 1463 y Fn(T)17 b Fq(;\032)1600 1449 y Fh(the)37 b(Banach)h(sp)-5 b(ac)g(e)38 b(of)f(holomorphic)j(and)e(b)-5 b(ounde)g(d)38 b(functions)75 1562 y(on)33 b Fp(T)43 b(\002)20 b Fs(D)s Ft(\(0)p Fs(;)15 b(\032)p Ft(\))34 b Fh(with)f(the)g(supr)-5 b(emum)34 b(norm,)g(the)f(holomorphic)j(and)d(b)-5 b(ounde)g(d)34 b(function)1142 1764 y Fs(g)1185 1779 y Fq(I)5 b(;I)1277 1760 y Fg(0)1303 1764 y Ft(\()p Fs(\017)p Ft(\))26 b(:=)f(\()p Fs(t;)15 b(z)t Ft(\))26 b Fp(7!)g Fs(X)1964 1778 y Fq(I)2004 1764 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))22 b Fp(\000)e Fs(X)2458 1779 y Fq(I)2494 1760 y Fg(0)2520 1764 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))75 1966 y Fh(with)33 b(values)g(in)g Fm(H)727 1980 y Fn(T)17 b Fq(;\032)875 1966 y Fh(has)34 b(the)f(formal)h(series)1735 1946 y Ft(^)1735 1966 y(0)f Fh(as)g Fs(q)s Fh(-asymptotic)h(exp)-5 b(ansion)34 b(on)g Fs(U)3074 1980 y Fq(I)3114 1966 y Fs(q)3158 1933 y Fn(\000)p Fj(N)3285 1966 y Fp(\\)19 b Fs(U)3427 1981 y Fq(I)3463 1962 y Fg(0)3490 1966 y Fs(q)3534 1933 y Fn(\000)p Fj(N)3641 1966 y Fh(.)75 2225 y Fi(Pro)s(of)47 b Ft(F)-8 b(or)48 b(all)f Fs(I)60 b Fp(2)52 b(I)7 b Ft(,)50 b(w)m(e)d(consider)g(the)g (Cauc)m(h)m(y)g(problem)f(\(47\),\(48\))k(for)c(the)h(initial)h (conditions)75 2338 y Fs(')134 2352 y Fq(U)182 2363 y Fk(I)217 2352 y Fq(V)258 2363 y Fk(I)294 2352 y Fq(;j)350 2338 y Ft(\()p Fs(\017;)15 b(t)p Ft(\))26 b(:=)f Fp(L)740 2305 y Fq(\025)781 2316 y Fk(I)740 2360 y Fq(q)821 2338 y Ft(\()p Fs(\034)35 b Fp(7!)25 b Fs(W)1133 2352 y Fq(U)1181 2363 y Fk(I)1216 2352 y Fq(V)1257 2363 y Fk(I)1293 2352 y Fq(;j)1349 2338 y Ft(\()p Fs(\034)5 b(;)15 b(\017)p Ft(\)\)\()p Fs(t)p Ft(\),)31 b(for)d(0)e Fp(\024)f Fs(j)30 b Fp(\024)25 b Fs(S)d Fp(\000)16 b Ft(1.)40 b(F)-8 b(rom)29 b(Theorem)f(4,)i(this)e(problem)g(has)75 2460 y(a)35 b(solution)h Fs(Y)557 2474 y Fq(U)605 2485 y Fk(I)640 2474 y Fq(V)681 2485 y Fk(I)720 2460 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))37 b(whic)m(h)d(is)h(holomorphic)h(and)e(b)s(ounded)f (on)i Fs(U)2659 2474 y Fq(I)2699 2460 y Fs(q)2743 2427 y Fn(\000)p Fj(N)2873 2460 y Fp(\002)23 b(T)3017 2475 y Fq(\025)3058 2486 y Fk(I)3093 2475 y Fq(;q)r(;\016)o(;r)3249 2484 y Fl(2)3309 2460 y Fp(\002)g Fs(D)s Ft(\(0)p Fs(;)15 b(\032)p Ft(\),)38 b(for)75 2573 y Fs(r)116 2587 y Fr(2)181 2573 y Fp(\024)25 b Fs(r)318 2587 y Fr(1)387 2573 y Ft(and)30 b(all)h Fs(\032)26 b(>)f Ft(0.)41 b(No)m(w,)31 b(w)m(e)g(put)1440 2775 y Fs(X)1515 2789 y Fq(I)1555 2775 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))27 b(:=)e Fs(Y)2022 2789 y Fq(U)2070 2800 y Fk(I)2105 2789 y Fq(V)2146 2800 y Fk(I)2185 2775 y Ft(\()p Fs(\017;)15 b(\017t;)g(z)t Ft(\))75 2977 y(for)34 b(all)h(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))33 b Fp(2)e Fs(U)801 2991 y Fq(I)841 2977 y Fs(q)885 2944 y Fn(\000)p Fj(N)1015 2977 y Fp(\002)22 b(T)45 b(\002)23 b Fs(D)s Ft(\(0)p Fs(;)15 b(\032)p Ft(\).)53 b(By)34 b(construction,)i Fs(X)2427 2991 y Fq(I)2468 2977 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))35 b(is)f(a)h(solution)f(of)h(the)f(Cauc)m(h)m(y)75 3090 y(problem)c(\(51\),)i(\(52\))g(whic)m(h)e(is)g(holomorphic)h(and)f (b)s(ounded)e(on)i Fs(U)2446 3104 y Fq(I)2486 3090 y Fs(q)2530 3057 y Fn(\000)p Fj(N)2657 3090 y Fp(\002)20 b(T)43 b(\002)20 b Fs(D)s Ft(\(0)p Fs(;)15 b(\032)p Ft(\).)216 3203 y(In)34 b(the)i(rest)f(of)g(the)g(pro)s(of,)h(w)m(e)f(sho)m(w)g (the)g(estimates)i(\(53\).)56 b(F)-8 b(rom)35 b(the)h(estimates)g (\(49\))h(there)e(exists)75 3316 y(a)d(sequence)g(\()p Fs(W)651 3330 y Fq(n)698 3316 y Ft(\()p Fs(\017;)15 b(z)t Ft(\)\))926 3330 y Fq(n)p Fn(\025)p Fr(0)1097 3316 y Ft(of)31 b(holomorphic)h(and)f(b)s(ounded)f(functions)h(on)g Fs(D)s Ft(\(0)p Fs(;)15 b(r)3033 3330 y Fr(0)3074 3316 y Ft(\))21 b Fp(\002)g Fs(D)s Ft(\(0)p Fs(;)15 b(\032)p Ft(\))33 b(and)e(t)m(w)m(o)75 3429 y(constan)m(ts)h Fs(C)q(;)15 b(A)26 b(>)f Ft(0)31 b(suc)m(h)f(that)75 3709 y(\(54\))583 b(sup)743 3792 y Fq(z)s Fn(2)p Fq(D)r Fr(\(0)p Fq(;\032)p Fr(\))1046 3709 y Fp(j)p Fs(X)1146 3723 y Fq(I)1187 3709 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))21 b Fp(\000)1575 3595 y Fq(n)p Fn(\000)p Fr(1)1576 3623 y Fo(X)1565 3817 y Fq(m)p Fr(=0)1733 3709 y Fs(W)1819 3723 y Fq(m)1886 3709 y Ft(\()p Fs(\017;)15 b(z)t Ft(\)\()p Fs(\017t)p Ft(\))2219 3672 y Fq(m)2287 3709 y Fp(j)25 b(\024)g Fs(C)7 b(A)2573 3672 y Fq(n)2620 3709 y Fp(j)p Fs(q)s Fp(j)2714 3672 y Fq(n)p Fr(\()p Fq(n)p Fn(\000)p Fr(1\))p Fq(=)p Fr(2)3020 3709 y Fp(j)p Fs(\017t)p Fp(j)3140 3672 y Fq(n)75 4002 y Ft(for)37 b(all)h Fs(\017)f Fp(2)f Fs(U)587 4016 y Fq(I)627 4002 y Fs(q)671 3969 y Fn(\000)p Fj(N)778 4002 y Ft(,)j(all)f Fs(t)e Fp(2)h(T)22 b Ft(,)40 b(for)c(all)j Fs(n)d Fp(\025)g Ft(1,)k(for)d(all)h Fs(I)44 b Fp(2)36 b(I)7 b Ft(.)60 b(F)-8 b(rom)38 b(the)f(estimates)i(\(54\),)i(w)m(e)d (get)g(a)75 4115 y(constan)m(t)32 b Fs(B)d(>)c Ft(0)31 b(suc)m(h)f(that)75 4317 y(\(55\))663 b(sup)748 4399 y Fq(t)p Fn(2T)18 b Fq(;z)s Fn(2)p Fq(D)r Fr(\(0)p Fq(;\032)p Fr(\))1200 4317 y Fp(j)p Fs(X)1300 4331 y Fq(I)1341 4317 y Ft(\()p Fs(\017;)d(t;)g(z)t Ft(\))21 b Fp(\000)f Fs(X)1794 4332 y Fq(I)1830 4313 y Fg(0)1857 4317 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))p Fp(j)27 b(\024)e Ft(2)p Fs(C)7 b Ft(\()p Fs(AB)e Ft(\))2600 4279 y Fq(n)2647 4317 y Fp(j)p Fs(q)s Fp(j)2741 4279 y Fq(n)p Fr(\()p Fq(n)p Fn(\000)p Fr(1\))p Fq(=)p Fr(2)3047 4317 y Fp(j)p Fs(\017)p Fp(j)3134 4279 y Fq(n)75 4601 y Ft(for)30 b(all)h Fs(I)7 b(;)15 b(I)474 4568 y Fn(0)524 4601 y Fp(2)24 b(I)37 b Ft(suc)m(h)30 b(that)h Fs(U)1160 4615 y Fq(I)1200 4601 y Fs(q)1244 4568 y Fn(\000)p Fj(N)1371 4601 y Fp(\\)20 b Fs(U)1514 4616 y Fq(I)1550 4598 y Fg(0)1576 4601 y Fs(q)1620 4568 y Fn(\000)p Fj(N)1752 4601 y Fp(6)p Ft(=)25 b Fp(;)p Ft(,)31 b(for)f(all)i Fs(\017)25 b Fp(2)g Fs(U)2425 4615 y Fq(I)2465 4601 y Fs(q)2509 4568 y Fn(\000)p Fj(N)2636 4601 y Fp(\\)20 b Fs(U)2779 4616 y Fq(I)2815 4598 y Fg(0)2841 4601 y Fs(q)2885 4568 y Fn(\000)p Fj(N)2992 4601 y Ft(,)30 b(for)g(all)i Fs(n)25 b Fp(\025)f Ft(1.)198 b Ff(2)75 4936 y Fu(5)135 b(Existence)50 b(of)f(formal)g(series)h(solutions)g (and)f Fe(q)t Fu(-asymptotic)h(expan-)277 5086 y(sion)45 b(in)g(the)g(parameter)h Fe(\017)75 5289 y Ft(In)35 b(this)g(section,)j (w)m(e)e(will)g(establish)g(the)g(existence)g(of)g(a)g(formal)g(series) f(solution)h(of)g(the)g(problem)f(\(51\),)75 5402 y(\(52\),)332 5379 y(^)306 5402 y Fs(X)8 b Ft(\()p Fs(\017)p Ft(\))44 b Fp(2)g Fm(H)716 5416 y Fn(T)16 b Fq(;\032)831 5402 y Ft([[)p Fs(\017)p Ft(]])43 b(with)e(co)s(e\016cien)m(ts)i(in)f Fm(H)1889 5416 y Fn(T)16 b Fq(;\032)2004 5402 y Ft(,)45 b(the)d(Banac)m(h)g(spaces)g(of)g(b)s(ounded)d(holomorphic)75 5514 y(functions)f(on)g Fp(T)48 b(\002)25 b Fs(D)s Ft(\(0)p Fs(;)15 b(\032)p Ft(\))40 b(with)e(the)g(suprem)m(um)f(norm,)j(suc)m(h) e(that)h(the)f(solutions)h Fs(X)3271 5528 y Fq(I)3311 5514 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))40 b(of)e(the)75 5627 y(problem)28 b(\(51\),\(52\))33 b(constructed)c(ab)s(o)m(v)m(e,)i (has)1763 5604 y(^)1737 5627 y Fs(X)36 b Ft(as)29 b Fs(q)s Ft(-asymptotic)i(expansion)e(on)g Fs(U)3107 5641 y Fq(I)3147 5627 y Fs(q)3191 5594 y Fn(\000)p Fj(N)3298 5627 y Ft(,)g(for)g(all)h Fs(I)i Fp(2)25 b(I)7 b Ft(.)75 5740 y(The)30 b(pro)s(of)f(is)i(based)f (on)g(a)h Fs(q)s Ft(-analog)h(of)f(the)f(Malgrange-Sibuy)m(a)i(theorem) f(obtained)g(in)f([10)q(].)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 3764 100 a Ft(17)75 399 y Fd(5.1)112 b(Construction)38 b(of)f(Banac)m(h)h(v)-6 b(alued)39 b(co)s(cyles)f(on)f(a)h(go)s(o)s(d)g(co)m(v)m(ering.)75 570 y Ft(W)-8 b(e)37 b(denote)g(b)m(y)f Fm(H)739 584 y Fn(T)17 b Fq(;\032)890 570 y Ft(the)37 b(Banac)m(h)g(space)f(of)h (holomorphic)f(and)f(b)s(ounded)f(functions)h(on)h(the)h(op)s(en)e(set) 75 683 y Fp(T)43 b(\002)20 b Fs(D)s Ft(\(0)p Fs(;)15 b(\032)p Ft(\))31 b(with)f(the)h(suprem)m(um)e(norm.)75 896 y Fi(De\014nition)35 b(6)46 b Fh(A)d(set)h(of)g(functions)g Fp(f)p Fs(g)1537 911 y Fq(I)5 b(;I)1629 892 y Fg(0)1655 896 y Ft(\()p Fs(\017)p Ft(\))p Fp(g)1807 915 y Fr(\()p Fq(I)g(;I)1926 896 y Fg(0)1949 915 y Fr(\))p Fn(2I)2067 896 y Fl(2)53 b Fh(is)43 b(c)-5 b(al)5 b(le)-5 b(d)45 b(a)f Fm(H)2681 910 y Fn(T)17 b Fq(;\032)2797 896 y Fh(-c)-5 b(o)g(cyle)44 b(attache)-5 b(d)46 b(to)f(a)f(go)-5 b(o)g(d)75 1019 y(c)g(overing)33 b Ft(\()p Fs(U)531 1033 y Fq(I)571 1019 y Fs(q)615 986 y Fn(\000)p Fj(N)722 1019 y Ft(\))757 1033 y Fq(I)5 b Fn(2I)920 1019 y Fh(if)33 b(the)g(fol)5 b(lowing)33 b(pr)-5 b(op)g(erties)36 b(ar)-5 b(e)33 b(satis\014e)-5 b(d)34 b(:)75 1182 y(1\))25 b(F)-7 b(or)25 b(al)5 b(l)25 b Ft(\()p Fs(I)7 b(;)15 b(I)635 1149 y Fn(0)660 1182 y Ft(\))25 b Fp(2)g(I)863 1149 y Fr(2)902 1182 y Fh(,)h(the)e(function) h Fs(g)1491 1197 y Fq(I)5 b(;I)1583 1178 y Fg(0)1633 1182 y Fh(is)25 b(a)f Fm(H)1864 1196 y Fn(T)17 b Fq(;\032)1980 1182 y Fh(-value)-5 b(d)25 b(holomorphic)j(function)c(on)h Fs(U)3323 1196 y Fq(I)3363 1182 y Fs(q)3407 1149 y Fn(\000)p Fj(N)3516 1182 y Fp(\\)r Fs(U)3641 1197 y Fq(I)3677 1178 y Fg(0)3704 1182 y Fs(q)3748 1149 y Fn(\000)p Fj(N)75 1295 y Fh(having)33 b(the)g(formal)h(series)1060 1275 y Ft(^)1060 1295 y(0)f Fh(as)g Fs(q)s Fh(-asymptotic)h(exp)-5 b(ansion)35 b(on)e Fs(U)2399 1309 y Fq(I)2439 1295 y Fs(q)2483 1262 y Fn(\000)p Fj(N)2610 1295 y Fp(\\)20 b Fs(U)2753 1310 y Fq(I)2789 1291 y Fg(0)2815 1295 y Fs(q)2859 1262 y Fn(\000)p Fj(N)2966 1295 y Fh(.)75 1407 y(2\))33 b(F)-7 b(or)34 b(al)5 b(l)33 b Fs(I)7 b(;)15 b(I)625 1374 y Fn(0)674 1407 y Fp(2)25 b(I)7 b Fh(,)32 b(we)g(have)h Fs(g)1263 1423 y Fq(I)5 b(;I)1355 1404 y Fg(0)1381 1407 y Ft(\()p Fs(\017)p Ft(\))26 b(=)f Fp(\000)p Fs(g)1724 1423 y Fq(I)1760 1404 y Fg(0)1782 1423 y Fq(;I)1842 1407 y Ft(\()p Fs(\017)p Ft(\))33 b Fh(for)g(al)5 b(l)33 b Fs(\017)26 b Fp(2)e Fs(U)2467 1421 y Fq(I)2507 1407 y Fs(q)2551 1374 y Fn(\000)p Fj(N)2679 1407 y Fp(\\)19 b Fs(U)2821 1423 y Fq(I)2857 1404 y Fg(0)2884 1407 y Fs(q)2928 1374 y Fn(\000)p Fj(N)3035 1407 y Fh(.)75 1520 y(3\))28 b(F)-7 b(or)29 b(al)5 b(l)28 b Fs(I)7 b(;)15 b(I)610 1487 y Fn(0)634 1520 y Fs(;)g(I)721 1487 