Content-Type: multipart/mixed; boundary="-------------0102261111605" This is a multi-part message in MIME format. ---------------0102261111605 Content-Type: text/plain; name="01-77.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-77.comments" 1991 MSC: 30D15, 35K55, 58F39 ---------------0102261111605 Content-Type: text/plain; name="01-77.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-77.keywords" Holomorphic Operators; Fixed Points; Stability; Convergence; Cauchy Problem. ---------------0102261111605 Content-Type: application/postscript; name="PAPER.PS" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="PAPER.PS" %!PS-Adobe-2.0 %%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: c:\user\tk\paper.dvi %%CreationDate: Sun Feb 25 11:16:04 2001 %%Pages: 34 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: C:\TEX\EMTEX\BIN\dvips32.exe 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48496F13FCA34819FEA4B518FFAD6C19FEA46C6D4B13FCA36C19F8A26C6D4B13F0A26C19 E06C6D4B13C0A26C6D4B13806C6D4B13006D6C4B5A6D6D495B6D6D495B010701F0010F13 E06D01FE017F5B010090B7C7FC023F15FC020715E0020092C8FC030713E048437CC151> I114 D<913A3FFF8007800107B5EAF81F011FEC FE7F017F91B5FC48B8FC48EBE0014890C7121FD80FFC1407D81FF0801600485A007F167F 49153FA212FF171FA27F7F7F6D92C7FC13FF14E014FF6C14F8EDFFC06C15FC16FF6C16C0 6C16F06C826C826C826C82013F1680010F16C01303D9007F15E0020315F0EC001F150004 1F13F81607007C150100FC81177F6C163FA2171F7EA26D16F0A27F173F6D16E06D157F6D 16C001FEEDFF806D0203130002C0EB0FFE02FCEB7FFC01DFB65A010F5DD8FE0315C026F8 007F49C7FC48010F13E035437BC140>II<902607FFC0ED3FFEB60207B5FCA6C6EE00076D826D82B3B3 A260A360A2607F60183E6D6D147E4E7F6D6D4948806D6DD907F0ECFF806D01FFEB3FE06D 91B55A6E1500021F5C020314F8DA003F018002F0C7FC51427BC05A>I 121 D E end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop 268 315 a Fx(A)54 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y(eac)m(h)f Fn(I)417 3060 y Fo(n)490 3046 y Fq(the)f(ev)m(olution)g(is)f(to)h(b)s (e)f(describ)s(ed)f(b)m(y)i(the)g(ab)s(o)m(v)m(e)h(equation)f(but)f(at) h(the)g(endp)s(oin)m(ts)166 3170 y(the)37 b(function)f Fp(f)10 b Fq(\()p Fp(t;)15 b(z)t Fq(\))38 b(is)e(c)m(hanged:)j Fp(f)10 b Fq(\()p Fp(n\034)5 b(;)15 b(z)t Fq(\))36 b Fn(!)1931 3097 y Fm(\002)1969 3170 y Fp(f)2038 3097 y Fm(\000)2080 3170 y Fp(n\034)5 b(;)15 b(z)t(\016)2309 3138 y Fl(\000)p Ft(1)p Fl(\000)p Fo(\025)2501 3097 y Fm(\001)o(\003)2580 3118 y Fo(\016)2618 3170 y Fq(,)37 b(with)f Fp(\025)h(>)f Fq(0)i(and)f(an)166 3283 y(in)m(teger)j Fp(\016)45 b Fn(\025)c Fq(2.)g(The)e(resolving)g(op)s(erator)h(of)g (suc)m(h)g(problem)e(preserv)m(es)i(the)g(set)h Fn(L)p Fq(.)f(It)g(is)166 3396 y(sho)m(wn)34 b(that)i(for)e Fp(t)f Fn(!)g Fq(+)p Fn(1)p Fq(,)i(the)g(asymptotic)g(prop)s(erties)e (of)i Fp(f)10 b Fq(\()p Fp(t;)15 b(z)t Fq(\))36 b(c)m(hange)g (considerably)166 3509 y(when)31 b(the)i(parameter)g Fp(\034)42 b Fq(reac)m(hes)34 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3315 y Fg(\()p Ff(g)t Fg(\)])2384 3286 y Ft(2)2460 3247 y Fe(\024)2605 3179 y Ff(\016)2652 3143 y Ft(1)p Fo(=)p Ft(2)p 2575 3224 218 4 v 2575 3315 a Ff(\022)j Fg(+)d(1)2802 3247 y Ff(:)451 b Fg(\(18\))166 3608 y(Thereb)m(y)-8 b(,)35 b(w)m(e)e(state)g(our)g(main)e(theorem.)166 3828 y Fi(Theorem)37 b(1.2)49 b Fh(F)-7 b(or)34 b(every)h Ff(\022)c Fe(\025)e Fg(0)34 b Fh(and)h Ff(g)c Fe(2)d(L)p Fg(\()p Ff(\025)p Fg(\))p Fh(,)35 b(ther)-5 b(e)35 b(exist)f(a)h(p)-5 b(ositive)35 b Ff(\034)3115 3843 y Fl(\003)3182 3828 y Fe(2)29 b Ff(I)8 b Fg(\()p Ff(g)t Fg(\))166 3948 y Fh(and)34 b(a)h(function)g Ff(C)f Fg(:)28 b([0)p Ff(;)17 b(\034)1145 3963 y Fl(\003)1184 3948 y Fg(])28 b Fe(!)f Ff(I)-25 b(R)1459 3963 y Ft(+)1553 3948 y Fh(such)35 b(that)216 4231 y Fg(\(i\))126 b Fh(for)34 b Ff(\034)39 b(<)28 b(\034)828 4246 y Fl(\003)868 4231 y Fh(,)34 b(the)h(se)-5 b(quenc)g(e)34 b(of)h(solutions)f(of)h(\(13\)) 446 4412 y Fe(f)p Ff(f)544 4427 y Fo(n)590 4412 y Fg(\()p Ff(t;)17 b(z)t Fg(\))64 b Fe(j)e 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Ff(\034)2991 4812 y Fl(\003)3030 4797 y Fg(\))p Ff(g)t Fg(\()p Ff(z)t Fg(\))p Fe(g)446 4977 y Fh(c)-5 b(onver)g(ges)33 b(in)i Fe(A)1080 5005 y Fo(\014)1123 4975 y Fa(\000)p Fc(1)1120 5018 y Fa(\003)1209 4977 y Fh(,)g(to)456 5280 y Ff(f)504 5295 y Fl(\003)544 5280 y Fg(\()p Ff(t;)17 b(z)t Fg(\))28 b(=)g Ff(\016)927 5239 y Fl(\000)p Fo(\016)r(\022)r(\025=)p Ft(\()p Fo(\016)r Fl(\000)p Ft(1\))1309 5280 y Fg([1)22 b Fe(\000)h Ff(t)p Fg(\(1)f Fe(\000)h Ff(\016)1798 5239 y Fl(\000)p Fo(\025)1898 5280 y Fg(\)])1963 5239 y Fl(\000)p Fo(\022)2074 5280 y Fg(exp)2239 5134 y Fd( )2331 5213 y Fg(1)p 2315 5257 82 4 v 2315 5349 a Ff(\034)2357 5364 y Fl(\003)2557 5213 y Fg(1)f Fe(\000)h Ff(\016)2775 5177 y Fl(\000)p Fo(\025)p 2417 5257 600 4 v 2417 5349 a Fg(1)e Fe(\000)i Ff(t)p Fg(\(1)f Fe(\000)h Ff(\016)2878 5320 y Fl(\000)p Fo(\025)2978 5349 y Fg(\))3026 5280 y Ff(z)3075 5134 y Fd(!)3158 5280 y Ff(:)95 b Fg(\(19\))1769 5712 y(6)p eop %%Page: 7 7 7 6 bop 166 83 a Fh(1.3)99 b(Some)34 b(Applic)-5 b(ations)34 b(and)h(F)-7 b(urther)34 b(R)-5 b(esults)166 433 y Fg(Let)23 b Fe(E)393 397 y Ft(\()p Fo(N)7 b Ft(\))514 433 y Fg(,)23 b Ff(N)38 b Fe(2)28 b Ff(I)-25 b(N)32 b Fg(b)s(e)23 b(the)g(set)g(of)f (analytic)f(functions)h Ff(F)42 b Fg(:)27 b Ff(I)-25 b(R)2449 391 y Fo(N)2544 433 y Fe(!)p 2694 433 5 50 v 26 w Ff(C)7 b Fg(.)22 b(F)-8 b(or)22 b(appropriate)166 553 y Ff(F)41 b Fe(2)29 b(E)427 517 y Ft(\()p Fo(N)7 b Ft(\))581 553 y Fg(and)32 b(some)h Ff(b)28 b(>)f Fg(0,)33 b(w)m(e)g(set)935 807 y Fe(k)p Ff(F)14 b Fe(k)1112 822 y Fo(b;N)1288 754 y Ft(def)1299 807 y Fg(=)103 b(sup)1446 899 y Fo(x)p Fl(2)p Fo(I)-18 b(R)1599 874 y Fb(N)1656 807 y Fe(fj)27 b Ff(F)14 b Fg(\()p Ff(x)p Fg(\))28 b Fe(j)g Fg(exp)q(\()p Fe(\000)p Ff(b)g Fe(j)f Ff(x)i Fe(j)2553 766 y Ft(2)2592 807 y Fg(\))p Fe(g)p Ff(;)573 b Fg(\(20\))166 1235 y(where)34 b Fe(j)27 b Ff(x)h Fe(j)33 b Fg(is)f(the)h(Euclidean)f (norm)f(of)i Ff(x)28 b Fe(2)g Ff(I)-25 b(R)1999 1193 y Fo(N)2067 1235 y Fg(.)32 b(Let)688 1494 y Fe(A)768 1453 y Ft(\()p Fo(N)7 b Ft(\))768 1518 y Fo(a)950 1441 y Ft(def)961 1494 y Fg(=)71 b Fe(f)p Ff(F)41 b Fe(2)28 b(E)1418 1453 y Ft(\()p Fo(N)7 b Ft(\))1600 1494 y Fe(j)60 b(k)p Ff(F)14 b Fe(k)1865 1509 y Fo(b;N)2009 1494 y Ff(<)27 b Fe(1)p Ff(;)17 b Fe(8)p Ff(b)28 b(>)g(a)p Fe(g)p Ff(;)81 b(a)28 b Fe(\025)g Fg(0)p Ff(:)327 b Fg(\(21\))166 1842 y(This)36 b(set)h(equipp)s(ed)g(with)f(the)g(top)s(ology)f(generated)i (b)m(y)g(the)f(family)e Fe(fk)p Ff(:)p Fe(k)3008 1857 y Fo(b;N)3124 1842 y Ff(;)17 b(b)34 b(>)g(a)p Fe(g)166 1963 y Fg(b)s(ecomes)48 b(a)f(F)-8 b(r)m(\023)-46 b(ec)m(het)48 b(space.)g(Let)f Ff(O)s Fg(\()p Ff(N)10 b Fg(\))47 b(stand)h(for)e(the) i(group)f(of)f(all)f(orthogonal)166 2083 y(transformations)25 b(of)h Ff(I)-25 b(R)1061 2041 y Fo(N)1128 2083 y Fg(.)26 b(A)h(function)f Ff(F)41 b Fe(2)28 b(E)1917 2047 y Ft(\()p Fo(N)7 b Ft(\))2065 2083 y Fg(is)26 b(said)g(to)g(b)s(e)h(isotropic)e (if)h Ff(F)14 b Fg(\()p Ff(U)c(x)p Fg(\))28 b(=)166 2203 y Ff(F)14 b Fg(\()p Ff(x)p 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b(b)m(y)1286 2950 y Fe(P)1363 2899 y Ft(\()p Fo(N)7 b Ft(\))1355 2973 y(isot)1514 2950 y Fe(3)28 b Ff(P)14 b Fg(\()p Ff(x)p Fg(\))27 b(=)h Ff(p)p Fg(\(\()p Ff(x;)17 b(x)p Fg(\)\))28 b Fe(2)g(P)8 b Ff(;)166 3298 y Fg(where)34 b(\()p Ff(:;)17 b(:)p Fg(\))32 b(is)g(the)h(scalar)f(pro)s(duct)h(in)f Ff(I)-25 b(R)1770 3256 y Fo(N)1837 3298 y Fg(.)32 b(Ob)m(viously)h(eac) m(h)g Ff(F)47 b Fg(ha)m(ving)32 b(the)h(form)1455 3537 y Ff(F)14 b Fg(\()p Ff(x)p Fg(\))28 b(=)f Ff(f)11 b Fg(\(\()p Ff(x;)17 b(x)p Fg(\)\))p Ff(;)1094 b Fg(\(22\))166 3899 y(with)33 b(certain)f Ff(f)40 b Fe(2)29 b(E)9 b Fg(,)32 b(b)s(elongs)h(to)f Fe(E)1550 3848 y Ft(\()p Fo(N)7 b Ft(\))1541 3923 y(isot)1672 3899 y Fg(.)33 b(Giv)m(en)g(a)g(subset)i Fe(X)43 b(\032)29 b(E)9 b Fg(,)32 b(w)m(e)i(write)f Fe(X)15 b Fg(\()p Ff(I)-25 b(R)3349 3857 y Fo(N)3416 3899 y Fg(\))166 4033 y(for)31 b(the)h(subset)h(of)e Fe(E)951 3982 y Ft(\()p Fo(N)7 b Ft(\))942 4057 y(isot)1105 4033 y Fg(consisting)31 b(of)g(the)h(functions)f(ob)s(eying)g(\(22\))g(with)h Ff(f)38 b Fe(2)28 b(X)15 b Fg(.)31 b(In)166 4167 y(this)h(notation)f Fe(P)823 4116 y Ft(\()p Fo(N)7 b Ft(\))815 4191 y(isot)974 4167 y Fg(=)27 b Fe(P)8 b Fg(\()p Ff(I)-25 b(R)1285 4125 y Fo(N)1353 4167 y Fg(\).)32 b(Consider)h(a)g(map)1005 4469 y Fe(E)1067 4418 y Ft(\()p Fo(N)7 b Ft(\))1058 4492 y(isot)1217 4469 y Fe(3)28 b Ff(F)41 b Fe(7!)1542 4322 y Fd( )1608 4469 y Fg(\001)23 b(+)1909 4401 y Ff(d)p 1820 4445 231 4 v 1820 4537 a Fg(\()p Ff(x;)17 b(x)p Fg(\))2060 4469 y(\()p Ff(x;)g Fe(r)p Fg(\))2318 4322 y Fd(!)2400 4469 y Ff(F)42 b Fe(2)28 b(E)2661 4418 y Ft(\()p Fo(N)7 b Ft(\))2652 4492 y(isot)2782 4469 y Ff(;)166 4892 y Fg(where)41 b(\001)g(and)f Fe(r)g Fg(stand)g(for)g(the)g (Laplacian)f(and)h(for)f(the)i(gradien)m(t)e(in)h Ff(I)-25 b(R)3088 4850 y Fo(N)3156 4892 y Fg(.)40 b(F)-8 b(or)39 b(a)166 5012 y(pair)32 b(of)g(functions)g Ff(F)14 b Fg(,)33 b Ff(f)43 b Fg(satisfying)31 b(\(22\),)h(one)h(has)g(\(c.f.)g(\(8\)\)) 887 5167 y Fd( )953 5313 y Fg(\001)22 b(+)1254 5246 y Ff(d)p 1164 5290 V 1164 5382 a Fg(\()p Ff(x;)17 b(x)p Fg(\))1405 5313 y(\()p Ff(x;)g Fe(r)p Fg(\))1663 5167 y Fd(!)1745 5313 y Ff(F)d Fg(\()p Ff(x)p Fg(\))28 b(=)f(4)17 b(\()o(\001)2268 5328 y Fo(\022)2308 5313 y Ff(f)11 b Fg(\))16 b(\(\()p Ff(x;)h(x)p Fg(\)\))p Ff(;)526 b Fg(\(23\))1769 5712 y(7)p eop %%Page: 8 8 8 7 bop 166 83 a Fg(where)34 b(\001)529 98 y Fo(\022)601 83 y Fg(is)e(de\014ned)i(b)m(y)f(\(7\))f(with)1578 374 y Ff(\022)f Fg(=)1767 307 y Ff(N)i Fg(+)22 b Ff(d)p 1767 351 260 4 v 1872 443 a Fg(2)2037 374 y Ff(:)1216 b Fg(\(24\))166 758 y(No)m(w)33 b(let)f(us)h(consider)g(the)g(follo)m(wing)d(Cauc)m(h)m (y)35 b(problem)c({)h(an)h(analog)e(of)h(\(13\))g({)371 1113 y Ff(@)5 b(F)490 1128 y Fo(n)538 1113 y Fg(\()p Ff(t;)17 b(x)p Fg(\))p 371 1157 378 4 v 514 1249 a Ff(@)5 b(t)775 1180 y Fg(=)16 b Ff(\034)938 1034 y Fd( )1003 1180 y Fg(\001)23 b(+)1305 1113 y Ff(d)p 1215 1157 231 4 v 1215 1249 a Fg(\()p Ff(x;)17 b(x)p Fg(\))1455 1180 y(\()p Ff(x;)g Fe(r)p Fg(\))1713 1034 y Fd(!)1795 1180 y Ff(F)1858 1195 y Fo(n)1906 1180 y Fg(\()p Ff(t;)g(x)p Fg(\))p Ff(;)49 b(t)28 b Fe(2)g Fg([0)p Ff(;)17 b Fg(1])p Ff(;)49 b(x)28 b Fe(2)g Ff(I)-25 b(R)2891 1139 y Fo(N)2958 1180 y Ff(;)424 1433 y(F)487 1448 y Fo(n)535 1433 y Fg(\(0)p Ff(;)17 b(x)p Fg(\))f(=)867 1336 y Fd(h)907 1433 y Ff(F)970 1448 y Fo(n)p Fl(\000)p Ft(1)1107 1433 y Fg(\(1)p Ff(;)h(x\016)1340 1392 y Fl(\000)p Ft(\(1+)p Fo(\025)p Ft(\))p Fo(=)p Ft(2)1656 1433 y Fg(\))1694 1336 y Fd(i)1733 1359 y Fo(\016)1787 1433 y Ff(;)49 b(n)28 b Fe(2)g Ff(I)-25 b(N)11 b(;)1103 b Fg(\(25\))432 1603 y Ff(F)495 1618 y Ft(0)535 1603 y Fg(\(1)p Ff(;)17 b(x)p Fg(\))f(=)g Ff(G)p Fg(\()p Ff(x)p Fg(\))p Ff(;)115 b(G)28 b Fe(2)g(L)1485 1562 y Ft(+)1543 1603 y Fg(\()p Ff(I)-25 b(R)1674 1561 y Fo(N)1741 1603 y Fg(\))p Ff(:)166 1845 y Fg(F)-8 b(or)27 b Ff(G)h Fe(2)g(L)604 1809 y Ft(+)663 1845 y Fg(\()p Ff(I)-25 b(R)793 1803 y Fo(N)860 1845 y Fg(\),)28 b(there)h(exists)g Ff(g)i Fe(2)d(L)1704 1809 y Ft(+)1790 1845 y Fg(suc)m(h)i(that)d Ff(G)h Fg(and)g Ff(g)j Fg(satisfy)d(\(22\),)f(th)m(us)i(the)166 1965 y(in)m(terv)-5 b(al)31 b(\(15\))h(is)g(de\014ned)i(for)e(suc)m(h)i Ff(G)p Fg(.)f(The)g(direct)g(corollary)e(of)h(Theorem)h(1.1)f(reads)166 2196 y Fi(Theorem)37 b(1.3)49 b Fh(F)-7 b(or)37 b(every)g Ff(d)32 b Fe(\025)h(\000)p Ff(N)10 b Fh(,)38 b Ff(G)32 b Fe(2)h(L)1981 2160 y Ft(+)2040 2196 y Fg(\()p Ff(I)-25 b(R)2171 2154 y Fo(N)2238 2196 y Fg(\))p Fh(,)37 b Ff(\034)44 b Fe(2)33 b Ff(I)8 b Fg(\()p Ff(g)t Fg(\))p Fh(,)36 b(and)h Ff(n)32 b Fe(2)h Ff(I)-25 b(N)10 b Fh(,)38 b(the)166 2316 y(pr)-5 b(oblem)34 b(\(25\))g(has)g(a)h(unique)g(solution)g Ff(F)1750 2331 y Fo(n)1797 2316 y Fh(,)f(which)g(also)h(b)-5 b(elongs)33 b(to)j Fe(L)2855 2280 y Ft(+)2913 2316 y Fg(\()p Ff(I)-25 b(R)3044 2275 y Fo(N)3111 2316 y Fg(\))p Fh(.)166 2547 y Fg(F)-8 b(or)32 b Ff(\025)27 b Fe(2)h Fg(\(0)p Ff(;)17 b Fg(1)p Ff(=)p Fg(2\),)32 b(one)h(has)g(the)g(follo)m (wing)c(analog)i(of)h(Theorem)h(1.2.)166 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Ff(F)424 4287 y Fl(\003)464 4272 y Fg(\()p Ff(t;)17 b(x)p Fg(\))28 b(=)f Ff(\016)852 4231 y Fl(\000)p Fo(\016)r(\022)r(\025=)p Ft(\()p Fo(\016)r Fl(\000)p Ft(1\))1235 4272 y Fg([1)22 b Fe(\000)h Ff(t)p Fg(\(1)f Fe(\000)g Ff(\016)1723 4231 y Fl(\000)p Fo(\025)1823 4272 y Fg(\)])1888 4231 y Fl(\000)p Fo(\022)1999 4272 y Fg(exp)2165 4126 y Fd( )2257 4205 y Fg(1)p 2240 4249 82 4 v 2240 4340 a Ff(\034)2282 4355 y Fl(\003)2483 4205 y Fg(1)g Fe(\000)g Ff(\016)2700 4168 y Fl(\000)p Fo(\025)p 2342 4249 600 4 v 2342 4340 a Fg(1)g Fe(\000)h Ff(t)p Fg(\(1)f Fe(\000)g Ff(\016)2803 4312 y Fl(\000)p Fo(\025)2903 4340 y Fg(\))2951 4272 y(\()p Ff(x;)17 b(x)p Fg(\))3181 4126 y Fd(!)3264 4272 y Ff(;)p Fg(\(26\))266 4738 y Fh(wher)-5 b(e)34 b Ff(\022)k Fh(is)d(given)f(by)h(\(24\).)166 4998 y Fg(Let)e Fe(M)f Fg(stand)h(for)f(the)h(set)g(of)f(probabilit)m(y)f (measures)i Ff(\026)g Fg(on)f Ff(I)-25 b(R)2576 4957 y Fo(N)2676 4998 y Fg(suc)m(h)34 b(that)1358 5147 y Fd(Z)1319 5405 y Fo(I)-18 b(R)1385 5380 y Fb(N)1478 5265 y Fg(exp)q(\()p Ff(")p Fg(\()p