Content-Type: multipart/mixed; boundary="-------------0205071010298" This is a multi-part message in MIME format. ---------------0205071010298 Content-Type: text/plain; name="02-214.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-214.keywords" QED, Feshbach map, renormalization group, spectral analysis ---------------0205071010298 Content-Type: application/postscript; name="BCFS-I-preprint-02-5-6.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="BCFS-I-preprint-02-5-6.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86d Copyright 1999 Radical Eye Software %%Title: BCFS-I-02-5-6.dvi %%Pages: 49 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Roman Times-Bold Times-Italic %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o BCFS-I-preprint-02-5-6.ps BCFS-I-02-5-6 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.05.07:1302 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: 8r.enc % @@psencodingfile@{ % author = "S. 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b Fs(\033)583 5352 y Fn(1)622 5388 y Fs(;)17 b(\033)725 5352 y Fn(2)764 5388 y Fs(;)g(\033)867 5352 y Fn(3)929 5388 y Fv(are)24 b(the)f(usual)f Fq(2)15 b Fr(\002)g Fq(2)23 b Fv(P)o(auli)f(matrices,)h(and)f Fs(")28 b(>)f Fq(0)c Fv(is)f(an)h(atomic)g(ener)n(gy)-6 b(.)p eop %%Page: 4 5 4 4 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2321 b Fv(4)404 573 y(This)36 b(idealized)h(atom)f(is)g(then)h(coupled)f(to) h(an)g(open,)j(dispersi)n(v)o(e)35 b(system)g(with)i(in-)257 693 y(\002nitely)32 b(man)o(y)f(quantum-mechanical)g(de)o(grees)g(of)i (freedom.)52 b(T)-8 b(o)32 b(be)g(speci\002c,)i(we)f(sup-)257 814 y(pose)j(that)f(these)h(de)o(grees)f(of)h(freedom)g(correspond)g (to)f(massless,)i(relati)n(vistic)d(bosonic)257 934 y(particles,)d(e.)f (g.,)h(photons)d(\(which)h(are)i(the)e(\002eld)h(quanta)g(of)g(the)f (quantized)g(electromag-)257 1054 y(netic)f(\002eld\).)39 b(In)27 b(order)h(to)f(k)o(eep)h(our)f(e)o(xposition)e(as)j(simple)e (as)h(possible,)g(we)h(ne)o(glect)e(the)257 1175 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5021 y Fq(\()p Fs(H)r(;)17 b(T)d Fq(\))955 4980 y Fl(\000)p Fn(1)1132 5021 y Fq(=)83 b Fs(\037)17 b(H)1458 4980 y Fl(\000)p Fn(1)1568 5021 y Fs(\037)50 b Fq(+)p 1805 4966 V 50 w Fs(\037)16 b(T)1953 4980 y Fl(\000)p Fn(1)p 2047 4966 V 2047 5021 a Fs(\037)23 b(:)1130 b Fv(\(II.9\))331 5268 y Ft(\(ii\))48 b(If)38 b Fs( )55 b Fr(2)d(H)33 b(n)f(f)p Fq(0)p Fr(g)37 b Ft(solves)g Fs(H)8 b( )55 b Fq(=)d(0)37 b Ft(then)g Fs(')52 b Fq(:=)f Fs(\037 )56 b Fr(2)c Fq(Ran)16 b Fs(\037)32 b Fr(n)g(f)p Fq(0)p Fr(g)37 b Ft(solves)501 