y Fn(00)789 1520 y Fp(2)25 b(I)7 b Fh(,)27 b(we)h(have)g Fs(g)1364 1536 y Fq(I)5 b(;I)1456 1517 y Fg(00)1500 1520 y Ft(\()p Fs(\017)p Ft(\))26 b(=)f Fs(g)1772 1536 y Fq(I)5 b(;I)1864 1517 y Fg(0)1890 1520 y Ft(\()p Fs(\017)p Ft(\))k(+)g Fs(g)2129 1536 y Fq(I)2165 1517 y Fg(0)2188 1536 y Fq(;I)2244 1517 y Fg(00)2288 1520 y Ft(\()p Fs(\017)p Ft(\))p Fh(,)29 b(for)f(al)5 b(l)28 b Fs(\017)e Fp(2)e Fs(U)2927 1534 y Fq(I)2967 1520 y Fs(q)3011 1487 y Fn(\000)p Fj(N)3127 1520 y Fp(\\)9 b Fs(U)3259 1536 y Fq(I)3295 1517 y Fg(0)3322 1520 y Fs(q)3366 1487 y Fn(\000)p Fj(N)3482 1520 y Fp(\\)g Fs(U)3614 1536 y Fq(I)3650 1517 y Fg(0)3676 1520 y Fs(q)3720 1487 y Fn(\000)p Fj(N)3827 1520 y Fh(.)75 1733 y Fi(Prop)s(osition)36 b(4)46 b Fh(We)27 b(c)-5 b(onsider)29 b(the)e(family)i(of)e (holomorphic)k(functions)d Fp(f)p Fs(X)2792 1747 y Fq(I)2833 1733 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))p Fp(g)3144 1747 y Fq(I)5 b Fn(2I)3303 1733 y Fh(c)-5 b(onstructe)g(d)30 b(in)75 1846 y(the)j(pr)-5 b(op)g(osition)36 b(3.)42 b(Then,)33 b(the)g(set)g(of)g(functions)g Fp(f)p Fs(g)1958 1861 y Fq(I)5 b(;I)2050 1842 y Fg(0)2076 1846 y Ft(\()p Fs(\017)p Ft(\))p Fp(g)2228 1866 y Fr(\()p Fq(I)g(;I)2347 1847 y Fg(0)2370 1866 y Fr(\))p Fn(2I)2488 1847 y Fl(2)2559 1846 y Fh(de\014ne)-5 b(d)34 b(by)1142 2050 y Fs(g)1185 2065 y Fq(I)5 b(;I)1277 2046 y Fg(0)1303 2050 y Ft(\()p Fs(\017)p Ft(\))26 b(:=)f(\()p Fs(t;)15 b(z)t Ft(\))26 b Fp(7!)g Fs(X)1964 2065 y Fq(I)2000 2046 y Fg(0)2026 2050 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))22 b Fp(\000)e Fs(X)2480 2064 y Fq(I)2520 2050 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))75 2254 y Fh(for)38 b(al)5 b(l)37 b Fs(I)7 b(;)15 b(I)493 2221 y Fn(0)550 2254 y Fp(2)33 b(I)43 b Fh(is)37 b(a)h Fm(H)994 2268 y Fn(T)17 b Fq(;\032)1110 2254 y Fh(-c)-5 b(o)g(cyle)37 b(attache)-5 b(d)39 b(to)f(the)f(go)-5 b(o)g(d)39 b(c)-5 b(overing)37 b Ft(\()p Fs(U)2709 2268 y Fq(I)2750 2254 y Fs(q)2794 2221 y Fn(\000)p Fj(N)2901 2254 y Ft(\))2936 2268 y Fq(I)5 b Fn(2I)3066 2254 y Fh(,)38 b(for)g Fp(T)56 b(\032)33 b Fm(C)3558 2221 y Fn(\003)3597 2254 y Fh(,)p Fs(\032)g(>)g Ft(0)75 2367 y Fh(given)f(in)g(the)h(pr)-5 b(op)g(osition)37 b(3.)75 2630 y Fi(Pro)s(of)29 b Ft(The)f(prop)s(ert)m (y)g(1\))h(in)f(the)h(de\014nition)f(6)g(is)h(a)f(consequence)i(of)e (the)h(prop)s(osition)f(3.)40 b(The)28 b(prop)s(erties)75 2742 y(2\))j(3\))g(are)g(ful\014lled)f(b)m(y)g(construction.)2340 b Ff(2)75 3036 y Fd(5.2)112 b(Splitting)30 b(of)g(the)f(co)s(cyles)h (in)f(spaces)i(of)e(Banac)m(h)i(v)-6 b(alued)30 b(Whitney)g Fc(C)3305 3000 y Fn(1)3409 3036 y Fd(functions)75 3207 y Ft(W)-8 b(e)32 b(recall)f(the)g(de\014nition)f(of)g(a)h Fs(C)1286 3174 y Fn(1)1391 3207 y Ft(function)f(the)g(the)h(sense)f(of) h(Whitney)-8 b(,)31 b(see)g([8)q(],)g([9],)g(p.67.)75 3420 y Fi(De\014nition)k(7)46 b Fh(L)-5 b(et)37 b Fs(f)k Ft(:)33 b Fs(A)f Fp(!)g Fm(H)k Fh(b)-5 b(e)36 b(a)h(c)-5 b(ontinuous)38 b(function)e(de\014ne)-5 b(d)38 b(on)f(a)g(c)-5 b(omp)g(act)38 b(set)f Fs(A)32 b Fp(\032)g Fm(R)3614 3387 y Fr(2)3690 3420 y Fh(with)75 3533 y(values)k(in)f(a)h(Banach)g (sp)-5 b(ac)g(e)37 b Ft(\()p Fm(H)p Fs(;)15 b Fp(jj)p Fs(:)p Fp(jj)1388 3547 y Fj(H)1446 3533 y Ft(\))p Fh(.)50 b(The)36 b(function)g Fs(f)44 b Fh(is)36 b(said)g(to)g(b)-5 b(e)35 b Fs(C)2790 3500 y Fn(1)2900 3533 y Fh(in)g(the)h(sense)f(of)h (Whitney)75 3646 y(on)f Fs(A)f Fh(if)g(ther)-5 b(e)35 b(exists)g(a)g(family)g(of)f(c)-5 b(ontinuous)36 b(functions)e Fs(f)2256 3613 y Fq(k)2327 3646 y Ft(:)28 b Fs(A)h Fp(!)f Fm(H)p Fh(,)34 b(for)h Fs(k)c Fp(2)d Fm(N)3109 3613 y Fr(2)3148 3646 y Fh(,)35 b(wher)-5 b(e)35 b Fs(f)3524 3613 y Fr(\(0)p Fq(;)p Fr(0\))3701 3646 y Ft(=)28 b Fs(f)75 3759 y Fh(satisfying)i(the)g(fol)5 b(lowing)31 b(pr)-5 b(op)g(erty)32 b Fi(\(W\))d Fh(:)h(F)-7 b(or)31 b(al)5 b(l)30 b Fs(m)25 b Fp(\025)g Ft(0)p Fh(,)30 b(for)g(al)5 b(l)31 b Fs(k)d Fp(2)d Fm(N)2731 3726 y Fr(2)2799 3759 y Fh(with)31 b Fp(j)p Fs(k)s Fp(j)26 b(\024)f Fs(m)p Fh(,)30 b(for)g(al)5 b(l)30 b Fs(\017)25 b(>)g Ft(0)p Fh(,)75 3872 y(ther)-5 b(e)34 b(exists)e Fs(\016)e(>)25 b Ft(0)32 b Fh(such)h(that)75 4144 y Ft(\(56\))85 b Fp(jj)p Fs(f)425 4106 y Fq(k)467 4144 y Ft(\()p Fs(x)554 4158 y Fr(1)594 4144 y Fs(;)15 b(y)679 4158 y Fr(1)718 4144 y Ft(\))21 b Fp(\000)1199 4058 y Fo(X)865 4260 y Fq(h)p Fr(=\()p Fq(h)1029 4269 y Fl(1)1063 4260 y Fq(;h)1124 4269 y Fl(2)1158 4260 y Fr(\))p Fn(2)p Fj(N)1280 4241 y Fl(2)1315 4260 y Fq(;)p Fn(j)p Fq(h)p Fn(j\024)p Fq(m)p Fn(\000j)p Fq(k)r Fn(j)1689 4082 y Fs(f)1744 4049 y Fr(\()p Fq(k)r Fr(+)p Fq(h)p Fr(\))1937 4082 y Ft(\()p Fs(x)2024 4096 y Fr(2)2064 4082 y Fs(;)15 b(y)2149 4096 y Fr(2)2188 4082 y Ft(\))p 1689 4123 534 4 v 1839 4206 a Fs(h)1891 4220 y Fr(1)1931 4206 y Ft(!)p Fs(h)2008 4220 y Fr(2)2048 4206 y Ft(!)2233 4144 y(\()p Fs(x)2320 4158 y Fr(1)2380 4144 y Fp(\000)20 b Fs(x)2523 4158 y Fr(2)2563 4144 y Ft(\))2598 4106 y Fq(h)2639 4115 y Fl(1)2677 4144 y Ft(\()p Fs(y)2757 4158 y Fr(1)2817 4144 y Fp(\000)g Fs(y)2953 4158 y Fr(2)2992 4144 y Ft(\))3027 4106 y Fq(h)3068 4115 y Fl(2)3107 4144 y Fp(jj)3157 4158 y Fj(H)2654 4417 y Fs(<)25 b(\017)p Fp(jj)p Ft(\()p Fs(x)2924 4431 y Fr(1)2985 4417 y Fp(\000)20 b Fs(x)3128 4431 y Fr(2)3167 4417 y Fs(;)15 b(y)3252 4431 y Fr(1)3312 4417 y Fp(\000)k Fs(y)3447 4431 y Fr(2)3486 4417 y Ft(\))p Fp(jj)3571 4369 y Fq(m)p Fn(\000j)p Fq(k)r Fn(j)3571 4451 y Fj(R)3619 4432 y Fl(2)75 4621 y Fh(for)33 b(al)5 b(l)33 b Ft(\()p Fs(x)437 4635 y Fr(1)477 4621 y Fs(;)15 b(y)562 4635 y Fr(1)601 4621 y Ft(\))p Fs(;)g Ft(\()p Fs(x)763 4635 y Fr(2)804 4621 y Fs(;)g(y)889 4635 y Fr(2)928 4621 y Ft(\))26 b Fp(2)f Fs(A)p Fh(,)32 b Ft(\()p Fs(x)1290 4635 y Fr(1)1330 4621 y Fs(;)15 b(y)1415 4635 y Fr(1)1455 4621 y Ft(\))25 b Fp(6)p Ft(=)g(\()p Fs(x)1698 4635 y Fr(2)1738 4621 y Fs(;)15 b(y)1823 4635 y Fr(2)1862 4621 y Ft(\))33 b Fh(with)h Fp(jj)p Ft(\()p Fs(x)2265 4635 y Fr(1)2325 4621 y Fp(\000)20 b Fs(x)2468 4635 y Fr(2)2507 4621 y Fs(;)15 b(y)2592 4635 y Fr(1)2652 4621 y Fp(\000)20 b Fs(y)2788 4635 y Fr(2)2827 4621 y Ft(\))p Fp(jj)2912 4641 y Fj(R)2960 4622 y Fl(2)3025 4621 y Fs(<)25 b(\016)s Fh(.)75 4833 y Fi(Remark:)45 b Ft(If)31 b(a)i(con)m(tin)m(uous)g(function)f Fs(f)41 b Ft(is)33 b Fs(C)1748 4800 y Fn(1)1854 4833 y Ft(in)f(the)g(sense)h(of)f(Whitney)h(on)f(a)g(compact)i(set)f Fs(A)p Ft(,)g(then)75 4946 y Fs(f)40 b Ft(is)30 b Fs(C)323 4913 y Fn(1)428 4946 y Ft(in)g(the)g(usual)g(sense)g(on)h(the)f(in)m (terior)h(In)m(t)q(\()p Fs(A)p Ft(\))g(of)f Fs(A)p Ft(.)41 b(Moreo)m(v)m(er,)33 b(w)m(e)e(ha)m(v)m(e)g(that)1476 5151 y Fs(f)1531 5113 y Fq(k)1573 5151 y Ft(\()p Fs(x;)15 b(y)s Ft(\))26 b(=)f Fs(@)1958 5113 y Fq(k)1995 5122 y Fl(1)1953 5173 y Fq(x)2034 5151 y Fs(@)2087 5113 y Fq(k)2124 5122 y Fl(2)2082 5173 y Fq(y)2163 5151 y Fs(f)10 b Ft(\()p Fs(x;)15 b(y)s Ft(\))p Fs(;)75 5355 y Ft(if)30 b Fs(k)f Ft(=)c(\()p Fs(k)412 5369 y Fr(1)452 5355 y Fs(;)15 b(k)539 5369 y Fr(2)579 5355 y Ft(\))26 b Fp(2)e Fm(N)791 5322 y Fr(2)831 5355 y Ft(,)30 b(for)g(all)i(\()p Fs(x;)15 b(y)s Ft(\))26 b Fp(2)f Ft(In)m(t\()p Fs(A)p Ft(\).)75 5518 y(In)e(the)i(follo)m(wing)g(w)m(e)g(will)g(denote)f(b)m (y)g Fp(E)8 b Ft(\()p Fs(A)p Ft(\))25 b(the)g(v)m(ector)g(space)g(of)f Fs(C)2452 5485 y Fn(1)2551 5518 y Ft(functions)f(in)h(the)h(sense)f(of) g(Whitney)75 5630 y(on)30 b Fs(A)p Ft(.)41 b(W)-8 b(e)32 b(recall)f(the)g(lemma)g(3.2)g(from)f([10)q(].)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 3764 100 a Ft(18)75 399 y Fi(Lemma)35 b(5)45 b Fh(L)-5 b(et)34 b Fs(U)43 b Fh(b)-5 b(e)33 b(an)g(op)-5 b(en)35 b(set)e(in)g Fm(C)1583 366 y Fn(\003)1622 399 y Fh(.)43 b(L)-5 b(et)34 b Fs(f)h Ft(:)27 b Fs(U)10 b(q)2099 366 y Fn(\000)p Fj(N)2231 399 y Fp(!)27 b Fm(H)32 b Fh(b)-5 b(e)33 b(a)h(holomorphic)i(function)d(having)h(a)75 511 y(formal)h(series)643 487 y Ft(^)623 511 y Fs(f)9 b Ft(\()p Fs(\017)p Ft(\))28 b(=)910 443 y Fo(P)1006 538 y Fq(n)p Fn(\025)p Fr(0)1158 511 y Fs(a)1206 525 y Fq(n)1253 511 y Fs(\017)1290 478 y Fq(n)1364 511 y Fp(2)f Fm(H)p Ft([[)p Fs(\017)p Ft(]])34 b Fh(as)g Fs(q)s Fh(-asymptotic)h(exp)-5 b(ansion)36 b(on)e Fs(U)10 b(q)3013 478 y Fn(\000)p Fj(N)3120 511 y Fh(.)44 b(Then,)35 b(the)f(family)75 624 y Fs(@)128 591 y Fq(n)123 647 y(\017)175 624 y Fs(f)10 b Ft(\()p Fs(\017)p Ft(\))35 b Fh(of)g Fs(n)p Fp(\000)p Fh(th)g(c)-5 b(omplex)37 b(derivatives)e(of)h Fs(f)10 b Fh(,)34 b(for)i Fs(n)29 b Fp(\025)g Ft(0)p Fh(,)36 b(satisfy)g(the)f(fol)5 b(lowing)37 b(pr)-5 b(op)g(erty)37 b Ft(\()p Fi(W)3517 638 y Fb(H)3585 624 y Ft(\))f Fh(:)46 b(L)-5 b(et)75 737 y Fs(K)38 b Fp(\032)31 b Fs(U)46 b Fh(b)-5 b(e)36 b(a)h(c)-5 b(omp)g(act)38 b(set.)52 b(Then,)37 b(for)g(al)5 b(l)36 b Fs(k)s(;)15 b(m)32 b Fp(\025)f Ft(0)p Fh(,)37 b(ther)-5 b(e)37 b(exists)g(a)f(c)-5 b(onstant)38 b Fs(C)3102 752 y Fq(K)q(;m;k)3342 737 y Fh(such)f(that)g(for)75 850 y(al)5 b(l)33 b Fs(\017)242 864 y Fq(a)284 850 y Fs(;)15 b(\017)361 865 y Fq(b)421 850 y Fp(2)24 b Fs(K)7 b(q)634 817 y Fn(\000)p Fj(N)761 850 y Fp([)20 b(f)p Ft(0)p Fp(g)p Fh(,)34 b(we)e(have)75 1108 y Ft(\(57\))551 b Fp(jj)p Fs(@)889 1071 y Fq(k)884 1131 y(\017)933 1108 y Fs(f)10 b Ft(\()p Fs(\017)1060 1122 y Fq(a)1101 1108 y Ft(\))21 b Fp(\000)1282 995 y Fq(m)1247 1022 y Fo(X)1248 1220 y Fq(h)p Fr(=0)1404 1047 y Fs(@)1457 1014 y Fq(k)r Fr(+)p Fq(h)1452 1069 y(\017)1595 1047 y Fs(f)10 b Ft(\()p Fs(\017)1722 1062 y Fq(b)1756 1047 y Ft(\))p 1404 1087 388 4 v 1559 1171 a Fs(h)p Ft(!)1802 1108 y(\()p Fs(\017)1874 1122 y Fq(a)1936 1108 y Fp(\000)20 b Fs(\017)2064 1123 y Fq(b)2098 1108 y Ft(\))2133 1071 y Fq(h)2178 1108 y Fp(jj)2228 1122 y Fj(H)2310 1108 y Fp(\024)25 b Fs(C)2471 1123 y Fq(K)q(;m;k)2675 1108 y Fp(j)p Fs(\017)2737 1122 y Fq(a)2799 1108 y Fp(\000)20 b Fs(\017)2927 1123 y Fq(b)2961 1108 y Fp(j)2986 1071 y Fq(m)p Fr(+1)75 1381 y Fh(wher)-5 b(e,)34 b(by)e(de\014nition,)i(we)e(put)h Fs(@)1258 1348 y Fq(n)1253 1403 y(\017)1306 1381 y Fs(f)10 b Ft(\(0\))25 b(=)g Fs(n)p Ft(!)p Fs(a)1725 1395 y Fq(n)1772 1381 y Fh(,)32 b(for)i(al)5 b(l)33 b Fs(n)25 b Fp(\025)g Ft(0)p Fh(.)216 1578 y Ft(The)30 