Ff(x;)17 b(x)p Fg(\)\))p Ff(\026)p Fg(\()p Ff(dx)p Fg(\))27 b Ff(<)h Fe(1)p Ff(;)1769 5712 y Fg(8)p eop %%Page: 9 9 9 8 bop 166 83 a Fg(with)32 b(a)h(certain)f Ff(")27 b(>)h Fg(0.)k(F)-8 b(or)32 b Ff(\026)27 b Fe(2)h(M)p Fg(,)k(the)h(function) 1164 325 y Ff(F)1227 340 y Fo(\026)1274 325 y Fg(\()p Ff(x)p Fg(\))1465 272 y Ft(def)1476 325 y Fg(=)1643 208 y Fd(Z)1604 465 y Fo(I)-18 b(R)1670 440 y Fb(N)1763 325 y Fg(exp)q(\(\()p Ff(x;)17 b(y)t Fg(\)\))p Ff(\026)p Fg(\()p Ff(dy)t Fg(\))p Ff(;)800 b Fg(\(27\))166 765 y(b)s(elongs)32 b(to)g Fe(E)697 729 y Ft(\()p Fo(N)7 b Ft(\))819 765 y Fg(.)33 b(F)-8 b(or)31 b(a)i(Borel)e(subset)k Ff(B)d Fe(\032)d Ff(I)-25 b(R)1997 724 y Fo(N)2064 765 y Fg(,)33 b(w)m(e)h(set)589 1085 y Ff(B)28 b Fe(\000)22 b Ff(x)28 b Fg(=)g Fe(f)p Ff(y)j Fe(2)d Ff(I)-25 b(R)1292 1043 y Fo(N)1420 1085 y Fe(j)27 b Ff(x)c Fg(+)f Ff(y)31 b Fe(2)d Ff(B)5 b Fe(g)p Ff(;)394 1230 y(U)10 b(B)34 b Fg(=)27 b Fe(f)p Ff(x)h Fe(2)g Ff(I)-25 b(R)1001 1189 y Fo(N)1129 1230 y Fe(j)59 b Ff(U)1292 1189 y Fl(\000)p Ft(1)1387 1230 y Ff(x)28 b Fe(2)h Ff(B)5 b Fe(g)p Ff(;)114 b(U)38 b Fe(2)28 b Ff(O)s Fg(\()p Ff(N)10 b Fg(\))p Ff(:)166 1448 y Fg(A)29 b(measure)g Ff(\026)e Fe(2)h(M)g Fg(is)g(said)h(to)f(b)s (e)h(isotropic)e(if)g(it)h(is)g Ff(O)s Fg(\()p Ff(N)10 b Fg(\){in)m(v)-5 b(arian)m(t)27 b(\(i.e.)h Ff(\026)p Fg(\()p Ff(U)10 b(B)5 b Fg(\))28 b(=)166 1568 y Ff(\026)p Fg(\()p Ff(B)5 b Fg(\)\).)42 b(The)h(subset)g Fe(M)1125 1583 y Ft(isot)1283 1568 y Fe(\032)h(M)e Fg(is)g(to)f(consist)i(of)e (the)i(isotropic)d(measures.)j(Ob-)166 1702 y(viously)-8 b(,)42 b Ff(F)589 1717 y Fo(\026)679 1702 y Fe(2)j(E)852 1651 y Ft(\()p Fo(N)7 b Ft(\))843 1726 y(isot)1016 1702 y Fg(for)41 b Ff(\026)j Fe(2)g(M)1507 1717 y Ft(isot)1621 1702 y Fg(.)f(No)m(w)f(let)g Fe(M)p Fg(\()p Ff(I)-25 b(R)2323 1660 y Fo(N)2390 1702 y Fg(\))42 b(b)s(e)h(the)f(subset)i(of)e Fe(M)3340 1717 y Ft(isot)166 1822 y Fg(consisting)37 b(of)f(the)i(measures)g(for)f(whic)m(h)h 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1549 y Fe(M)1041 1564 y Fo(\025)1086 1549 y Fg(\()p Ff(I)-25 b(R)1217 1507 y Fo(N)1284 1549 y Fg(\))1382 1496 y Ft(def)1393 1549 y Fg(=)71 b Fe(f)p Ff(\027)34 b Fe(2)28 b(M)p Fg(\()p Ff(I)-25 b(R)2016 1507 y Fo(N)2084 1549 y Fg(\))60 b Fe(j)g Ff(g)31 b Fe(2)d(L)p Fg(\()p Ff(\025)p Fg(\))p Fe(g)p Ff(:)559 b Fg(\(31\))166 1902 y(The)34 b(follo)m(wing)29 b(assertion)k(is)f(a)g(corollary)f(of)h(Theorem)h(1.4)f(for)g Ff(d)c Fg(=)f(0.)166 2133 y Fi(Theorem)37 b(1.5)49 b Fh(F)-7 b(or)34 b(every)h Ff(N)k Fe(2)29 b Ff(I)-25 b(N)45 b Fh(and)35 b Ff(\027)f Fe(2)29 b(M)2113 2148 y Fo(\025)2158 2133 y Fg(\()p Ff(I)-25 b(R)2289 2091 y Fo(N)2356 2133 y Fg(\))p Fh(,)35 b(ther)-5 b(e)35 b(exists)g Ff(\034)3011 2148 y Fl(\003)3079 2133 y Ff(>)28 b Fg(0)35 b Fh(such)166 2254 y(that)216 2523 y Fg(\(i\))126 b Fh(for)34 b Ff(\034)39 b(<)28 b(\034)828 2538 y Fl(\003)868 2523 y Fh(,)34 b(the)h(se)-5 b(quenc)g(e)34 b(of)h(me)-5 b(asur)g(es)34 b(de\014ne)-5 b(d)34 b(by)h(\(29\))446 2703 y Fe(f)p Ff(\026)555 2718 y Fo(n)664 2703 y Fe(j)62 b Ff(n)28 b Fe(2)g Ff(I)-25 b(N)1040 2718 y Ft(0)1079 2703 y Ff(;)52 b(\026)1217 2718 y Ft(0)1284 2703 y Fg(=)27 b Ff(\027)6 b Fe(g)35 b Fh(c)-5 b(onver)g(ges)34 b(we)-5 b(akly)34 b(to)i(the)e(me)-5 b(asur)g(e)446 2884 y(de)g(gener)g(ate)34 b(at)h(zer)-5 b(o;)216 3088 y Fg(\(ii\))98 b Fh(for)34 b Ff(\034)39 b Fg(=)28 b Ff(\034)828 3103 y Fl(\003)868 3088 y Fh(,)34 b(this)h(se)-5 b(quenc)g(e)34 b(c)-5 b(onver)g(ges)34 b(we)-5 b(akly)34 b(to)h(the)g(isotr)-5 b(opic)446 3269 y(Gaussian)34 b(me)-5 b(asur)g(e)34 b(with)h(varianc)-5 b(e)34 b Fg(2)p Ff(N)10 b Fg(\()p Ff(\016)2073 3232 y Fo(\025)2141 3269 y Fe(\000)22 b Fg(1\))p Ff(=\034)2418 3284 y Fl(\003)2458 3269 y Fh(.)166 3511 y Fg(It)29 b(should)f(b)s(e)h (p)s(oin)m(ted)g(out)f(that)h(the)g(con)m(v)m(ergence)i(to)e(a)f (nondegenerate)i(measure)f(for)166 3631 y(the)35 b(abnormal)e (normalization)e(describ)s(ed)k(b)m(y)h(claim)c(\(ii\))h(means)i(the)g (app)s(earance)g(of)166 3752 y(a)e(strong)g(dep)s(endence)j(b)s(et)m(w) m(een)g(the)d(random)g(v)m(ectors)i(considered.)f(F)-8 b(or)33 b Ff(\034)40 b(<)29 b(\034)3218 3767 y Fl(\003)3258 3752 y Fg(,)k(the)166 3872 y(dep)s(endence)j(is)c(w)m(eak)j(and)e(the)h (classical)d(cen)m(tral)i(limit)d(theorem)j(ough)m(t)g(to)g(hold.)g(T) -8 b(o)166 3992 y(sho)m(w)37 b(this)e(w)m(e)i(in)m(tro)s(duce)f(the)g (classical)e(normalization)e(instead)k(of)f(\(30\).)g(So)h(w)m(e)h(set) 166 4113 y(along)31 b(with)h(\(29\):)369 4519 y(~)-57 b Ff(\026)420 4534 y Fo(n)467 4519 y Fg(\()p Ff(dy)t Fg(\))15 b(=)875 4451 y(1)p 764 4495 271 4 v 800 4571 a(~)764 4597 y Ff(M)858 4612 y Fo(n)905 4597 y Fg(\()p Ff(\034)c Fg(\))1061 4519 y(exp)1226 4422 y Fd(\020)1276 4519 y Ff(\016)1323 4477 y Fl(\000)p Fo(n\025=)p Ft(2)1537 4519 y Ff(\034)g Fg(\()p Ff(y)t(;)17 b(y)t Fg(\))1814 4422 y Fd(\021)1886 4519 y Fg(~)-57 b Ff(\026)1937 4477 y Fo(?\016)1937 4543 y(n)p Fl(\000)p Ft(1)2074 4519 y Fg(\()2112 4429 y Fe(p)p 2195 4429 47 4 v 90 x Ff(\016)t(dy)t Fg(\))p Ff(;)48 b(\026)2517 4534 y Ft(0)2584 4519 y Fg(=)27 b Ff(\027)34 b Fe(2)28 b(M)p Fg(\()p Ff(I)-25 b(R)3114 4477 y Fo(N)3181 4519 y Fg(\))p Ff(;)410 4747 y Fg(~)374 4772 y Ff(M)468 4787 y Fo(n)515 4772 y Fg(\()p Ff(\034)11 b Fg(\))17 b(=)774 4655 y Fd(Z)735 4912 y Fo(I)-18 b(R)801 4887 y Fb(N)894 4772 y Fg(exp)1059 4676 y Fd(\020)1109 4772 y Ff(\016)1156 4731 y Fl(\000)p Fo(n\025=)p Ft(2)1369 4772 y Ff(\034)11 b Fg(\()p Ff(y)t(;)17 b(y)t Fg(\))1646 4676 y Fd(\021)1719 4772 y Fg(~)-57 b Ff(\026)1770 4731 y Fo(?\016)1770 4797 y(n)p Fl(\000)p Ft(1)1907 4772 y Fg(\()1945 4682 y Fe(p)p 2028 4682 V 90 x Ff(\016)t(dy)t Fg(\))p Ff(:)1037 b Fg(\(32\))166 5120 y Fi(Theorem)37 b(1.6)49 b Fh(L)-5 b(et)38 b Ff(N)10 b Fh(,)38 b Ff(\027)6 b Fh(,)38 b(and)e Ff(\034)1515 5135 y Fl(\003)1593 5120 y Fh(b)-5 b(e)37 b(as)g(in)g(The)-5 b(or)g(em)36 b(1.5.)h(Then)g(for)g Ff(\034)44 b(<)33 b(\034)3220 5135 y Fl(\003)3259 5120 y Fh(,)38 b(the)166 5240 y(se)-5 b(quenc)g(e)44 b(of)g(me)-5 b(asur)g(es)45 b Fe(f)8 b Fg(~)-57 b Ff(\026)1233 5255 y Fo(n)1370 5240 y Fe(j)90 b Ff(n)47 b Fe(2)f Ff(I)-25 b(N)1811 5255 y Ft(0)1851 5240 y Ff(;)69 b Fg(~)-57 b Ff(\026)1998 5255 y Ft(0)2083 5240 y Fg(=)46 b Ff(\027)6 b Fe(g)45 b Fh(de\014ne)-5 b(d)44 b(by)h(\(32\))f(c)-5 b(onver)g(ges)166 5360 y(we)g(akly)34 b(to)h(an)g(isotr)-5 b(opic)34 b(Gaussian)h(me)-5 b(asur)g(e.)1745 5712 y Fg(10)p eop %%Page: 11 11 11 10 bop 166 83 a Fi(2)112 b(Preliminaries)166 423 y Fh(2.1)99 b(L)-5 b(aguerr)g(e)35 b(Entir)-5 b(e)35 b(F)-7 b(unctions)34 b(and)g(Evolution)h(Op)-5 b(er)g(ator)166 764 y Fg(W)d(e)36 b(start)g(with)g(the)h(description)e(of)h(the)g(F)-8 b(r)m(\023)-46 b(ec)m(het)37 b(spaces)g Fe(A)2491 779 y Fo(a)2533 764 y Fg(.)f(A)g(subset)h Ff(B)i Fe(\032)34 b(A)3311 779 y Fo(a)3388 764 y Fg(is)166 884 y(said)39 b(to)g(b)s(e)h(b)s(ounded)g(in)e Fe(A)1245 899 y Fo(a)1326 884 y Fg(if)g(for)h(ev)m(ery)i Ff(b)f(>)f(a)p Fg(,)h(there)g(exists)g Ff(K)2771 899 y Fo(b)2845 884 y Ff(>)f Fg(0)g(suc)m(h)i(that)166 1005 y Fe(k)p Ff(f)11 b Fe(k)324 1034 y Fo(b)386 1005 y Fe(\024)29 b Ff(K)575 1020 y Fo(b)641 1005 y Fg(for)j(all)f Ff(f)38 b Fe(2)28 b Ff(B)5 b Fg(.)166 1225 y Fi(Prop)s(osition)36 b(2.1)49 b Fh(F)-7 b(or)30 b(every)h Ff(a)d Fe(\025)g Fg(0)p Fh(,)j(the)g(sp)-5 b(ac)g(e)30 b Fe(A)2178 1240 y Fo(a)2251 1225 y Fh(p)-5 b(ossesses)29 b(the)j(fol)5 b(lowing)29 b(pr)-5 b(op-)166 1345 y(erties:)216 1603 y Fg(\(i\))153 b Fh(the)35 b(r)-5 b(elative)34 b(top)-5 b(olo)g(gy)35 b(on)f(b)-5 b(ounde)g(d)34 b(subsets)h(of)g Fe(A)2385 1618 y Fo(a)2461 1603 y Fh(c)-5 b(oincides)33 b(with)473 1783 y(the)i(top)-5 b(olo)g(gy)34 b(of)h(uniform)f(c)-5 b(onver)g(genc)g(e)33 b(on)i(c)-5 b(omp)g(act)34 b(subsets)h(of)p 3023 1783 5 50 v 45 w Ff(C)7 b Fh(;)216 1988 y Fg(\(ii\))125 b Fh(multiplic)-5 b(ation,)34 b(i.e.,)g Fg(\()p Ff(f)5 b(;)17 b(g)t Fg(\))27 b Fe(7!)g Ff(f)11 b(g)38 b Fh(is)d(a)f(c)-5 b(ontinuous)35 b(map)473 2168 y(fr)-5 b(om)34 b Fe(A)783 2183 y Fo(a)846 2168 y Fe(\002)23 b(A)1026 2183 y Fo(b)1095 2168 y Fh(into)34 b Fe(A)1376 2183 y Fo(a)p Ft(+)p Fo(b)1503 2168 y Fh(;)216 2372 y Fg(\(iii\))97 b Fh(di\013er)-5 b(entiation,)33 b(i.e.)i Ff(f)j Fe(7!)27 b Ff(f)1578 2336 y Fl(0)1636 2372 y Fh(is)35 b(a)g(c)-5 b(ontinuous)34 b(self-map)g(of)g Fe(A)2899 2387 y Fo(a)2941 2372 y Fh(.)166 2604 y Fi(Remark)j(2.1)49 b Fh(It)35 b(c)-5 b(an)34 b(b)-5 b(e)35 b(e)-5 b(asily)34 b(shown)g(that,)h(for)g(p)-5 b(ositive)34 b Ff(a)h Fh(and)f Ff(b)p Fh(,)1381 2823 y Fe(k)p Ff(f)11 b(g)t Fe(k)1591 2838 y Fo(a)p Ft(+)p Fo(b)1744 2823 y Fe(\024)28 b(k)p Ff(f)11 b Fe(k)2008 2838 y Fo(a)2049 2823 y Fe(k)p Ff(g)t Fe(k)2200 2838 y Fo(b)2233 2823 y Ff(;)1020 b Fg(\(33\))166 3142 y Fh(thus)38 b Fg(\()p Ff(f)5 b(;)17 b(g)t Fg(\))31 b Fe(7!)g Ff(f)11 b(g)40 b Fh(is)d(a)g(c)-5 b(ontinuous)37 b(map)g(fr)-5 b(om)36 b Fe(B)2111 3157 y Fo(a)2177 3142 y Fe(\002)24 b(B)2343 3157 y Fo(b)2415 3142 y Fh(into)37 b Fe(B)2684 3157 y Fo(a)p Ft(+)p Fo(b)2812 3142 y Fh(,)g(which)f(implies)166 3262 y(claim)e Fg(\(ii\))f Fh(of)i(the)f(latter)i(statement.)166 3482 y Fi(Prop)s(osition)g(2.2)49 b Fh(Every)29 b(se)-5 b(quenc)g(e)28 b Fe(f)p Ff(f)1725 3497 y Fo(n)1772 3482 y Ff(;)17 b(n)28 b Fe(2)g Ff(I)-25 b(N)10 b Fe(g)28 b(\032)g(L)2354 3497 y Fo(a)2395 3482 y Fh(,)h Ff(a)f Fe(\025)g Fg(0)p Fh(,)h(that)h(c)-5 b(onver)g(ges)28 b(in)166 3603 y Fe(E)41 b Fh(to)32 b(a)g(function)f Ff(f)39 b Fe(2)28 b(A)1097 3618 y Fo(a)1138 3603 y Fh(,)k(which)f(do)-5 b(es)31 b(not)h(vanish)f(identic)-5 b(al)5 b(ly,)32 b(is)g(a)f(b)-5 b(ounde)g(d)32 b(subset)166 3723 y(of)g Fe(A)358 3738 y Fo(a)432 3723 y Fh(and)g(henc)-5 b(e,)31 b(by)h(claim)g(\(i\))g(of)g (Pr)-5 b(op)g(osition)32 b(2.1,)g(it)g(c)-5 b(onver)g(ges)31 b(in)h Fe(A)2986 3738 y Fo(a)3060 3723 y Fh(to)h Ff(f)38 b Fe(2)28 b(L)p Fh(.)166 3943 y Fg(F)-8 b(or)25 b Ff(f)39 b Fe(2)28 b(L)584 3907 y Ft(+)643 3943 y Fg(,)e(one)g(has)h Ff(f)11 b Fg(\(0\))27 b Ff(>)h Fg(0)d(\(see)j(\(5\)\).)e(Therefore,)h (there)g(exists)g(a)f(neigh)m(b)s(orho)s(o)s(d)166 4063 y Fe(D)31 b Fg(of)c(the)i(origin,)d(in)h(whic)m(h)h Ff(f)39 b Fe(6)p Fg(=)27 b(0.)h(Hence)h(the)g(follo)m(wing)c(holomorphic)h (function)h(can)166 4184 y(b)s(e)33 b(de\014ned)1238 4403 y Ff(')p Fg(\()p Ff(z)t Fg(\))c(=)e(log)17 b Ff(f)11 b Fg(\()p Ff(z)t Fg(\))p Ff(;)212 b(z)32 b Fe(2)c(D)s Ff(:)877 b Fg(\(34\))166 4722 y(In)33 b(the)g(sequel)g(w)m(e)h(use)g (the)f(notation)835 4941 y Ff(')899 4900 y Ft(\()p Fo(k)r Ft(\))1024 4941 y Fg(=)27 b(\()p Ff(D)1249 4900 y Fo(k)1308 4941 y Fg(log)17 b Ff(f)11 b Fg(\)\(0\))p Ff(;)211 b(k)31 b Fe(2)d Ff(I)-25 b(N)2193 4956 y Ft(0)2233 4941 y Ff(;)211 b(f)39 b Fe(2)28 b(L)2721 4900 y Ft(+)2780 4941 y Ff(:)473 b Fg(\(35\))166 5161 y Fi(Prop)s(osition)36 b(2.3)49 b Fg([The)33 b(sign)f(rule])j Fh(L)-5 b(et)35 b Ff(f)j Fe(2)28 b(L)2034 5124 y Ft(+)2093 5161 y Ff(;)35 b Fh(then)1255 5380 y Fg(\()p Fe(\000)p Fg(1\))1457 5339 y Fo(k)r Fl(\000)p Ft(1)1590 5380 y Ff(')1654 5339 y Ft(\()p Fo(k)r Ft(\))1780 5380 y Fe(\025)28 b Fg(0)p Ff(;)116 b(k)31 b Fe(2)d Ff(I)-25 b(N)10 b(:)894 b Fg(\(36\))1745 5712 y(11)p eop %%Page: 12 12 12 11 bop 166 83 a Fh(Equalities)35 b(hold)f(simultane)-5 b(ously)34 b(for)h(al)5 b(l)35 b Ff(k)30 b Fe(\025)f Fg(2)69 b Fh(and)34 b(only)h(for)g Ff(f)11 b Fg(\()p Ff(z)t Fg(\))28 b(=)f Ff(C)d Fg(exp)q(\()p Ff(az)t Fg(\))p Ff(:)166 303 y Fi(Lemma)37 b(2.1)49 b Fh(F)-7 b(or)45 b(a)h(se)-5 b(quenc)g(e)45 b Fe(f)p Ff(f)1549 318 y Fo(n)1641 303 y Fe(j)h Ff(n)i Fe(2)h Ff(I)-25 b(N)2042 318 y Ft(0)2173 303 y Ff(f)2221 318 y Fo(n)2316 303 y Fe(2)48 b(L)2499 267 y Ft(+)2558 303 y Fe(g)p Ff(;)e Fh(let)g(the)g(derivatives)166 423 y(\(35\))h(ob)-5 b(ey)47 b(the)g(fol)5 b(lowing)46 b(c)-5 b(onditions:)47 b Fg(\(i\))92 b Fe(f)p Ff(')2041 387 y Ft(\()p Fo(k)r Ft(\))2041 448 y Fo(n)2138 423 y Fe(g)47 b Fh(c)-5 b(onver)g(ges)47 b(to)g Ff(')2876 387 y Ft(\()p Fo(k)r Ft(\))2974 423 y Fh(,)g Ff(k)54 b Fg(=)d(0)p Ff(;)17 b Fg(1)p Fh(;)166 544 y Fg(\(ii\))59 b Fe(f)p Ff(')471 508 y Ft(\(2\))471 568 y Fo(n)565 544 y Fe(g)31 b Fh(c)-5 b(onver)g(ges)31 b(to)h(zer)-5 b(o.)31 b(Then)g Fe(f)p Ff(f)1773 559 y Fo(n)1820 544 y Fe(g)g Fh(c)-5 b(onver)g(ges)31 b(to)g Fg(exp)q(\()p Ff(')2697 508 y Ft(\(0\))2807 544 y Fg(+)15 b Ff(')2962 508 y Ft(\(1\))3056 544 y Ff(z)t Fg(\))32 b Fh(in)g Fe(A)3372 559 y Fo( )3424 544 y Fh(,)166 680 y Ff( )g Fg(=)27 b(sup)528 583 y Fd(n)583 680 y Ff(')647 643 y Ft(\(1\))647 704 y Fo(n)769 680 y Fg(:)h Ff(n)g Fe(2)g Ff(I)-25 b(N)1110 695 y Ft(0)1149 583 y Fd(o)1205 680 y Fh(.)166 912 y Fi(Pro)s(of.)35 b Fg(In)g(view)g(of)g(claim)d(\(ii\))h(of)h(Prop)s(osition)g(2.1)g(and) h(Prop)s(osition)e(2.2,)i(to)f(pro)m(v)m(e)166 1032 y(this)22 b(statemen)m(t)h(w)m(e)h(only)e(need)h(to)f(sho)m(w)i(that)e(the)h (sequence)i Fe(f)p Ff(f)2536 1047 y Fo(n)2583 1032 y Fg(\()p Ff(z)t Fg(\))p Ff(=f)2805 1047 y Fo(n)2853 1032 y Fg(\(0\))p Fe(g)c Fg(con)m(v)m(erges)166 1153 y(to)27 b(exp)q(\()p Ff(')531 1117 y Ft(\(1\))625 1153 y Ff(z)t Fg(\))g(uniformly)e(on)i(compact)f(subsets)j(of)p 2157 1153 5 50 v 29 w Ff(C)7 b Fg(.)27 b(By)h(Vitali's)c(theorem)j(and)g (due)166 1273 y(to)35 b(the)h(fact)f(that,)h(for)f(the)h(functions)f (considered,)i Ff(M)2244 