5388 y Fs(F)564 5403 y Fm(\037)612 5388 y Fq(\()p Fs(H)r(;)17 b(T)d Fq(\))j Fs(')27 b Fq(=)g(0)p Ft(.)p eop %%Page: 12 13 12 12 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(12)303 573 y Ft(\(iii\))48 b(If)22 b Fs(')27 b Fr(2)h Fq(Ran)17 b Fs(\037)9 b Fr(n)g(f)p Fq(0)p Fr(g)21 b Ft(solves)g Fs(F)1588 588 y Fm(\037)1636 573 y Fq(\()p Fs(H)r(;)c(T)d Fq(\))j Fs(')26 b Fq(=)i(0)21 b Ft(then)g Fs( )32 b Fq(:=)27 b Fs(Q)2686 588 y Fm(\037)2734 573 y Fq(\()p Fs(H)r(;)17 b(T)d Fq(\))p Fs(')27 b Fr(2)i(H)10 b(n)f(f)p Fq(0)p Fr(g)501 693 y Ft(solves)25 b Fs(H)8 b( )31 b Fq(=)c(0)p Ft(.)314 892 y(\(iv\))49 b(The)40 b(multiplicity)c(of)j(the)g(spectr)o (al)e(value)i Fr(f)p Fq(0)p Fr(g)f Ft(is)h(conserved)f(in)h(the)f (sense)h(that)501 1012 y Fq(dim)15 b(Ker)q Fs(H)35 b Fq(=)27 b(dim)15 b(Ker)q Fs(F)1457 1027 y Fm(\037)1505 1012 y Fq(\()p Fs(H)r(;)i(T)d Fq(\))p Ft(.)342 1211 y(\(v\))49 b(Assume)o(,)29 b(in)g(addition,)f(that)f Fs(H)43 b Fq(=)35 b Fs(H)1860 1175 y Fl(\003)1927 1211 y Ft(and)29 b Fs(T)48 b Fq(=)35 b Fs(T)2393 1175 y Fl(\003)2461 1211 y Ft(ar)l(e)29 b(selfadjoint,)e(and)i(intr)l(o-)501 1331 y(duce)c(the)g(bounded)f (oper)o(ator)o(s)1412 1534 y Fs(M)94 b Fq(:=)83 b Fs(H)1875 1493 y Fl(\000)p Fn(1)p 1867 1522 48 3 v 1867 1560 a Fm(\037)p 1985 1480 62 4 v 1985 1534 a Fs(\037)17 b(W)30 b(\037)189 b Ft(and)653 b Fv(\(II.10\))1428 1714 y Fs(N)94 b Fq(:=)1786 1633 y Fo(\000)1831 1714 y Fp(1)39 b Fq(+)g Fs(M)2145 1673 y Fl(\003)2185 1714 y Fs(M)2289 1633 y Fo(\001)2335 1656 y Fl(\000)p Fn(1)p Fm(=)p Fn(2)2522 1714 y Fs(:)689 b Fv(\(II.11\))501 1917 y Ft(Then,)25 b(for)f(any)h Fs( )32 b Fr(2)c(H)q Ft(,)534 2120 y Fq(lim)532 2183 y Fm(")p Fl(&)p Fn(0)715 2120 y Fq(Im)831 2040 y Fo(\012)878 2120 y Fs( )945 2036 y Fo(\014)945 2096 y(\014)1001 2120 y Fq(\()p Fs(H)h Fr(\000)23 b Fs(i")p Fq(\))1366 2079 y Fl(\000)p Fn(1)1477 2120 y Fs( )1544 2040 y Fo(\013)1643 2120 y Fq(=)1519 b Fv(\(II.12\))700 2354 y Fq(lim)698 2417 y Fm(")p Fl(&)p Fn(0)881 2354 y Fq(Im)997 2243 y Fo(D)1058 2354 y Fs(N)27 b(Q)1240 2369 y Fm(\037)1289 2354 y Fq(\()p Fs(H)r(;)17 b(T)d Fq(\))1563 2313 y Fl(\003)1618 2354 y Fs( )1685 2239 y Fo(\014)1685 2299 y(\014)1685 2359 y(\014)1740 2273 y(\000)1786 2354 y Fs(N)27 b(F)1954 2369 y Fm(\037)2002 2354 y Fq(\()p Fs(H)r(;)17 b(T)d Fq(\))j Fs(N)48 b Fr(\000)40 b Fs(i")2615 2273 y Fo(\001)2660 2295 y Fl(\000)p Fn(1)2777 2354 y Fs(N)27 b(Q)2959 2369 y Fm(\037)3007 2354 y Fq(\()p Fs(H)r(;)17 b(T)d Fq(\))3281 2313 y Fl(\003)3337 2354 y Fs( )3404 2243 y Fo(E)501 2599 y Ft(and)1058 2802 y Fq(lim)1057 2865 y Fm(")p Fl(&)p Fn(0)1239 