b(next)i(lemma)f(sho)m(ws)g(that)g(all)h (holomorphic)f(functions)f(ha)m(ving)i Fs(q)s Ft(-asymptotic)g (expansion)f(m)m(ust)75 1691 y(satisfy)g(the)f(prop)s(ert)m(y)g Fi(\(W\))p Ft(.)75 1888 y Fi(Lemma)35 b(6)45 b Fh(L)-5 b(et)50 b Fs(f)65 b Ft(:)56 b Fs(U)10 b(q)1024 1855 y Fn(\000)p Fj(N)1187 1888 y Fp(!)55 b Fm(H)49 b Fh(b)-5 b(e)50 b(a)g(holomorphic)i(function)e(having)g(a)g(formal)h(series)3586 1864 y Ft(^)3566 1888 y Fs(f)9 b Ft(\()p Fs(\017)p Ft(\))57 b(=)75 1933 y Fo(P)171 2028 y Fq(n)p Fn(\025)p Fr(0)323 2001 y Fs(a)371 2015 y Fq(n)418 2001 y Fs(\017)455 1968 y Fq(n)541 2001 y Fp(2)39 b Fm(H)p Ft([[)p Fs(\017)p Ft(]])h Fh(as)h Fs(q)s Fh(-asymptotic)h(exp)-5 b(ansion)42 b(on)f Fs(U)10 b(q)2235 1968 y Fn(\000)p Fj(N)2342 2001 y Fh(.)64 b(L)-5 b(et)41 b Fs(K)k Fp(\032)39 b Fs(U)50 b Fh(b)-5 b(e)40 b(c)-5 b(omp)g(act)43 b(set.)64 b(Then,)75 2114 y(the)40 b(function)g Ft(\()p Fs(\017)670 2128 y Fr(1)709 2114 y Fs(;)15 b(\017)786 2128 y Fr(2)826 2114 y Ft(\))38 b Fp(7!)g Fs(\036)p Ft(\()p Fs(\017)1154 2128 y Fr(1)1194 2114 y Fs(;)15 b(\017)1271 2128 y Fr(2)1310 2114 y Ft(\))39 b(=)e Fs(f)10 b Ft(\()p Fs(\017)1619 2128 y Fr(1)1683 2114 y Ft(+)25 b Fs(i\017)1847 2128 y Fr(2)1887 2114 y Ft(\))40 b Fh(is)f(a)h Fs(C)2224 2081 y Fn(1)2338 2114 y Fh(function)f(in)h(the)g(Whitney)g(sense)f(on)h(the) 75 2227 y(c)-5 b(omp)g(act)35 b(set)1123 2340 y Fs(A)25 b Ft(=)g Fp(f)p Ft(\()p Fs(\017)1429 2354 y Fr(1)1469 2340 y Fs(;)15 b(\017)1546 2354 y Fr(2)1586 2340 y Ft(\))26 b Fp(2)f Fm(R)1799 2302 y Fr(2)1838 2340 y Fs(=\017)1920 2354 y Fr(1)1980 2340 y Ft(+)20 b Fs(i\017)2139 2354 y Fr(2)2203 2340 y Fp(2)25 b Fs(K)7 b(q)2417 2302 y Fn(\000)p Fj(N)2544 2340 y Fp([)20 b(f)p Ft(0)p Fp(gg)75 2581 y Fi(Pro)s(of)31 b Ft(F)-8 b(or)31 b(all)h Fs(\017)25 b Fp(2)g Fs(K)7 b(q)925 2548 y Fn(\000)p Fj(N)1031 2581 y Ft(,)31 b(w)m(e)g(ha)m(v)m(e)g(that)987 2770 y(\()p Fs(@)1070 2784 y Fq(\017)1099 2793 y Fl(1)1138 2770 y Fs(\036)p Ft(\)\()p Fs(\017)1299 2784 y Fr(1)1340 2770 y Fs(;)15 b(\017)1417 2784 y Fr(2)1456 2770 y Ft(\))26 b(=)f Fs(@)1661 2784 y Fq(\017)1694 2770 y Fs(f)10 b Ft(\()p Fs(\017)p Ft(\))60 b Fs(;)76 b Ft(\()p Fs(@)2100 2784 y Fq(\017)2129 2793 y Fl(2)2168 2770 y Fs(\036)p Ft(\)\()p Fs(\017)2329 2784 y Fr(1)2369 2770 y Fs(;)15 b(\017)2446 2784 y Fr(2)2486 2770 y Ft(\))26 b(=)f Fs(i@)2722 2784 y Fq(\017)2755 2770 y Fs(f)10 b Ft(\()p Fs(\017)p Ft(\))p Fs(:)75 2959 y Ft(Then,)30 b(w)m(e)g(de\014ne)g(a)h(set)g(of)f (functions)g Fs(\036)1501 2926 y Fr(\()p Fq(k)1565 2935 y Fl(1)1600 2926 y Fq(;k)1657 2935 y Fl(2)1691 2926 y Fr(\))1723 2959 y Ft(\()p Fs(\017)1795 2973 y Fr(1)1834 2959 y Fs(;)15 b(\017)1911 2973 y Fr(2)1951 2959 y Ft(\),)31 b(\()p Fs(k)2124 2973 y Fr(1)2164 2959 y Fs(;)15 b(k)2251 2973 y Fr(2)2291 2959 y Ft(\))26 b Fp(2)f Fm(N)2504 2926 y Fr(2)2573 2959 y Ft(as)31 b(follo)m(ws)g(:)1360 3147 y Fs(\036)1414 3110 y Fr(\()p Fq(k)1478 3119 y Fl(1)1513 3110 y Fq(;k)1570 3119 y Fl(2)1604 3110 y Fr(\))1636 3147 y Ft(\()p Fs(\017)1708 3161 y Fr(1)1747 3147 y Fs(;)15 b(\017)1824 3161 y Fr(2)1864 3147 y Ft(\))26 b(:=)f Fs(i)2077 3110 y Fq(k)2114 3119 y Fl(2)2153 3147 y Fs(@)2206 3110 y Fq(k)2243 3119 y Fl(1)2277 3110 y Fr(+)p Fq(k)2369 3119 y Fl(2)2201 3170 y Fq(\017)2408 3147 y Fs(f)10 b Ft(\()p Fs(\017)p Ft(\))75 3336 y(for)30 b(all)h(\()p Fs(k)422 3350 y Fr(1)463 3336 y Fs(;)15 b(k)550 3350 y Fr(2)590 3336 y Ft(\))25 b Fp(2)g Fm(N)802 3303 y Fr(2)841 3336 y Ft(,)31 b(all)g(\()p Fs(\017)1095 3350 y Fr(1)1135 3336 y Fs(;)15 b(\017)1212 3350 y Fr(2)1251 3336 y Ft(\))26 b Fp(2)f Fs(A)p Ft(.)216 3449 y(In)40 b(the)g(next)g(step,)j(w)m(e)e (sho)m(w)f(that)h(the)f(family)h Fp(f)p Fs(\036)2089 3416 y Fr(\()p Fq(k)2153 3425 y Fl(1)2188 3416 y Fq(;k)2245 3425 y Fl(2)2279 3416 y Fr(\))2311 3449 y Fp(g)2356 3469 y Fr(\()p Fq(k)2420 3478 y Fl(1)2455 3469 y Fq(;k)2512 3478 y Fl(2)2546 3469 y Fr(\))p Fn(2)p Fj(N)2668 3450 y Fl(2)2747 3449 y Ft(satis\014es)g(the)f(prop)s(ert)m(y)g Fi(\(W\))p Ft(.)75 3562 y(F)-8 b(rom)32 b(the)f(lemma)h(5,)g(the)f (function)g Fs(f)41 b Ft(satis\014es)32 b(the)f(inequalit)m(y)h(\(57\)) h(whic)m(h)e(can)h(b)s(e)e(rewritten)i(in)f(terms)75 3675 y(of)25 b(the)g(functions)g Fs(\036)765 3642 y Fr(\()p Fq(k)829 3651 y Fl(1)864 3642 y Fq(;k)921 3651 y Fl(2)955 3642 y Fr(\))1011 3675 y Ft(as)g(follo)m(ws.)40 b(Let)26 b Fs(K)32 b Fp(\032)25 b Fs(U)35 b Ft(b)s(e)24 b(a)i(compact)g(set.)39 b(F)-8 b(or)26 b(all)g(\()p Fs(k)2992 3689 y Fr(1)3032 3675 y Fs(;)15 b(k)3119 3689 y Fr(2)3159 3675 y Ft(\))26 b Fp(2)f Fm(N)3372 3642 y Fr(2)3411 3675 y Ft(,)h(all)g Fs(m)f Fp(\025)g Ft(0,)75 3788 y(there)31 b(exists)g(a)f(constan)m(t)i Fs(C)1066 3803 y Fq(K)q(;m;k)1266 3812 y Fl(1)1299 3803 y Fq(;k)1356 3812 y Fl(2)1420 3788 y Fs(>)25 b Ft(0)30 b(suc)m(h)g(that)158 4076 y Fp(jj)249 4014 y Ft(1)p 218 4055 107 4 v 218 4138 a Fs(i)249 4112 y Fq(k)286 4121 y Fl(2)335 4076 y Fs(\036)389 4038 y Fr(\()p Fq(k)453 4047 y Fl(1)488 4038 y Fq(;k)545 4047 y Fl(2)579 4038 y Fr(\))611 4076 y Ft(\()p Fs(x)698 4090 y Fr(1)737 4076 y Fs(;)15 b(y)822 4090 y Fr(1)862 4076 y Ft(\))20 b Fp(\000)1043 3962 y Fq(m)1008 3990 y Fo(X)1011 4184 y Fq(p)p Fr(=0)1237 3990 y Fo(X)1155 4187 y Fq(h)1196 4196 y Fl(1)1230 4187 y Fr(+)p Fq(h)1326 4196 y Fl(2)1360 4187 y Fr(=)p Fq(p)1476 4014 y Fs(\036)1530 3981 y Fr(\()p Fq(k)1594 3990 y Fl(1)1629 3981 y Fr(+)p Fq(h)1725 3990 y Fl(1)1759 3981 y Fq(;k)1816 3990 y Fl(2)1850 3981 y Fr(+)p Fq(h)1946 3990 y Fl(2)1980 3981 y Fr(\))2012 4014 y Ft(\()p Fs(x)2099 4028 y Fr(2)2138 4014 y Fs(;)15 b(y)2223 4028 y Fr(2)2263 4014 y Ft(\))p Fs(=)p Ft(\()p Fs(i)2409 3981 y Fq(k)2446 3990 y Fl(2)2482 3981 y Fr(+)p Fq(h)2578 3990 y Fl(2)2616 4014 y Ft(\))p 1476 4055 1176 4 v 2028 4138 a Fs(p)p Ft(!)880 4367 y Fp(\002)1062 4306 y Fs(p)p Ft(!)p 981 4346 235 4 v 981 4429 a Fs(h)1033 4443 y Fr(1)1073 4429 y Ft(!)p Fs(h)1150 4443 y Fr(2)1190 4429 y Ft(!)1225 4367 y(\()p Fs(x)1312 4381 y Fr(1)1372 4367 y Fp(\000)20 b Fs(x)1515 4381 y Fr(2)1554 4367 y Ft(\))1589 4329 y Fq(h)1630 4338 y Fl(1)1669 4367 y Fs(i)1700 4329 y Fq(h)1741 4338 y Fl(2)1780 4367 y Ft(\()p Fs(y)1860 4381 y Fr(1)1919 4367 y Fp(\000)g Fs(y)2055 4381 y Fr(2)2094 4367 y Ft(\))2129 4329 y Fq(h)2170 4338 y Fl(2)2209 4367 y Fp(jj)2259 4381 y Fj(H)2340 4367 y Fp(\024)25 b Fs(C)2501 4382 y Fq(K)q(;m;k)2701 4391 y Fl(1)2735 4382 y Fq(;k)2792 4391 y Fl(2)2830 4367 y Fp(jj)p Ft(\()p Fs(x)2967 4381 y Fr(1)3027 4367 y Fp(\000)20 b Fs(x)3170 4381 y Fr(2)3209 4367 y Fs(;)15 b(y)3294 4381 y Fr(1)3354 4367 y Fp(\000)20 b Fs(y)3490 4381 y Fr(2)3529 4367 y Ft(\))p Fp(jj)3614 4329 y Fq(m)p Fr(+1)3614 4401 y Fj(R)3662 4382 y Fl(2)75 4595 y Ft(for)30 b(all)h(\()p Fs(x)427 4609 y Fr(1)467 4595 y Fs(;)15 b(y)552 4609 y Fr(1)592 4595 y Ft(\))p Fs(;)g Ft(\()p Fs(x)754 4609 y Fr(2)794 4595 y Fs(;)g(y)879 4609 y Fr(2)918 4595 y Ft(\))26 b Fp(2)f Fs(A)p Ft(.)41 b(In)29 b(particular,)i(w)m(e)g(get)h (that)f(the)f(inequalities)i(\(56\))g(hold.)493 b Ff(2)216 4753 y Ft(In)31 b(the)i(next)f(prop)s(osition,)g(w)m(e)h(will)f(see)h (that)g(the)f(co)s(cycle)i(constructed)e(in)g(the)g(prop)s(osition)g(4) g(splits)75 4866 y(in)e(the)h(space)g(of)f(Whitney)h Fp(C)1111 4833 y Fn(1)1216 4866 y Ft(functions.)75 5063 y Fi(Prop)s(osition)36 b(5)46 b Fh(L)-5 b(et)34 b Ft(\()p Fs(U)993 5077 y Fq(I)1033 5063 y Fs(q)1077 5030 y Fn(\000)p Fj(N)1184 5063 y Ft(\))1219 5077 y Fq(I)5 b Fn(2I)1384 5063 y Fh(b)-5 b(e)34 b(a)g(go)-5 b(o)g(d)35 b(c)-5 b(overing)35 b(and)f(let)g Fp(f)p Fs(g)2537 5078 y Fq(I)5 b(;I)2629 5059 y Fg(0)2656 5063 y Ft(\()p Fs(\017)p Ft(\))p Fp(g)2808 5082 y Fr(\()p Fq(I)g(;I)2927 5064 y Fg(0)2950 5082 y Fr(\))p Fn(2I)3068 5064 y Fl(2)3140 5063 y Fh(b)-5 b(e)34 b(the)g Fm(H)3476 5077 y Fn(T)17 b Fq(;\032)3592 5063 y Fh(-c)-5 b(o)g(cyle)75 5176 y(c)g(onstructe)g(d)27 b(in)f(the)g(pr)-5 b(op)g(osition)29 b(4.)40 b(We)25 b(cho)-5 b(ose)27 b(a)f(family)g(of)g(c)-5 b(omp)g(act)27 b(sets)f Fs(K)2841 5190 y Fq(I)2907 5176 y Fp(\032)f Fs(U)3065 5190 y Fq(I)3105 5176 y Fh(,)h(for)g(al)5 b(l)26 b Fs(I)33 b Fp(2)25 b(I)7 b Fh(,)26 b(with)75 5289 y Ft(In)m(t\()p Fs(K)303 5303 y Fq(I)344 5289 y Ft(\))f Fp(6)p Ft(=)g Fp(;)p Fh(,)31 b(in)g(such)f(a)h(way)g(that)h Fp([)1421 5303 y Fq(I)5 b Fn(2I)1551 5289 y Ft(\()p Fs(K)1663 5303 y Fq(I)1704 5289 y Fs(q)1748 5256 y Fn(\000)p Fj(N)1854 5289 y Ft(\))31 b Fh(is)g(of)f(the)h(form)g Fp(U)25 b(n)16 b(f)p Ft(0)p Fp(g)p Fh(,)32 b(wher)-5 b(e)31 b Fp(U)40 b Fh(is)30 b(a)h(neighb)-5 b(orho)g(o)g(d)75 5402 y(of)33 b(0)g(in)f Fm(C)p Fh(.)75 5514 y(Then,)d(for)f(al)5 b(l)28 b Fs(I)33 b Fp(2)24 b(I)7 b Fh(,)28 b(ther)-5 b(e)29 b(exists)f(a)g Fs(C)1484 5481 y Fn(1)1585 5514 y Fh(function)g Fs(f)1983 5528 y Fq(I)2023 5514 y Ft(\()p Fs(\017)2095 5528 y Fr(1)2134 5514 y Fs(;)15 b(\017)2211 5528 y Fr(2)2251 5514 y Ft(\))28 b Fh(in)g(the)g(sense)f(of)h(Whitney)g(on)g(the)g(c)-5 b(omp)g(act)75 5627 y(set)1074 5740 y Fs(A)1142 5754 y Fq(I)1207 5740 y Ft(=)25 b Fp(f)p Ft(\()p Fs(\017)1420 5754 y Fr(1)1460 5740 y Fs(;)15 b(\017)1537 5754 y Fr(2)1577 5740 y Ft(\))25 b Fp(2)g Fm(R)1789 5703 y Fr(2)1828 5740 y Fs(=\017)1910 5754 y Fr(1)1970 5740 y Ft(+)20 b Fs(i\017)2129 5754 y Fr(2)2194 5740 y Fp(2)25 b Fs(K)2357 5754 y Fq(I)2397 5740 y Fs(q)2441 5703 y Fn(\000)p Fj(N)2568 5740 y Fp([)20 b(f)p Ft(0)p Fp(gg)p Fs(;)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 3764 100 a Ft(19)75 399 y Fh(with)33 b(values)g(in)g(the)g(Banach)h(sp)-5 b(ac)g(e)34 b Fm(H)1444 413 y Fn(T)16 b Fq(;\032)1559 399 y Fh(,)33 b(such)g(that)75 601 y Ft(\(58\))991 b Fs(g)1269 617 y Fq(I)5 b(;I)1361 598 y Fg(0)1387 601 y Ft(\()p Fs(\017)1459 615 y Fr(1)1519 601 y Ft(+)20 b Fs(i\017)1678 615 y Fr(2)1717 601 y Ft(\))26 b(=)f Fs(f)1919 617 y Fq(I)1955 598 y Fg(0)1981 601 y Ft(\()p Fs(\017)2053 615 y Fr(1)2092 601 y Fs(;)15 b(\017)2169 615 y Fr(2)2209 601 y Ft(\))21 b Fp(\000)f Fs(f)2401 615 y Fq(I)2440 601 y Ft(\()p Fs(\017)2512 615 y Fr(1)2552 601 y Fs(;)15 b(\017)2629 615 y Fr(2)2668 601 y Ft(\))75 804 y Fh(for)33 b(al)5 b(l)33 b Fs(I)7 b(;)15 b(I)484 771 y Fn(0)533 804 y Fp(2)25 b(I)39 b Fh(such)33 b(that)h