1288 y Fo(f)2289 1273 y Fg(\()p Ff(r)s Fg(\))c(=)f Ff(f)11 b Fg(\()p Ff(r)s Fg(\),)35 b(w)m(e)h(ma)m(y)g(do)f(this)166 1394 y(b)m(y)42 b(pro)m(ving)f(the)h (p)s(oin)m(t)m(wise)f(con)m(v)m(ergence)j(of)d Fe(f)p Ff(f)2054 1409 y Fo(n)2101 1394 y Fg(\()p Ff(z)t Fg(\))p Ff(=f)2323 1409 y Fo(n)2370 1394 y Fg(\(0\))p Fe(g)g Fg(on)g Ff(I)-25 b(R)2823 1409 y Ft(+)2882 1394 y Fg(.)42 b(T)-8 b(o)41 b(this)g(end)166 1514 y(w)m(e)h(use)f(the)g(sp)s (eci\014c)g(form)e(of)h Ff(f)52 b Fe(2)42 b(L)1651 1478 y Ft(\(1\))1785 1514 y Fg(giv)m(en)f(b)m(y)g(\(4\).)f(F)-8 b(or)40 b(eac)m(h)h Ff(\015)47 b Fe(\025)41 b Fg(0,)g(one)f(has)166 1634 y(exp)q(\()p Ff(\015)27 b Fe(\000)541 1595 y Ft(1)p 541 1611 36 4 v 541 1669 a(2)586 1634 y Ff(\015)642 1598 y Ft(2)681 1634 y Fg(\))h Fe(\024)g Fg(1)22 b(+)g Ff(\015)33 b Fe(\024)28 b Fg(exp)q(\()p Ff(\015)5 b Fg(\).)33 b(Hence)g(for)f Ff(z)h Fe(2)28 b Ff(I)-25 b(R)2254 1649 y Ft(+)2313 1634 y Ff(;)903 1906 y Fg(exp)q(\()p Ff(z)t(')1203 1865 y Ft(\(1\))1203 1930 y Fo(n)1320 1906 y Fg(+)1428 1838 y(1)p 1428 1883 49 4 v 1428 1974 a(2)1487 1906 y Ff(z)1536 1865 y Ft(2)1576 1906 y Ff(')1640 1865 y Ft(\(2\))1640 1930 y Fo(n)1734 1906 y Fg(\))28 b Fe(\024)1915 1838 y Ff(f)1963 1853 y Fo(n)2010 1838 y Fg(\()p Ff(z)t Fg(\))p 1915 1883 221 4 v 1915 1974 a Ff(f)1963 1989 y Fo(n)2011 1974 y Fg(\(0\))2173 1906 y Fe(\024)g Fg(exp)q(\()p Ff(z)t(')2578 1865 y Ft(\(1\))2578 1930 y Fo(n)2673 1906 y Fg(\))p Ff(;)542 b Fg(\(37\))166 2263 y(whic)m(h)33 b(yields)f(the)h(stated)h (con)m(v)m(ergence.)p 2661 2255 42 42 v 166 2483 a(F)-8 b(or)32 b(an)g(en)m(tire)h(function)f Ff(f)38 b Fe(2)29 b(E)41 b Fg(and)32 b Ff(t)c Fe(\025)g Fg(0,)33 b(w)m(e)g(set)1063 2747 y(\(exp)q(\()p Ff(t)p Fg(\001)1404 2762 y Fo(\022)1443 2747 y Fg(\))p Ff(f)11 b Fg(\)\()p Ff(z)t Fg(\))1764 2694 y Ft(def)1774 2747 y Fg(=)1950 2639 y Fl(1)1925 2664 y Fd(X)1921 2848 y Fo(k)r Ft(=0)2078 2680 y Ff(t)2113 2643 y Fo(k)p 2076 2724 82 4 v 2076 2815 a Ff(k)s Fg(!)2167 2747 y(\(\001)2286 2706 y Fo(k)2286 2772 y(\022)2329 2747 y Ff(f)g Fg(\)\()p Ff(z)t Fg(\))p Ff(:)702 b Fg(\(38\))166 3021 y Fi(Prop)s(osition)36 b(2.4)49 b Fh(F)-7 b(or)36 b(every)h(p)-5 b(ositive)37 b Ff(a)g Fh(and)g Ff(t)h Fh(ob)-5 b(eying)36 b Ff(at)d(<)f Fg(1)p Fh(,)37 b(and)g Ff(\022)f Fe(\025)c Fg(0)p Fh(,)37 b(one)166 3141 y(has)860 3348 y Fe(k)17 b Fg(exp\()p Ff(t)p Fg(\001)1229 3363 y Fo(\022)1269 3348 y Fg(\))p Ff(f)11 b Fe(k)1416 3363 y Fo(b)1477 3348 y Fe(\024)28 b Fg(\(1)22 b Fe(\000)h Ff(at)p Fg(\))1915 3307 y Fl(\000)p Fo(\022)2009 3348 y Fe(k)p Ff(f)11 b Fe(k)2168 3363 y Fo(a)2209 3348 y Ff(;)87 b(b)28 b Fg(=)f Ff(a=)p Fg(\(1)22 b Fe(\000)h Ff(at)p Fg(\))p Ff(;)166 3655 y Fh(which)34 b(me)-5 b(ans)34 b(that)h(\(38\))f(de\014nes)g(a)h(c)-5 b(ontinuous)34 b(line)-5 b(ar)35 b(map)694 3882 y Fe(A)774 3897 y Fo(a)843 3882 y Fe(3)28 b Ff(f)38 b Fe(7!)28 b 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4529 y Fo(t)p 1301 4558 135 4 v 1322 4650 a Ff(@)g(t)1473 4581 y Fg(=)28 b(\001)1658 4596 y Fo(\022)1697 4581 y Ff(f)1745 4596 y Fo(t)1775 4581 y Ff(;)121 b(t)28 b Fe(2)g Fg(\(0)p Ff(;)17 b(t)2246 4596 y Ft(0)2285 4581 y Fg(\))p Ff(:)930 b Fg(\(40\))166 4919 y(One)30 b(of)g(the)g(main)e(results)i (of)g([6])f(is)h(Theorem)g(1.3)f(whic)m(h)i(asserts)g(that)f(the)g(op)s (erators)166 5039 y(of)i(the)h(t)m(yp)s(e)h(of)e(\(39\))g(preserv)m(es) j(the)e(class)g Fe(L)p Fg(.)f(In)h(our)f(case)i(it)d(reads)j(as)e (follo)m(ws)166 5259 y Fi(Prop)s(osition)k(2.5)49 b Fh(L)-5 b(et)26 b Ff(a)p Fh(,)g Ff(b)p Fh(,)h Ff(t)p Fh(,)f(and)g Ff(\022)j Fh(b)-5 b(e)26 b(as)g(in)g(Pr)-5 b(op)g(osition)25 b(2.4.)h(Then)f(the)h(op)-5 b(er)g(ator)166 5380 y(\(38\),)34 b(\(39\))g(maps)g Fe(L)947 5395 y Fo(a)1024 5380 y Fh(into)g Fe(L)1294 5395 y Fo(b)1328 5380 y Fh(.)1745 5712 y Fg(12)p eop %%Page: 13 13 13 12 bop 166 83 a Fg(The)34 b(follo)m(wing)29 b(statemen)m(ts)34 b(ha)m(v)m(e)g(also)e(b)s(een)h(pro)m(v)m(en)h(in)e([6].)166 303 y Fi(Prop)s(osition)k(2.6)49 b Fh(F)-7 b(or)33 b Ff(t)28 b(>)g Fg(0)p Fh(,)34 b(the)h(ab)-5 b(ove)34 b(op)-5 b(er)g(ator)35 b(has)f(the)h(inte)-5 b(gr)g(al)35 b(form)361 712 y Fg(\(exp)q(\()p Ff(t)p Fg(\001)702 727 y Fo(\022)742 712 y Fg(\))p Ff(f)11 b Fg(\)\()p Ff(z)t Fg(\))27 b(=)h(exp)1299 591 y Fd(\022)1360 712 y Fe(\000)1447 645 y Ff(z)p 1447 689 50 4 v 1454 781 a(t)1507 591 y Fd(\023)1603 579 y Ft(+)p Fl(1)1606 595 y Fd(Z)1611 836 y Ft(0)1727 712 y Ff(s)1773 671 y Fo(\022)r Fl(\000)p Ft(1)1902 712 y Ff(w)1972 727 y Fo(\022)2027 591 y Fd(\022)2098 645 y Ff(z)t(s)p 2098 689 96 4 v 2129 781 a(t)2204 591 y Fd(\023)2282 712 y Ff(e)2327 671 y Fl(\000)p Fo(s)2419 712 y Ff(f)11 b Fg(\()p Ff(ts)p Fg(\))p Ff(ds;)521 b Fg(\(41\))166 1019 y Fh(wher)-5 b(e)34 b Ff(w)511 1034 y Fo(\022)585 1019 y Fh(is)h(de\014ne)-5 b(d)33 b(by)i(\(12\).)166 1239 y Fi(Remark)i(2.2)49 b Fh(Setting)f(in)f(\(41\))h Ff(z)56 b Fg(=)c(0)p Fh(,)47 b(one)h(obtains)f Fg(\(exp)q(\()p Ff(t)p Fg(\001)2728 1254 y Fo(\022)2768 1239 y Fg(\))p Ff(f)11 b Fg(\)\(0\))51 b Ff(>)g Fg(0)p Fh(,)d(for)166 1360 y Ff(f)38 b Fe(2)29 b(L)416 1324 y Ft(+)509 1360 y Fh(and)34 b Ff(\022)d(>)d Fg(0)p Fh(.)34 b(On)h(the)g(other)f(hand,)g (one)h(has)f(fr)-5 b(om)34 b(\(38\))361 1742 y Fg(\(exp)q(\()p Ff(t)p Fg(\001)702 1757 y Fo(\022)742 1742 y Fg(\))p Ff(f)11 b Fg(\)\(0\))27 b(=)1161 1634 y Fl(1)1136 1659 y Fd(X)1132 1844 y Fo(k)r Ft(=0)1289 1675 y Ff(t)1324 1639 y Fo(k)p 1287 1719 82 4 v 1287 1811 a Ff(k)s Fg(!)1378 1742 y Ff(f)1437 1701 y Ft(\()p Fo(k)r Ft(\))1535 1742 y Fg(\(0\))1670 1675 y(\000\()p Ff(\022)e Fg(+)d Ff(k)s Fg(\))p 1670 1719 360 4 v 1756 1811 a(\000\()p Ff(\022)s Fg(\))2039 1742 y Ff(:)1214 b Fg(\(42\))166 2021 y Fh(Passing)34 b(her)-5 b(e)35 b(to)g(the)f(limit)h Ff(\022)c Fe(!)c Fg(0)35 b Fh(one)f(gets)361 2329 y Fg(\(exp)q(\()p Ff(t)p Fg(\001)702 2344 y Ft(0)742 2329 y Fg(\))p Ff(f)11 b Fg(\)\(0\))27 b(=)g Ff(f)11 b Fg(\(0\))28 b Ff(>)f Fg(0)p Ff(:)1757 b Fg(\(43\))166 2541 y Fh(Below)34 b(the)h(c)-5 b(ase)34 b Ff(\022)d Fg(=)c(0)35 b Fh(is)g(always)f(understo)-5 b(o)g(d)34 b(as)h(the)g(ab)-5 b(ove)34 b(limit.)166 2761 y Fi(Prop)s(osition)i(2.7)49 b Fh(L)-5 b(et)41 b Ff(v)h Fe(2)e Ff(I)-25 b(R)41 b Fh(and)f Fg(exp)r(\()p Ff(v)t(z)t Fg(\))p Ff(h)p Fg(\()p Ff(z)t Fg(\))f Fe(2)h(A)2393 2776 y Fo(b)2427 2761 y Fh(,)g Ff(b)g Fe(\025)f Fg(0)p Fh(.)i(F)-7 b(or)39 b(any)i Ff(u)e Fe(\025)g Fg(0)166 2881 y Fh(ob)-5 b(eying)34 b(the)h(c)-5 b(ondition)34 b Ff(ub)28 b(<)f Fg(1)p Fh(,)865 3125 y Fg(exp)q(\()p Ff(u)p Fg(\001)1189 3140 y Fo(\022)1228 3125 y Fg(\))17 b(exp)q(\()p Ff(v)t(z)t Fg(\))p Ff(h)p Fg(\()p Ff(z)t Fg(\))29 b(=)e(exp)2087 3004 y Fd(\022)2246 3058 y Ff(v)t(z)p 2158 3102 277 4 v 2158 3193 a Fg(1)22 b Fe(\000)g Ff(uv)2445 3004 y Fd(\023)2522 3125 y Ff(h)2578 3140 y Fo(u)2623 3125 y Fg(\()p Ff(z)t Fg(\))p Ff(;)505 b Fg(\(44\))166 3470 y Fh(wher)-5 b(e)703 3720 y Ff(h)759 3735 y Fo(u)805 3720 y Fg(\()p Ff(z)t 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Fl(\000)p Ft(1)2048 294 y Fg(\))p Ff(:)1167 b Fg(\(49\))166 605 y(The)52 b(expression)h(\(47\)) e(ma)m(y)g(b)s(e)h(used)g(to)f(de\014ne)i(op)s(erators)e(acting)g(b)s (et)m(w)m(een)i(the)166 725 y(F)-8 b(r)m(\023)-46 b(ec)m(het)42 b(spaces)h Fe(A)908 740 y Fo(a)991 725 y Fg(and)e(b)s(et)m(w)m(een)j (the)e(Banac)m(h)g(spaces)h Fe(B)2482 740 y Fo(b)2516 725 y Fg(.)f(In)g(all)d(suc)m(h)k(cases)g(w)m(e)166 846 y(denote)32 b(these)h(op)s(erators)f(b)m(y)g Ff(T)1350 861 y Fo(t)1412 846 y Fg(p)s(oin)m(ting)d(out,)j(if)e(necessary)-8 b(,)34 b(b)s(et)m(w)m(een)g(whic)m(h)e(spaces)166 966 y(acts)39 b(giv)m(en)f Ff(T)690 981 y Fo(t)720 966 y Fg(.)h(Com)m(bining)d(claim)g(\(ii\))h(of)g(Prop)s(osition)g(2.1)h (with)g(Prop)s(ositions)f(2.4)166 1087 y(and)c(2.5)f(one)h(has)166 1307 y Fi(Prop)s(osition)j(2.8)49 b Fh(F)-7 b(or)42 b(every)i Ff(a)g(<)g(\016)1683 1270 y Fo(\025)1728 1307 y Ff(=t\034)11 b Fh(,)44 b(the)g(op)-5 b(er)g(ator)43 b Ff(T)2559 1322 y Fo(t)2633 1307 y Fh(c)-5 b(ontinuously)44 b(maps:)166 1427 y Fe(B)231 1442 y Fo(a)301 1427 y Fe(!)27 b(B)493 1442 y Fo(b)563 1427 y Fh(,)p Fe(A)673 1442 y Fo(a)741 1427 y Fe(!)h(A)949 1442 y Fo(b)983 1427 y Fh(,)34 b(and)h Fe(L)1306 1442 y Fo(a)1375 1427 y Fe(!)27 b(L)1571 1442 y Fo(b)1605 1427 y Fh(,)35 b(wher)-5 b(e)34 b Ff(b)28 b Fg(=)g Ff(a\016)2216 1391 y Fl(\000)p Fo(\025)2316 1427 y Ff(=)p Fg(\(1)22 b Fe(\000)g Ff(at\034)11 b(\016)2759 1391 y Fl(\000)p Fo(\025)2860 1427 y Fg(\))p Fh(.)166 1647 y Fi(Prop)s(osition)36 b(2.9)49 b Fh(L)-5 b(et)40 b(the)h(se)-5 b(quenc)g(e)40 b Fe(f)p Ff(f)1805 1662 y Fo(n)1851 1647 y Fg(\()p Ff(z)t Fg(\))80 b Fe(j)e Ff(n)38 b Fe(2)h Ff(I)-25 b(N)2469 1662 y Ft(0)2508 1647 y Ff(;)57 b(f)2640 1662 y Ft(0)2680 1647 y Fg(\()p Ff(z)t Fg(\))39 b(=)f Ff(g)t Fg(\()p Ff(z)t Fg(\))g Fe(2)g(L)3345 1611 y Ft(+)3404 1647 y Fe(g)166 1767 y Fh(de\014ne)-5 b(d)37 b(by)g(\(49\))g(c)-5 b(onver)g(ge)37 b(in)g Fe(A)1444 1782 y Fo(a)1485 1767 y Fh(,)h Ff(a)33 b Fe(\025)g Fg(0)38 b Fh(to)g(a)f(function)g Ff(f)11 b Fh(.)38 b(Then)f(the)g(se)-5 b(quenc)g(e)37 b(of)166 1888 y(solutions)31 b(of)g(\(13\))g Fe(f)p Ff(f)991 1903 y Fo(n)1038 1888 y Fg(\()p Ff(t;)17 b(z)t Fg(\))32 b Fe(j)f Ff(n)c Fe(2)i Ff(I)-25 b(N)1619 1903 y Ft(0)1658 1888 y Ff(;)48 b(f)1781 1903 y Ft(0)1821 1888 y Fg(\()p Ff(z)t Fg(\))28 b(=)g Ff(g)t Fg(\()p Ff(z)t Fg(\))p Fe(g)p Fh(,)j(de\014ne)-5 b(d)30 b(by)h(\(47\),)g(c)-5 b(onver)g(ges)166 2008 y(in)35 b Fe(A)366 2023 y Fo(a)442 2008 y Fh(to)g(the)g(function)1436 2219 y Ff(f)11 b Fg(\()p Ff(t;)17 b(z)t Fg(\))28 b(=)g(\()p Ff(T)1926 2234 y Fo(t)1956 2219 y Ff(f)11 b Fg(\)\()p Ff(z)t Fg(\))p Ff(:)1075 b Fg(\(50\))166 2530 y(T)-8 b(o)39 b(establish)g(the)h(existence)h(and)f (con)m(v)m(ergence)i(of)d Fe(f)p Ff(f)2281 2545 y Fo(n)2327 2530 y Fe(g)h Fg(w)m(e)g(use)g(an)g(analog)d(of)i(the)166 2650 y(F)-8 b(r)m(\023)-46 b(ec)m(het)26 b(deriv)-5 b(ativ)m(e)25 b(of)g Ff(T)39 b Fg(and)26 b(then)g(study)h(the)e(\014xed)i(p)s(oin)m (ts)e(of)g Ff(T)39 b Fg(and)25 b(their)g(stabilit)m(y)-8 b(.)166 2771 y(The)30 b(follo)m(wing)d(corollary)h(of)h(Prop)s(osition) e(2.8)i(allo)m(ws)g(to)g(de\014ne)i(the)e(di\013eren)m(tiabilit)m(y)166 2891 y(of)j Ff(T)46 b Fg(acting)32 b(b)s(et)m(w)m(een)j(the)e(F)-8 b(r)m(\023)-46 b(ec)m(het)33 b(spaces.)h(F)-8 b(or)32 b Ff(a)c Fe(2)g Fg([0)p Ff(;)17 b(\034)11 b(\016)2457 2855 y Fl(\000)p Fo(\025)2557 2891 y Fg(\),)33 b(w)m(e)h(set)1409 3165 y Ff(b)p Fg(\()p Ff(a)p Fg(\))1638 3112 y Ft(def)1649 3165 y Fg(=)1885 3097 y Ff(a\016)1983 3061 y Fl(\000)p Fo(\025)p 1773 3142 423 4 v 1773 3233 a Fg(1)22 b Fe(\000)g Ff(a\034)11 b(\016)2094 3204 y Fl(\000)p Fo(\025)2205 3165 y Ff(:)1048 b Fg(\(51\))166 3423 y Fi(Corollary)36 b(2.2)49 b Fh(L)-5 b(et)40 b Ff(a)c(<)g(\016)1275 3387 y Fo(\025)1320 3423 y Ff(=\034)11 b Fh(,)40 b(then)f(ther)-5 b(e)39 b(exists)g Ff(")d(>)g Fg(0)j Fh(such)g(that,)h(for)f(al)5 b(l)39 b Ff(a)3328 3387 y Fl(0)3387 3423 y Fe(2)166 3543 y Fg(\()p Ff(a;)17 b(a)22 b Fg(+)g Ff(")p Fg(\))p Fh(,)35 b(the)f(op)-5 b(er)g(ator)35 b Ff(T)49 b Fh(c)-5 b(ontinuously)35 b(maps)f Fe(B)2152 3558 y Fo(a)2189 3539 y Fa(0)2251 3543 y Fh(into)h Fe(B)2518 3559 y Fo(b)p Ft(\()p Fo(a)2612 3540 y Fa(0)2636 3559 y Ft(\))2667 3543 y Fh(.)166 3763 y Fi(De\014nition)h(2.1)49 b Fh(The)40 b(op)-5 b(er)g(ator)40 b Ff(T)52 b Fg(:)39 b Fe(A)1730 3778 y Fo(a)1809 3763 y Fe(!)f(A)2027 3779 y Fo(b)p Ft(\()p Fo(a)p Ft(\))2194 3763 y Fh(is)i(said)g(to)h(b)-5 b(e)40 b(di\013er)-5 b(entiable)39 b(on)166 3884 y Fe(A)246 3899 y Fo(a)329 3884 y Fh(if)j(for)g(every)g Ff(f)51 b Fe(2)42 b(A)1146 3899 y Fo(a)1187 3884 y Fh(,)g(ther)-5 b(e)42 b(exist)g Ff(")e(>)h Fg(0)h Fh(and)f(a)h(c)-5 b(ontinuous)42 b(line)-5 b(ar)41 b(op)-5 b(er)g(ator)166 4004 y Ff(T)237 3968 y Fl(0)260 4004 y Fg([)p Ff(f)11 b Fg(])40 b(:)g Fe(A)560 4019 y Fo(a)642 4004 y Fe(!)g(A)862 4020 y Fo(b)p Ft(\()p Fo(a)p Ft(\))1029 4004 y Fh(such)i(that,)g(for)f(every)h Ff(a)1972 3968 y Fl(0)2035 4004 y Fe(2)f Fg(\()p Ff(a;)17 b(a)27 b Fg(+)g Ff(")p Fg(\))p Fh(,)41 b Ff(T)2682 3968 y Fl(0)2705 4004 y Fg([)p Ff(f)11 b Fg(])42 b Fh(is)f(the)h(F)-7 b(r)n(\023)-47 b(echet)166 4124 y(derivative)36 b(of)g Ff(T)50 b Fh(c)-5 b(onsider)g(e)g(d)35 b(as)i(an)f(op)-5 b(er)g(ator)36 b(b)-5 b(etwe)g(en)36 b(the)h(Banach)e(sp)-5 b(ac)g(es)35 b Fe(B)3198 4139 y Fo(a)3235 4120 y Fa(0)3299 4124 y Fh(and)166 4245 y Fe(B)231 4260 y Fo(b)p Ft(\()p Fo(a)325 4241 y Fa(0)349 4260 y Ft(\))381 4245 y Fh(.)f(The)h(op)-5 b(er)g(ator)34 b Ff(T)1099 4209 y Fl(0)1122 4245 y Fg([)p Ff(f)11 b Fg(])28 b(:)g Fe(A)1398 4260 y Fo(a)1466 4245 y Fe(!)g(A)1674 4260 y Fo(b)p Ft(\()p Fo(a)p Ft(\))1835 4245 y Fh(is)34 b(said)h(to)g(b)-5 b(e)34 b(a)h(derivative)f(of)h Ff(T)48 b Fh(at)35 b Ff(f)11 b Fh(.)166 4465 y Fi(Lemma)37 b(2.2)49 b Fh(F)-7 b(or)38 b Ff(a)d(<)f(\016)1184 4429 y Fo(\025)1229 4465 y Ff(=\034)11 b Fh(,)39 b(the)g(op)-5 b(er)g(ator)38 b Ff(T)49 b Fg(:)34 b Fe(A)2200 4480 y Fo(a)2276 4465 y Fe(!)g(A)2490 4480 y Fo(b)p Ft(\()p Fo(a)p Ft(\))2655 4465 y Fh(is)39 b(di\013er)-5 b(entiable)37 b(on)166 4585 y Fe(A)246 4600 y Fo(a)322 4585 y Fh(and)d(its)h (derivative)f Ff(T)1166 4549 y Fl(0)1189 4585 y Fg([)p Ff(f)11 b Fg(])35 b Fh(is)g(the)g(fol)5 b(lowing)33 b(line)-5 b(ar)34 b(op)-5 b(er)g(ator)871 