Fs(A)1169 818 y Fq(I)1229 804 y Fp(\\)20 b Fs(A)1378 819 y Fq(I)1414 800 y Fg(0)1466 804 y Fp(6)p Ft(=)k Fp(;)p Fh(,)33 b(for)h(al)5 b(l)33 b Ft(\()p Fs(\017)2015 818 y Fr(1)2054 804 y Fs(;)15 b(\017)2131 818 y Fr(2)2171 804 y Ft(\))26 b Fp(2)f Fs(A)2386 818 y Fq(I)2446 804 y Fp(\\)20 b Fs(A)2595 819 y Fq(I)2631 800 y Fg(0)2657 804 y Fh(.)75 1065 y Fi(Pro)s(of)35 b Ft(The)e(pro)s(of)g(follo)m(ws)j (the)e(argumen)m(ts)g(giv)m(en)h(in)f(Lemma)g(3.12)h(from)f([10)q(].)52 b(First)34 b(of)g(all,)i(for)e(ev)m(ery)75 1177 y Fs(I)7 b(;)15 b(I)209 1145 y Fn(0)268 1177 y Fp(2)35 b(I)42 b Ft(suc)m(h)36 b(that)h Fs(A)938 1191 y Fq(I)1002 1177 y Fp(\\)24 b Fs(A)1155 1193 y Fq(I)1191 1174 y Fg(0)1253 1177 y Fp(6)p Ft(=)35 b Fp(;)p Ft(,)j(the)f(function)f(\()p Fs(\017)2064 1191 y Fr(1)2103 1177 y Fs(;)15 b(\017)2180 1191 y Fr(2)2220 1177 y Ft(\))36 b Fp(7!)f Fs(g)2460 1193 y Fq(I)5 b(;I)2552 1174 y Fg(0)2578 1177 y Ft(\()p Fs(\017)2650 1191 y Fr(1)2714 1177 y Ft(+)23 b Fs(i\017)2876 1191 y Fr(2)2916 1177 y Ft(\))37 b(is)f Fs(C)3157 1145 y Fn(1)3268 1177 y Ft(in)g(the)g(sense)g(of)75 1290 y(Whitney)d(on)f Fs(A)647 1304 y Fq(I)709 1290 y Fp(\\)21 b Fs(A)859 1306 y Fq(I)895 1287 y Fg(0)922 1290 y Ft(.)47 b(Indeed,)32 b(b)m(y)g(construction,)i Fs(g)2037 1306 y Fq(I)5 b(;I)2129 1287 y Fg(0)2155 1290 y Ft(\()p Fs(\017)p Ft(\))33 b(has)f(the)2618 1271 y(^)2618 1290 y(0)h(formal)g(series)f(as)h Fs(q)s Ft(-asymptotic)75 1403 y(expansion)d(on)g Fs(K)700 1417 y Fq(I)741 1403 y Fs(q)785 1370 y Fn(\000)p Fj(N)912 1403 y Fp(\\)19 b Fs(K)1069 1419 y Fq(I)1105 1400 y Fg(0)1132 1403 y Fs(q)1176 1370 y Fn(\000)p Fj(N)1283 1403 y Ft(,)30 b(so)h(that,)g(b)m(y)g(the)f(lemma)h(6,)g(w)m(e)g(get)g(the)g(result.) 75 1566 y(Let)i(one)f Fs(I)k Fp(2)28 b(I)7 b Ft(.)45 b(Consider)31 b(a)i Fs(C)1231 1533 y Fn(1)1337 1566 y Ft(function)f Fs(f)1740 1580 y Fq(I)1812 1566 y Ft(in)g(the)g(sense)g (of)h(Whitney)f(on)g Fs(A)2990 1580 y Fq(I)3031 1566 y Ft(.)46 b(W)-8 b(e)33 b(will)g(see)f(ho)m(w)h(to)75 1678 y(construct)e(the)f(functions)g Fs(f)1068 1694 y Fq(I)1104 1675 y Fg(0)1160 1678 y Ft(on)h(all)g(the)f Fs(I)1616 1645 y Fn(0)1670 1678 y Ft(suc)m(h)g(that)h Fs(A)2140 1694 y Fq(I)2176 1675 y Fg(0)2223 1678 y Fp(\\)20 b Fs(A)2372 1692 y Fq(I)2437 1678 y Fp(6)p Ft(=)25 b Fp(;)p Ft(.)75 1841 y(Case)30 b(1.)41 b(There)29 b(is)h(at)h(least)g (one)f Fs(I)1292 1808 y Fn(0)1341 1841 y Fp(2)24 b(I)7 b Ft(,)30 b Fs(I)1585 1808 y Fn(0)1633 1841 y Fp(6)p Ft(=)25 b Fs(I)7 b Ft(,)31 b(suc)m(h)e(that)i Fs(A)2301 1856 y Fq(I)2337 1837 y Fg(0)2382 1841 y Fp(\\)19 b Fs(A)2530 1855 y Fq(I)2595 1841 y Fp(6)p Ft(=)25 b Fp(;)30 b Ft(and)g(there)g(is) f(no)h Fs(I)3438 1808 y Fn(00)3506 1841 y Fp(2)25 b(I)36 b Ft(with)75 1954 y Fs(I)122 1921 y Fn(00)190 1954 y Fp(6)p Ft(=)25 b Fs(I)333 1921 y Fn(0)381 1954 y Fp(6)p Ft(=)g Fs(I)36 b Ft(suc)m(h)28 b(that)h Fs(A)1019 1968 y Fq(I)1076 1954 y Fp(\\)16 b Fs(A)1221 1969 y Fq(I)1257 1950 y Fg(0)1300 1954 y Fp(\\)g Fs(A)1445 1969 y Fq(I)1481 1950 y Fg(00)1551 1954 y Fp(6)p Ft(=)25 b Fp(;)p Ft(.)40 b(W)-8 b(e)30 b(put)e Fs(e)2119 1969 y Fq(I)5 b(;I)2211 1950 y Fg(0)2237 1954 y Ft(\()p Fs(\017)2309 1968 y Fr(1)2348 1954 y Fs(;)15 b(\017)2425 1968 y Fr(2)2465 1954 y Ft(\))26 b(=)f Fs(f)2667 1968 y Fq(I)2706 1954 y Ft(\()p Fs(\017)2778 1968 y Fr(1)2818 1954 y Fs(;)15 b(\017)2895 1968 y Fr(2)2935 1954 y Ft(\))h(+)h Fs(g)3117 1969 y Fq(I)5 b(;I)3209 1950 y Fg(0)3235 1954 y Ft(\()p Fs(\017)3307 1968 y Fr(1)3363 1954 y Ft(+)16 b Fs(i\017)3518 1968 y Fr(2)3558 1954 y Ft(\))28 b(for)h(all)75 2067 y(\()p Fs(\017)147 2081 y Fr(1)187 2067 y Fs(;)15 b(\017)264 2081 y Fr(2)303 2067 y Ft(\))32 b Fp(2)g Fs(A)531 2081 y Fq(I)593 2067 y Fp(\\)23 b Fs(A)745 2082 y Fq(I)781 2063 y Fg(0)807 2067 y Ft(.)52 b(F)-8 b(rom)35 b(the)f(Whitney)h(extension)g(theorem,)h (see)e([8)q(],)i([9],)g(p.67{68,)h(w)m(e)e(can)f(extend)75 2179 y(the)d Fs(C)304 2146 y Fn(1)408 2179 y Ft(function)f Fs(e)806 2195 y Fq(I)5 b(;I)898 2176 y Fg(0)954 2179 y Ft(as)31 b(a)g Fs(C)1214 2146 y Fn(1)1318 2179 y Ft(function)f Fs(f)1719 2195 y Fq(I)1755 2176 y Fg(0)1781 2179 y Ft(\()p Fs(\017)1853 2193 y Fr(1)1893 2179 y Fs(;)15 b(\017)1970 2193 y Fr(2)2010 2179 y Ft(\))30 b(in)g(the)h(sense)f(of)h(Whitney)g (on)f Fs(A)3243 2195 y Fq(I)3279 2176 y Fg(0)3305 2179 y Ft(.)41 b(So)30 b(that)1226 2382 y Fs(g)1269 2398 y Fq(I)5 b(;I)1361 2379 y Fg(0)1387 2382 y Ft(\()p Fs(\017)1459 2396 y Fr(1)1519 2382 y Ft(+)20 b Fs(i\017)1678 2396 y Fr(2)1717 2382 y Ft(\))26 b(=)f Fs(f)1919 2398 y Fq(I)1955 2379 y Fg(0)1981 2382 y Ft(\()p Fs(\017)2053 2396 y Fr(1)2092 2382 y Fs(;)15 b(\017)2169 2396 y Fr(2)2209 2382 y Ft(\))21 b Fp(\000)f Fs(f)2401 2396 y Fq(I)2440 2382 y Ft(\()p Fs(\017)2512 2396 y Fr(1)2552 2382 y Fs(;)15 b(\017)2629 2396 y Fr(2)2668 2382 y Ft(\))75 2585 y(for)30 b(all)h(\()p Fs(\017)412 2599 y Fr(1)452 2585 y Fs(;)15 b(\017)529 2599 y Fr(2)569 2585 y Ft(\))25 b Fp(2)g Fs(A)783 2599 y Fq(I)844 2585 y Fp(\\)19 b Fs(A)992 2600 y Fq(I)1028 2581 y Fg(0)1055 2585 y Ft(.)75 2747 y(Case)34 b(2.)49 b(There)33 b(are)h(t)m(w)m(o)g Fs(I)1057 2714 y Fn(0)1081 2747 y Fs(;)15 b(I)1168 2714 y Fn(00)1241 2747 y Fp(2)29 b(I)40 b Ft(with)33 b Fs(I)1678 2714 y Fn(0)1731 2747 y Fp(6)p Ft(=)d Fs(I)1879 2714 y Fn(00)1951 2747 y Fp(6)p Ft(=)g Fs(I)7 b Ft(,)34 b(suc)m(h)f(that)h Fs(A)2634 2761 y Fq(I)2696 2747 y Fp(\\)22 b Fs(A)2847 2763 y Fq(I)2883 2744 y Fg(0)2932 2747 y Fp(\\)f Fs(A)3082 2763 y Fq(I)3118 2744 y Fg(00)3193 2747 y Fp(6)p Ft(=)30 b Fp(;)p Ft(.)50 b(First,)34 b(as)g(in)75 2860 y(case)d(1,)g(w)m(e)g(construct)g(a)g Fs(C)1050 2827 y Fn(1)1154 2860 y Ft(function)f Fs(f)1555 2876 y Fq(I)1591 2857 y Fg(0)1617 2860 y Ft(\()p Fs(\017)1689 2874 y Fr(1)1729 2860 y Fs(;)15 b(\017)1806 2874 y Fr(2)1846 2860 y Ft(\))30 b(in)g(the)h(sense)f(of)h(Whitney)f(on)h Fs(A)3079 2876 y Fq(I)3115 2857 y Fg(0)3172 2860 y Ft(suc)m(h)f(that)75 3063 y(\(59\))991 b Fs(g)1269 3078 y Fq(I)5 b(;I)1361 3059 y Fg(0)1387 3063 y Ft(\()p Fs(\017)1459 3077 y Fr(1)1519 3063 y Ft(+)20 b Fs(i\017)1678 3077 y Fr(2)1717 3063 y Ft(\))26 b(=)f Fs(f)1919 3078 y Fq(I)1955 3059 y Fg(0)1981 3063 y Ft(\()p Fs(\017)2053 3077 y Fr(1)2092 3063 y Fs(;)15 b(\017)2169 3077 y Fr(2)2209 3063 y Ft(\))21 b Fp(\000)f Fs(f)2401 3077 y Fq(I)2440 3063 y Ft(\()p Fs(\017)2512 3077 y Fr(1)2552 3063 y Fs(;)15 b(\017)2629 3077 y Fr(2)2668 3063 y Ft(\))75 3266 y(for)30 b(all)h(\()p Fs(\017)412 3280 y Fr(1)452 3266 y Fs(;)15 b(\017)529 3280 y Fr(2)569 3266 y Ft(\))25 b Fp(2)g Fs(A)783 3280 y Fq(I)844 3266 y Fp(\\)19 b Fs(A)992 3281 y Fq(I)1028 3262 y Fg(0)1055 3266 y Ft(.)41 b(Then,)29 b(w)m(e)i(put)75 3469 y(\(60\))946 b Fs(e)1223 3484 y Fq(I)5 b(;I)1315 3465 y Fg(00)1359 3469 y Ft(\()p Fs(\017)1431 3483 y Fr(1)1471 3469 y Fs(;)15 b(\017)1548 3483 y Fr(2)1588 3469 y Ft(\))25 b(=)g Fs(f)1789 3483 y Fq(I)1829 3469 y Ft(\()p Fs(\017)1901 3483 y Fr(1)1940 3469 y Fs(;)15 b(\017)2017 3483 y Fr(2)2057 3469 y Ft(\))21 b(+)f Fs(g)2247 3484 y Fq(I)5 b(;I)2339 3465 y Fg(00)2383 3469 y Ft(\()p Fs(\017)2455 3483 y Fr(1)2515 3469 y Ft(+)20 b Fs(i\017)2674 3483 y Fr(2)2713 3469 y Ft(\))75 3671 y(for)30 b(all)h(\()p Fs(\017)412 3685 y Fr(1)452 3671 y Fs(;)15 b(\017)529 3685 y Fr(2)569 3671 y Ft(\))25 b Fp(2)g Fs(A)783 3685 y Fq(I)844 3671 y Fp(\\)19 b Fs(A)992 3687 y Fq(I)1028 3668 y Fg(00)1098 3671 y Fp(\032)25 b Fs(A)1262 3687 y Fq(I)1298 3668 y Fg(00)1373 3671 y Ft(and)75 3874 y(\(61\))912 b Fs(e)1189 3890 y Fq(I)1225 3871 y Fg(0)1248 3890 y Fq(;I)1304 3871 y Fg(00)1348 3874 y Ft(\()p Fs(\017)1420 3888 y Fr(1)1460 3874 y Fs(;)15 b(\017)1537 3888 y Fr(2)1576 3874 y Ft(\))26 b(=)f Fs(f)1778 3890 y Fq(I)1814 3871 y Fg(0)1840 3874 y Ft(\()p Fs(\017)1912 3888 y Fr(1)1951 3874 y Fs(;)15 b(\017)2028 3888 y Fr(2)2068 3874 y Ft(\))21 b(+)f Fs(g)2258 3890 y Fq(I)2294 3871 y Fg(0)2316 3890 y Fq(;I)2372 3871 y Fg(00)2416 3874 y Ft(\()p Fs(\017)2488 3888 y Fr(1)2548 3874 y Ft(+)g Fs(i\017)2707 3888 y Fr(2)2747 3874 y Ft(\))75 4077 y(for)30 b(all)h(\()p Fs(\017)412 4091 y Fr(1)452 4077 y Fs(;)15 b(\017)529 4091 y Fr(2)569 4077 y Ft(\))25 b Fp(2)g Fs(A)783 4092 y Fq(I)819 4073 y Fg(0)866 4077 y Fp(\\)20 b Fs(A)1015 4092 y Fq(I)1051 4073 y Fg(00)1121 4077 y Fp(\032)25 b Fs(A)1285 4092 y Fq(I)1321 4073 y Fg(00)1365 4077 y Ft(.)41 b(No)m(w,)31 b(w)m(e)g(sho)m(w)f(that)75 4280 y(\(62\))1216 b Fs(e)1493 4295 y Fq(I)5 b(;I)1585 4276 y Fg(00)1629 4280 y Ft(\()p Fs(\017)1701 4294 y Fr(1)1741 4280 y Fs(;)15 b(\017)1818 4294 y Fr(2)1858 4280 y Ft(\))25 b(=)g Fs(e)2056 4295 y Fq(I)2092 4276 y Fg(0)2115 4295 y Fq(;I)2171 4276 y Fg(00)2215 4280 y Ft(\()p Fs(\017)2287 4294 y Fr(1)2326 4280 y Fs(;)15 b(\017)2403 4294 y Fr(2)2443 4280 y Ft(\))75 4483 y(for)30 b(all)h(\()p Fs(\017)412 4497 y Fr(1)452 4483 y Fs(;)15 b(\017)529 4497 y Fr(2)569 4483 y Ft(\))25 b Fp(2)g Fs(A)783 4497 y Fq(I)844 4483 y Fp(\\)19 b Fs(A)992 4498 y Fq(I)1028 4479 y Fg(0)1075 4483 y Fp(\\)h Fs(A)1224 4498 y Fq(I)1260 4479 y Fg(00)1305 4483 y Ft(.)40 b(Indeed,)30 b(from)g(the)g(relation)i(\(59\),)g(w)m(e)f (get)h(that)840 4686 y Fs(e)882 4701 y Fq(I)918 4682 y Fg(0)940 4701 y Fq(;I)996 4682 y Fg(00)1040 4686 y Ft(\()p Fs(\017)1112 4700 y Fr(1)1152 4686 y Fs(;)15 b(\017)1229 4700 y Fr(2)1269 4686 y Ft(\))25 b(=)g Fs(f)1470 4700 y Fq(I)1510 4686 y Ft(\()p Fs(\017)1582 4700 y Fr(1)1621 4686 y Fs(;)15 b(\017)1698 4700 y Fr(2)1738 4686 y Ft(\))21 b(+)f Fs(g)1928 4701 y Fq(I)5 b(;I)2020 4682 y Fg(0)2046 4686 y Ft(\()p Fs(\017)2118 4700 y Fr(1)2177 4686 y Ft(+)20 b Fs(i\017)2336 4700 y Fr(2)2376 4686 y Ft(\))h(+)e Fs(g)2565 4701 y Fq(I)2601 4682 y Fg(0)2624 4701 y Fq(;I)2680 4682 y Fg(00)2724 4686 y Ft(\()p Fs(\017)2796 4700 y Fr(1)2856 4686 y Ft(+)h Fs(i\017)3015 4700 y Fr(2)3055 4686 y Ft(\))75 4888 y(for)42 b(all)g(\()p Fs(\017)435 4902 y Fr(1)475 4888 y Fs(;)15 b(\017)552 4902 y Fr(2)592 4888 y Ft(\))44 b Fp(2)g Fs(A)844 4902 y Fq(I)912 4888 y Fp(\\)27 b Fs(A)1068 4904 y Fq(I)1104 4885 y Fg(0)1159 4888 y Fp(\\)g Fs(A)1315 4904 y Fq(I)1351 4885 y Fg(00)1438 4888 y Ft(But,)45 b(from)c(the)h(the)g(co)s(cycle)h(relation)g Fs(g)2932 4904 y Fq(I)5 b(;I)3024 4885 y Fg(0)3078 4888 y Ft(+)28 b Fs(g)3220 4904 y Fq(I)3256 4885 y Fg(0)3278 4904 y Fq(;I)3334 4885 y Fg(00)3423 4888 y Ft(=)44 b Fs(g)3581 4904 y Fq(I)5 b(;I)3673 4885 y Fg(00)3759 4888 y Ft(on)75 5001 y Fs(A)143 5015 y Fq(I)203 5001 y Fp(\\)20 b Fs(A)352 5017 y Fq(I)388 4998 y Fg(0)435 5001 y Fp(\\)g Fs(A)584 5017 y Fq(I)620 4998 y Fg(00)664 5001 y Ft(,)31 b(w)m(e)g(deduce)f (\(62\).)42 b(No)m(w,)31 b(w)m(e)g(put)936 5258 y Fs(e)978 5273 y Fq(I)1014 5254 y Fg(00)1059 5258 y Ft(\()p Fs(\017)1131 5272 y Fr(1)1170 5258 y Fs(;)15 b(\017)1247 5272 y Fr(2)1287 5258 y Ft(\))26 b(=)1444 5130 y Fo(\032)1553 5201 y Fs(e)1595 5217 y Fq(I)5 b(;I)1687 5198 y Fg(00)1732 5201 y Ft(\()p Fs(\017)1804 5215 y Fr(1)1843 5201 y Fs(;)15 b(\017)1920 5215 y Fr(2)1960 5201 y Ft(\))106 b(if)30 b(\()p Fs(\017)2256 5215 y Fr(1)2296 5201 y Fs(;)15 b(\017)2373 5215 y Fr(2)2413 5201 y Ft(\))25 b Fp(2)g Fs(A)2627 5215 y Fq(I)2687 5201 y Fp(\\)20 b Fs(A)2836 5217 y Fq(I)2872 5198 y Fg(00)2917 5201 y Ft(,)1553 5314 y Fs(e)1595 5329 y Fq(I)1631 5311 y Fg(0)1654 5329 y Fq(;I)1710 5311 y Fg(00)1754 5314 y Ft(\()p Fs(\017)1826 5328 y Fr(1)1866 5314 y Fs(;)15 b(\017)1943 5328 y Fr(2)1982 5314 y Ft(\))84 b(if)30 b(\()p Fs(\017)2256 5328 y Fr(1)2296 5314 y Fs(;)15 b(\017)2373 5328 y Fr(2)2413 5314 y Ft(\))25 b Fp(2)g Fs(A)2627 5328 y Fq(I)2687 5314 y Fp(\\)20 b Fs(A)2836 5329 y Fq(I)2872 5311 y Fg(00)2917 5314 y Ft(.)75 5514 y(F)-8 b(rom)27 b(Lemma)f(3.10)i(in)e([10)q(],)h(w)m(e)g(kno)m(w)f(that)h(the)f(sets)h Fs(A)2063 5528 y Fq(I)2115 5514 y Fp(\\)12 b Fs(A)2256 5530 y Fq(I)2292 5511 y Fg(00)2362 5514 y Ft(and)26 b Fs(A)2603 5530 y Fq(I)2639 5511 y Fg(0)2677 5514 y Fp(\\)12 b Fs(A)2818 5530 y Fq(I)2854 5511 y Fg(00)2924 5514 y Ft(are)27 b(regularly)g(separated,)75 5627 y(so)i(that)g(from)f(the)h (Lo)5 b(jasiewicz)30 b(theorem,)f([8)q(],)g([9)q(])g(p.83,)h(the)e (function)h Fs(e)2654 5643 y Fq(I)2690 5624 y Fg(00)2734 5627 y Ft(\()p Fs(\017)2806 5641 y Fr(1)2846 5627 y Fs(;)15 b(\017)2923 5641 y Fr(2)2963 5627 y Ft(\))29 b(is)f Fs(C)3188 5594 y Fn(1)3291 5627 y Ft(in)g(the)h(sense)f(of)75 5740 y(Whitney)j(on)g(\()p Fs(A)679 5754 y Fq(I)740 5740 y Fp(\\)20 b Fs(A)889 5756 y Fq(I)925 5737 y Fg(00)969 5740 y Ft(\))h Fp([)f Ft(\()p Fs(A)1209 5756 y Fq(I)1245 5737 y Fg(0)1293 5740 y Fp(\\)g Fs(A)1442 5756 y Fq(I)1478 5737 y Fg(00)1522 5740 y Ft(\))26 b Fp(\032)g Fs(A)1748 5756 y Fq(I)1784 5737 y Fg(00)1829 5740 y Ft(.)41 b(F)-8 b(rom)32 b(the)f(Whitney)g(extension)g(theorem,)h(w)m(e)f(deduce)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 3764 100 a Ft(20)75 399 y(that)37 b(the)g(function)f Fs(e)845 414 y Fq(I)881 395 y Fg(00)926 399 y Ft(\()p Fs(\017)998 413 y Fr(1)1037 399 y Fs(;)15 b(\017)1114 413 y Fr(2)1154 399 y Ft(\))37 b(extends)f(to)h(a)g Fs(C)1834 366 y Fn(1)1944 399 y Ft(function)g(in)f(the)g(sense)h(of)f (Whitney)h(on)f Fs(A)3510 414 y Fq(I)3546 395 y Fg(00)3627 399 y Ft(called)75 511 y Fs(f)120 527 y Fq(I)156 508 y Fg(00)200 511 y Ft(\()p Fs(\017)272 525 y Fr(1)312 511 y Fs(;)15 b(\017)389 525 y Fr(2)428 511 y Ft(\).)41 b(Finally)-8 b(,)32 b(w)m(e)f(ha)m(v)m(e)h(the)e(relation)i(\(59\),)g (from)e(\(60\))i(w)m(e)f(get)1208 706 y Fs(f)1253 721 y Fq(I)1289 702 y Fg(00)1333 706 y Ft(\()p Fs(\017)1405 720 y Fr(1)1444 706 y Fs(;)15 b(\017)1521 720 y Fr(2)1561 706 y Ft(\))26 b(=)f Fs(f)1763 720 y Fq(I)1802 706 y Ft(\()p Fs(\017)1874 720 y Fr(1)1914 706 y Fs(;)15 b(\017)1991 720 y Fr(2)2030 706 y Ft(\))21 b(+)f Fs(g)2220 721 y Fq(I)5 b(;I)2312 702 y Fg(00)2356 706 y Ft(\()p Fs(\017)2428 720 y Fr(1)2488 706 y Ft(+)20 b Fs(i\017)2647 720 y Fr(2)2687 706 y Ft(\))75 900 y(for)30 b(all)h(\()p Fs(\017)412 914 y Fr(1)452 900 y Fs(;)15 b(\017)529 914 y Fr(2)569 900 y Ft(\))25 b Fp(2)g Fs(A)783 914 y Fq(I)844 900 y Fp(\\)19 b Fs(A)992 915 y Fq(I)1028 896 y Fg(00)1103 900 y Ft(and)30 b(from)g(\(61\))i(w)m(e)f(get)1185 1094 y Fs(f)1230 1109 y Fq(I)1266 1090 y Fg(00)1310 1094 y Ft(\()p Fs(\017)1382 1108 y Fr(1)1422 1094 y Fs(;)15 b(\017)1499 1108 y Fr(2)1539 1094 y Ft(\))25 b(=)g Fs(f)1740 1109 y Fq(I)1776 1090 y Fg(0)1802 1094 y Ft(\()p Fs(\017)1874 1108 y Fr(1)1914 1094 y Fs(;)15 b(\017)1991 1108 y Fr(2)2030 1094 y Ft(\))21 b(+)f Fs(g)2220 1109 y Fq(I)2256 1090 y Fg(0)2278 1109 y Fq(;I)2334 1090 y Fg(00)2379 1094 y Ft(\()p Fs(\017)2451 1108 y Fr(1)2510 1094 y Ft(+)g Fs(i\017)2669 1108 y Fr(2)2709 1094 y Ft(\))75 1288 y(for)30 b(all)h(\()p Fs(\017)412 1302 y Fr(1)452 1288 y Fs(;)15 b(\017)529 1302 y Fr(2)569 1288 y Ft(\))25 b Fp(2)g Fs(A)783 1304 y Fq(I)819 1285 y Fg(0)866 1288 y Fp(\\)20 b Fs(A)1015 1304 y Fq(I)1051 1285 y Fg(00)1095 1288 y Ft(.)75 1448 y(By)35 b(construction)h(of)f(the)h(go)s(o)s(d)f(co)m(v)m(ering,)j(w)m (e)e(cannot)g(ha)m(v)m(e)g Fs(A)2356 1462 y Fq(I)2419 1448 y Fp(\\)23 b Fs(A)2571 1463 y Fq(I)2607 1444 y Fg(0)2657 1448 y Fp(\\)g Fs(A)2809 1463 y Fq(I)2845 1444 y Fg(00)2913 1448 y Fp(\\)g Fs(A)3065 1463 y Fq(I)3101 1444 y Fg(000)3197 1448 y Fp(6)p Ft(=)33 b Fp(;)p Ft(,)k(if)e Fs(I)40 b Fp(6)p Ft(=)33 b Fs(I)3727 1415 y Fn(0)3784 1448 y Fp(6)p Ft(=)75 1561 y Fs(I)122 1528 y Fn(00)190 1561 y Fp(6)p Ft(=)25 b Fs(I)333 1528 y Fn(000)394 1561 y Ft(.)40 b(So)27 b(only)g(the)h(case)g(1)f(or)g(2)h(can)f(app)s(ear.)39 b(Starting)28 b(from)e(one)i(function)f Fs(f)2996 1575 y Fq(I)3062 1561 y Ft(on)g Fs(A)3253 1575 y Fq(I)3320 1561 y Ft(and)g(using)f(the)75 1674 y(pro)s(cedures)g(describ)s(ed)f (in)i(cases)h(1)f(and)f(2,)i(w)m(e)g(can)f(construct)g(a)g(set)h(of)f (functions)f Fs(f)3033 1688 y Fq(J)3082 1674 y Ft(,)i(for)e(all)i Fs(J)34 b Fp(2)25 b(I)20 b(n)13 b(f)p Fs(I)7 b Fp(g)p Ft(,)75 1786 y Fs(C)147 1753 y Fn(1)252 1786 y Ft(in)30 b(the)g(sense)g(of)h(Whitney)g(on)f Fs(A)1419 1800 y Fq(J)1468 1786 y Ft(,)h(whic)m(h)f(satisfy)h(the)f(relations)i(\(58\).) 977 b Ff(2)75 2075 y Fd(5.3)112 b(Existence)38 b(of)f(a)h(formal)h (series)f(and)g Fc(q)t Fd(-asymptotic)g(expansions)75 2246 y Ft(In)g(the)g(next)h(prop)s(osition,)i(w)m(e)e(will)f(giv)m(e)i (a)f(useful)f(decomp)s(osition)h(of)g(the)f(functions)g Fs(X)3315 2260 y Fq(I)3394 2246 y Ft(constructed)75 2359 y(in)f(Prop)s(osition)f(3)i(in)e(terms)h(of)g(the)g(functions)g Fs(f)1855 2373 y Fq(I)1894 2359 y Ft(,)i(constructed)e(in)g(Prop)s (osition)f(5,)k Fs(C)3245 2326 y Fn(1)3356 2359 y Ft(and)c(analytic)75 2472 y(functions)30 b(on)g(some)h(neigh)m(b)s(orho)s(o)s(d)e(of)h(the)h (origin)g(in)f Fm(R)2082 2439 y Fr(2)2121 2472 y Ft(.)75 2675 y Fi(Prop)s(osition)36 b(6)46 b Fh(F)-7 b(or)36 b(al)5 b(l)35 b Fs(I)h Fp(2)29 b(I)7 b Fh(,)34 b(we)h(denote)g(by)g Fs(X)1952 2689 y Fq(I)1992 2675 y Ft(\()p Fs(\017)p Ft(\))p Fh(,)h(the)f(holomorphic)j(function)c(fr)-5 b(om)36 b Fs(U)3473 2689 y Fq(I)3513 2675 y Fs(q)3557 2642 y Fn(\000)p Fj(N)3699 2675 y Fh(into)75 2788 y Fm(H)146 2802 y Fn(T)17 b Fq(;\032)294 2788 y Fh(de\014ne)-5 b(d)34 b(by)1410 2901 y Fs(X)1485 2915 y Fq(I)1526 2901 y Ft(\()p Fs(\017)p Ft(\))25 b(:=)h(\()p Fs(t;)15 b(z)t Ft(\))26 b Fp(7!)f Fs(X)2186 2915 y Fq(I)2226 2901 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))p Fs(:)75 3062 y Fh(Then,)27 b(ther)-5 b(e)26 b(exist)g(a)g Fs(C)902 3029 y Fn(1)1001 3062 y Fh(function)g Fs(u)p Ft(\()p Fs(\017)1476 3076 y Fr(1)1516 3062 y Fs(;)15 b(\017)1593 3076 y Fr(2)1632 3062 y Ft(\))26 b Fh(and)g(an)g(analytic)h(function)e Fs(a)p Ft(\()p Fs(\017)2792 3076 y Fr(1)2837 3062 y Ft(+)5 b Fs(i\017)2981 3076 y Fr(2)3020 3062 y Ft(\))25 b Fh(on)h(the)g(neighb)-5 b(orho)g(o)g(d)75 3175 y Ft(In)m(t\()p Fp([)287 3189 y Fq(I)5 b Fn(2I)418 3175 y Fs(A)486 3189 y Fq(I)526 3175 y Ft(\))32 b Fh(of)h(the)g(origin)g(in)g Fm(R)1295 3142 y Fr(2)1367 3175 y Fh(such)f(that)75 3370 y Ft(\(63\))779 b Fs(X)1089 3384 y Fq(I)1129 3370 y Ft(\()p Fs(\017)1201 3384 y Fr(1)1261 3370 y Ft(+)20 b Fs(i\017)1420 3384 y Fr(2)1459 3370 y Ft(\))26 b(=)f Fs(f)1661 3384 y Fq(I)1700 3370 y Ft(\()p Fs(\017)1772 3384 y Fr(1)1812 3370 y Fs(;)15 b(\017)1889 3384 y Fr(2)1929 3370 y Ft(\))20 b(+)g Fs(u)p Ft(\()p Fs(\017)2199 3384 y Fr(1)2239 3370 y Fs(;)15 b(\017)2316 3384 y Fr(2)2356 3370 y Ft(\))20 b(+)g Fs(a)p Ft(\()p Fs(\017)2622 3384 y Fr(1)2682 3370 y Ft(+)g Fs(i\017)2841 3384 y Fr(2)2881 3370 y Ft(\))75 3564 y Fh(for)33 b(al)5 b(l)33 b Fs(I)g Fp(2)25 b(I)7 b Fh(,)31 b(for)i(al)5 b(l)33 b Ft(\()p Fs(\017)972 3578 y Fr(1)1012 3564 y Fs(;)15 b(\017)1089 3578 y Fr(2)1129 3564 y Ft(\))25 b Fp(2)g Ft(In)m(t\()p Fs(A)1494 3578 y Fq(I)1535 3564 y Ft(\))p Fh(.)75 3813 y Fi(Pro)s(of)31 b Ft(F)-8 b(rom)31 b(the)g(de\014nition)f(of)g(the)h Fm(H)1490 3827 y Fn(T)17 b Fq(;\032)1606 3813 y Ft(-co)s(cyle)32 b Fs(g)1952 3828 y Fq(I)5 b(;I)2044 3809 y Fg(0)2070 3813 y Ft(\()p Fs(\017)2142 3827 y Fr(1)2202 3813 y Ft(+)19 b Fs(i\017)2360 3827 y Fr(2)2400 3813 y Ft(\))31 b(and)f(the)g(prop)s(osition)g(5,)h(w)m(e)g (get)g(that)75 4007 y(\(64\))706 b Fs(X)1016 4021 y Fq(I)1057 4007 y Ft(\()p Fs(\017)1129 4021 y Fr(1)1189 4007 y Ft(+)19 b Fs(i\017)1347 4021 y Fr(2)1387 4007 y Ft(\))i Fp(\000)f Fs(f)1579 4021 y Fq(I)1618 4007 y Ft(\()p Fs(\017)1690 4021 y Fr(1)1730 4007 y Fs(;)15 b(\017)1807 4021 y Fr(2)1846 4007 y Ft(\))26 b(=)f Fs(X)2078 4022 y Fq(I)2114 4003 y Fg(0)2141 4007 y Ft(\()p Fs(\017)2213 4021 y Fr(1)2273 4007 y Ft(+)19 b Fs(i\017)2431 4021 y Fr(2)2471 4007 y Ft(\))i Fp(\000)f Fs(f)2663 4022 y Fq(I)2699 4003 y Fg(0)2724 4007 y Ft(\()p Fs(\017)2796 4021 y Fr(1)2836 4007 y Fs(;)15 b(\017)2913 4021 y Fr(2)2953 4007 y Ft(\))75 4201 y(for)31 b(all)i Fs(I)7 b(;)15 b(I)477 4168 y Fn(0)528 4201 y Fp(2)27 b(I)38 b Ft(suc)m(h)32 b(that)g Fs(A)1177 4215 y Fq(I)1238 4201 y Fp(\\)21 b Fs(A)1388 4217 y Fq(I)1424 4198 y Fg(0)1478 4201 y Fp(6)p Ft(=)27 b Fp(;)p Ft(,)32 b(for)g(all)h(\()p Fs(\017)2019 4215 y Fr(1)2058 4201 y Fs(;)15 b(\017)2135 4215 y Fr(2)2175 4201 y Ft(\))28 b Fp(2)f Fs(A)2394 4215 y Fq(I)2455 4201 y Fp(\\)21 b Fs(A)2605 4217 y Fq(I)2641 4198 y Fg(0)2688 4201 y Fp(n)h(f)p Ft(\(0)p Fs(;)15 b Ft(0\))p Fp(g)p Ft(.)47 b(F)-8 b(rom)33 b(the)e(relations)75 4314 y(\(64\),)h(w)m(e)f(see)g(that)g(the)g(follo) m(wing)g(function)f(de\014ned)g(b)m(y)1130 4508 y(\()p Fs(X)e