4809 y Ff(T)942 4768 y Fl(0)965 4809 y Fg([)p Ff(f)11 b Fg(]\()p Ff(h)p Fg(\)\()p Ff(z)t Fg(\))28 b(=)g Ff(\016)20 b Fg(exp)q(\()p Ff(\034)11 b Fg(\001)1851 4824 y Fo(\022)1891 4809 y Fg(\))1946 4712 y Fd(\020)1995 4809 y Fg(\()p Ff(f)2092 4768 y Fo(\016)r Fl(\000)p Ft(1)2220 4809 y Ff(h)p Fg(\)\()p Ff(\016)2399 4768 y Fl(\000)p Ft(1)p Fl(\000)p Fo(\025)2589 4809 y Ff(z)t Fg(\))2676 4712 y Fd(\021)2743 4809 y Ff(:)510 b Fg(\(52\))166 5140 y Fi(Pro)s(of.)32 b Fg(F)-8 b(or)32 b Ff(a)727 5103 y Fl(0)778 5140 y Fe(2)d Fg(\()p Ff(a;)17 b(\016)1053 5103 y Fo(\025)1098 5140 y Ff(=\034)11 b Fg(\))32 b(and)h Ff(f)5 b(;)33 b(h)28 b Fe(2)g(B)1816 5155 y Fo(a)1853 5136 y Fa(0)1880 5140 y Ff(;)33 b Fg(one)g(has)644 5363 y Ff(T)14 b Fg(\()p Ff(f)32 b Fg(+)22 b Ff(h)p Fg(\))28 b(=)g Ff(T)14 b Fg(\()p Ff(f)d Fg(\))21 b(+)h Ff(\016)f Fg(exp)q(\()p Ff(\034)11 b Fg(\001)1867 5378 y Fo(\022)1906 5363 y Fg(\))1961 5267 y Fd(\020)2011 5363 y Fg(\()p Ff(f)2108 5322 y Fo(\016)r Fl(\000)p Ft(1)2235 5363 y Ff(h)p Fg(\)\()p Ff(\016)2414 5322 y Fl(\000)p Ft(1)p Fl(\000)p Fo(\025)2605 5363 y Fe(\001)p Fg(\))2671 5267 y Fd(\021)2742 5363 y Fg(+)22 b Ff(R)q Fg(\()p Ff(f)5 b(;)17 b(h)p Fg(\))p Ff(;)1745 5712 y Fg(14)p eop %%Page: 15 15 15 14 bop 361 162 a Ff(R)q Fg(\()p Ff(f)5 b(;)17 b(h)p Fg(\))g(=)822 54 y Fo(\016)779 79 y Fd(X)774 264 y Fo(k)r Ft(=2)920 16 y Fd( )989 95 y Ff(\016)985 231 y(k)1039 16 y Fd(!)1122 66 y(\020)1171 162 y Fg(exp)q(\()p Ff(\034)11 b Fg(\001)1492 177 y Fo(\022)1532 162 y Fg(\))p Ff(f)1629 121 y Fo(\016)r Fl(\000)p Fo(k)1760 162 y Ff(h)1816 121 y Fo(k)1859 66 y Fd(\021)1925 162 y Fg(\()p Ff(\016)2010 121 y Fl(\000)p Ft(1)p Fl(\000)p Fo(\025)2200 162 y Fe(\001)p Fg(\))p Ff(:)166 453 y Fg(By)33 b(means)g(of)f(Remark)g(2.1,)g(\(33\),) g(and)h(Prop)s(osition)e(2.4,)h(one)h(obtains)361 691 y Fd(\015)361 741 y(\015)361 791 y(\015)p Fg(exp)q(\()p Ff(\034)11 b Fg(\001)728 806 y Fo(\022)768 791 y Fg(\))823 695 y Fd(\020)872 791 y Ff(f)931 750 y Fo(\016)r Fl(\000)p Fo(k)1062 791 y Ff(h)1118 750 y Fo(k)1161 695 y Fd(\021)1227 791 y Fg(\()p Ff(\016)1312 750 y Fl(\000)p Ft(1)p Fl(\000)p Fo(\025)1502 791 y Fe(\001)p Fg(\))1568 691 y Fd(\015)1568 741 y(\015)1568 791 y(\015)1614 845 y Fo(b)p Ft(\()p Fo(a)1708 826 y Fa(0)1732 845 y Ft(\))1780 791 y Fe(\024)17 b Fg(\(1)22 b Fe(\000)g Ff(a)2133 750 y Fl(0)2157 791 y Ff(\034)11 b(\016)2257 750 y Fl(\000)p Fo(\025)2358 791 y Fg(\))2396 750 y Fl(\000)p Fo(\022)2489 791 y Fe(k)p Ff(f)g Fe(k)2648 750 y Fo(\016)r Fl(\000)p Fo(k)2648 816 y(a)2685 798 y Fa(0)2779 791 y Fe(k)p Ff(h)p Fe(k)2935 750 y Fo(k)2935 816 y(a)2972 797 y Fa(0)2999 791 y Ff(;)166 1050 y Fg(for)32 b Ff(k)f Fg(=)c(1)p Ff(;)17 b Fg(2)p Ff(;)g(:)g(:)g(:)e(;)i(\016)37 b Fg(.)32 b(This)h(giv)m(es)g(for)f(all) e Ff(a)1796 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2120 y Fg(By)31 b(the)g(latter)e(estimate,)h Ff(T)1233 2084 y Fl(0)1256 2120 y Fg([)p Ff(f)11 b Fg(])30 b(con)m(tin)m(uously)h(maps)f Fe(B)2282 2135 y Fo(a)2319 2116 y Fa(0)2377 2120 y Fg(in)m(to)g Fe(B)2638 2136 y Fo(b)p Ft(\()p Fo(a)2732 2117 y Fa(0)2756 2136 y Ft(\))2818 2120 y Fg(whereas)i(b)m(y)f(the)166 2241 y(former)h(one,)h(this)f(op)s(erator)g(is)g(the)h(F)-8 b(r)m(\023)-46 b(ec)m(het)33 b(deriv)-5 b(ativ)m(e)32 b(of)g Ff(T)42 b Fg(:)28 b Fe(B)2655 2256 y Fo(a)2692 2237 y Fa(0)2747 2241 y Fe(!)f(B)2939 2256 y Fo(b)p Ft(\()p Fo(a)3033 2237 y Fa(0)3057 2256 y Ft(\))3089 2241 y Fg(.)p 3378 2233 42 42 v 166 2361 a(The)d(case)g(of)e Ff(\034)39 b Fg(=)28 b(0)22 b(w)m(as)i(considered)g(ab)s(o)m(v)m(e)f(\(see)h (\(16\)\),)f(th)m(us)g(from)f(no)m(w)i(on)e(w)m(e)i(assume)166 2481 y Ff(\034)39 b(>)28 b Fg(0.)j(It)g(turns)h(out)e(that)h(it)f(is)h (con)m(v)m(enien)m(t)i(to)d(remo)m(v)m(e)i(the)g(explicit)d(dep)s (endence)34 b(on)166 2602 y Ff(\034)44 b Fg(from)31 b(the)i(op)s (erator)f Ff(T)14 b Fg(.)33 b(T)-8 b(o)32 b(this)g(end)i(w)m(e)f(set) 1488 2834 y Ff(\034)1602 2781 y Ft(def)1613 2834 y Fg(=)38 b Ff(\014)6 b Fg(\()p Ff(\016)1873 2793 y Fo(\025)1940 2834 y Fe(\000)22 b Fg(1\))p Ff(;)1127 b Fg(\(53\))166 3151 y(and)30 b(include)f Ff(\014)36 b Fg(in)m(to)29 b Ff(z)t Fg(.)h(Then)h(w)m(e)g(consider)f(the)g(sequence)j Fe(f)p Ff(g)2512 3166 y Fo(n)2558 3151 y Fg(\()p Ff(z)t Fg(\))p Ff(;)47 b(n)28 b Fe(2)g Ff(I)-25 b(N)11 b Fe(g)p Fg(,)29 b(de\014ned)166 3272 y(as)k(follo)m(ws)361 3585 y Ff(g)408 3600 y Fo(n)455 3585 y Fg(\()p Ff(z)t Fg(\))28 b(=)f Ff(Q)p Fg(\()p Ff(g)873 3600 y Fo(n)p Fl(\000)p Ft(1)1010 3585 y Fg(\)\()p Ff(z)t Fg(\))2107 b(\(54\))597 3710 y Ft(def)608 3763 y Fg(=)27 b(exp)877 3666 y Fd(\020)926 3763 y Fg(\()p Ff(\016)1011 3722 y Fo(\025)1079 3763 y Fe(\000)22 b Fg(1\)\001)1346 3778 y Fo(\022)1385 3666 y Fd(\021)16 b(h)1491 3763 y Ff(g)1538 3778 y Fo(n)p Fl(\000)p Ft(1)1674 3763 y Fg(\()p Ff(\016)1759 3722 y Fl(\000)p Ft(1)p Fl(\000)p Fo(\025)1950 3763 y Ff(z)t Fg(\))2037 3666 y Fd(i)2077 3689 y Fo(\016)2131 3763 y Ff(;)49 b(g)2254 3778 y Ft(0)2293 3763 y Fg(\()p Ff(z)t Fg(\))29 b(=)e Ff(g)t Fg(\()p Ff(\014)6 b(z)t Fg(\))p Ff(;)166 3993 y Fg(where)45 b Ff(g)i Fg(is)c(the)h(function)f(whic)m(h) i(determines)f(the)g(starting)e(elemen)m(t)i(of)f Fe(f)p Ff(f)3175 4008 y Fo(n)3222 3993 y Fe(g)p Fg(.)h(T)-8 b(o)166 4113 y(pro)m(v)m(e)50 b(Theorem)f(1.6)g(w)m(e)h(shall)d(also)h (use)i(the)f(sequence)j(of)c(functions)h(from)f Fe(L)3333 4077 y Ft(\(1\))3427 4113 y Fg(,)166 4234 y Fe(f)s Fg(~)-52 b Ff(g)263 4249 y Fo(n)310 4234 y Fg(\()p Ff(z)t Fg(\))33 b Fe(j)f Ff(n)c Fe(2)g Ff(I)-25 b(N)814 4249 y Ft(0)853 4234 y Ff(;)53 b Fg(~)-52 b Ff(g)977 4249 y Ft(0)1016 4234 y Fg(\()p Ff(z)t Fg(\))28 b(=)g Ff(g)t Fg(\()p Ff(\014)6 b(z)t Fg(\))p Fe(g)p Fg(,)32 b(where)h Ff(g)j Fg(is)c(as)h(ab)s(o)m(v)m (e,)h(and)365 4564 y(~)-53 b Ff(g)408 4579 y Fo(n)455 4564 y Fg(\()p Ff(z)t Fg(\))28 b(=)733 4539 y(~)711 4564 y Ff(Q)788 4579 y Fo(n)835 4564 y Fg(\()s(~)-52 b Ff(g)920 4579 y Fo(n)p Fl(\000)p Ft(1)1057 4564 y Fg(\)\()p Ff(z)t Fg(\))2060 b(\(55\))597 4701 y Ft(def)608 4754 y Fg(=)748 4686 y(1)p 721 4730 104 4 v 736 4806 a(~)721 4832 y Ff(Y)778 4847 y Fo(n)851 4632 y Fd(\032)913 4754 y Fg(exp)1079 4657 y Fd(\020)1129 4754 y Ff(\016)1176 4712 y Fl(\000)p Fo(n\025)1319 4754 y Fg(\()p Ff(\016)1404 4712 y Fo(\025)1471 4754 y Fe(\000)23 b Fg(1\)\001)1739 4769 y Fo(\022)1778 4657 y Fd(\021)16 b(h)1886 4754 y Fg(~)-52 b Ff(g)1930 4769 y Fo(n)p Fl(\000)p Ft(1)2067 4754 y Fg(\()p Ff(\016)2152 4712 y Fl(\000)p Ft(1)2246 4754 y Fe(\001)p Fg(\))2312 4657 y Fd(i)2351 4680 y Fo(\016)2389 4632 y Fd(\033)2468 4754 y Fg(\()p Ff(z)t Fg(\))p Ff(;)492 4979 y Fg(~)477 5004 y Ff(Y)534 5019 y Fo(n)597 4952 y Ft(def)608 5004 y Fg(=)711 4883 y Fd(\032)773 5004 y Fg(exp)939 4908 y Fd(\020)989 5004 y Ff(\016)1036 4963 y Fl(\000)p Fo(n\025)1178 5004 y Fg(\()p Ff(\016)1263 4963 y Fo(\025)1331 5004 y Fe(\000)22 b Fg(1\)\001)1598 5019 y Fo(\022)1637 4908 y Fd(\021)17 b(h)1746 5004 y Fg(~)-52 b Ff(g)1790 5019 y Fo(n)p Fl(\000)p Ft(1)1927 5004 y Fg(\()p Ff(\016)2012 4963 y Fl(\000)p Ft(1)2106 5004 y Fe(\001)p Fg(\))2172 4908 y Fd(i)2211 4931 y Fo(\016)2249 4883 y Fd(\033)2327 5004 y Fg(\(0\))p Ff(:)166 5259 y Fg(Comparing)29 b(\(47\),)h(\(48\))g (with)g(\(54\))g(one)h(obtains)f(from)f(Prop)s(osition)g(2.8)h(and)h (Lemma)166 5380 y(2.2.)1745 5712 y(15)p eop %%Page: 16 16 16 15 bop 166 83 a Fi(Prop)s(osition)36 b(2.10)49 b Fh(F)-7 b(or)27 b(every)i Ff(a)f(<)f(\016)1676 47 y Fo(\025)1721 83 y Ff(=)p Fg(\()p Ff(\016)1855 47 y Fo(\025)1908 83 y Fe(\000)8 b Fg(1\))p Fh(,)29 b Ff(Q)g Fh(is)f(a)h(di\013er)-5 b(entiable)27 b(\(and)h(henc)-5 b(e)166 203 y(c)g(ontinuous\))39 b(op)-5 b(er)g(ator,)40 b(which)f(maps:)f Fe(A)1767 218 y Fo(a)1845 203 y Fe(!)f(A)2062 218 y Fo(b)2092 199 y Fa(0)2118 203 y Fh(,)j Fe(L)2257 167 y Ft(+)2257 228 y Fo(a)2352 203 y Fe(!)c(L)2557 162 y Ft(+)2557 229 y Fo(b)2587 210 y Fa(0)2616 203 y Fh(,)k(wher)-5 b(e)39 b Ff(b)3007 167 y Fl(0)3067 203 y Fg(=)e Ff(a)p Fg([)p Ff(\016)3305 167 y Fo(\025)3376 203 y Fe(\000)166 324 y Ff(a)p Fg(\()p Ff(\016)302 288 y Fo(\025)370 324 y Fe(\000)22 b Fg(1\)])583 288 y Fl(\000)p Ft(1)677 324 y Fh(.)35 b(Its)g(derivative)f(is)361 686 y Ff(Q)438 645 y Fl(0)462 686 y Fg([)p Ff(g)t Fg(]\()p Ff(h)p Fg(\)\()p Ff(z)t Fg(\))16 b(=)h(exp)1099 590 y Fd(\020)1148 686 y Fg(\()p Ff(\016)1233 645 y Fo(\025)1300 686 y Fe(\000)23 b Fg(1\)\001)1568 701 y Fo(\022)1607 590 y Fd(\021)16 b(\020h)1762 686 y Ff(g)1813 645 y Fo(\016)r Fl(\000)p Ft(1)1940 686 y Ff(h)1996 590 y Fd(i)2052 686 y Fg(\()p Ff(\016)2137 645 y Fl(\000)p Ft(1)p Fl(\000)p Fo(\025)2328 686 y Ff(z)t Fg(\))2415 590 y Fd(\021)2482 686 y Ff(:)771 b Fg(\(56\))166 1073 y(F)-8 b(or)34 b Ff(\034)44 b Fe(2)33 b Ff(I)8 b Fg(\()p Ff(g)t Fg(\),)34 b Ff(\014)41 b Fg(v)-5 b(aries)34 b(in)h Ff(J)9 b Fg(\()p Ff(g)t Fg(\))1477 1020 y Ft(def)1488 1073 y Fg(=)43 b(\(0)p Ff(;)17 b Fg(1)p Ff(=\013)q Fg(])34 b(\(see)i(\(53\))e(and)i(\(15\)\).)e(Let)i Ff(g)f Fe(2)e(L)3260 1037 y Ft(+)3353 1073 y Fg(b)s(e)166 1193 y(c)m(hosen.)f(Then)g(it)e(ma)m(y)g(b)s(e)h(written)g(in)e(the)i (form)f(\(4\))g(with)g Ff(\013)f Fe(\025)f Fg(0,)i(whic)m(h)i (determines)166 1313 y(the)h(in)m(terv)-5 b(als)32 b Ff(I)8 b Fg(\()p Ff(g)t Fg(\))31 b(\(15\))h(and)h Ff(J)9 b Fg(\()p Ff(g)t Fg(\),)32 b(and)h Ff(g)d Fe(2)f(L)2010 1277 y Ft(+)2010 1338 y Fo(\013)2096 1313 y Fe(\032)f(A)2281 1328 y Fo(\013)2330 1313 y Fg(.)166 1541 y Fi(Lemma)37 b(2.3)49 b Fh(F)-7 b(or)35 b Ff(\034)41 b Fe(2)31 b Ff(I)8 b Fg(\()p Ff(g)t Fg(\))p Fh(,)34 b(al)5 b(l)36 b(the)g(elements)f(of)h Fe(f)p Ff(f)2280 1556 y Fo(n)2363 1541 y Fe(j)f Ff(n)30 b Fe(2)g Ff(I)-25 b(N)2717 1556 y Ft(0)2756 1541 y Ff(;)53 b(f)2884 1556 y Ft(0)2953 1541 y Fg(=)30 b Ff(g)t Fe(g)35 b Fh(b)-5 b(elong)166 1661 y(to)34 b Fe(L)351 1625 y Ft(+)351 1686 y Fo(\013)438 1661 y Fe(\032)28 b(A)623 1676 y Fo(\013)672 1661 y Fh(,)33 b(al)5 b(l)34 b(the)g(elements)f(of)g Fe(f)p Ff(g)1645 1676 y Fo(n)1726 1661 y Fe(j)g Ff(n)28 b Fe(2)g Ff(I)-25 b(N)2073 1676 y Ft(0)2113 1661 y Ff(;)50 b(g)2237 1676 y Ft(0)2277 1661 y Fg(\()p Ff(z)t Fg(\))28 b(=)f Ff(g)t Fg(\()p Ff(\014)6 b(z)t Fg(\))p Fe(g)34 b Fh(b)-5 b(elong)33 b(to)h Fe(L)3332 1620 y Ft(+)3332 1686 y Fo(\014)s(\013)3424 1661 y Fh(.)166 1889 y Fi(Pro)s(of.)h Fg(F)-8 b(or)34 b Ff(\034)44 b Fe(2)32 b Ff(I)8 b Fg(\()p Ff(g)t Fg(\),)34 b Ff(\013)f Fe(\024)g Fg(\()p Ff(\016)1394 1852 y Fo(\025)1463 1889 y Fe(\000)24 b Fg(1\))p Ff(=\034)43 b(<)32 b(\016)1940 1852 y Fo(\025)1985 1889 y Ff(=\034)11 b Fg(,)36 b(th)m(us)g(b)m(y)g(Corollary)d(2.2,)i Ff(T)49 b Fg(maps)166 2009 y Fe(A)246 2024 y Fo(\013)328 2009 y Fg(in)m(to)31 b Fe(A)605 2024 y Fo(b)p Ft(\()p Fo(\013)p Ft(\))772 2009 y Fg(with)1095 2313 y Ff(b)p Fg(\()p Ff(\013)q Fg(\))c(=)1528 2246 y Ff(\013)q(\016)1638 2210 y Fl(\000)p Fo(\025)p 1416 2290 434 4 v 1416 2382 a Fg(1)22 b Fe(\000)g Ff(\013)q(\034)11 b(\016)1749 2353 y Fl(\000)p Fo(\025)1887 2313 y Fe(\024)2141 2246 y Ff(\013)q(\016)2251 2210 y Fl(\000)p Fo(\025)p 2003 2290 487 4 v 2003 2382 a Fg(1)21 b Fe(\000)i Fg(1)f(+)g Ff(\016)2389 2353 y Fl(\000)p Fo(\025)2527 2313 y Fg(=)27 b Ff(\013)q(;)166 2689 y Fg(whic)m(h)44 b(means)g Ff(T)60 b Fg(:)47 b Fe(A)1037 2704 y Fo(\013)1133 2689 y Fe(!)f(A)1359 2704 y Fo(\013)1408 2689 y Fg(.)e(By)g(Prop)s(osition)e(2.8,)h Ff(T)58 b Fg(maps)43 b Fe(L)g Fg(in)m(to)g(itself;)g(b)m(y)166 2809 y(Remark)h(2.2,)g(\()p Ff(T)14 b(f)d Fg(\)\(0\))47 b Ff(>)h Fg(0)d(whenev)m(er)i Ff(f)11 b Fg(\(0\))47 b Ff(>)h Fg(0.)c(The)i(second)g(part)e(of)g(Lemma)166 2930 y(concerning)33 b Fe(f)p Ff(g)751 2945 y Fo(n)797 2930 y Fe(g)g Fg(directly)f(follo)m(ws)f(from)g(the)i(\014rst)g(one.)p 2984 2922 42 42 v 166 3050 a(Since)c(the)g(starting)f(elemen)m(t)h(of)f Fe(f)p Ff(g)1502 3065 y Fo(n)1549 3050 y Fe(g)g Fg(has)h(the)h(form)d Ff(g)2235 3065 y Ft(0)2274 3050 y Fg(\()p Ff(z)t Fg(\))i(=)e Ff(g)t Fg(\()p Ff(\014)6 b(z)t Fg(\),)28 b(all)f(its)h(elemen)m(ts)166 3170 y(dep)s(end)k(on)e Ff(\014)6 b Fg(.)31 b(Therefore,)g(one)g(ma)m (y)g(consider)g Ff(g)2033 3185 y Fo(n)2110 3170 y Fg(as)g(a)f(map)g (from)f Ff(J)9 b Fg(\()p Ff(g)t Fg(\))30 b(in)m(to)g Fe(A)3246 3185 y Ft(1)3285 3170 y Fg(.)h(T)-8 b(o)166 3291 y(emphasize)33 b(this)f(fact)g(w)m(e)i(write)e(sometimes)g Ff(g)1928 3306 y Fo(n)1974 3291 y Fg(\()p Fe(\001)p Ff(;)17 b(\014)6 b Fg(\))32 b(instead)g(of)g Ff(g)2709 3306 y Fo(n)2756 3291 y Fg(.)166 3518 y Fi(Lemma)37 b(2.4)49 b Fh(F)-7 b(or)34 b(every)h Ff(n)28 b Fe(2)g Ff(I)-25 b(N)1481 3533 y Ft(0)1520 3518 y Fh(,)35 b(the)g(map)1501 3753 y Ff(g)1548 3768 y Fo(n)1622 3753 y Fg(:)28 b Ff(J)9 b Fg(\()p Ff(g)t Fg(\))27 b Fe(!)g(A)2101 3768 y Ft(1)3280 3753 y Fg(\(57\))166 4095 y Fh(is)35 b(di\013er)-5 b(entiable)33 b(on)i Fg(In)m(t)p Ff(J)9 b Fg(\()p Ff(g)t Fg(\))p Fh(,)34 b(its)h(derivative)49 b Fg(_)-42 b Ff(g)2001 4110 y Fo(n)2083 4095 y Fh(at)35 b Ff(\014)41 b Fh(is)34 b(an)h(element)f(of)g Fe(A)3095 4110 y Ft(1)3134 4095 y Fh(.)166 4322 y 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b(\014)6 b Fg(\))16 b(=)g(\001)p Ff(\014)6 b(Q)1414 796 y Fl(0)1437 837 y Fg([)p Ff(g)1511 852 y Fo(n)p Fl(\000)p Ft(1)1648 837 y Fg(])17 b(\()d(_)-41 b Ff(g)1777 852 y Fo(n)p Fl(\000)p Ft(1)1914 837 y Fg(\))22 b(+)g Ff(Q)2149 796 y Fl(0)2173 837 y Fg([)p Ff(g)2247 852 