Fp(\000)20 b Fs(f)10 b Ft(\)\()p Fs(\017)1521 4522 y Fr(1)1560 4508 y Fs(;)15 b(\017)1637 4522 y Fr(2)1677 4508 y Ft(\))26 b(:=)f Fs(X)1934 4522 y Fq(I)1974 4508 y Ft(\()p Fs(\017)2046 4522 y Fr(1)2106 4508 y Ft(+)20 b Fs(i\017)2265 4522 y Fr(2)2305 4508 y Ft(\))g Fp(\000)g Fs(f)2496 4522 y Fq(I)2536 4508 y Ft(\()p Fs(\017)2608 4522 y Fr(1)2647 4508 y Fs(;)15 b(\017)2724 4522 y Fr(2)2764 4508 y Ft(\))75 4703 y(for)31 b(all)i(\()p Fs(\017)415 4717 y Fr(1)455 4703 y Fs(;)15 b(\017)532 4717 y Fr(2)571 4703 y Ft(\))28 b Fp(2)f Fs(A)790 4717 y Fq(I)851 4703 y Fp(n)22 b(f)p Ft(\(0)p Fs(;)15 b Ft(0\))p Fp(g)p Ft(,)35 b(for)c(all)i Fs(I)h Fp(2)27 b(I)7 b Ft(,)31 b(is)h(w)m(ell)g (de\014ned)f(on)g Fs(W)j Fp(n)21 b(f)p Ft(\(0)p Fs(;)15 b Ft(0\))p Fp(g)p Ft(,)35 b(where)d Fs(W)39 b Ft(=)27 b Fp([)3616 4717 y Fq(I)5 b Fn(2I)3746 4703 y Fs(A)3814 4717 y Fq(I)75 4816 y Ft(is)30 b(a)h(closed)g(neigh)m(b)s(orho)s(o)s(d) e(of)i(\(0)p Fs(;)15 b Ft(0\).)75 4975 y(W)-8 b(e)32 b(will)e(denote)h(b)m(y)f Fs(@)871 4989 y Fr(\026)-36 b Fq(\017)934 4975 y Ft(the)30 b(op)s(erator)h(de\014ned)e(b)m(y)1596 5202 y Fs(@)1645 5216 y Fr(\026)-36 b Fq(\017)1702 5202 y Ft(=)1808 5141 y(1)p 1808 5181 46 4 v 1808 5265 a(2)1864 5202 y(\()p Fs(@)1947 5216 y Fq(\017)1976 5225 y Fl(1)2035 5202 y Ft(+)20 b Fs(i@)2205 5216 y Fq(\017)2234 5225 y Fl(2)2273 5202 y Ft(\))p Fs(:)75 5433 y Ft(F)-8 b(rom)25 b(the)g(fact)g(that)h Fs(X)899 5447 y Fq(I)963 5433 y Ft(is)f(a)g(holomorphic)g(function)f(on)g Fs(U)2166 5447 y Fq(I)2206 5433 y Fs(q)2250 5400 y Fn(\000)p Fj(N)2357 5433 y Ft(,)i(the)f(Cauc)m(h)m(y-Riemann)g(equations)g(yield)75 5546 y Fs(@)124 5560 y Fr(\026)-36 b Fq(\017)156 5546 y Fs(X)231 5560 y Fq(I)271 5546 y Ft(\()p Fs(\017)343 5560 y Fr(1)403 5546 y Ft(+)20 b Fs(i\017)562 5560 y Fr(2)602 5546 y Ft(\))25 b(=)g(0)31 b(for)f(all)h(\()p Fs(\017)1171 5560 y Fr(1)1211 5546 y Fs(;)15 b(\017)1288 5560 y Fr(2)1328 5546 y Ft(\))25 b Fp(2)g Fs(A)1542 5560 y Fq(I)1603 5546 y Fp(n)20 b(f)p Ft(\(0)p Fs(;)15 b Ft(0\))p Fp(g)p Ft(,)34 b(for)c(all)h Fs(I)h Fp(2)25 b(I)7 b Ft(.)40 b(W)-8 b(e)32 b(deduce)e(that)75 5740 y(\(65\))1088 b Fs(@)1372 5754 y Fr(\026)-36 b Fq(\017)1404 5740 y Ft(\()p Fs(X)28 b Fp(\000)19 b Fs(f)10 b Ft(\)\()p Fs(\017)1794 5754 y Fr(1)1834 5740 y Fs(;)15 b(\017)1911 5754 y Fr(2)1951 5740 y Ft(\))25 b(=)g Fp(\000)p Fs(@)2227 5754 y Fr(\026)-36 b Fq(\017)2259 5740 y Fs(f)2304 5754 y Fq(I)2343 5740 y Ft(\()p Fs(\017)2415 5754 y Fr(1)2455 5740 y Fs(;)15 b(\017)2532 5754 y Fr(2)2572 5740 y Ft(\))p eop end %%Page: 21 21 TeXDict begin 21 20 bop 3764 100 a Ft(21)75 399 y(for)35 b(all)g(\()p Fs(\017)421 413 y Fr(1)461 399 y Fs(;)15 b(\017)538 413 y Fr(2)578 399 y Ft(\))33 b Fp(2)f Ft(In)m(t\()p Fs(A)958 413 y Fq(I)999 399 y Ft(\).)54 b(F)-8 b(rom)36 b(the)f(fact)h(that)f Fs(f)1948 413 y Fq(I)2022 399 y Ft(is)g Fs(C)2190 366 y Fn(1)2299 399 y Ft(in)g(the)g(sense)g(of)g (Whitney)g(on)g Fs(A)3494 413 y Fq(I)3534 399 y Ft(,)h(w)m(e)f(get)75 511 y(that)28 b Fp(\000)p Fs(@)389 525 y Fr(\026)-36 b Fq(\017)421 511 y Fs(f)466 525 y Fq(I)505 511 y Ft(\()p Fs(\017)577 525 y Fr(1)617 511 y Fs(;)15 b(\017)694 525 y Fr(2)734 511 y Ft(\))27 b(can)h(b)s(e)f(extended)h(to)g(a)g Fs(C)1718 478 y Fn(1)1820 511 y Ft(function)f(in)g(the)h(sense)f(of)h (Whitney)g(on)f Fs(A)3323 525 y Fq(I)3363 511 y Ft(.)40 b(W)-8 b(e)29 b(deduce)75 624 y(that)75 829 y(\(66\))1357 b Fs(@)1641 843 y Fr(\026)-36 b Fq(\017)1673 829 y Ft(\()p Fs(X)28 b Fp(\000)20 b Fs(f)10 b Ft(\))25 b Fp(2)g(E)8 b Ft(\()p Fs(A)2262 843 y Fq(I)2302 829 y Ft(\))75 1033 y(for)30 b(all)h Fs(I)h Fp(2)25 b(I)7 b Ft(.)40 b(On)29 b(the)i(other)f(hand,)g(from)f(the)i(fact)g(that)g Fs(g)2208 1048 y Fq(I)5 b(;I)2300 1029 y Fg(0)2326 1033 y Ft(\()p Fs(\017)p Ft(\))30 b(is)h(holomorphic)f(on)g Fs(U)3263 1047 y Fq(I)3303 1033 y Fs(q)3347 1000 y Fn(\000)p Fj(N)3474 1033 y Fp(\\)19 b Fs(U)3616 1048 y Fq(I)3652 1029 y Fg(0)3678 1033 y Fs(q)3722 1000 y Fn(\000)p Fj(N)3829 1033 y Ft(,)75 1146 y(w)m(e)31 b(get)g(from)f(\(58\))i(that)75 1350 y(\(67\))1229 b Fs(@)1513 1364 y Fr(\026)-36 b Fq(\017)1545 1350 y Fs(f)1590 1364 y Fq(I)1629 1350 y Ft(\()p Fs(\017)1701 1364 y Fr(1)1741 1350 y Fs(;)15 b(\017)1818 1364 y Fr(2)1858 1350 y Ft(\))25 b(=)g Fs(@)2063 1364 y Fr(\026)-36 b Fq(\017)2095 1350 y Fs(f)2140 1365 y Fq(I)2176 1346 y Fg(0)2202 1350 y Ft(\()p Fs(\017)2274 1364 y Fr(1)2314 1350 y Fs(;)15 b(\017)2391 1364 y Fr(2)2430 1350 y Ft(\))75 1554 y(for)37 b(all)h(\()p Fs(\017)426 1568 y Fr(1)466 1554 y Fs(;)15 b(\017)543 1568 y Fr(2)582 1554 y Ft(\))37 b Fp(2)f Ft(In)m(t\()p Fs(A)970 1568 y Fq(I)1036 1554 y Fp(\\)24 b Fs(A)1189 1569 y Fq(I)1225 1551 y Fg(0)1251 1554 y Ft(\),)40 b(for)d(all)h Fs(I)7 b(;)15 b(I)1764 1521 y Fn(0)1824 1554 y Fp(2)36 b(I)7 b Ft(.)60 b(F)-8 b(rom)38 b(the)f(fact)h(that)g Fs(f)2907 1568 y Fq(I)2984 1554 y Ft(is)f Fs(C)3154 1521 y Fn(1)3265 1554 y Ft(in)g(the)g(sense)g (of)75 1667 y(Whitney)31 b(on)f Fs(A)643 1681 y Fq(I)683 1667 y Ft(,)h(w)m(e)g(get)g(that)g(the)g(relation)g(\(67\))h(can)f(b)s (e)e(extended)i(to)g Fs(A)2759 1681 y Fq(I)2819 1667 y Fp(\\)20 b Fs(A)2968 1682 y Fq(I)3004 1663 y Fg(0)3031 1667 y Ft(.)216 1780 y(F)-8 b(rom)32 b(Lemma)f(3.10)i(in)e([10)q(],)h (w)m(e)f(kno)m(w)g(that)h(the)f(sets)h Fs(A)2253 1794 y Fq(I)2324 1780 y Ft(and)f Fs(A)2570 1795 y Fq(I)2606 1776 y Fg(0)2663 1780 y Ft(are)h(regularly)f(separated.)44 b(F)-8 b(rom)75 1893 y(\(66\))32 b(and)e(\(67\),)i(w)m(e)f(deduce)f (with)g(the)g(Lo)5 b(jasiewicz)32 b(extension)f(theorem)g(\([9)q(])g (p.83\),)g(that)1484 2097 y Fs(@)1533 2111 y Fr(\026)-36 b Fq(\017)1565 2097 y Ft(\()p Fs(X)28 b Fp(\000)20 b Fs(f)10 b Ft(\))25 b Fp(2)f(E)8 b Ft(\()p Fp([)2146 2111 y Fq(I)d Fn(2I)2277 2097 y Fs(A)2345 2111 y Fq(I)2385 2097 y Ft(\))p Fs(:)75 2301 y Ft(In)30 b(particular,)h(the)f(function)g Fs(@)1195 2315 y Fr(\026)-36 b Fq(\017)1227 2301 y Ft(\()p Fs(X)28 b Fp(\000)20 b Fs(f)10 b Ft(\))30 b(de\014nes)g(a)g Fs(C)2022 2268 y Fn(1)2127 2301 y Ft(function)g(in)g(the)g(usual)g (sense)h(on)f(In)m(t\()p Fs(W)13 b Ft(\).)216 2414 y(No)m(w,)39 b(from)d([4)q(],)j(theorem)e(14.4,)j(w)m(e)d(kno)m(w)g(that)g(there)g (exists)g(a)g Fs(C)2659 2381 y Fn(1)2770 2414 y Ft(function)f(in)h(the) f(usual)g(sense)75 2527 y Fs(u)p Ft(\()p Fs(\017)199 2541 y Fr(1)239 2527 y Fs(;)15 b(\017)316 2541 y Fr(2)355 2527 y Ft(\))31 b(on)f(In)m(t)q(\()p Fs(W)13 b Ft(\))30 b(suc)m(h)g(that)75 2731 y(\(68\))1139 b Fs(@)1423 2745 y Fr(\026)-36 b Fq(\017)1455 2731 y Fs(u)p Ft(\()p Fs(\017)1579 2745 y Fr(1)1619 2731 y Fs(;)15 b(\017)1696 2745 y Fr(2)1736 2731 y Ft(\))25 b(=)g Fs(@)1941 2745 y Fr(\026)-36 b Fq(\017)1973 2731 y Ft(\()p Fs(X)28 b Fp(\000)20 b Fs(f)10 b Ft(\)\()p Fs(\017)2364 2745 y Fr(1)2403 2731 y Fs(;)15 b(\017)2480 2745 y Fr(2)2520 2731 y Ft(\))75 2936 y(for)37 b(all)g(\()p Fs(\017)425 2950 y Fr(1)465 2936 y Fs(;)15 b(\017)542 2950 y Fr(2)582 2936 y Ft(\))36 b Fp(2)g Ft(In)m(t\()p Fs(W)13 b Ft(\).)60 b(F)-8 b(rom)37 b(this)g(latter)h(equalit)m(y)-8 b(,)41 b(w)m(e)c(deduce)g(that)g Fs(u)p Ft(\()p Fs(\017)2952 2950 y Fr(1)2992 2936 y Fs(;)15 b(\017)3069 2950 y Fr(2)3109 2936 y Ft(\))24 b Fp(\000)h Ft(\()p Fs(X)32 b Fp(\000)24 b Fs(f)10 b Ft(\)\()p Fs(\017)3663 2950 y Fr(1)3702 2936 y Fs(;)15 b(\017)3779 2950 y Fr(2)3819 2936 y Ft(\))75 3049 y(de\014nes)42 b(a)g(holomorphic)h(function)f(on)g(In)m(t\()p Fs(W)13 b Ft(\))29 b Fp(n)f(f)p Ft(\(0)p Fs(;)15 b Ft(0\))p Fp(g)p Ft(.)80 b(F)-8 b(rom)43 b(the)f(fact)i(that)f(for)f(all)h Fs(I)53 b Fp(2)44 b(I)7 b Ft(,)45 b(the)75 3161 y(functions)28 b Fs(X)540 3175 y Fq(I)609 3161 y Ft(are)i(b)s(ounded)c(\(as)k Fm(H)1346 3175 y Fn(T)17 b Fq(;\032)1490 3161 y Ft(v)-5 b(alued)29 b(functions\),)h(w)m(e)f(deduce)f(that)i(the)f(holomorphic)g (function)75 3274 y Fs(u)p Ft(\()p Fs(\017)199 3288 y Fr(1)239 3274 y Fs(;)15 b(\017)316 3288 y Fr(2)355 3274 y Ft(\))28 b Fp(\000)e Ft(\()p Fs(X)35 b Fp(\000)26 b Fs(f)10 b Ft(\)\()p Fs(\017)919 3288 y Fr(1)958 3274 y Fs(;)15 b(\017)1035 3288 y Fr(2)1075 3274 y Ft(\))41 b(is)f(b)s(ounded)e(on)i(In)m(t\()p Fs(W)13 b Ft(\))27 b Fp(n)h(f)p Ft(\(0)p Fs(;)15 b Ft(0\))p Fp(g)p Ft(.)73 b(Finally)-8 b(,)44 b(from)c(the)g(classical)j(result)75 3387 y(on)h(remo)m(v)-5 b(able)44 b(singularities,)49 b(w)m(e)44 b(deduce)f(that)h Fs(u)p Ft(\()p Fs(\017)2003 3401 y Fr(1)2043 3387 y Fs(;)15 b(\017)2120 3401 y Fr(2)2160 3387 y Ft(\))29 b Fp(\000)g Ft(\()p Fs(X)37 b Fp(\000)29 b Fs(f)10 b Ft(\)\()p Fs(\017)2733 3401 y Fr(1)2772 3387 y Fs(;)15 b(\017)2849 3401 y Fr(2)2889 3387 y Ft(\))44 b(can)f(b)s(e)h(extended)f(as)h(a)75 3500 y(holomorphic)30 b(function)h(on)f(In)m(t\()p Fs(W)13 b Ft(\).)41 b(This)29 b(yields)i(the)f(decomp)s(osition)h(\(63\).)924 b Ff(2)75 3663 y Ft(In)30 b(the)g(follo)m(wing,)i(w)m(e)f(state)h(the)e(main)h (result)f(of)h(the)f(pap)s(er.)75 3875 y Fi(Theorem)35 b(5)46 b Fh(Under)32 b(the)h(same)h(hyp)-5 b(otheses)35 b(as)e(in)g(Pr)-5 b(op)g(osition)35 b(3,)e(ther)-5 b(e)33 b(exists)g(a)g(formal)h(series)1260 4099 y Ft(^)1234 4122 y Fs(X)8 b Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))26 b(=)1705 4036 y Fo(X)1707 4234 y Fq(k)r Fn(\025)p Fr(0)1862 4061 y Fs(X)1937 4076 y Fq(k)1980 4061 y Ft(\()p Fs(t;)15 b(z)t Ft(\))p 1862 4102 309 4 v 1978 4185 a Fs(k)s Ft(!)2180 4122 y Fs(\017)2217 4085 y Fq(k)2285 4122 y Fp(2)25 b Fm(H)2442 4136 y Fn(T)17 b Fq(;\032)2557 4122 y Ft([[)p Fs(\017)p Ft(]])75 4417 y Fh(which)34 b(satis\014es)f(the)g(singularly) h(p)-5 b(erturb)g(e)g(d)34 b(pr)-5 b(oblem)75 4702 y Ft(\(69\))312 b Fs(\017t@)670 4665 y Fq(S)665 4725 y(z)747 4679 y Ft(^)721 4702 y Fs(X)7 b Ft(\()p Fs(\017;)15 b(q)s(t;)g(z)t Ft(\))22 b(+)e Fs(@)1279 4665 y Fq(S)1274 4725 y(z)1356 4679 y Ft(^)1330 4702 y Fs(X)7 b Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))27 b(=)1801 4589 y Fq(S)t Fn(\000)p Fr(1)1804 4616 y Fo(X)1805 4813 y Fq(k)r Fr(=0)1953 4702 y Fs(b)1992 4717 y Fq(k)2035 4702 y Ft(\()p Fs(\017;)15 b(z)t Ft(\)\()p Fs(\017t\033)2385 4716 y Fq(q)2424 4702 y Ft(\))2459 4665 y Fq(m)2521 4677 y Fl(0)p Fk(;k)2613 4702 y Ft(\()p Fs(@)2701 4665 y Fq(k)2696 4725 y(z)2771 