y Fo(n)p Fl(\000)p Ft(1)2384 837 y Fg(])17 b(\()o Ff(r)2509 852 y Fo(n)p Fl(\000)p Ft(1)2646 837 y Fg(\()p Fe(\001)p Ff(;)g Fg(\001)p Ff(\014)6 b Fg(\)\))21 b(+)h Ff(R)3167 852 y Fo(n)p Fl(\000)p Ft(1)3305 837 y Ff(;)166 1080 y Fg(where)34 b(for)e(all)e Ff(a)e(>)g Fg(1,)361 1450 y Fe(k)p Ff(R)485 1465 y Fo(n)p Fl(\000)p Ft(1)622 1450 y Fe(k)672 1465 y Fo(a)730 1450 y Fg(=)17 b Ff(o)887 1354 y Fd(\020)936 1450 y Fg(\001)p Ff(\014)6 b Fe(k)14 b Fg(_)-41 b Ff(g)1175 1465 y Fo(n)p Fl(\000)p Ft(1)1312 1450 y Fe(k)1362 1466 y Fo(c)p Ft(\()p Fo(a)p Ft(\))1511 1450 y Fg(+)22 b Fe(k)p Ff(r)1703 1465 y Fo(n)p Fl(\000)p Ft(1)1839 1450 y Fg(\()p Fe(\001)p Ff(;)17 b Fg(\001)p Ff(\014)6 b Fg(\))p Fe(k)2179 1466 y Fo(c)p Ft(\()p Fo(a)p Ft(\))2306 1354 y Fd(\021)730 1683 y Fg(=)17 b Ff(o)p Fg(\(\001)p Ff(\014)6 b Fg(\))p Ff(;)179 b(c)p Fg(\()p Ff(a)p Fg(\))1523 1631 y Ft(def)1534 1683 y Fg(=)1866 1616 y Ff(a\016)1964 1580 y Fo(\025)p 1658 1660 559 4 v 1658 1751 a Fg(1)22 b(+)g Ff(a)p Fg(\()p Ff(\016)1963 1723 y Fo(\025)2031 1751 y Fe(\000)g Fg(1\))2227 1683 y Ff(:)166 1982 y Fg(Since)33 b(the)g(op)s(erator)f Ff(Q)1059 1946 y Fl(0)1082 1982 y Fg([)p Ff(g)1156 1997 y Fo(n)p Fl(\000)p Ft(1)1293 1982 y Fg(])h(is)f(linear)f(and)i(con)m (tin)m(uous,)g(the)g(function)1291 2226 y Ff(Q)1368 2185 y Fl(0)1392 2226 y Fg([)p Ff(g)1466 2241 y Fo(n)p Fl(\000)p Ft(1)1603 2226 y Fg(])17 b(\()o Ff(r)1728 2241 y Fo(n)p Fl(\000)p Ft(1)1865 2226 y Fg(\()p Fe(\001)p Ff(;)g Fg(\001)p Ff(\014)6 b Fg(\)\))21 b(+)h Ff(R)2386 2241 y Fo(n)p Fl(\000)p Ft(1)166 2581 y Fg(ob)s(eys)34 b(the)f(conditions)e(imp)s (osed)h(on)g Ff(r)1634 2596 y Fo(n)1681 2581 y Fg(,)h(th)m(us)48 b(_)-42 b Ff(g)2002 2596 y Fo(n)2082 2581 y Fg(exists)33 b(and)1393 2824 y(_)-41 b Ff(g)1426 2839 y Fo(n)1500 2824 y Fg(=)28 b Ff(Q)1681 2783 y Fl(0)1704 2824 y Fg([)p Ff(g)1778 2839 y Fo(n)p Fl(\000)p Ft(1)1915 2824 y Fg(])17 b(\()d(_)-41 b Ff(g)2044 2839 y Fo(n)p Fl(\000)p Ft(1)2181 2824 y Fg(\))16 b Ff(:)1018 b Fg(\(60\))p 1887 3171 42 42 v 166 3299 a(Let)33 b Ff(g)392 3263 y Ft(\()p Fo(k)r Ft(\))388 3324 y Fo(n)516 3247 y Ft(def)527 3299 y Fg(=)38 b Ff(D)725 3263 y Fo(k)722 3324 y(z)768 3299 y Ff(g)815 3314 y Fo(n)861 3299 y Fg(,)33 b Ff(k)e Fe(2)d Ff(I)-25 b(N)10 b Fg(,)33 b(then)g(claim)d(\(iii\))g(of)i(Prop)s(osition)f(2.1)h (implies)166 3531 y Fi(Corollary)k(2.3)49 b Fh(F)-7 b(or)43 b(every)g Ff(n)h Fe(2)h Ff(I)-25 b(N)1632 3546 y Ft(0)1715 3531 y Fh(and)43 b Ff(k)k Fe(2)e Ff(I)-25 b(N)10 b Fh(,)43 b(the)h(map)f Ff(g)2746 3495 y Ft(\()p Fo(k)r Ft(\))2742 3556 y Fo(n)2887 3531 y Fg(:)h Ff(J)9 b Fg(\()p Ff(g)t Fg(\))43 b Fe(!)h(A)3415 3546 y Ft(1)166 3651 y Fh(is)38 b(di\013er)-5 b(entiable)38 b(on)g Fg(In)m(t)q Ff(J)9 b Fg(\()p Ff(g)t Fg(\))p Fh(,)38 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y(men)m(ts)i(of)e Fe(f)p Ff(g)664 5154 y Fo(n)746 5139 y Fe(j)h Ff(g)856 5154 y Ft(0)895 5139 y Fg(\()p Ff(z)t Fg(\))e(=)e Ff(C)7 b Fg(\()p Ff(\014)f Fg(\))p Ff(g)t Fg(\()p Ff(\014)g(z)t Fg(\))p Ff(;)51 b(\014)38 b(<)32 b(\014)1945 5154 y Fl(\003)1984 5139 y Fe(g)j Fg(remain)f(in)g(the)i (stable)f(manifold)d(of)166 5259 y Ff(g)213 5274 y Ft(1)p Fo(;)p Fl(\003)307 5259 y Fg(.)e(The)g(con)m(v)m(ergence)i(of)d(the)h (corresp)s(onding)g(sequences)i Fe(f)p Ff(f)2531 5274 y Fo(n)2578 5259 y Fe(g)e Fg(will)d(b)s(e)i(obtained)g(as)166 5380 y(a)j(direct)h(corollary)-8 b(.)1745 5712 y(19)p eop %%Page: 20 20 20 19 bop 166 83 a Fi(3)112 b(Pro)s(ofs)166 426 y Fh(3.1)99 b(Main)35 b(L)-5 b(emmas)166 769 y Fg(The)33 b(case)g(where)h(the)e (initial)d(elemen)m(ts)j(of)g(the)g(sequences)k(considered)d(are)f(c)m (hosen)h(in)166 890 y Fe(G)j Fg(has)29 b(already)g(b)s(een)h(discussed) h(ab)s(o)m(v)m(e.)g(Th)m(us)g(from)d(no)m(w)i(on)f(w)m(e)i(supp)s(ose)f (that)f(these)166 1010 y(functions)k(are)f(c)m(hosen)i(outside)f(of)f Fe(G)6 b Fg(.)33 b(W)-8 b(e)33 b(set)g(\(see)h(\(34\),)e(\(35\)\))429 1235 y Ff(g)476 1250 y Fo(n)522 1235 y Fg(\()p Ff(z)t Fg(\))d(=)e Ff(C)849 1250 y Fo(n)913 1235 y Fg(exp)q(\()p Ff(')1164 1250 y Fo(n)1210 1235 y Fg(\()p Ff(z)t Fg(\)\))p Ff(;)86 b Fg(~)-52 b Ff(g)1530 1250 y Fo(n)1576 1235 y Fg(\()p Ff(z)t Fg(\))29 b(=)e(exp)q(\()16 b(~)-65 b Ff(')2084 1250 y Fo(n)2131 1235 y Fg(\()p Ff(z)t Fg(\)\))p Ff(;)82 b(')2467 1250 y Fo(n)2514 1235 y Fg(\(0\))28 b(=)43 b(~)-65 b Ff(')2834 1250 y Fo(n)2881 1235 y Fg(\(0\))27 b(=)h(0)p Ff(;)67 b Fg(\(73\))166 1563 y(and)33 b(for)f Ff(k)f Fe(2)d Ff(I)-25 b(N)10 b Fg(,)361 1914 y Ff(')425 1872 y Ft(\()p Fo(k)r Ft(\))425 1938 y Fo(n)583 1861 y Ft(def)594 1914 y Fg(=)70 b(\()p Ff(D)862 1872 y Fo(k)905 1914 y Ff(')969 1929 y Fo(n)1016 1914 y Fg(\)\(0\))p Ff(;)97 b Fg(~)-65 b Ff(')1351 1872 y Ft(\()p Fo(k)r Ft(\))1351 1938 y Fo(n)1508 1861 y Ft(def)1519 1914 y Fg(=)71 b(\()p Ff(D)1788 1872 y Fo(k)1846 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4524 y Fo(n)p Fl(\000)p Ft(1)2848 4381 y Fd(\033)2927 4502 y Ff(:)326 b Fg(\(79\))166 4868 y Fh(Equalities)35 b(in)f(\(77\){\(79\))g(hold)g(only)h(in)f(the)h(c)-5 b(ase)34 b Ff(\022)d Fg(=)d(0)p Fh(.)34 b(F)-7 b(urther)361 5217 y Ff(')425 5176 y Ft(\(2\))425 5242 y Fo(n)536 5217 y Ff(>)16 b(\016)675 5176 y Ft(2)p Fo(\025)p Fl(\000)p Ft(1)846 5217 y Ff(\024)902 5176 y Ft(4)902 5242 y Fo(n)949 5217 y Ff(')1013 5166 y Ft(\(2\))1013 5239 y Fo(n)p Fl(\000)p Ft(1)1150 5217 y Fg(;)2103 b(\(80\))361 5380 y Ff(')425 5339 y Ft(\(1\))425 5404 y Fo(n)536 5380 y Ff(<)16 b(\024)684 5395 y Fo(n)732 5380 y Ff(')796 5329 y Ft(\(1\))796 5402 y Fo(n)p Fl(\000)p Ft(1)933 5380 y Fg(;)2320 b(\(81\))1745 5712 y(20)p eop %%Page: 21 21 21 20 bop 846 101 a Ff(')910 60 y Ft(\(1\))910 125 y Fo(n)1032 101 y Ff(>)28 b(\024)1192 116 y Fo(n)1239 101 y Ff(')1303 50 y Ft(\(1\))1303 123 y Fo(n)p Fl(\000)p Ft(1)1462 101 y Fg(+)22 b(\()p Ff(\022)j Fg(+)d(1\)\(1)g Fe(\000)g Ff(\016)2108 60 y Fl(\000)p Fo(\025)2208 101 y 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Ft(2)2133 5345 y Fg(\()p Ff(\022)j Fg(+)d(1\)\()p Ff(\014)2525 5304 y Ft(+)2519 5369 y(0)2584 5345 y Fg(\))2622 5304 y Ft(2)2661 5345 y Ff(m)2746 5360 y Ft(2)2786 5345 y Fg(\()p Ff(g)t Fg(\)])2940 5304 y Ft(1)p Fo(=)p Ft(2)3049 5345 y Fe(g)1745 5712 y Fg(24)p eop %%Page: 25 25 25 24 bop 1007 91 a Fg(=)16 b(1)22 b(+)g([)p Ff(\016)1342 50 y Fl(\000)p Ft(1)p Fo(=)p Ft(2)1507 91 y Fg(\()p Ff(\022)k Fg(+)c(1\)\()p Ff(\014)1900 50 y Ft(+)1894 116 y(0)1958 91 y Fg(\))1996 50 y Ft(2)2035 91 y Ff(m)2120 106 y Ft(2)2160 91 y Fg(\()p Ff(g)t Fg(\)])2314 50 y Ft(1)p Fo(=)p Ft(2)2451 91 y Fe(\024)28 b Fg(1)22 b(+)g Ff(\014)2786 50 y Ft(+)2780 116 y(0)2845 91 y Ff(m)2930 106 y Ft(1)2969 91 y Fg(\()p Ff(g)t Fg(\))p Ff(:)166 326 y Fg(Therefore,)37 b Ff(\014)699 285 y Ft(+)693 348 y(0)790 326 y Fe(2)d Ff(J)9 b Fg(\()p Ff(g)t Fg(\))35 b(and)g Ff(s)1353 290 y Ft(1)1353 351 y(0)1393 326 y Ff(;)h(s)1502 290 y Ft(2)1502 351 y(0)1577 326 y Fg(are)f(true.)h(T)-8 b(o)36 b(pro)m(v)m(e)h Ff(s)2437 290 y Ft(3)2437 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Ff(\033)2789 3820 y Fl(\000)p Ft(1)2883 3856 y Fg(,)h(then)g(w)m(e)h(get)166 3976 y(from)31 b(\(80\))1147 4225 y Ff(')1211 4184 y Ft(\(2\))1211 4250 y Fo(n)1333 4225 y Ff(>)d(\033)1496 4184 y Fl(\000)p Ft(4)1590 4225 y Ff(\016)1637 4184 y Ft(2)p Fo(\025)p Fl(\000)p Ft(1)1808 4225 y Ff(')1872 4174 y Ft(\(2\))1872 4247 y Fo(n)p Fl(\000)p Ft(1)2037 4225 y Fe(\025)g Ff(')2206 4174 y Ft(\(2\))2206 4247 y Fo(n)p Fl(\000)p Ft(1)2370 4225 y Fe(\025)h Fg(\010)2546 4184 y Ft(\(2\))2640 4225 y Ff(;)166 4568 y Fg(where)k(the)g(follo)m (wing)c(estimates)j(w)m(ere)h(used:)g Ff(')1992 4532 y Ft(\(2\))1992 4593 y Fo(n)2114 4568 y Ff(<)28 b Fg(0,)j Fe(8)p Ff(n)e Fe(2)f Ff(I)-25 b(N)2667 4583 y Ft(0)2738 4568 y Fg(;)32 b Ff(\033)2856 4532 y Fl(\000)p Ft(1)2978 4568 y Fe(\024)c Ff(\016)3130 4532 y Ft(\(1)p Fl(\000)p Ft(2)p Fo(\025)p Ft(\))p Fo(=)p Ft(4)3427 4568 y Fg(.)p 166 4681 42 42 v 166 4917 a Fi(Corollary)36 b(3.1)49 b Fh(The)34 b(ine)-5 b(quality)35 b(\(75\))f(holds)g(for)h Ff(\014)e Fe(\024)28 b Ff(\014)2369 4880 y Ft(+)2363 4941 y Fo(n)2428 4917 y Fh(.)35 b(Thus)f Ff(J)2793 4932 y Fo(n)2875 4917 y Fh(is)h(nonempty.)166 5144 y Fi(Lemma)i(3.3)49 b Fh(Ther)-5 b(e)34 b(exists)h Ff(\014)1357 5159 y Fl(\003)1424 5144 y Fe(2)28 b Ff(J)9 b Fg(\()p Ff(g)t Fg(\))34 b Fh(such)h(that,)g (for)g Ff(\014)e Fg(=)28 b Ff(\014)2596 5159 y Fl(\003)2635 5144 y Fh(,)1236 5380 y Fg(1)f Ff(<)h(')1480 5339 y Ft(\(1\))1480 5404 y Fo(n)1602 5380 y Ff(<)f Fg(\010)1775 5339 y Ft(\(1\))1870 5380 y Ff(;)52 b Fe(8)p Ff(n)28 b Fe(2)g Ff(I)-25 b(N)2290 5395 y Ft(0)2330 5380 y Ff(:)875 b Fg(\(107\))1745 5712 y(25)p eop %%Page: 26 26 26 25 bop 166 83 a Fh(F)-7 b(or)41 b Ff(\014)46 b(<)40 b(\014)627 98 y Fl(\003)667 83 y Fh(,)i(the)g(upp)-5 b(er)41 b(estimate)h(also)f(holds)g(and,)g(mor)-5 b(e)g(over,)41 b(ther)-5 b(e)42 b(exists)f Ff(K)48 b Fg(=)166 203 y Ff(K)7 b Fg(\()p Ff(\014)f Fg(\))27 b Ff(>)h Fg(0)p Fh(,)34 b(such)h(that)1268 454 y Ff(')1332 413 y Ft(\(1\))1332 479 y Fo(n)1454 454 y Ff(<)28 b(K)7 b(\016)1695 413 y Fl(\000)p Fo(\025n)1838 454 y Ff(;)51 b Fe(8)p Ff(n)29 b Fe(2)f Ff(I)-25 b(N)2258 469 y Ft(0)2297 454 y Ff(:)908 b Fg(\(108\))166 840 y Fi(Pro)s(of.)76 b Fg(Consider)36 b(the)f(set)h(\001)1361 855 y Fo(n)1476 787 y Ft(def)1487 840 y Fg(=)43 b Fe(f)p Ff(\014)38 b Fe(2)33 b Fg(\(0)p Ff(;)17 b(\014)2040 804 y Ft(+)2034 864 y Fo(n)2098 840 y Fg(\))32 b Fe(j)g Fg(1)h Ff(<)f(')2482 804 y Ft(\(1\))2482 864 y Fo(n)2609 840 y Ff(<)g Fg(\010)2787 804 y Ft(\(1\))2882 840 y Fe(g)p Fg(.)j(Just)h(ab)s(o)m(v)m(e)166 960 y(w)m(e)g(ha)m(v)m(e) g(sho)m(wn)g(that)f(\001)1132 975 y Fo(n)1211 960 y Fe(\022)d Fg(\()p Ff(\014)1419 924 y Fl(\000)1413 985 y Fo(n)1478 960 y Ff(;)17 b(\014)1583 924 y Ft(+)1577 985 y Fo(n)1641 960 y Fg(\),)35 b(\001)1822 975 y Fo(n)1904 960 y Fg(is)f(nonempt)m(y)i (and)f(op)s(en.)g(Let)g(us)g(pro)m(v)m(e)166 1080 y(that)j(\001)464 1095 y Fo(n)549 1080 y Fe(\022)g Fg(\001)745 1095 y Fo(n)p Fl(\000)p Ft(1)882 1080 y Fg(.)h(Supp)s(ose)g(there)g(exists)g(some)f 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y Fo(n)2359 1452 y Fg(b)s(e)h(the)g(closure)f(of)g(\001) 3161 1467 y Fo(n)3209 1452 y Fg(,)g(then)983 1703 y Ff(D)1064 1718 y Fo(n)1139 1703 y Fg(=)g Fe(f)p Ff(\014)33 b Fe(2)28 b Fg([)p Ff(\014)1563 1662 y Fl(\000)1557 1728 y Fo(n)1621 1703 y Ff(;)17 b(\014)1726 1662 y Ft(+)1720 1728 y Fo(n)1785 1703 y Fg(])28 b Fe(j)f Fg(1)g Fe(\024)i Ff(')2141 1662 y Ft(\(1\))2141 1728 y Fo(n)2262 1703 y Fe(\024)g Fg(\010)2438 1662 y Ft(\(1\))2532 1703 y Fe(g)p Ff(:)623 b Fg(\(109\))166 2069 y Ff(D)247 2084 y Fo(n)330 2069 y Fg(is)36 b(nonempt)m(y)g(and)g Ff(D)1164 2084 y Fo(n)1245 2069 y Fe(\022)e Ff(D)1437 2084 y Fo(n)p Fl(\000)p Ft(1)1608 2069 y Fe(\022)g Ff(:::)g Fe(\022)g Ff(D)2026 2084 y Ft(0)2099 2069 y Fe(\022)g Ff(J)9 b Fg(\()p Ff(g)t Fg(\).)35 b(Let)i Ff(D)2722 2084 y Fl(\003)2795 2069 y Fg(=)2904 2002 y Fd(T)2973 2089 y Fo(n)3037 2069 y Ff(D)3118 2084 y Fo(n)3201 2069 y Fg(,)f(then)166 2189 y Ff(D)247 2204 y Fl(\003)325 2189 y Fg(is)j(also)f(nonempt)m(y)h(and)g(closed,)h(and)f Ff(D)1888 2204 y Fl(\003)1966 2189 y Fe(\032)g Ff(J)9 b Fg(\()p Ff(g)t Fg(\))p Ff(:)38 b Fg(No)m(w)h(let)g(us)g(sho)m(w)h (that,)f(for)166 2309 y(ev)m(ery)31 b Ff(\014)i Fe(2)28 b Ff(D)683 2324 y Fl(\003)723 2309 y Ff(;)h Fg(the)h(estimates)e (\(107\))h(hold.)f(Indeed,)j(directly)d(from)g(the)i(de\014nition)e(of) 166 2430 y Ff(D)247 2445 y Fl(\003)319 2430 y Fg(one)33 b(has)1347 2681 y(1)27 b Fe(\024)h Ff(')1592 2639 y Ft(\(1\))1592 2705 y Fo(n)1714 2681 y Fe(\024)g Fg(\010)1889 2639 y Ft(\(1\))1984 2681 y Ff(;)49 b Fe(8)p Ff(n)28 b Fe(2)h Ff(I)-25 b(N)2402 2696 y Ft(0)2441 2681 y Ff(:)166 3046 y Fg(Supp)s(ose)36 b Ff(')615 3010 y Ft(\(1\))615 3071 y Fo(n)742 3046 y Fg(=)c(1)k(for)e(some)i Ff(n)c Fe(2)h Ff(I)-25 b(N)1629 3061 y Ft(0)1704 3046 y Fg(and)35 b Ff(\014)k Fe(2)32 b Ff(D)2169 3061 y Fl(\003)2209 3046 y Fg(,)j(then)h Ff(')2560 3010 y Ft(\(1\))2560 3071 y Fo(m)2687 3046 y Ff(<)c Fg(1)j(for)g(all)e Ff(m)g(>)f(n)166 3167 y Fg(\(see)42 b(\(81\)\))e(\)\).)h(The)h(latter)d(means)i(that)g (this)f Ff(\014)47 b Fg(do)s(es)41 b(not)g(b)s(elong)e(to)i(all)e Ff(D)3157 3182 y Fo(m)3264 3167 y Fg(with)166 3287 y Ff(m)29 b(>)f(n)p Fg(.)33 b(This)g(con)m(tradicts)g(the)h(supp)s (osition)d Ff(\014)j Fe(2)29 b Ff(D)2179 3302 y Fl(\003)2218 3287 y Ff(:)k Fg(The)h(case)g Ff(')2750 3251 y Ft(\(1\))2750 3312 y Fo(n)2873 3287 y Fg(=)28 b(\010)3047 3251 y Ft(\(1\))3174 3287 y Fg(can)33 b(b)s(e)166 3407 y(excluded)40 b(similarly)-8 b(.)36 b(Set)j Ff(\014)1230 3422 y Fl(\003)1308 3407 y Fg(=)g(min)15 b Ff(D)1683 3422 y Fl(\003)1723 3407 y Fg(.)39 b(W)-8 b(e)39 b(ha)m(v)m(e)i(just)e(pro)m(v)m(ed)i(that,)e (for)f Ff(\014)44 b Fg(=)39 b Ff(\014)3387 3422 y Fl(\003)3427 3407 y Fg(,)166 3528 y(\(107\))29 b(holds,)g(th)m(us)i(it)e(remains)g (to)g(pro)m(v)m(e)i(the)f(second)h(part)f(of)f(Lemma.)f(T)-8 b(ak)m(e)31 b Ff(\014)j(<)27 b(\014)3387 3543 y Fl(\003)3427 3528 y Ff(:)166 3648 y Fg(If)42 b Ff(')337 3612 y Ft(\(1\))337 3673 y Fo(n)475 3648 y Ff(>)h Fg(1)f(for)f(all)f Ff(n)k Fe(2)g Ff(I)-25 b(N)1306 3663 y Ft(0)1346 3648 y Ff(;)42 b Fg(then)g(either)g(\(107\))f(holds)h(or)f(there)i(exists)g(suc)m(h)g Ff(n)3414 3663 y Ft(0)166 3769 y Fg(that)37 b Ff(')446 3732 y Ft(\(1\))446 3793 y Fo(n)489 3802 y Fc(0)576 3769 y Fe(\025)g Fg(\010)760 3732 y Ft(\(1\))855 3769 y Ff(:)g Fg(This)h(means)f(either)g Ff(\014)42 b Fe(2)36 b Ff(D)2011 3784 y Fl(\003)2088 3769 y Fg(or)h Ff(\014)42 b(>)35 b Fg(inf)23 b Ff(\014)2616 3732 y Ft(+)2610 3793 y Fo(n)2675 3769 y Ff(:)37 b Fg(Both)h(these)g(cases)166 3911 y(con)m(tradict)26 b(the)g(de\014nition)f(of)h Ff(\014)1371 3926 y Fl(\003)1410 3911 y Fg(.)g(Hence)i(there)e(exists)h Ff(n)2310 3926 y Ft(0)2376 3911 y Fg(suc)m(h)g(that)f Ff(')2858 3861 y Ft(\(1\))2858 3933 y Fo(n)2901 3942 y Fc(0)2935 3933 y Fl(\000)p Ft(1)3057 3911 y Fe(\024)i Fg(1)p Ff(;)e Fg(then)166 4032 y Ff(')230 3996 y Ft(\(1\))230 4056 y Fo(n)356 4032 y Ff(<)32 b Fg(1)j(for)f(all)f Ff(n)g Fe(\025)f Ff(n)1095 4047 y Ft(0)1135 4032 y Ff(:)j Fg(In)g(what)h (follo)m(ws,)e(the)h(de\014nition)f(\(76\))h(and)g(the)g(estimate)166 4152 y(\(81\))g(imply)e(for)i(the)g(sequences)k Fe(f)p Ff(')1533 4116 y Ft(\(1\))1533 4177 y Fo(n)1627 4152 y Ff(;)52 b(n)32 b Fe(\025)h Ff(n)1964 4167 y Ft(0)2003 4152 y Fe(g)i Fg(and)h Fe(f)p Ff(\024)2387 4167 y Fo(n)2434 4152 y Ff(;)51 b(n)33 b Fe(\025)f Ff(n)2770 4167 y Ft(0)2810 4152 y Fe(g)j Fg(to)g(b)s(e)g(strictly)166 4273 y(decreasing.)e(Then)h (for)e(all)e Ff(n)e(>)g(n)1452 4288 y Ft(0)1492 4273 y Fg(,)k(one)h(has)g(\(see)h(\(81\)\))736 4537 y Ff(')800 4496 y Ft(\(1\))800 4562 y Fo(n)922 4537 y Ff(<)28 b(\024)1082 4552 y Fo(n)1129 4537 y Ff(')1193 4486 y Ft(\(1\))1193 4559 y Fo(n)p Fl(\000)p Ft(1)1357 4537 y Ff(<)g(:::)g(<)g(\024)1730 4552 y Fo(n)1777 4537 y Ff(\024)1833 4552 y Fo(n)p Fl(\000)p Ft(1)1970 4537 y Ff(:::\024)2107 4552 y Fo(n)2150 4561 y Fc(0)2185 4552 y Ft(+1)2279 4537 y Ff(')2343 4496 y Ft(\(1\))2343 4562 y Fo(n)2386 4571 y Fc(0)2465 4537 y Ff(<)g Fg(\()p Ff(\024)2663 4552 y Fo(n)2706 4561 y Fc(0)2740 4552 y Ft(+1)2834 4537 y Fg(\))2872 4496 y Fo(n)p Fl(\000)p Fo(n)3013 4505 y Fc(0)3052 4537 y Ff(:)166 4903 y Fg(Since)33 b Ff(\024)477 4918 y Fo(n)520 4927 y Fc(0)554 4918 y Ft(+1)677 4903 y Ff(<)28 b Fg(1)p Ff(;)k Fg(one)h(has)1242 4836 y Fd(P)1330 4863 y Fl(1)1330 4927 y Fo(n)p Ft(=0)1484 4903 y Ff(')1548 4867 y Ft(\(1\))1548 4927 y Fo(n)1670 4903 y Ff(<)28 b Fe(1)p Ff(:)k Fg(Th)m(us)i(there)g (exists)f(0)28 b Ff(<)g(K)2963 4918 y Ft(0)3030 4903 y Ff(<)g Fe(1)k Fg(suc)m(h)166 5023 y(that)1554 5199 y Fl(1)1536 5224 y Fd(Y)1523 5405 y Fo(n)p Ft(=1)1672 5307 y Ff(\027)1720 5322 y Fo(n)1795 5254 y Ft(def)1806 5307 y Fg(=)38 b Ff(K)2003 5322 y Ft(0)2043 5307 y Ff(:)1162 b Fg(\(110\))1745 5712 y(26)p eop %%Page: 27 27 27 26 bop 166 83 a Fg(Finally)-8 b(,)30 b(w)m(e)k(apply)e(\(81\))g (once)h(again)e(and)i(obtain)363 453 y Ff(')427 412 y Ft(\(1\))427 477 y Fo(n)549 453 y Ff(<)28 b(\016)700 412 y Fl(\000)p Fo(\025n)843 453 y Ff(\027)891 468 y Fo(n)938 453 y Ff(\027)986 468 y Fo(n)p Fl(\000)p Ft(1)1123 453 y Ff(:::\027)1252 468 y Ft(1)1292 453 y Ff(')1356 402 y Ft(\(1\))1356 475 y(0)1478 453 y Ff(<)g(\016)1629 412 y Fl(\000)p Fo(\025n)1772 453 y Ff(K)1855 468 y Ft(0)1894 453 y Ff(')1958 402 y Ft(\(1\))1958 475 y(0)2112 400 y(def)2123 453 y Fg(=)71 b Ff(K)7 b(\016)2407 412 y Fl(\000)p Fo(\025n)2550 453 y Ff(;)49 b Fe(8)p Ff(n)29 b Fe(2)f Ff(I)-25 b(N)2968 468 y Ft(0)3007 453 y Ff(:)198 b Fg(\(111\))p 1789 959 42 42 v 166 1088 a(No)m(w)28 b(w)m(e)h(state)f(the)g(lemmas)e (the)i(pro)s(of)f(of)g(our)h(theorems)g(directly)f(follo)m(ws)f(from.)h (The)166 1208 y(\014rst)32 b(four)f(lemmas)e(describ)s(e)j(the)g (sequences)i Fe(f)p Ff(g)2000 1223 y Fo(n)2047 1208 y Fe(g)d Fg(de\014ned)h(b)m(y)g(\(54\))f(whose)h(elemen)m(ts)166 1328 y(ha)m(v)m(e)i(the)f(form)e(\(73\).)166 1613 y Fi(Lemma)37 b(3.4)49 b Fh(F)-7 b(or)34 b(every)h Ff(\022)c Fe(\025)d Fg(0)35 b Fh(and)f Ff(g)d Fe(2)d(L)p Fg(\()p Ff(\025)p Fg(\))p Fh(,)34 b(ther)-5 b(e)35 b(exists)f Ff(\014)2651 1628 y Fl(\003)2719 1613 y Fe(2)28 b Ff(J)9 b Fg(\()p Ff(g)t Fg(\))34 b Fh(such)h(that,)216 1961 y Fg(\(i\))126 b Fh(for)34 b Ff(\014)f Fg(=)28 b Ff(\014)848 1976 y Fl(\003)888 1961 y Fh(,)34 b Fg(lim)1088 1976 y Fo(n)p Fl(!1)1292 1961 y Ff(')1356 1925 y Ft(\(1\))1356 1986 y Fo(n)1478 1961 y Fg(=)28 b(1)35 b Fh(and)69 b Fg(lim)2025 1976 y Fo(n)p Fl(!1)2230 1961 y Ff(')2294 1925 y Ft(\(2\))2294 1986 y Fo(n)2416 1961 y Fg(=)27 b(0)p Fh(;)216 2165 y Fg(\(ii\))98 b Fh(for)34 b Ff(\014)f(<)28 b(\014)848 2180 y Fl(\003)888 2165 y Fh(,)34 b Fg(lim)1088 2180 y Fo(n)p Fl(!1)1292 2165 y Ff(')1356 2129 y Ft(\(1\))1356 2190 y Fo(n)1478 2165 y Fg(=)28 b(0)35 b Fh(and)69 b Fg(lim)2025 2180 y Fo(n)p Fl(!1)2230 2165 y Ff(')2294 2129 y Ft(\(2\))2294 2190 y Fo(n)2416 2165 y Fg(=)27 b(0)p Fh(.)166 2467 y Fi(Lemma)37 b(3.5)49 b Fh(L)-5 b(et)35 b Ff(\022)s Fh(,)f Ff(g)k Fh(and)c Ff(\014)1364 2482 y Fl(\003)1437 2467 y Fh(b)-5 b(e)34 b(as)g(ab)-5 b(ove.)34 b(Then)f(ther)-5 b(e)35 b(exists)f Ff(C)g Fg(:)28 b(\(0)p Ff(;)17 b(\014)3081 2482 y Fl(\003)3120 2467 y Fg(])28 b Fe(!)f Ff(I)-25 b(R)3395 2482 y Ft(+)166 2588 y Fh(such)41 b(that)h(the)f(se)-5 b(quenc)g(e)40 b Fe(f)p Ff(C)1290 2603 y Fo(n)1378 2588 y Fe(j)h Ff(n)f Fe(2)f Ff(I)-25 b(N)1757 2603 y Ft(0)1796 2588 y Ff(;)58 b(C)1951 2603 y Fo(n)2037 2588 y Fg(=)39 b Ff(g)2199 2603 y Fo(n)2246 2588 y Fg(\(0\))p Ff(;)58 b(C)2526 2603 y Ft(0)2604 2588 y Fg(=)40 b Ff(C)7 b Fg(\()p Ff(\014)f Fg(\))p Fe(g)p Fh(,)40 b(c)-5 b(onver)g(ges)166 2708 y(to)37 b Ff(C)355 2723 y Ft(2)p Fo(;)p Fl(\003)486 2708 y Fh(\(r)-5 b(esp.)36 b(to)i Ff(C)953 2723 y Ft(1)p Fo(;)p Fl(\003)1047 2708 y Fh(\))e(given)h(by)g(\(64\))f(for)h Ff(\014)g Fg(=)31 b Ff(\014)2137 2723 y Fl(\003)2213 2708 y Fh(\(r)-5 b(esp.)37 b Ff(\014)g(<)31 b(\014)2745 2723 y Fl(\003)2785 2708 y Fh(\).)36 b(The)h(se)-5 b(quenc)g(e)166 2828 y Fe(f)p Ff(C)286 2843 y Fo(n)367 2828 y Fe(j)35 b Ff(C)500 2843 y Ft(0)567 2828 y Ff(>)27 b(C)7 b Fg(\()p Ff(\014)f Fg(\))p Fe(g)34 b Fh(is)h(diver)-5 b(gent,)34 b(the)h(se)-5 b(quenc)g(e)34 b Fe(f)p Ff(C)2198 2843 y Fo(n)2279 2828 y Fe(j)g Ff(C)2411 2843 y Ft(0)2478 2828 y Ff(<)28 b(C)7 b Fg(\()p Ff(\014)f Fg(\))p Fe(g)34 b Fh(tends)g(to)h(zer)-5 b(o.)166 3113 y Fi(Lemma)37 b(3.6)49 b Fh(L)-5 b(et)29 b Ff(\022)s Fh(,)f Ff(g)t Fh(,)f Ff(\014)1186 3128 y Fl(\003)1254 3113 y Fh(and)h Ff(C)7 b Fg(\()p Ff(\014)f Fg(\))27 b Fh(b)-5 b(e)28 b(as)g(ab)-5 b(ove.)27 b(Then)h(for)g Ff(\014)33 b Fg(=)28 b Ff(\014)2840 3128 y Fl(\003)2879 3113 y Fh(,)g(the)h(se)-5 b(quenc)g(e)166 3234 y Fe(f)p Ff(g)263 3249 y Fo(n)347 3234 y Fe(j)37 b Ff(n)c Fe(2)h Ff(I)-25 b(N)709 3249 y Ft(0)748 3234 y Ff(;)55 b(g)877 3249 y Ft(0)916 3234 y Fg(\()p Ff(z)t Fg(\))33 b(=)g Ff(C)7 b Fg(\()p Ff(\014)1353 3249 y Fl(\003)1392 3234 y Fg(\))p Ff(g)t Fg(\()p Ff(\014)f(z)t Fg(\))p Fe(g)37 b Fh(c)-5 b(onver)g(ges)36 b(in)i Fe(A)2394 3249 y Ft(1)2470 3234 y Fh(to)g Ff(g)2637 3249 y Ft(2)p Fo(;)p Fl(\003)2731 3234 y Fg(\()p Ff(z)t Fg(\))c(=)e Ff(C)3068 3249 y Ft(2)p Fo(;)p Fl(\003)3179 3234 y Fg(exp)q(\()p Ff(z)t Fg(\))166 3354 y Fh(de\014ne)-5 b(d)34 b(by)h(\(69\).)166 3639 y Fi(Lemma)i(3.7)49 b Fh(L)-5 b(et)35 b Ff(\022)s Fh(,)g Ff(g)t Fh(,)f Ff(\014)1206 3654 y Fl(\003)1281 3639 y Fh(and)g Ff(C)7 b Fg(\()p Ff(\014)f Fg(\))34 b Fh(b)-5 b(e)35 b(as)f(ab)-5 b(ove.)34 b(Then)g(for)h(every)g Ff(\014)e(<)28 b(\014)3171 3654 y Fl(\003)3210 3639 y Fh(,)216 3987 y Fg(\(i\))126 b Fh(the)34 b(se)-5 b(quenc)g(e)34 b Fe(f)p Ff(g)1100 4002 y Fo(n)1182 3987 y Fe(j)g Ff(n)28 b Fe(2)g Ff(I)-25 b(N)1531 4002 y Ft(0)1570 3987 y Ff(;)52 b(g)1696 4002 y Ft(0)1735 3987 y Fg(\()p Ff(z)t Fg(\))28 b(=)g Ff(C)7 b Fg(\()p Ff(\014)f Fg(\))p Ff(g)t Fg(\()p Ff(\014)g(z)t Fg(\))p Fe(g)33 b Fh(c)-5 b(onver)g(ges)34 b(in)g Fe(A)3160 4002 y Ft(1)446 4168 y Fh(to)h Ff(g)610 4183 y Ft(1)p Fo(;)p Fl(\003)704 4168 y Fg(\()p Ff(z)t Fg(\))28 b Fe(\021)g Fg(1)35 b Fh(;)216 4372 y Fg(\(ii\))98 b Fh(the)34 b(se)-5 b(quenc)g(e)34 b Fe(f)s Fg(~)-52 b Ff(g)1100 4387 y Fo(n)1182 4372 y Fe(j)34 b Ff(n)28 b Fe(2)g Ff(I)-25 b(N)1531 4387 y Ft(0)1570 4372 y Ff(;)55 b Fg(~)-52 b Ff(g)1696 4387 y Ft(0)1735 4372 y Fg(\()p Ff(z)t Fg(\))28 b(=)g Ff(g)t Fg(\()p Ff(\014)6 b(z)t Fg(\))p Fe(g)34 b Fh(de\014ne)-5 b(d)33 b(by)j(\(55\))446 4552 y(c)-5 b(onver)g(ges)33 b(in)i Fe(A)1080 4567 y Fo(')1164 4552 y Fh(to)k Fg(~)-53 b Ff(g)1328 4567 y Fl(\003)1367 4552 y Fg(\()p Ff(z)t Fg(\))29 b(=)e(exp)q(\()p Ff('z)t Fg(\))36 b Fh(with)e(c)-5 b(ertain)35 b Ff(')28 b Fg(=)f Ff(')p Fg(\()p Ff(\014)6 b Fg(\))27 b Ff(>)h Fg(0)p Fh(.)166 4854 y Fg(Directly)21 b(from)g(the)i(de\014nitions)e (\(47\),)h(\(48\),)g(and)g(\(54\))g(one)h(has)f(the)h(follo)m(wing)c (corollary)166 4974 y(of)32 b(the)h(ab)s(o)m(v)m(e)h(lemmas.)166 5259 y Fi(Lemma)j(3.8)49 b Fh(F)-7 b(or)46 b(every)h Ff(\022)53 b Fe(\025)e Fg(0)46 b Fh(and)h Ff(g)53 b Fe(2)d(L)p Fg(\()p Ff(\025)p Fg(\))p Fh(,)d(ther)-5 b(e)47 b(exist)f Ff(\034)2772 5274 y Fl(\003)2862 5259 y Fe(2)51 b Ff(I)8 b Fg(\()p Ff(g)t Fg(\))46 b Fh(and)g(a)166 5380 y(function)35 b Ff(C)f Fg(:)28 b([0)p Ff(:\034)854 5395 y Fl(\003)894 5380 y Fg(])g Fe(!)f Ff(I)-25 b(R)1169 5395 y Ft(+)1228 5380 y Fh(,)34 b(such)h(that:)1745 5712 y Fg(27)p eop %%Page: 28 28 28 27 bop 216 126 a Fg(\(i\))126 b Fh(for)34 b Ff(\034)39 b(<)28 b(\034)828 141 y Fl(\003)868 126 y Fh(,)34 b(the)h(se)-5 b(quenc)g(e)34 b Fe(f)p Ff(f)1588 141 y Fo(n)1670 126 y Fe(j)h Ff(n)28 b Fe(2)g Ff(I)-25 b(N)2019 141 y Ft(0)2058 126 y Ff(;)52 b(f)2185 141 y Ft(0)2224 126 y Fg(\()p Ff(z)t Fg(\))29 b(=)e Ff(C)7 b Fg(\()p Ff(\034)k Fg(\))p Ff(g)t Fg(\()p Ff(z)t Fg(\))p Fe(g)446 307 y Fh(c)-5 b(onver)g(ges)33 b(in)i Fe(A)1080 335 y Fo(\014)1123 305 y Fa(\000)p Fc(1)1120 348 y Fa(\003)1244 307 y Fh(to)g Ff(f)1409 322 y Ft(1)p Fo(;)p Fl(\003)1503 307 y Fg(\()p Ff(z)t Fg(\))29 b Fe(\021)f Fg(1)p Fh(;)216 511 y Fg(\(ii\))98 b Fh(for)34 b Ff(\034)39 b Fg(=)28 b Ff(\034)828 526 y Fl(\003)868 511 y Fh(,)34 b(the)h(se)-5 b(quenc)g(e)34 b Fe(f)p Ff(f)1588 526 y Fo(n)1670 511 y Fe(j)h Ff(n)28 b Fe(2)g Ff(I)-25 b(N)2019 526 y Ft(0)2058 511 y Ff(;)52 b(f)2185 526 y Ft(0)2224 511 y Fg(\()p Ff(z)t Fg(\))29 b(=)e Ff(C)7 b Fg(\()p Ff(\034)2638 526 y Fl(\003)2678 511 y Fg(\))p Ff(g)t Fg(\()p Ff(z)t Fg(\))p Fe(g)446 692 y Fh(de\014ne)-5 b(d)33 b(by)i(\(47\),)f(\(48\),)h(and)f(\(49\))g (c)-5 b(onver)g(ges)34 b(in)g Fe(A)2427 719 y Fo(\014)2470 690 y Fa(\000)p Fc(1)2467 733 y Fa(\003)2556 692 y Fh(,)446 872 y Ff(\014)501 887 y Fl(\003)568 872 y Fg(=)27 b Ff(\034)713 887 y Fl(\003)753 872 y Fg(\()p Ff(\016)838 836 y Fo(\025)905 872 y Fe(\000)c Fg(1\))1092 836 y Fl(\000)p Ft(1)1221 872 y Fh(to)35 b Ff(f)1386 887 y Ft(2)p Fo(;)p Fl(\003)1480 872 y Fg(\()p Ff(z)t Fg(\))29 b(=)e Ff(\016)1784 836 y Fl(\000)p Fo(\025\022)r(=)p Ft(\()p Fo(\016)r Fl(\000)p Ft(1\))2150 872 y Fg(exp)q(\()p Ff(\014)2398 836 y Fl(\000)p Ft(1)2392 897 y Fl(\003)2492 872 y Ff(z)t Fg(\))p Fh(.)166 1233 y(3.2)99 b(Pr)-5 b(o)g(of)35 b(of)f(The)-5 b(or)g(ems)166 1575 y Fi(Pro)s(of)50 b(of)h(Theorem)e(1.1.)i Fg(As)44 b(it)f(has)h(already)g(b)s(een)g(established,)g(the)g(function)166 1696 y Ff(f)214 1711 y Fo(n)261 1696 y Fg(\()p Ff(t;)17 b(z)t Fg(\))30 b(de\014ned)h(b)m(y)f(\(47\))f(giv)m(es)h(the)g (solution)e(of)h(the)h(problem)e(\(13\))h(pro)m(vided)h(all)d Ff(f)3360 1711 y Fo(m)3427 1696 y Fg(,)166 1816 y Ff(m)38 b Fg(=)g(0)p Ff(;)17 b Fg(1)p Ff(;)g(:)g(:)g(:)e(n)27 b Fe(\000)g Fg(1)38 b(b)s(elong)g(to)g(the)h(domain)e(of)i(the)g(op)s (erators)f Ff(T)2751 1831 y Fo(t)2781 1816 y Fg(,)h Ff(t)f Fe(2)h Fg([0)p Ff(;)17 b Fg(1].)38 b(The)166 1937 y(latter)32 b(fact)g(follo)m(ws)f(from)h(Lemma)f(2.3)h(and)h(Prop)s(osition)d(2.8.) p 2995 1929 42 42 v 166 2057 a Fi(Pro)s(of)25 b(of)h(Theorem)f(1.2.)h Fg(Prop)s(osition)21 b(2.9)h(and)g(Lemma)f(3.8)h(yield)f(that,)h(for)g Ff(\034)39 b(<)28 b(\034)3387 2072 y Fl(\003)3427 2057 y Fg(,)166 2177 y(the)g(sequence)i Fe(f)p Ff(f)826 2192 y Fo(n)873 2177 y Fg(\()p Ff(t;)17 b(z)t Fg(\))p Fe(g)28 b Fg(con)m(v)m(erges)i(in)d Fe(A)1776 2205 y Fo(\014)1819 2175 y Fa(\000)p Fc(1)1816 2219 y Fa(\003)1933 2177 y Fg(to)g Ff(T)2104 2192 y Fo(t)2134 2177 y Fg(\()p Ff(f)2220 2192 y Ft(1)p Fo(;)p Fl(\003)2314 2177 y Fg(\),)g(whic)m(h)i(ma)m(y)e (b)s(e)h(iden)m(tically)166 2298 y(one.)48 b(F)-8 b(or)47 b Ff(\034)65 b Fg(=)54 b Ff(\034)856 2313 y Fl(\003)895 2298 y Fg(,)48 b(one)g(has)g(the)h(same)e(con)m(v)m(ergence)k(to)c Ff(T)2542 2313 y Fo(t)2572 2298 y Fg(\()p Ff(f)2658 2313 y Ft(2)p Fo(;)p Fl(\003)2752 2298 y Fg(\),)h(whic)m(h)g(can)g(b)s(e)166 2418 y(calculated)32 b(explicitly)e(b)m(y)k(means)f(of)f(Prop)s (osition)e(2.7.)p 2858 2410 V 166 2539 a(The)f(pro)s(of)e(of)g (Theorems)i(1.3)f(and)g(1.4)f(follo)m(ws)g(directly)g(from)g(Theorems)i (1.1)e(and)h(1.2)166 2659 y(resp)s(ectiv)m(ely)34 b(on)e(the)h(base)g (of)g(the)g(iden)m(tit)m(y)f(\(23\).)166 2779 y Fi(Pro)s(of)i(of)h (Theorem)f(1.5)g(and)i(Theorem)e(1.6.)g Fg(By)d(the)f(con)m(tin)m(uit)m (y)g(theorem)g(\(see)166 2900 y(e.g.)38 b([2],)f(p.27\),)g(the)h(con)m 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3649 y Fo(\026)1486 3658 y Fb(n)p Fa(\000)p Fc(1)1611 3634 y Fg(\)\()p Ff(z)t Fg(\))1791 3537 y Fd(h)1831 3634 y Ff(T)g Fg(\()p Ff(f)1988 3649 y Fo(\026)2030 3658 y Fb(n)p Fa(\000)p Fc(1)2155 3634 y Fg(\)\(0\))2318 3537 y Fd(i)2357 3560 y Fl(\000)p Ft(1)2467 3634 y Ff(;)115 