4679 y Ft(^)2745 4702 y Fs(X)7 b Ft(\)\()p Fs(\017;)15 b(t;)g(z)t(q)3137 4665 y Fn(\000)p Fq(m)3254 4677 y Fl(1)p Fk(;k)3347 4702 y Ft(\))75 5006 y Fh(and)47 b(which)g(is)f(the)h Fs(q)s Fh(-asymptotic)h(exp)-5 b(ansion)48 b(on)1958 4983 y Ft(~)1934 5006 y Fs(K)2011 5020 y Fq(I)2051 5006 y Fs(q)2095 4973 y Fn(\000)p Fj(N)2248 5006 y Fh(of)e(the)h (holomorphic)i(functions)e Fs(X)3547 5020 y Fq(I)3587 5006 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))75 5119 y Fh(c)-5 b(onstructe)g(d)34 b(in)f(the)g(pr)-5 b(op)g(osition)36 b(3,)d(wher)-5 b(e)1671 5096 y Ft(~)1647 5119 y Fs(K)1724 5133 y Fq(I)1796 5119 y Fh(is)33 b(any)g(c)-5 b(omp)g(act)35 b(subset)d(of)h Ft(In)m(t\()p Fs(K)3023 5133 y Fq(I)3064 5119 y Ft(\))p Fh(,)g(for)g(al)5 b(l)33 b Fs(I)f Fp(2)25 b(I)7 b Fh(.)75 5381 y Fi(Pro)s(of)30 b Ft(F)-8 b(rom)30 b(the)g(decomp)s(osition)g(\(63\),)h(w)m(e)f(get)h(that)f(for)f(all)h Fs(I)i Fp(2)25 b(I)7 b Ft(,)29 b(the)h(function)f Fs(X)3164 5395 y Fq(I)3204 5381 y Ft(\()p Fs(\017)3276 5395 y Fr(1)3334 5381 y Ft(+)18 b Fs(i\017)3491 5395 y Fr(2)3531 5381 y Ft(\))29 b(can)h(b)s(e)75 5494 y(extended)g(to)h(a)g Fs(C)720 5461 y Fn(1)825 5494 y Ft(function)f(in)g(the)g(sense)h(of)f (Whitney)h(on)f(ev)m(ery)h(compact)h(set)f(of)g(the)f(form)705 5675 y(~)681 5698 y Fs(A)749 5712 y Fq(I)814 5698 y Ft(=)25 b Fp(f)p Ft(\()p Fs(\017)1027 5712 y Fr(1)1068 5698 y Fs(;)15 b(\017)1145 5712 y Fr(2)1184 5698 y Ft(\))26 b Fp(2)f Fm(R)1397 5661 y Fr(2)1436 5698 y Fs(=\017)1518 5712 y Fr(1)1578 5698 y Ft(+)20 b Fs(i\017)1737 5712 y Fr(2)1802 5698 y Fp(2)1912 5675 y Ft(~)1888 5698 y Fs(K)1965 5712 y Fq(I)2005 5698 y Fs(q)2049 5661 y Fn(\000)p Fj(N)2176 5698 y Fp([)g(f)p Ft(0)p Fp(gg)27 b(\032)e Ft(In)m(t\()p Fs(A)2779 5712 y Fq(I)2819 5698 y Ft(\))c Fp([)f(f)p Ft(\(0)p Fs(;)15 b Ft(0\))p Fp(g)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 3764 100 a Ft(22)75 399 y(where)356 376 y(~)332 399 y Fs(K)409 413 y Fq(I)473 399 y Ft(is)25 b(a)g(compact)g(set)g(in)f(In)m(t\()p Fs(K)1450 413 y Fq(I)1491 399 y Ft(\).)39 b(Let)25 b Fs(I)32 b Fp(2)25 b(I)7 b Ft(.)38 b(W)-8 b(e)26 b(consider)e(the)g(family)h(of)g(con)m (tin)m(uous)g(functions)75 527 y Fs(X)157 479 y Fr(\()p Fq(h)225 488 y Fl(1)260 479 y Fq(;h)321 488 y Fl(2)355 479 y Fr(\))150 556 y Fq(I)387 527 y Ft(\()p Fs(\017)459 541 y Fr(1)498 527 y Fs(;)15 b(\017)575 541 y Fr(2)615 527 y Ft(\))28 b(on)825 504 y(~)801 527 y Fs(A)869 541 y Fq(I)936 527 y Ft(asso)s(ciated)h(to)f Fs(X)1549 541 y Fq(I)1616 527 y Ft(from)f(the)h(de\014nition)f(7,)h(for)f(all)i(\()p Fs(h)2828 541 y Fr(1)2868 527 y Fs(;)15 b(h)2960 541 y Fr(2)3000 527 y Ft(\))26 b Fp(2)e Fm(N)3212 494 y Fr(2)3252 527 y Ft(.)39 b(F)-8 b(rom)28 b(the)g(fact)75 640 y(that)j Fs(X)347 654 y Fq(I)387 640 y Ft(\()p Fs(\017)p Ft(\))g(is)g (holomorphic)f(on)g(In)m(t\()p Fs(K)1491 654 y Fq(I)1532 640 y Ft(\))p Fs(q)1611 607 y Fn(\000)p Fj(N)1718 640 y Ft(,)h(w)m(e)g(ha)m(v)m(e)g(the)g(relations)884 845 y Fs(X)966 797 y Fr(\()p Fq(h)1034 806 y Fl(1)1070 797 y Fq(;h)1131 806 y Fl(2)1164 797 y Fr(\))959 873 y Fq(I)1196 845 y Ft(\()p Fs(\017)1268 859 y Fr(1)1308 845 y Fs(;)15 b(\017)1385 859 y Fr(2)1424 845 y Ft(\))26 b(=)f Fs(@)1634 807 y Fq(h)1675 816 y Fl(1)1629 867 y Fq(\017)1658 876 y Fl(1)1714 845 y Fs(@)1767 807 y Fq(h)1808 816 y Fl(2)1762 867 y Fq(\017)1791 876 y Fl(2)1846 845 y Fs(X)1921 859 y Fq(I)1962 845 y Ft(\()p Fs(\017)2034 859 y Fr(1)2094 845 y Ft(+)19 b Fs(i\017)2252 859 y Fr(2)2292 845 y Ft(\))26 b(=)f Fs(i)2480 807 y Fq(h)2521 816 y Fl(2)2559 845 y Fs(@)2612 807 y Fq(h)2653 816 y Fl(1)2688 807 y Fr(+)p Fq(h)2784 816 y Fl(2)2607 867 y Fq(\017)2822 845 y Fs(X)2897 859 y Fq(I)2938 845 y Ft(\()p Fs(\017)p Ft(\))75 1058 y(for)i(all)358 1035 y(~)334 1058 y Fs(A)402 1072 y Fq(I)455 1058 y Fp(n)14 b(f)p Ft(\(0)p Fs(;)h Ft(0\))p Fp(g)p Ft(.)43 b(By)27 b(con)m(tin)m(uit)m(y)i(of)f(the)f(functions)f Fs(X)2161 1011 y Fr(\()p Fq(h)2229 1020 y Fl(1)2265 1011 y Fq(;h)2326 1020 y Fl(2)2359 1011 y Fr(\))2154 1087 y Fq(I)2418 1058 y Ft(at)i(\(0)p Fs(;)15 b Ft(0\),)30 b(w)m(e)d(deduce)g(that)h(there)f(exists)75 1171 y(a)k(sequence)g Fs(X)603 1186 y Fq(k)r(;I)726 1171 y Fp(2)25 b Fm(H)883 1185 y Fn(T)17 b Fq(;\032)999 1171 y Ft(,)30 b Fs(k)f Fp(\025)c Ft(0,)31 b(suc)m(h)f(that)75 1445 y(\(70\))1561 1381 y Fs(X)1643 1333 y Fr(\()p Fq(h)1711 1342 y Fl(1)1746 1333 y Fq(;h)1807 1342 y Fl(2)1841 1333 y Fr(\))1636 1410 y Fq(I)1872 1381 y Ft(\(0)p Fs(;)15 b Ft(0\))p 1561 1424 514 4 v 1762 1507 a Fs(i)1793 1481 y Fq(h)1834 1490 y Fl(2)2110 1445 y Ft(=)24 b Fs(X)2280 1460 y Fq(k)r(;I)75 1667 y Ft(for)30 b(all)h(\()p Fs(h)427 1681 y Fr(1)468 1667 y Fs(;)15 b(h)560 1681 y Fr(2)600 1667 y Ft(\))25 b Fp(2)g Fm(N)812 1634 y Fr(2)882 1667 y Ft(suc)m(h)30 b(that)h Fs(h)1336 1681 y Fr(1)1396 1667 y Ft(+)20 b Fs(h)1539 1681 y Fr(2)1604 1667 y Ft(=)25 b Fs(k)s Ft(.)216 1780 y(W)-8 b(e)23 b(no)m(w)e(apply)g(the)g(estimates)i(\(56\))g (satis\014ed)e(b)m(y)g(the)g(family)h Fs(X)2461 1747 y Fq(h)2454 1807 y(I)2527 1780 y Ft(for)f Fs(k)29 b Ft(=)c(\(0)p Fs(;)15 b Ft(0\))23 b(and)e(\()p Fs(x)3307 1794 y Fr(2)3346 1780 y Fs(;)15 b(y)3431 1794 y Fr(2)3471 1780 y Ft(\))25 b(=)g(\(0)p Fs(;)15 b Ft(0\).)75 1893 y(Using)31 b(the)f(relations)i (\(70\),)g(w)m(e)f(get)g(that)g(for)f(all)h Fs(m)26 b Fp(\025)e Ft(0,)32 b(all)f Fs(\017)25 b(>)g Ft(0,)31 b(there)f(exists)h Fs(\016)f(>)25 b Ft(0)30 b(suc)m(h)g(that)75 2166 y(\(71\))588 b Fp(jj)p Fs(X)948 2180 y Fq(I)989 2166 y Ft(\()p Fs(\017)1061 2180 y Fr(1)1121 2166 y Ft(+)20 b Fs(i\017)1280 2180 y Fr(2)1320 2166 y Ft(\))g Fp(\000)1466 2052 y Fq(m)p Fr(+1)1477 2079 y Fo(X)1478 2277 y Fq(k)r Fr(=0)1644 2102 y Fs(X)1719 2117 y Fq(k)r(;I)p 1644 2145 174 4 v 1693 2228 a Fs(k)s Ft(!)1827 2166 y(\()p Fs(\017)1899 2180 y Fr(1)1959 2166 y Ft(+)g Fs(i\017)2118 2180 y Fr(2)2158 2166 y Ft(\))2193 2128 y Fq(k)2236 2166 y Fp(jj)2286 2180 y Fj(H)2338 2191 y Fg(T)14 b Fk(;\032)2471 2166 y Fs(<)25 b(\017)p Fp(j)p Fs(\017)2666 2180 y Fr(1)2726 2166 y Ft(+)19 b Fs(i\017)2884 2180 y Fr(2)2924 2166 y Fp(j)2949 2128 y Fq(m)p Fr(+1)75 2454 y Ft(for)34 b(all)h Fs(\017)385 2468 y Fr(1)447 2454 y Ft(+)22 b Fs(i\017)608 2468 y Fr(2)679 2454 y Fp(2)796 2431 y Ft(~)772 2454 y Fs(K)849 2468 y Fq(I)889 2454 y Fs(q)933 2421 y Fn(\000)p Fj(N)1074 2454 y Ft(with)33 b Fp(j)p Fs(\017)1346 2468 y Fr(1)1409 2454 y Ft(+)22 b Fs(i\017)1570 2468 y Fr(2)1610 2454 y Fp(j)31 b Fs(<)h(\016)s Ft(.)52 b(This)34 b(implies,)h(in)f (particular,)i(estimates)f(of)g(the)f(form)75 2567 y(\(45\).)47 b(This)31 b(sho)m(ws)h(that)h(the)f(formal)g(series)1695 2544 y(^)1669 2567 y Fs(X)1744 2581 y Fq(I)1784 2567 y Ft(\()p Fs(\017)p Ft(\))c(=)2018 2499 y Fo(P)2114 2594 y Fq(k)r Fn(\025)p Fr(0)2262 2567 y Fs(X)2337 2582 y Fq(k)r(;I)2435 2567 y Fs(\017)2472 2534 y Fq(k)2515 2567 y Fs(=k)s Ft(!)33 b(is)f(the)g Fs(q)s Ft(-asymptotic)i(expansion)75 2697 y(of)d Fs(X)254 2711 y Fq(I)324 2697 y Ft(on)475 2674 y(~)450 2697 y Fs(K)527 2711 y Fq(I)567 2697 y Fs(q)611 2664 y Fn(\000)p Fj(N)718 2697 y Ft(,)g(for)f(ev)m(ery)h Fs(I)i Fp(2)24 b(I)7 b Ft(.)216 2810 y(No)m(w,)28 b(w)m(e)d(sho)m(w)h (that)1012 2787 y(^)986 2810 y Fs(X)1061 2824 y Fq(I)1126 2810 y Ft(is)g(indep)s(enden)m(t)e(of)i Fs(I)7 b Ft(.)39 b(Indeed,)26 b(let)g Fs(I)7 b(;)15 b(I)2510 2777 y Fn(0)2559 2810 y Fp(2)25 b(I)32 b Ft(suc)m(h)25 b(that)3143 2787 y(~)3119 2810 y Fs(K)3196 2824 y Fq(I)3236 2810 y Fs(q)3280 2777 y Fn(\000)p Fj(N)3397 2810 y Fp(\\)3492 2787 y Ft(~)3468 2810 y Fs(K)3545 2825 y Fq(I)3581 2806 y Fg(0)3608 2810 y Fs(q)3652 2777 y Fn(\000)p Fj(N)3784 2810 y Fp(6)p Ft(=)75 2923 y Fp(;)p Ft(.)39 b(W)-8 b(e)26 b(get)f(that)g(the)g (formal)f(series)1365 2900 y(^)1339 2923 y Fs(X)1414 2938 y Fq(I)1450 2919 y Fg(0)1476 2923 y Ft(\()p Fs(\017)p Ft(\))8 b Fp(\000)1697 2900 y Ft(^)1670 2923 y Fs(X)1745 2937 y Fq(I)1786 2923 y Ft(\()p Fs(\017)p Ft(\))25 b(is)g(the)f Fs(q)s Ft(-asymptotic)i(expansion)e(of)h Fs(X)3279 2938 y Fq(I)3315 2919 y Fg(0)3341 2923 y Ft(\()p Fs(\017)p Ft(\))8 b Fp(\000)g Fs(X)3610 2937 y Fq(I)3651 2923 y Ft(\()p Fs(\017)p Ft(\))26 b(=)75 3036 y Fs(g)118 3051 y Fq(I)154 3032 y Fg(0)177 3051 y Fq(;I)236 3036 y Ft(\()p Fs(\017)p Ft(\))41 b(on)544 3013 y(~)520 3036 y Fs(K)597 3050 y Fq(I)637 3036 y Fs(q)681 3003 y Fn(\000)p Fj(N)814 3036 y Fp(\\)926 3013 y Ft(~)901 3036 y Fs(K)978 3051 y Fq(I)1014 3032 y Fg(0)1041 3036 y Fs(q)1085 3003 y Fn(\000)p Fj(N)1192 3036 y Ft(.)70 b(But,)42 b(from)e(the)g(prop)s (osition)g(3,)j(w)m(e)e(kno)m(w)f(that)g Fs(g)3136 3051 y Fq(I)3172 3032 y Fg(0)3195 3051 y Fq(;I)3254 3036 y Ft(\()p Fs(\017)p Ft(\))h(has)3574 3016 y(^)3574 3036 y(0)f(as)g Fs(q)s Ft(-)75 3149 y(asymptotic)31 b(expansion)e(on)1113 3126 y(~)1088 3149 y Fs(K)1165 3163 y Fq(I)1206 3149 y Fs(q)1250 3116 y Fn(\000)p Fj(N)1375 3149 y Fp(\\)1478 3126 y Ft(~)1454 3149 y Fs(K)1531 3164 y Fq(I)1567 3145 y Fg(0)1593 3149 y Fs(q)1637 3116 y Fn(\000)p Fj(N)1744 3149 y Ft(.)40 b(F)-8 b(rom)30 b(the)g(unicit)m(y)g(of)g(the)f Fs(q)s Ft(-asymptotic)i(expansion,)f(w)m(e)75 3261 y(deduce)g(that)601 3238 y(^)575 3261 y Fs(X)650 3277 y Fq(I)686 3258 y Fg(0)713 3261 y Ft(\()p Fs(\017)p Ft(\))25 b(=)968 3238 y(^)941 3261 y Fs(X)1016 3275 y Fq(I)1057 3261 y Ft(\()p Fs(\017)p Ft(\).)75 3419 y(In)34 b(conclusion,)k(there)d(exist)h(a)g(formal)f (series)1768 3396 y(^)1742 3419 y Fs(X)7 b Ft(\()p Fs(\017)p Ft(\))34 b(=)2069 3351 y Fo(P)2165 3446 y Fq(k)r Fn(\025)p Fr(0)2313 3419 y Fs(X)2388 3434 y Fq(k)2431 3419 y Fs(\017)2468 3386 y Fq(k)2511 3419 y Fs(=k)s Ft(!)i(whic)m(h)f(is)g(the)g Fs(q)s Ft(-asymptotic)i(ex-)75 3549 y(pansion)30 b(of)g Fs(X)587 3563 y Fq(I)658 3549 y Ft(on)808 3526 y(~)784 3549 y Fs(K)861 3563 y Fq(I)901 3549 y Fs(q)945 3516 y Fn(\000)p Fj(N)1052 3549 y Ft(,)h(for)f(ev)m(ery)h Fs(I)h Fp(2)25 b(I)7 b Ft(.)75 3707 y(In)32 b(the)g(last)h(step)g(of)f (the)h(pro)s(of,)f(w)m(e)h(will)g(c)m(hec)m(k)h(that)2029 3684 y(^)2003 3707 y Fs(X)7 b Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))34 b(satis\014es)f(the)g(equation)g(\(51\).)48 b(F)-8 b(rom)33 b(the)75 3820 y(remark)d(after)h(the)g(de\014nition)f (2,)h(w)m(e)f(get)i(that)75 4008 y(\(72\))1059 b(lim)1134 4085 y Fq(\017)p Fn(!)p Fr(0)p Fq(;\017)p Fn(2)1382 4068 y Fr(~)1365 4085 y Fq(K)1425 4096 y Fk(I)1459 4085 y Fq(q)1493 4066 y Fg(\000)p Fa(N)1596 4008 y Fs(@)1649 3971 y Fq(l)1644 4031 y(\017)1677 4008 y Fs(X)1752 4022 y Fq(I)1792 4008 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))27 b(=)e Fs(X)2256 4023 y Fq(l)2282 4008 y Ft(\()p Fs(t;)15 b(z)t Ft(\))26 b Fp(2)f Fm(H)2654 4022 y Fn(T)17 b Fq(;\032)2770 4008 y Fs(;)75 4259 y Ft(for)27 b(all)i Fs(l)e Fp(\025)e Ft(0,)k(all)g Fs(I)j Fp(2)25 b(I)7 b Ft(.)39 b(Let)28 b Fs(I)k Fp(2)25 b(I)7 b Ft(.)39 b(By)28 b(construction,)h(the)f (function)f Fs(X)2692 4273 y Fq(I)2733 4259 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))29 b(satis\014es)f(the)g(equation)75 4372 y(\(51\).)48 b(T)-8 b(aking)33 b(the)g Fs(l)r Fp(\000)p Ft(deriv)-5 b(ativ)m(e)33 b(of)g(the)f(righ)m(t)h(and)f(left)h (handsides)e(of)i(the)f(equation)i(\(51\))f(with)f(resp)s(ect)75 4485 y(to)f Fs(\017)p Ft(,)g(w)m(e)f(deduce)g(that)h Fs(@)965 4452 y Fq(l)960 4507 y(\017)993 4485 y Fs(X)1068 4499 y Fq(I)1109 4485 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))31 b(satis\014es)g(the)g(follo)m(wing)h(equation)75 4699 y(\(73\))85 b Fs(\017t@)443 4661 y Fq(S)438 4721 y(z)493 4699 y Fs(@)546 4661 y Fq(l)541 4721 y(\017)574 4699 y Fs(X)649 4713 y Fq(I)690 4699 y Ft(\()p Fs(\017;)15 b(q)s(t;)g(z)t Ft(\))21 b(+)f Fs(t@)1198 4661 y Fq(S)1193 4721 y(z)1249 4699 y Fs(l)r(@)1331 4661 y Fq(l)q Fn(\000)p Fr(1)1326 4721 y Fq(\017)1447 4699 y Fs(X)1522 4713 y Fq(I)1563 4699 y Ft(\()p Fs(\017;)15 b(q)s(t;)g(z)t Ft(\))21 b(+)f Fs(@)2038 4661 y Fq(S)2033 4721 y(z)2089 4699 y Fs(@)2142 4661 y Fq(l)2137 4721 y(\017)2170 4699 y Fs(X)2245 4713 y Fq(I)2285 4699 y Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))1113 4934 y(=)1209 4820 y Fq(S)t Fn(\000)p Fr(1)1212 4848 y Fo(X)1213 5045 y Fq(k)r Fr(=0)1417 4848 y Fo(X)1361 5045 y Fq(l)1382 5054 y Fl(1)1417 5045 y Fr(+)p Fq(l)1493 5054 y Fl(2)1527 5045 y Fr(=)p Fq(l)1694 4872 y Fs(l)r Ft(!)p 1629 4913 184 4 v 1629 4996 a Fs(l)1656 5010 y Fr(1)1696 4996 y Ft(!)p Fs(l)1748 5010 y Fr(2)1787 4996 y Ft(!)1823 4934 y Fs(@)1876 4896 y Fq(l)1897 4905 y Fl(1)1871 4956 y Fq(\017)1936 4934 y Ft(\()p Fs(b)2010 4949 y Fq(k)2053 4934 y Ft(\()p Fs(\017;)g(z)t Ft(\))p Fs(\017)2283 4896 y Fq(m)2345 4908 y Fl(0)p Fk(;k)2438 4934 y Ft(\))p Fs(@)2526 4896 y Fq(l)2547 4905 y Fl(2)2521 4956 y Fq(\017)2586 4934 y Ft(\(\()p Fs(t\033)2741 4948 y Fq(q)2780 4934 y Ft(\))2815 4896 y Fq(m)2877 4908 y Fl(0)p Fk(;k)2969 4934 y Fs(@)3022 4896 y Fq(k)3017 4956 y(z)3065 4934 y Fs(X)3140 4948 y Fq(I)3181 4934 y Ft(\)\()p Fs(\017;)g(t;)g(z)t(q)3491 4896 y Fn(\000)p Fq(m)3608 4908 y Fl(1)p Fk(;k)3701 4934 y Ft(\)\))75 5231 y(for)39 b(all)g Fs(l)j Fp(\025)d Ft(1,)j(all)e(\()p Fs(t;)15 b(z)t Ft(\))40 b Fp(2)f(T)49 b(\002)26 b Fs(D)s Ft(\(0)p Fs(;)15 b(\032)p Ft(\),)42 b(all)e Fs(\017)f Fp(2)1991 5208 y Ft(~)1966 5231 y Fs(K)2043 5245 y Fq(I)2084 5231 y Fs(q)2128 5198 y Fn(\000)p Fj(N)2234 5231 y Ft(.)67 b(Letting)40 b Fs(\017)f Ft(tend)f(to)i(0)f(in)g(\(73\))h(yields)f(the) 75 5344 y(recursion)75 5457 y(\(74\))179 5656 y Fs(t@)265 5619 y Fq(S)260 5679 y(z)326 5595 y Fs(X)401 5610 y Fq(l)q Fn(\000)p Fr(1)517 5595 y Ft(\()p Fs(q)s(t;)15 b(z)t Ft(\))p 326 5635 426 4 v 398 5719 a(\()p Fs(l)23 b Fp(\000)c Ft(1\)!)782 5656 y(+)h Fs(@)926 5619 y Fq(S)921 5679 y(z)987 5595 y Fs(X)1062 5610 y Fq(l)1088 5595 y Ft(\()p Fs(t;)15 b(z)t Ft(\))p 987 5635 292 4 v 1105 5719 a Fs(l)r Ft(!)1313 5656 y(=)1409 5543 y Fq(S)t Fn(\000)p Fr(1)1412 5570 y Fo(X)1413 5768 y Fq(k)r Fr(=0)1617 5570 y Fo(X)1561 5768 y Fq(l)1582 5777 y Fl(1)1617 5768 y Fr(+)p Fq(l)1693 5777 y Fl(2)1728 5768 y Fr(=)p Fq(l)1829 5595 y Fs(@)1882 5562 y Fq(l)1903 5571 y Fl(1)1877 5617 y Fq(\017)1942 5595 y Ft(\()p Fs(\017)2014 5562 y Fq(m)2076 5574 y Fl(0)p Fk(;k)2169 5595 y Fs(b)2208 5610 y Fq(k)2250 5595 y Ft(\()p Fs(\017;)g(z)t Ft(\)\))p Fp(j)2503 5609 y Fq(\017)p Fr(=0)p 1829 5635 799 4 v 2183 5719 a Fs(l)2210 5733 y Fr(1)2249 5719 y Ft(!)2648 5594 y(\(\()p Fs(t\033)2803 5608 y Fq(q)2842 5594 y Ft(\))2877 5561 y Fq(m)2939 5573 y Fl(0)p Fk(;k)3031 5594 y Fs(@)3084 5561 y Fq(k)3079 5617 y(z)3127 5594 y Fs(X)3202 5609 y Fq(l)3223 5618 y Fl(2)3262 5594 y Ft(\)\()p Fs(t;)g(z)t(q)3495 5561 y Fn(\000)p Fq(m)3612 5573 y Fl(1)p Fk(;k)3705 5594 y Ft(\))p 2648 5635 1093 4 v 3148 5719 a Fs(l)3175 5733 y Fr(2)3215 5719 y Ft(!)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 3764 100 a Ft(23)75 399 y(for)31 b(all)g Fs(l)d Fp(\025)e Ft(1,)31 b(all)h(\()p Fs(t;)15 b(z)t Ft(\))27 b Fp(2)f(T)43 b(\002)20 b Fs(D)s Ft(\(0)p Fs(;)15 b(\032)p Ft(\).)44 b(F)-8 b(rom)31 b(the)g(fact)h(that)f Fs(b)2369 414 y Fq(k)2412 399 y Ft(\()p Fs(\017;)15 b(z)t Ft(\))32 b(is)f(analytic)h(with)f(resp)s(ect)g(to)g Fs(\017)g Ft(at)75 511 y(0,)g(w)m(e)g(ha)m(v)m(e)g(that)75 769 y(\(75\))938 b Fs(\017)1210 732 y Fq(m)1272 744 y Fl(0)p Fk(;k)1364 769 y Fs(b)1403 784 y Fq(k)1445 769 y Ft(\()p Fs(\017;)15 b(z)t Ft(\))27 b(=)1761 683 y Fo(X)1770 881 y Fq(l)q Fn(\025)p Fr(0)1917 708 y Fs(@)1970 675 y Fq(l)1965 730 y(\017)1998 708 y Ft(\()p Fs(\017)2070 675 y Fq(m)2132 687 y Fl(0)p Fk(;k)2225 708 y Fs(b)2264 723 y Fq(k)2306 708 y Ft(\()p Fs(\017;)15 b(z)t Ft(\)\))p Fp(j)2559 722 y Fq(\017)p Fr(=0)p 1917 748 767 4 v 2274 832 a Fs(l)r Ft(!)2694 769 y Fs(\017)2731 732 y Fq(l)75 1069 y Ft(for)28 b(all)h Fs(\017)f Ft(on)g(a)h(neigh)m(b)s(orho)s(o)s(d)d(of)j(0,)g(all) g Fs(z)h Fp(2)25 b Fm(C)p Ft(.)39 b(Finally)-8 b(,)30 b(one)f(c)m(hec)m(ks)h(from)d(the)i(recursion)e(\(74\))j(and)e(\(75\),) 75 1182 y(that)j(the)g(formal)f(series)984 1159 y(^)958 1182 y Fs(X)7 b Ft(\()p Fs(\017;)15 b(t;)g(z)t Ft(\))27 b(=)1429 1113 y Fo(P)1525 1208 y Fq(k)r Fn(\025)p Fr(0)1673 1182 y Fs(X)1748 1197 y Fq(k)1791 1182 y Ft(\()p Fs(t;)15 b(z)t Ft(\))p Fs(\017)2017 1149 y Fq(k)2061 1182 y Fs(=k)s Ft(!)31 b(satis\014es)g(the)f(equation)i(\(51\).)496 b Ff(2)75 1518 y Fu(References)120 1721 y Ft([1])47 b(M.)d (Canalis-Durand,)i(J.)d(Mozo-F)-8 b(ernandez,)49 b(R.)43 b(Sc)m(h\177)-45 b(afk)m(e,)48 b Fh(Monomial)e(summability)f(and)h (doubly)262 1834 y(singular)33 b(di\013er)-5 b(ential)34 b(e)-5 b(quations.)31 b Ft(J.)g(Di\013eren)m(tial)h(Equations)f(233)g (\(2007\),)j(no.)c(2,)h(485{511.)120 2021 y([2])47 b(L.)29 b(Di)g(Vizio,)i(J.-P)-8 b(.)29 b(Ramis,)h(J.)e(Saulo)m(y)-8 b(,)31 b(C.)d(Zhang,)2113 1998 y Fh(\023)2100 2021 y(Equations)k(aux)f Fs(q)s Fh(-di\013)n(\023)-44 b(er)-5 b(enc)g(es)p Ft(.)29 b(Gaz.)h(Math.)g(No.)262 2134 y(96)h(\(2003\),)i(20{49.)120 2322 y([3])47 b(L.)32 b(Di)h(Vizio,)i(C.)d(Zhang,)h Fh(On)h Fs(q)s Fh(-summation)i(and)f(c)-5 b(on\015uenc)g(e.)33 b Ft(Ann.)f(Inst.)g(F)-8 b(ourier)33 b(\(Grenoble\))h(59)262 2435 y(\(2009\),)f(no.)d(1,)h(347{392.)120 2622 y([4])47 b(L.)42 b(H\177)-45 b(ormander,)44 b Fh(A)n(n)e(intr)-5 b(o)g(duction)46 b(to)d(c)-5 b(omplex)45 b(analysis)g(in)e(sever)-5 b(al)44 b(variables.)e Ft(Second)g(revised)262 2735 y(edition.)31 b(North-Holland)g(Mathematical)j(Library)-8 b(,)30 b(V)-8 b(ol.)32 b(7,)f(1973.)120 2923 y([5])47 b(S.)35 b(Malek,)k Fh(On)e(c)-5 b(omplex)39 b(singularity)f(analysis)h(for)f(line)-5 b(ar)39 b Fs(q)s Fh(-di\013er)-5 b(enc)g(e-di\013er)g(ential)38 b(e)-5 b(quations.)37 b Ft(J.)262 3036 y(Dyn.)30 b(Con)m(trol)h(Syst.)g (15)g(\(2009\),)i(no.)d(1,)h(83{98.)120 3223 y([6])47 b(S.)23 b(Malek,)j Fh(On)g(functional)h(line)-5 b(ar)27 b(p)-5 b(artial)29 b(di\013er)-5 b(ential)27 b(e)-5 b(quations)27 b(in)f(Gevr)-5 b(ey)27 b(sp)-5 b(ac)g(es)28 b(of)e(holomorphic)262 3336 y(functions.)k Ft(Ann.)g(F)-8 b(ac.)32 b(Sci.)f(T)-8 b(oulouse)31 b(Math.)g(\(6\))h(16)f(\(2007\),)i(no.)d(2,)h(285{302.)120 3524 y([7])47 b(S.)31 b(Malek,)i Fh(Singularly)i(p)-5 b(erturb)g(e)g(d)35 b(p)-5 b(artial)36 b(inte)-5 b(gr)g(o-di\013er)g (ential)35 b(e)-5 b(quations)35 b(with)f(irr)-5 b(e)g(gular)35 b(singular-)262 3637 y(ity,)30 b Ft(J.)h(Dynam.)f(Con)m(trol.)i(Syst)e (13)h(\(2007\),)i(No.)e(3.)120 3825 y([8])47 b(B.)39 b(Malgrange,)j Fh(Ide)-5 b(als)42 b(of)e(di\013er)-5 b(entiable)41 b(functions.)d Ft(T)-8 b(ata)40 b(Institute)f(of)f(F)-8 b(undamen)m(tal)39 b(Researc)m(h)262 3937 y(Studies)34 b(in)g(Mathematics,)j(No.)f(3)e(T)-8 b(ata)36 b(Institute)f(of)f(F)-8 b(undamen)m(tal)36 b(Researc)m(h,)g(Bom)m(ba)m(y;)i(Oxford)262 4050 y(Univ)m(ersit)m(y)31 b(Press,)g(London)e(1967.)120 4238 y([9])47 b(V.)22 b(P)m(o)m(\023)-43 b(enaru,)24 b Fh(A)n(nalyse)h(di\013)n(\023)-44 b(er)-5 b(entiel)5 b(le.)21 b Ft(Lecture)h(Notes)g(in)f(Mathematics,)26 b(V)-8 b(ol.)22 b(371.)h(Springer-V)-8 b(erlag,)262 4351 y(Berlin-New)31 b(Y)-8 b(ork,)31 b(1974.)75 4539 y([10])47 b(J-P)-8 b(.)45 b(Ramis,)k(J.)c(Saulo)m(y)-8 b(,)50 b(C.)44 b(Zhang,)k Fh(L)-5 b(o)g(c)g(al)48 b(analytic)f(classi\014c)-5 b(ation)48 b(of)e Fs(q)s Fh(-di\013er)-5 b(enc)g(e)46 b(e)-5 b(quations)p Ft(,)262 4651 y(preprin)m(t)29 b(ArXiv,)i(2009.)75 4839 y([11])47 b(J-P)-8 b(.)34 b(Ramis,)i(C.)e(Zhang,)h Fh(D)n(\023)-44 b(evelopp)-5 b(ement)37 b(asymptotique)h Fs(q)s Fh(-Gevr)-5 b(ey)36 b(et)g(fonction)g(th)n(^)-44 b(eta)38 b(de)e(Jac)-5 b(obi)p Ft(.)262 4952 y(C.)30 b(R.)h(Math.)g(Acad.)g(Sci.)g(P)m(aris)f(335)i(\(2002\),)h(no.)e(11,)g (899{902.)75 5140 y([12])47 b(C.)28 b(Zhang,)h Fh(Une)h(sommation)k (discr)n(\022)-44 b(ete)31 b(p)-5 b(our)32 b(des)d(\023)-44 b(equations)32 b(aux)f Fs(q)s Fh(-di\013)n(\023)-44 b(er)-5 b(enc)g(es)31 b(lin)n(\023)-44 b(eair)-5 b(es)32 b(et)f(\022)-46 b(a)31 b(c)-5 b(o)g(ef-)262 5253 y(\014cients)34 b(analytiques:)47 b(th)n(\023)-44 b(eorie)35 b(g)n(\023)-44 b(en)n(\023)g(er)-5 b(ale)35 b(et)g(exemples.)e Ft(\(F)-8 b(renc)m(h\))33 b([Discrete)i(summation)d(for)g(linear)262 5365 y Fs(q)s Ft(-di\013erence)h(equations)g(with)f(analytic)i(co)s(e\016cien)m(ts:) 46 b(general)34 b(theory)e(and)g(examples])h(Di\013eren)m(tial)262 5478 y(equations)e(and)f(the)g(Stok)m(es)i(phenomenon,)d(309{329,)34 b(W)-8 b(orld)31 b(Sci.)g(Publ.,)f(Riv)m(er)g(Edge,)h(NJ,)g(2002.)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------1010050442937--