b(n)28 b Fe(2)g Ff(I)-25 b(N)10 b(:)166 3959 y Fg(If)33 b(one)f(c)m(ho)s(oses)i(the)f(starting)f(elemen)m(t)g Ff(\026)1741 3974 y Ft(0)1808 3959 y Fg(=)c Ff(\027)39 b Fg(suc)m(h)34 b(that)1259 4089 y Fd(Z)1220 4347 y Fo(I)-18 b(R)1286 4322 y Fb(N)1378 4207 y Fg(exp)q(\(\()p Ff(x;)17 b(y)t Fg(\)\))p Ff(\027)6 b Fg(\()p Ff(dy)t Fg(\))26 b(=)i Ff(g)t Fg(\(\()p Ff(x;)17 b(x)p Fg(\)\))p Ff(;)166 4642 y Fg(where)44 b Ff(g)j Fg(is)c(the)g(starting)f(elemen)m(t)h(of)g Fe(f)p Ff(f)1804 4657 y Fo(n)1851 4642 y Fe(g)g Fg(describ)s(ed)h(b)m (y)g(Lemma)e(3.8,)g(then)i(the)166 4763 y(v)-5 b(alidit)m(y)27 b(of)i(Theorem)h(1.5)f(follo)m(ws)f(from)g(this)h(Lemma.)f(The)j (assertion)e(regarding)f(the)166 4883 y(v)-5 b(ariance)37 b(in)g(claim)e(\(ii\))g(ma)m(y)i(b)s(e)h(c)m(hec)m(k)m(ed)i(directly)-8 b(.)37 b(Similarly)-8 b(,)34 b(the)j(transforms)g Ff(F)3370 4898 y Ft(~)-41 b Fo(\026)3406 4906 y Fb(n)166 5004 y Fg(\(27\))30 b(of)37 b(~)-56 b Ff(\026)538 5019 y Fo(n)584 5004 y Fg(,)31 b(de\014ned)h(b)m(y)f(\(32\),)f(and)g(the)h(elemen)m(ts) g(of)e(the)i(sequence)i Fe(f)s Fg(~)-52 b Ff(g)2863 5019 y Fo(n)2910 5004 y Fe(g)p Fg(,)30 b(de\014ned)i(b)m(y)166 5124 y(\(55\))g(and)h(describ)s(ed)g(b)m(y)h(claim)c(\(ii\))g(of)i (Lemma)g(3.7,)g(ob)s(ey)h(the)g(relation)765 5372 y Ff(F)834 5387 y Ft(~)-41 b Fo(\026)870 5395 y Fb(n)917 5372 y Fg(\()955 5267 y Fd(q)p 1038 5267 61 4 v 105 x Ff(\014)6 b(x)p Fg(\))28 b(=)j(~)-53 b Ff(g)1370 5387 y Fo(n)1417 5372 y Fg(\(\()p Ff(x;)17 b(x)p Fg(\)\))p Ff(;)82 b(n)28 b Fe(2)g Ff(I)-25 b(N)10 b(;)114 b(F)2328 5387 y Ft(~)-41 b Fo(\026)2364 5396 y Fc(0)2404 5372 y Fg(\()p Ff(x)p Fg(\))28 b(=)f Ff(g)t Fg(\(\()p Ff(x;)17 b(x)p Fg(\)\))p Ff(:)1745 5712 y Fg(28)p eop %%Page: 29 29 29 28 bop 166 83 a Fg(Then)40 b(the)g(v)-5 b(alidit)m(y)37 b(of)i(Theorem)h(1.6)f(follo)m(ws)f(directly)g(from)g(claim)f(\(ii\))g (of)i(Lemma)166 203 y(3.7.)p 1881 195 42 42 v 166 546 a Fh(3.3)99 b(Pr)-5 b(o)g(of)35 b(of)f(L)-5 b(emmas)166 887 y Fi(Pro)s(of)50 b(of)h(Lemma)g(3.4.)44 b Fg(Consider)g(the)h(case) g Ff(\014)53 b Fg(=)47 b Ff(\014)2360 902 y Fl(\003)2399 887 y Ff(;)d Fg(where)i(\(107\))d(holds)h(and)166 1021 y Ff(')230 970 y Ft(\(2\))230 1043 y(0)364 1021 y Fe(\025)c Fg(\010)551 985 y Ft(\(2\))685 1021 y Fg(b)m(y)h(statemen)m(t)f Ff(s)1334 985 y Ft(3)1334 1046 y(0)1413 1021 y Fg(of)f(Lemma)f(3.2)i(.) f(First)g(w)m(e)i(pro)m(v)m(e)g(that)e Ff(')3065 985 y Ft(\(2\))3065 1046 y Fo(n)3199 1021 y Fe(!)g Fg(0)h(.)166 1142 y(Here)33 b(w)m(e)h(ha)m(v)m(e)g(suc)m(h)g(t)m(w)m(o)f(p)s (ossibilities:)166 1362 y(\(a\))97 b Ff(\033)32 b(>)27 b(\016)625 1326 y Ft(\(2)p Fo(\025)p Fl(\000)p Ft(1\))p Fo(=)p Ft(4)922 1362 y Ff(:)97 b Fg(F)-8 b(rom)31 b(\(76\))h(and)h (\(107\))f(w)m(e)h(obtain)f Ff(\024)2456 1377 y Fo(n)2531 1362 y Ff(<)27 b(\033)2693 1326 y Fl(\000)p Ft(1)2787 1362 y Fg(.)33 b(Th)m(us)1295 1583 y Ff(\016)1342 1542 y Ft(2)p Fo(\025)p Fl(\000)p Ft(1)1512 1583 y Ff(\024)1568 1542 y Ft(4)1568 1608 y Fo(n)1643 1583 y Ff(<)28 b(\016)1794 1542 y Ft(2)p Fo(\025)p Fl(\000)p Ft(1)1965 1583 y Ff(\033)2024 1542 y Fl(\000)p Ft(4)2146 1583 y Ff(<)f Fg(1)934 b(\(112\))166 1904 y(Applying)32 b(this)g(estimate)g(in)f(\(80\))h(one)h(gets)362 2139 y Fe(j)27 b Ff(')481 2098 y Ft(\(2\))481 2163 y Fo(n)603 2139 y Fe(j)p Ff(<)g(\016)781 2098 y Ft(2)p Fo(\025)p Fl(\000)p Ft(1)952 2139 y Ff(\033)1011 2098 y Fl(\000)p Ft(4)1105 2139 y Fe(j)16 b Fg(^)-65 b Ff(')1197 2088 y Ft(\(2\))1197 2160 y Fo(n)p Fl(\000)p Ft(1)1334 2139 y Fe(j)27 b Ff(<)h(:::)g(<)g Fg(\()p Ff(\016)1791 2098 y Ft(2)p Fo(\025)p Fl(\000)p Ft(1)1961 2139 y Ff(\033)2020 2098 y Fl(\000)p Ft(4)2115 2139 y Fg(\))2153 2098 y Fo(n)2227 2139 y Fe(j)43 b Fg(^)-65 b Ff(')2346 2088 y Ft(\(2\))2346 2160 y(0)2468 2139 y Fe(j\024)28 b Fg(\()p Ff(\016)2686 2098 y Ft(2)p Fo(\025)p Fl(\000)p Ft(1)2857 2139 y Ff(\033)2916 2098 y Fl(\000)p Ft(4)3010 2139 y Fg(\))3048 2098 y Fo(n)3123 2139 y Fe(j)f Fg(\010)3248 2098 y Ft(\(2\))3371 2139 y Fe(j)g Ff(:)166 2460 y Fg(In)33 b(view)g(of)f(\(112\),)g(this)g(giv)m (es)1464 2681 y Ff(')1528 2640 y Ft(\(2\))1528 2705 y Fo(n)1650 2681 y Fe(!)27 b Fg(0)p Ff(;)82 b(n)28 b Fe(!)f Fg(+)p Fe(1)p Ff(:)166 3002 y Fg(\(b\))98 b Ff(\033)31 b Fg(=)d Ff(\016)631 2966 y Ft(\(2)p Fo(\025)p Fl(\000)p Ft(1\))p Fo(=)p Ft(4)927 3002 y Ff(:)98 b Fg(In)33 b(this)f(case)h(w)m (e)h(ha)m(v)m(e)g(only)1281 3223 y Ff(\016)1328 3182 y Ft(2)p Fo(\025)p Fl(\000)p Ft(1)1499 3223 y Ff(\024)1555 3182 y Ft(4)1555 3247 y Fo(n)1630 3223 y Ff(<)27 b(\016)1780 3182 y Ft(2)p Fo(\025)p Fl(\000)p Ft(1)1951 3223 y Ff(\033)2010 3182 y Fl(\000)p Ft(4)2132 3223 y Fg(=)h(1)p Ff(:)920 b Fg(\(113\))166 3544 y(Making)32 b(use)i(of)e(\(80\))g(one)h(obtains) 832 3778 y(0)28 b Ff(>)f(')1076 3737 y Ft(\(2\))1076 3803 y Fo(n)1198 3778 y Ff(>)h(\016)1349 3737 y Ft(2)p Fo(\025)p Fl(\000)p Ft(1)1519 3778 y Ff(\024)1575 3737 y Ft(4)1575 3803 y Fo(n)1623 3778 y Ff(')1687 3727 y Ft(\(2\))1687 3800 y Fo(n)p Fl(\000)p Ft(1)1851 3778 y Ff(>)g(')2019 3727 y Ft(\(2\))2019 3800 y Fo(n)p Fl(\000)p Ft(1)2184 3778 y Ff(>)f(:::)h(>)g(')2564 3727 y Ft(\(2\))2564 3800 y(0)2686 3778 y Fe(\025)g Fg(\010)2861 3737 y Ft(\(2\))2955 3778 y Ff(:)166 4099 y Fg(Hence)k Fe(f)p Ff(')568 4063 y Ft(\(2\))568 4124 y Fo(n)662 4099 y Fe(g)e Fg(is)g(strictly)f (increasing)h(and)g(b)s(ounded.)i(Then)f(it)e(is)h(con)m(v)m(ergen)m(t) j(and)d(its)166 4220 y(limit,)36 b(sa)m(y)41 b Ff(')673 4184 y Ft(\(2\))767 4220 y Ff(;)f Fg(ob)s(eys)g(the)g(condition)e Ff(')1786 4184 y Ft(\(2\))1920 4220 y Ff(>)i Fg(\010)2106 4184 y Ft(\(2\))2201 4220 y Fg(.)f(Assume)i(no)m(w)f(that)f Ff(')3128 4184 y Ft(\(2\))3262 4220 y Fe(6)p Fg(=)h(0)p Ff(:)166 4340 y Fg(Com)m(bining)31 b(\(80\))h(and)h(\(113\))e(one)i (obtains)f(\(recall)f(that)h Ff(')2406 4304 y Ft(\(2\))2406 4365 y Fo(n)2528 4340 y Ff(<)c Fg(0\))1499 4560 y Ff(')1563 4524 y Ft(\(2\))1563 4585 y Fo(n)p 1477 4604 202 4 v 1477 4715 a Ff(')1541 4664 y Ft(\(2\))1541 4737 y Fo(n)p Fl(\000)p Ft(1)1716 4627 y Ff(<)g(\016)1867 4586 y Ft(2)p Fo(\025)p Fl(\000)p Ft(1)2037 4627 y Ff(\024)2093 4586 y Ft(4)2093 4652 y Fo(n)2168 4627 y Ff(<)g Fg(1)p Ff(;)166 5038 y Fg(whic)m(h)40 b(means)f Ff(\024)813 5053 y Fo(n)899 5038 y Fe(!)g Ff(\016)1085 5002 y Ft(\(1)p Fl(\000)p Ft(2)p Fo(\025)p Ft(\))p Fo(=)p Ft(4)1381 5038 y Ff(:)g Fg(The)h(latter)e(as)i(w)m(ell)e(as)h(the)h(de\014nitions)f(of)f Ff(\024)3210 5053 y Fo(n)3297 5038 y Fg(and)166 5159 y(\010)236 5123 y Ft(\(1\))363 5159 y Fg(immediately)30 b(yield)1655 5380 y Ff(')1719 5339 y Ft(\(1\))1719 5404 y Fo(n)1841 5380 y Fe(!)d Fg(\010)2038 5339 y Ft(\(1\))2133 5380 y Ff(:)1745 5712 y Fg(29)p eop %%Page: 30 30 30 29 bop 166 83 a Fg(P)m(assing)33 b(to)f(the)h(limit)c Ff(n)f Fe(!)f Fg(+)p Fe(1)33 b Fg(in)e(\(82\))h(one)h(obtains)711 316 y(\010)781 275 y Ft(\(1\))903 316 y Fe(\025)28 b Ff(\016)1055 275 y Ft(\(1)p Fl(\000)p Ft(2)p Fo(\025)p Ft(\))p Fo(=)p Ft(4)1351 316 y Fg(\010)1421 275 y Ft(\(1\))1538 316 y Fg(+)22 b(\()p Ff(\022)k Fg(+)c(1\)\(1)f Fe(\000)i Ff(\016)2185 275 y Fl(\000)p Fo(\025)2285 316 y Fg(\))p Ff(\016)2370 275 y Ft(2)p Fo(\025)p Fl(\000)p Ft(1)2541 316 y Ff(\016)2588 275 y Ft(3\(1)p Fl(\000)p Ft(2)p Fo(\025)p Ft(\))p Fo(=)p Ft(4)2919 316 y Ff(')2983 275 y Ft(\(2\))3077 316 y Ff(;)166 656 y Fg(whic)m(h)33 b(yields)f(in)g(turn)1153 946 y Ff(')1217 905 y Ft(\(2\))1339 946 y Fe(\024)d(\000)p Fg(\010)1592 905 y Ft(\(1\))1719 879 y Fg(\(1)22 b Fe(\000)h Ff(\033)t Fg(\))p Ff(\033)2084 842 y Ft(2)2123 879 y Ff(\016)2170 842 y Ft(1)p Fl(\000)p Fo(\025)p 1697 923 632 4 v 1697 1014 a Fg(\()p Ff(\022)i Fg(+)d(1\)\()p Ff(\016)2075 986 y Fo(\025)2142 1014 y Fe(\000)h Fg(1\))2366 946 y(=)28 b(\010)2540 905 y Ft(\(2\))2634 946 y Ff(:)166 1336 y Fg(The)34 b(latter)d(giv)m(es)i(the)g(follo)m(wing)d(con)m (tradictory)j(inequalities)1518 1569 y(\010)1588 1528 y Ft(\(2\))1710 1569 y Ff(<)28 b(')1878 1528 y Ft(\(2\))2000 1569 y Fe(\024)g Fg(\010)2175 1528 y Ft(\(2\))2270 1569 y Ff(:)166 1909 y Fg(Th)m(us)34 b Ff(')477 1872 y Ft(\(2\))599 1909 y Fg(=)28 b(0.)k(T)-8 b(o)33 b(sho)m(w)g(that)g Ff(')1472 1872 y Ft(\(1\))1472 1933 y Fo(n)1594 1909 y Fe(!)27 b Fg(1,)32 b(w)m(e)i(set)1069 2155 y Ff(b)1110 2170 y Fo(n)1185 2155 y Fg(=)27 b(\()p Ff(\022)f Fg(+)c(1\)\(1)f Fe(\000)i Ff(\016)1837 2114 y Fl(\000)p Fo(\025)1937 2155 y Fg(\))p Ff(\016)2022 2114 y Ft(2)p Fo(\025)p Fl(\000)p Ft(1)2193 2155 y Ff(\024)2249 2114 y Ft(3)2249 2180 y Fo(n)2296 2155 y Ff(')2360 2104 y Ft(\(2\))2360 2177 y Fo(n)p Fl(\000)p Ft(1)2497 2155 y Ff(:)708 b Fg(\(114\))166 2495 y(Since)34 b Fe(f)p Ff(\024)528 2510 y Fo(n)575 2495 y Fe(g)g Fg(is)f(b)s(ounded)i(and)f Ff(')1413 2459 y Ft(\(2\))1413 2519 y Fo(n)1537 2495 y Fe(!)29 b Fg(0)p Ff(;)34 b Fg(one)g(has)g Ff(b)2172 2510 y Fo(n)2250 2495 y Fe(!)29 b Fg(0)p Ff(:)34 b 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3585 y Fl(\000)p Ft(1)2205 3564 y Fg(\))28 b(=)f(0)p Ff(:)166 3932 y Fg(The)34 b(latter)d(can)i(b)s(e)g(rewritten) f(as)h(\(see)h(76\))1400 4228 y Ff(a)22 b Fe(\000)1791 4160 y Ff(\016)1838 4124 y Fl(\000)p Fo(\025)1938 4160 y Ff(a)p 1583 4204 616 4 v 1583 4296 a Fg(1)g Fe(\000)g Fg(\(1)g Fe(\000)h Ff(\016)2009 4267 y Fl(\000)p Fo(\025)2109 4296 y Fg(\))p Ff(a)2236 4228 y Fg(=)k(0)p Ff(:)166 4648 y Fg(Since)41 b Ff(\025)h(>)g Fg(0)p Ff(;)f Fg(the)g(ab)s(o)m(v)m(e)h (equation)e(has)i(only)e(one)i(solution)d(on)2744 4552 y Fd(h)2783 4648 y Fg(1)p Ff(;)17 b Fg(\010)2946 4612 y Ft(\(1\))3040 4552 y Fd(i)3080 4648 y Fg(,)41 b(namely)166 4792 y Ff(a)28 b Fg(=)f(1.)c(In)g(what)f(follo)m(ws,)g(the)h(b)s (ounded)g(sequence)i Fe(f)p Ff(')2184 4756 y Ft(\(1\))2184 4817 y Fo(n)2278 4792 y Fe(g)e Fg(has)g(only)f(one)g(accum)m(ulation) 166 4912 y(p)s(oin)m(t,)29 b(hence)h(it)f(con)m(v)m(erges)i(to)e Ff(a)f Fg(=)g(1)h(itself.)f(In)i(the)f(case)i Ff(\014)i(<)27 b(\014)2610 4927 y Fl(\003)2679 4912 y Fg(the)j(estimate)f(\(108\))166 5033 y(yields)h Ff(')502 4997 y Ft(\(1\))502 5057 y Fo(n)624 5033 y Fe(!)e Fg(0.)i(Then)i Ff(\024)1167 5048 y Fo(n)1245 5033 y Fg(giv)m(en)f(b)m(y)g(\(76\))f(tends)i(to)e Ff(\016)2258 4997 y Fl(\000)p Fo(\025)2359 5033 y Fg(,)g(whic)m(h)i(immediately)27 b(giv)m(es)166 5153 y(in)32 b(\(80\))g Ff(')550 5117 y Ft(\(2\))550 5178 y Fo(n)672 5153 y Fe(!)27 b Fg(0.)p 2257 5145 42 42 v 166 5380 a Fi(Pro)s(of)37 b(of)h(Lemma)f(3.5.)c Fg(F)-8 b(rom)31 b(the)i(de\014nitions)f(\(54\))g(and)h(\(73\))e(one)i (obtains)1745 5712 y(30)p eop %%Page: 31 31 31 30 bop 504 87 a Ff(C)574 102 y Fo(n)648 87 y Fg(=)27 b Ff(C)828 46 y Fo(\016)821 112 y(n)p Fl(\000)p Ft(1)958 87 y Fg(\011)1034 102 y Fo(n)p Fl(\000)p Ft(1)1171 87 y Fg(\()p Ff(\014)6 b Fg(\))p Ff(;)361 259 y Fg(\011)437 274 y Fo(n)484 259 y Fg(\()p Ff(\014)g Fg(\))637 206 y Ft(def)648 259 y Fg(=)751 163 y Fd(n)806 259 y Fg(exp)972 163 y Fd(\020)1022 259 y Fg(\()p Ff(\016)1107 218 y Fo(\025)1174 259 y Fe(\000)23 b Fg(1\)\001)1442 274 y Fo(\022)1481 163 y Fd(\021)1547 259 y Fg(exp)1712 163 y Fd(\020)1762 259 y Ff(\016)t(')1873 274 y Fo(n)1920 259 y Fg(\()p Ff(\016)2005 218 y Fl(\000)p Ft(1)p Fl(\000)p Fo(\025)2195 259 y Fe(\001)p Fg(\))2261 163 y Fd(\021)o(o)2382 259 y Fg(\(0\))p Ff(:)166 503 y Fg(F)-8 b(or)33 b Ff(\022)f Fg(=)c(0,)34 b(\011)709 518 y Fo(n)755 503 y Fg(\()p Ff(\014)6 b Fg(\))29 b(=)g(1)k(\(see)h(Remark)f(2.2\))g(and)g(the)h (situation)e(with)g Ff(C)2927 518 y Fo(n)3008 503 y Fg(is)g(ob)m (vious.)166 624 y(Consider)h(the)g(case)h Ff(\022)c(>)e Fg(0.)k(Then)560 856 y Ff(C)630 871 y Fo(n)704 856 y Fg(=)c Ff(C)885 814 y Fo(\016)918 791 y Fb(n)878 880 y Ft(0)965 856 y Fg(\004)1030 871 y Fo(n)1077 856 y Fg(\()p Ff(\014)6 b Fg(\))p Ff(;)81 b Fg(\004)1387 871 y Fo(n)1434 856 y Fg(\()p Ff(\014)6 b Fg(\))1631 803 y Ft(def)1641 856 y Fg(=)71 b(\011)1864 871 y Fo(n)p Fl(\000)p Ft(1)2001 856 y Fg(\()p Ff(\014)6 b Fg(\)\011)2214 814 y Fo(\016)2214 880 y(n)p Fl(\000)p Ft(2)2351 856 y Fg(\()p 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Fo(n)2345 1589 y Ff(\020)2396 1548 y Fl(\000)p Ft(1)2489 1589 y Ff(C)2559 1604 y Fo(n)2606 1589 y Ff(:)599 b Fg(\(116\))166 1988 y(By)35 b(Lemma)e(3.3,)h Ff(')921 1952 y Ft(\(1\))921 2013 y Fo(n)1046 1988 y Ff(<)d Fg(\010)1223 1952 y Ft(\(1\))1352 1988 y Fg(for)j(all)e Ff(n)f Fe(2)g Ff(I)-25 b(N)1932 2003 y Ft(0)2006 1988 y Fg(and)34 b Ff(\014)j Fe(2)31 b Fg(\(0)p Ff(;)17 b(\014)2572 2003 y Fl(\003)2611 1988 y Fg(].)34 b(This)h(giv)m(es)g(in)e(\(76\))166 2108 y Ff(\024)222 2123 y Fo(n)297 2108 y Ff(<)27 b(\033)459 2072 y Fl(\000)p Ft(1)581 2108 y Fe(\024)i Ff(\016)734 2072 y Ft(\(1)p Fl(\000)p Ft(2)p Fo(\025)p Ft(\))p Fo(=)p Ft(4)1062 2108 y Fg(for)j(suc)m(h)i Ff(\014)k Fg(and)33 b Ff(n)p Fg(.)g(W)-8 b(e)33 b(set)361 2452 y Ff(\020)412 2411 y Fl(\000)487 2452 y Fg(=)16 b Ff(\016)626 2411 y Fl(\000)p Fo(\022)r Ft(\(1+2)p Fo(\025)p Ft(\))p Fo(=)p Ft(4\()p Fo(\016)r Fl(\000)p Ft(1\))1191 2452 y Ff(;)2014 b Fg(\(117\))394 2592 y(\007)17 b(=)f([)p Ff(\020)657 2551 y Fl(\000)716 2592 y Ff(;)h Fg(1])27 b Fe(\032)h Ff(I)-25 b(R)1061 2607 y Ft(+)1120 2592 y Ff(:)2085 b Fg(\(118\))166 2824 y(F)-8 b(or)39 b(a)g(\014xed)i Ff(\014)k Fe(2)40 b Fg(\(0)p Ff(;)17 b(\014)1071 2839 y Fl(\003)1110 2824 y Fg(],)39 b(let)g(us)h(pro)m(v)m(e)h(that)e(the)h(follo)m(wing)d (inductiv)m(e)j(statemen)m(ts)166 2944 y(hold)32 b(true)h(for)f(all)e Ff(n)e Fe(2)g Ff(I)-25 b(N)1159 2959 y Ft(0)361 3277 y Ff(i)394 3236 y Ft(+)394 3302 y Fo(n)470 3277 y Fg(=)17 b Fe(f9)p Ff(\020)719 3236 y Ft(+)711 3302 y Fo(n)805 3277 y Fe(2)28 b Fg(\007)f(:)h Ff(C)1127 3292 y Fo(n)1174 3277 y Fg(\()p Ff(\020)1263 3236 y Ft(+)1255 3302 y Fo(n)1321 3277 y Ff(;)17 b(\014)6 b Fg(\))27 b(=)g(1)p Fe(g)p Ff(;)1512 b Fg(\(119\))361 3417 y Ff(i)394 3375 y Fl(\000)394 3441 y Fo(n)470 3417 y Fg(=)17 b Fe(f9)p Ff(\020)719 3375 y Fl(\000)711 3441 y Fo(n)805 3417 y Fe(2)28 b Fg(\007)f(:)h Ff(C)1127 3432 y Fo(n)1174 3417 y Fg(\()p Ff(\020)1263 3375 y Fl(\000)1255 3441 y Fo(n)1321 3417 y Ff(;)17 b(\014)6 b Fg(\))27 b(=)g 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b(\014)6 b Fg(\))27 b Ff(>)g Fg(1)p Ff(;)114 b(C)2004 4136 y Fo(n)2051 4121 y Fg(\()p Ff(\020)2140 4080 y Fl(\000)2132 4146 y Fo(n)p Fl(\000)p Ft(1)2269 4121 y Ff(;)17 b(\014)6 b Fg(\))27 b Ff(<)g(\020)2593 4080 y Fl(\000)2651 4121 y Ff(:)166 4458 y Fg(T)-8 b(aking)40 b(in)m(to)f(accoun)m(t)i(that)f Ff(C)1361 4473 y Fo(n)1448 4458 y Fg(dep)s(ends)h(on)f Ff(\020)48 b Fg(as)40 b(giv)m(en)g(b)m(y)h (\(116\))e(one)i(concludes)166 4579 y(that)32 b(there)i(exist)f Ff(\020)909 4543 y Fl(\006)901 4603 y Fo(n)999 4579 y Fg(suc)m(h)h(that)1297 4811 y Ff(\020)1348 4769 y Fl(\000)1340 4835 y Fo(n)p Fl(\000)p Ft(1)1504 4811 y Ff(<)28 b(\020)1659 4769 y Fl(\000)1651 4835 y Fo(n)1745 4811 y Ff(<)f(\020)1899 4769 y Ft(+)1891 4835 y Fo(n)1985 4811 y Ff(<)h(\020)2140 4769 y Ft(+)2132 4835 y Fo(n)p Fl(\000)p Ft(1)2269 4811 y Ff(;)936 b Fg(\(120\))166 5148 y(and)33 b(the)g(statemen)m(ts)g Ff(i)1048 5112 y Fl(\006)1048 5173 y Fo(n)1140 5148 y Fg(hold)f(true.)h(Set)1498 5380 y(\007)1574 5395 y Fo(n)1648 5380 y Fg(=)28 b([)p Ff(\020)1830 5339 y Fl(\000)1822 5404 y Fo(n)1888 5380 y Ff(;)17 b(\020)1983 5339 y Ft(+)1975 5404 y Fo(n)2041 5380 y Fg(])p Ff(:)1137 b Fg(\(121\))1745 5712 y(31)p eop %%Page: 32 32 32 31 bop 166 83 a Fg(Then)1432 327 y(\007)1508 342 y Fo(n)1582 327 y Fe(\032)28 b Fg(\007)1763 342 y Fo(n)p Fl(\000)p Ft(1)1928 327 y Fe(\032)g Ff(:)17 b(:)g(:)27 b Fe(\032)h Fg(\007)p Ff(;)166 682 y Fg(and)33 b(there)g(exists)883 656 y(~)874 682 y Ff(\020)917 697 y Fo(n)992 682 y Fe(2)28 b Fg(\007)1162 697 y Fo(n)1241 682 y Fg(suc)m(h)34 b(that)1137 1009 y Ff(\020)1188 968 y Ft(+)1180 1034 y Fo(n)1268 1009 y Fe(\000)22 b Ff(\020)1418 968 y Fl(\000)1410 1034 y Fo(n)1504 1009 y Fg(=)28 b(\(1)22 b Fe(\000)g Ff(\020)1867 968 y Fl(\000)1926 1009 y Fg(\))1981 863 y Fd(")2038 942 y Ff(@)5 b(C)2164 957 y Fo(n)2212 942 y Fg(\()2259 916 y(~)2250 942 y Ff(\020)2293 957 y Fo(n)2340 942 y Ff(;)17 b(\014)6 b Fg(\))p 2038 986 444 4 v 2207 1078 a Ff(@)f(\020)2492 863 y Fd(#)2540 876 y Fl(\000)p Ft(1)2651 1009 y Ff(:)166 1420 y Fg(Let)1601 1664 y(\007)1677 1623 y Fl(\003)1744 1664 y Fg(=)1901 1581 y Fd(\\)1848 1765 y Fo(n)p Fl(2)p Fo(I)-18 b(N)2013 1774 y Fc(0)2064 1664 y Fg(\007)2140 1679 y Fo(n)2187 1664 y Ff(;)166 2095 y Fg(whic)m(h)43 b(is)f(closed)g(and)h(nonempt)m(y)-8 b(.)43 b(F)-8 b(or)41 b Ff(\020)52 b Fe(2)45 b Fg(\007)2010 2059 y Fl(\003)2049 2095 y Fg(,)d(all)f Ff(C)2334 2110 y Fo(n)2423 2095 y Fg(b)s(elong)g(to)h(\007.)h(Hence)g(the)166 2215 y(sequence)27 b Fe(f)p Ff(C)682 2230 y Fo(n)729 2215 y Fe(g)c Fg(is)h(separated)h(from)e(zero)i(for)e(suc)m(h)j Ff(\020)8 b Fg(.)23 b(This)i(yields)f(that)g(the)g(deriv)-5 b(ativ)m(e)166 2336 y(giv)m(en)30 b(b)m(y)g(\(116\))f(tends)i(to)e(+)p Fe(1)g Fg(when)h Ff(n)e Fe(!)g(1)p Fg(.)h(T)-8 b(aking)29 b(in)m(to)g(accoun)m(t)h(all)e(these)i(facts)166 2456 y(one)j(concludes)1367 2700 y(\007)1443 2659 y Fl(\003)1510 2700 y Fg(=)27 b Fe(f)p Ff(\020)1714 2659 y Fl(\003)1753 2700 y Fe(g)p Ff(;)81 b(\020)1962 2659 y Fl(\003)2029 2700 y Fe(2)28 b Fg(\007)p Ff(;)1006 b Fg(\(122\))166 3055 y(and,)33 b(for)f(all)e Ff(n)e Fe(2)g Ff(I)-25 b(N)954 3070 y Ft(0)993 3055 y Fg(,)1491 3299 y Ff(C)1561 3314 y Fo(n)1607 3299 y Fg(\()p Ff(\020)1696 3258 y Fl(\003)1735 3299 y Ff(;)17 b(\014)6 b Fg(\))27 b Fe(2)h Fg(\007)p Ff(:)1130 b Fg(\(123\))166 3654 y(It)34 b(should)f(b)s(e)h(p)s(oin)m (ted)f(out)h(that)f Ff(\020)1508 3618 y Fl(\003)1576 3654 y Fg(=)c Ff(\020)1732 3618 y Fl(\003)1771 3654 y Fg(\()p Ff(\014)6 b Fg(\).)33 b(Cho)s(ose)h Ff(\020)j Fg(=)29 b Ff(\020)2544 3618 y Fl(\003)2583 3654 y Fg(.)k(Then)i(b)m(y)f (\(123\),)f(the)166 3775 y(sequence)42 b Fe(f)p Ff(C)697 3790 y Fo(n)744 3775 y Fe(g)d Fg(is)g(b)s(ounded,)i(hence)g(it)d(con)m (tains)i(a)f(con)m(v)m(ergen)m(t)j(subsequence.)h(F)-8 b(or)166 3895 y Ff(\014)47 b Fg(=)41 b Ff(\014)440 3910 y Fl(\003)480 3895 y Fg(,)f(b)m(y)i(means)f(of)f(\(78\))g(and)g(\(79\)) g(one)h(ma)m(y)g(sho)m(w)g(that)g(suc)m(h)h(a)e(subsequence)166 4015 y(con)m(v)m(erges)35 b(to)e Ff(C)793 4030 y Ft(2)p Fo(;)p Fl(\003)916 4015 y Fg(=)28 b Ff(\016)1067 3979 y Fl(\000)p Fo(\025\022)r(=)p Ft(\()p Fo(\016)r Fl(\000)p Ft(1\))1417 4015 y Fg(.)33 b(As)g(in)g(the)g(case)h(of)f Fe(f)p Ff(')2337 3979 y Ft(\(1\))2337 4040 y Fo(n)2431 4015 y Fe(g)g Fg(considered)h(ab)s(o)m(v)m(e,)g(this)166 4136 y(fact)h(implies)e(the)i(con)m(v)m(ergence)j(of)d(the)g(whole)g (sequence)j(to)d(this)g(limit.)c(F)-8 b(or)35 b Ff(\014)i(<)32 b(\014)3387 4151 y Fl(\003)3427 4136 y Fg(,)166 4256 y(one)42 b(emplo)m(ys)g(\(77\))f(and)h(\(78\))f(and)h(sho)m(ws)h (similarly)38 b(the)k(con)m(v)m(ergence)j(of)c Fe(f)p Ff(C)3229 4271 y Fo(n)3275 4256 y Fe(g)h Fg(to)166 4376 y Ff(C)236 4391 y Ft(1)p Fo(;)p Fl(\003)358 4376 y Fg(=)27 b(1.)33 b(Th)m(us)h(w)m(e)g(c)m(ho)s(ose)f(the)g(function)f Ff(C)7 b Fg(\()p Ff(\014)f Fg(\))32 b(to)g(b)s(e)h Ff(C)7 b Fg(\()p Ff(\014)f Fg(\))27 b(=)h Ff(\020)2714 4340 y Fl(\003)2752 4376 y Fg(\()p Ff(\014)6 b Fg(\).)p 3278 4368 42 42 v 166 4608 a Fi(Pro)s(of)37 b(of)h(Lemma)f(3.6.)c Fg(It)f(follo)m(ws)g(from)f(Lemmas)h(2.1,)g(3.4,)g(and)h(3.5.)p 3300 4600 V 166 4840 a Fi(Pro)s(of)46 b(of)h(Lemma)f(3.7.)41 b Fg(Claim)d(\(i\))i(follo)m(ws)f(from)g(the)i(lemmas)e(just)i(men)m (tioned.)166 4973 y(T)-8 b(o)32 b(pro)m(v)m(e)h(claim)c(\(ii\))g(w)m(e) k(\014x)f Ff(\014)i(<)27 b(\014)1518 4988 y Fl(\003)1589 4973 y Fg(and)32 b(sho)m(w)h(the)f(con)m(v)m(ergence)i(of)2834 4877 y Fd(n)2905 4973 y Fg(~)-64 b Ff(')2954 4937 y Ft(\(2\))2954 4998 y Fo(n)3048 4877 y Fd(o)3135 4973 y Fg(to)31 b(zero.)166 5106 y(Indeed,)j(\(83\))e(and)h(\(110\))e(imply)808 5264 y Fd(\014)808 5313 y(\014)808 5363 y(\014)16 b Fg(~)-65 b Ff(')900 5322 y Ft(\(2\))900 5388 y Fo(n)994 5264 y Fd(\014)994 5313 y(\014)994 5363 y(\014)28 b Ff(<)f(\016)1200 5322 y Fl(\000)p Fo(n)1302 5363 y Fg(\()p Ff(\027)1388 5378 y Fo(n)1435 5363 y Ff(\027)1483 5378 y Fo(n)p Fl(\000)p Ft(1)1620 5363 y Ff(:::\027)1749 5378 y Ft(1)1789 5363 y Fg(\))1827 5322 y Ft(4)1883 5264 y Fd(\014)1883 5313 y(\014)1883 5363 y(\014)16 b Fg(~)-65 b Ff(')1975 5312 y Ft(\(2\))1975 5385 y(0)2069 5264 y Fd(\014)2069 5313 y(\014)2069 5363 y(\014)27 b Ff(<)h(\016)2275 5322 y Fl(\000)t Fo(n)2381 5363 y Ff(K)2471 5322 y Ft(4)2464 5388 y(0)2527 5264 y Fd(\014)2527 5313 y(\014)2527 5363 y(\014)15 b Fg(~)-64 b Ff(')2619 5312 y Ft(\(2\))2619 5385 y(0)2713 5264 y Fd(\014)2713 5313 y(\014)2713 5363 y(\014)16 b Ff(:)448 b Fg(\(124\))1745 5712 y(32)p eop %%Page: 33 33 33 32 bop 166 86 a Fg(Th)m(us)45 b(to)d(complete)g(the)i(pro)s(of)e(w)m (e)i(ha)m(v)m(e)g(only)e(to)h(sho)m(w)h(that)f Fe(f)16 b Fg(~)-65 b Ff(')2750 50 y Ft(\(1\))2750 111 y Fo(n)2844 86 y Fe(g)43 b Fg(is)f(a)h(Cauc)m(h)m(y)166 206 y(sequence.)35 b(T)-8 b(o)33 b(this)f(end,)i(for)e Ff(n)c Fe(2)g Ff(I)-25 b(N)42 b Fg(and)33 b Ff(p)28 b Fe(2)g Ff(I)-25 b(N)10 b Fg(,)33 b(w)m(e)g(set)1277 437 y Ff(a)1328 452 y Fo(n;p)1458 437 y Fg(=)27 b Ff(\027)1609 452 y Fo(n)p Ft(+)p Fo(p)1747 437 y Ff(\027)1795 452 y Fo(n)p Ft(+)p Fo(p)p Fl(\000)p Ft(1)2022 437 y Ff(:::\027)2151 452 y 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Fd(1)1741 1440 y(A)1836 1462 y Fe(\000)c Fg(1)k Fe(!)h Fg(0)p Ff(;)81 b(n)28 b Fe(!)f Fg(+)p Fe(1)p Ff(:)519 b Fg(\(127\))166 1884 y(On)33 b(the)g(other)f(hand,)h(\(110\))f(and)h (\(124\))e(giv)m(e)429 2172 y Fe(j)p Ff(b)498 2187 y Fo(n;p)600 2172 y Fe(j)d Ff(<)f Fg(\()p Ff(\022)f Fg(+)c(1\)\(1)f Fe(\000)i Ff(\016)1308 2131 y Fl(\000)p Fo(\025)1408 2172 y Fg(\))p Ff(\016)1493 2131 y Fl(\000)p Ft(1)1587 2172 y Ff(K)1677 2131 y Ft(7)1670 2197 y(0)1733 2073 y Fd(\014)1733 2122 y(\014)1733 2172 y(\014)16 b Fg(~)-65 b Ff(')1825 2121 y Ft(\(2\))1825 2194 y(0)1919 2073 y Fd(\014)1919 2122 y(\014)1919 2172 y(\014)1993 2064 y Fl(1)1969 2089 y Fd(X)1963 2263 y Fo(s)p Ft(=)p Fo(n)2110 2172 y Ff(\016)2157 2131 y Fl(\000)p Ft(\(1+)p Fo(\025)p Ft(\))p Fo(s)2463 2172 y Fe(!)27 b Fg(0)p Ff(;)82 b(n)28 b Fe(!)f Fg(+)p Fe(1)p Ff(:)68 b Fg(\(128\))166 2569 y(The)34 b(estimates)e(\(84\))g(and)g(\(85\))g(yield)g(resp)s(ectiv)m (ely)903 2813 y(~)-65 b Ff(')951 2762 y Ft(\(1\))951 2832 y Fo(n)p Ft(+)p 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b(in)g(mind)f(\(74\))h(and)h(\(108\),)f(one)h(gets)1293 3944 y(0)28 b Ff(<)43 b Fg(~)-65 b Ff(')1537 3903 y Ft(\(1\))1537 3969 y Fo(n)1659 3944 y Fg(=)28 b Ff(\016)1810 3903 y Fo(\025n)1898 3944 y Ff(')1962 3903 y Ft(\(1\))1962 3969 y Fo(n)2084 3944 y Ff(<)f(K)r(:)933 b Fg(\(131\))166 4279 y(No)m(w)36 b(it)e(su\016ces)k(to)d(apply)g(the)h(latter)e (estimate)h(together)h(with)f(\(127\))f(and)i(\(128\))e(in)166 4399 y(\(130\))d(and)g(conclude)h(that)f Fe(f)16 b Fg(~)-65 b Ff(')1332 4363 y Ft(\(1\))1332 4424 y Fo(n)1426 4399 y Fe(g)31 b Fg(is)g(a)g(Cauc)m(h)m(y)j(sequence.)g(Th)m(us,)f(for)e(ev) m(ery)i Ff(\014)h(<)27 b(\014)3387 4414 y Fl(\003)3427 4399 y Fg(,)166 4520 y(there)33 b(exists)49 b(~)-65 b Ff(')28 b Fg(=)43 b(~)-65 b Ff(')p Fg(\()p Ff(\014)6 b Fg(\))27 b Ff(>)h Fg(0)k(suc)m(h)h(that)48 b(~)-65 b Ff(')1786 4484 y Ft(\(1\))1786 4544 y Fo(n)1908 4520 y Fe(!)43 b Fg(~)-64 b Ff(':)32 b Fg(No)m(w)h(w)m(e)g(apply)f(Lemma)f (2.1)h(and)166 4640 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5384 y Fg(\))p Ff(')2735 5333 y Ft(\(1\))2735 5406 y Fo(n)p Fl(\000)p Ft(1)2882 5297 y Ff(:)323 b Fg(\(132\))1745 5712 y(33)p eop %%Page: 34 34 34 33 bop 166 83 a Fh(This)37 b(b)-5 b(ound)38 b(is)g(achieve)-5 b(d)37 b(for)h Ff(f)1385 98 y Ft(0)1424 83 y Fg(\()p Ff(z)t Fg(\))d(=)e Ff(C)24 b Fg(exp)q(\()p Ff(\013)q(z)t Fg(\))38 b Fh(\(in)g(this)g(c)-5 b(ase)37 b(we)h(may)g(c)-5 b(alculate)170 203 y Fg(~)-53 b Ff(g)213 218 y Fo(n)293 203 y Fh(explicitly,)32 b(se)-5 b(e)33 b(\(66\)\).)f(It)g(is)h(quite)g (likely)g(that)g(this)g(b)-5 b(ound)32 b(is)h(achieve)-5 b(d)32 b(also)g(in)h(the)166 324 y(gener)-5 b(al)44 b(c)-5 b(ase,)44 b(but)h(to)g(pr)-5 b(ove)44 b(this)g(c)-5 b(onje)g(ctur)g(e) 44 b(we)h(would)f(ne)-5 b(e)g(d)44 b(mor)-5 b(e)44 b(sophistic)-5 b(ate)g(d)166 444 y(estimates)34 b(than)h(\(85\))f(or)h(\(82\).)166 830 y Fi(References)166 1163 y Fq([1])c(V.N.)d(Deniso)m(v,)g(On)e (Stabilization)g(of)h(solutions)f(of)h(the)h(Cauc)m(h)m(y)f(problem)f (for)h(parab)s(olic)292 1276 y(equations,)j(Nonlinear)f(Anal.:)h (Theory)-8 b(,)31 b(Metho)s(ds,)f(Appl.,)g(30)h(\(1997\))i(123{127.)166 1462 y([2])e(I.A)23 b(Ibragimo)m(v,)g(Y)-8 b(u.V.)24 b(Linnik,)c(Indep)s(enden)m(t)i(and)g(Stationary)h(Sequences)g(of)g(V) -8 b(ariables)292 1575 y(Random,)30 b(W)-8 b(olters{No)s(ordho\013,)32 b(Groningen,)d(1971.)166 1762 y([3])i(L.Iliev,)e(Laguerre)i(En)m(tire)f (F)-8 b(unctions,)30 b(Bulgarian)f(Academ)m(y)i(of)g(Sciences,)f (So\014a,)h(1987.)166 1948 y([4])g(S.)51 b(Kamin)g(\(Kamenomostsk)-5 b(a)m(y)m(a\),)55 b(On)50 b(stabilization)g(of)i(solutions)e(of)i(the)g (Cauc)m(h)m(y)292 2061 y(problem)26 b(for)h(parab)s(olic)f(equation,)h (Pro)s(ceed.)h(Ro)m(y)m(al)g(So)s(c.)f(Edin)m(burgh,)e(76A)k(\(1977\))g (43{)292 2174 y(53.)166 2360 y([5])i(Y)-8 b(u.V.)22 b(Kozitsky)-8 b(,)21 b(Hierarc)m(hical)g(ferromagnetic)g(v)m(ector)i(spin)c(mo)s(del) h(p)s(ossessing)f(the)j(Lee{)292 2473 y(Y)-8 b(ang)29 b(prop)s(ert)m(y)-8 b(.)27 b(Thermo)s(dynamic)f(limit)f(at)j(the)g (critical)f(p)s(oin)m(t)f(and)h(ab)s(o)m(v)m(e,)i(Journal)d(of)292 2586 y(Statistical)k(Ph)m(ysics,)g(87)h(\(1997\))i(799{820.)166 2773 y([6])e(Y)-8 b(u.)30 b(Kozitsky)-8 b(,)30 b(L.)g(W)-8 b(o )-25 b(lo)m(wski,)30 b(Laguerre)g(en)m(tire)f(functions)f(and)h (related)h(lo)s(cally)e(con)m(v)m(ex)292 2885 y(spaces.)33 b(Los)g(Alamos)g(Electronic)f(Preprin)m(t)g(CV/9812111,)k(1998;)f (Preprin)m(t)c(BiBoS)i(No)292 2998 y(839/5/99,)28 b(Univ)m(ersit\177) -45 b(at)23 b(Bielefeld,)g(Bielefeld,)g(1999)j(\(to)f(app)s(ear)e(in)g (Complex)g(V)-8 b(ariables:)292 3111 y(Theory)30 b(and)g (Applications\).)166 3298 y([7])h(B.J.)g(Levin,)e(Distribution)f(of)j (Zeros)f(of)g(En)m(tire)g(F)-8 b(unctions.)30 b(Amer.)h(Math.)g(So)s (c.)f(1964.)166 3484 y([8])h(D.)56 b(Luna,)e(F)-8 b(onctions)56 b(di\013)m(\023)-43 b(eren)m(tiables)54 b(in)m(v)-5 b(arian)m(tes)54 b(sous)h(l'op)m(\023)-43 b(eration)56 b(d'un)d(group)s(e)292 3597 y(r)m(\023)-43 b(eductif,)30 b(Ann.)g(Inst.)g(F)-8 b(ourier,)30 b(Grenoble,)g(26)i(\(1976\))g(33-49.)166 3783 y([9])f(H.)e(W)-8 b(eyl,)29 b(The)g(Classical)e(Groups)h(their)f (In)m(v)-5 b(arian)m(ts)29 b(and)f(Represen)m(tations,)h(Princeton,)292 3896 y(1938.)1745 5712 y Fg(34)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0102261111605--