2802 y Fq(Im)1356 2721 y Fo(\012)1403 2802 y Fs( )1470 2717 y Fo(\014)1470 2777 y(\014)1525 2721 y(\000)1571 2802 y Fs(N)27 b(F)1739 2817 y Fm(\037)1787 2802 y Fq(\()p Fs(H)r(;)17 b(T)d Fq(\))j Fs(N)48 b Fr(\000)40 b Fs(i")2400 2721 y Fo(\001)2445 2744 y Fl(\000)p Fn(1)2556 2802 y Fs( )2623 2721 y Fo(\013)2723 2802 y Fq(=)439 b Fv(\(II.13\))1224 3035 y Fq(lim)1223 3098 y Fm(")p Fl(&)p Fn(0)1405 3035 y Fq(Im)1522 2955 y Fo(\012)1569 3035 y Fs(\037)17 b(N)1735 2994 y Fl(\000)p Fn(1)1846 3035 y Fs( )1913 2951 y Fo(\014)1913 3011 y(\014)1969 3035 y Fq(\()p Fs(H)46 b Fr(\000)39 b Fs(i")2329 2955 y Fo(\001)2375 2977 y Fl(\000)p Fn(1)2485 3035 y Fs(\037)17 b(N)2651 2994 y Fl(\000)p Fn(1)2763 3035 y Fs( )2830 2925 y Fo(E)2913 3035 y Fs(:)257 3320 y Ff(Remark)39 b(II.2.)48 b Ft(W)-9 b(e)38 b(r)l(emark)f(that)g(the)h(original)e(F)-7 b(eshbac)o(h)36 b(map)i([2)o(,)g(3],)i(or)e(F)-7 b(eshbac)o(h)257 3440 y(pr)l(ojection)28 b(method,)h(is)g(included)f(in)h(the)g(fr)o (ame)o(work)f(of)g(the)h(smooth)f(F)-7 b(eshbac)o(h)28 b(map)h(as)257 3561 y(the)e(special)g(case)g(in)g(whic)o(h)g Fs(\037)32 b Fq(=)g Fs(\037)1551 3525 y Fn(2)1617 3561 y Ft(and)p 1794 3506 V 27 w Fs(\037)g Fq(=)g Fp(1)24 b Fr(\000)g Fs(\037)k Ft(ar)l(e)f(pr)l(ojections.)36 b(Namely)-5 b(,)27 b(given)257 3681 y Fs(H)46 b Ft(and)38 b(pr)l(ojections)f Fs(P)14 b Ft(,)p 1198 3601 77 4 v 41 w Fs(P)67 b Fq(=)52 b Fp(1)33 b Fr(\000)f Fs(P)14 b Ft(,)42 b(and)c(assuming)f(that)p 2594 3601 V 37 w Fs(P)14 b(H)p 2760 3601 V 8 w(P)51 b Ft(is)38 b(in)l(vertible)g(on)257 3802 y Fq(Ran)p 432 3722 V Fs(P)14 b Ft(,)26 b(we)h(set)f Fs(T)44 b Fq(:=)30 b Fs(P)14 b(H)8 b(P)36 b Fq(+)p 1432 3722 V 23 w Fs(P)14 b(H)p 1598 3722 V 8 w(P)39 b Ft(and)26 b(observe)g(that)f Fq(\()p Fs(H)r(;)17 b(\013)q(T)d Fq(\))26 b Ft(is)f(a)h(F)-7 b(eshbac)o(h)25 b(pair)257 3922 y(corr)l(esponding)i (to)g Fs(P)14 b Ft(,)28 b(for)f(any)h Fq(0)33 b Fr(\024)g Fs(\013)h Fr(\024)g Fq(1)p Ft(.)39 b(W)-9 b(e)28 b(then)g(obtain)e(the) i(\(original\))e Fv(Feshbach)257 4042 y(Map)f Ft(as)386 4256 y Fs(F)463 4205 y Fn(\(0\))449 4283 y Fm(P)557 4256 y Fq(\()p Fs(H)8 b Fq(\))52 b(:=)g Fs(P)14 b(H)8 b(P)51 b Fr(\000)40 b Fs(P)14 b(H)p 1492 4176 V 8 w(P)1567 4176 y Fo(\000)p 1613 4176 V 80 x Fs(P)f(H)p 1778 4176 V 8 w(P)1854 4176 y Fo(\001)1900 4198 y Fl(\000)p Fn(1)p 1994 4176 V 1994 4256 a Fs(P)g(H)8 b(P)66 b Fq(=)52 b Fs(P)30 b(F)2572 4271 y Fm(P)2631 4256 y Fq(\()p Fs(H)r(;)17 b(\013)q(T)d Fq(\))j Fs(P)35 b(;)128 b Fv(\(II.14\))257 4460 y Ft(independently)24 b(of)h Fs(\013)q Ft(.)257 4649 y(Pr)l(oof:)i Fv(Throughout)17 b(the)i(proof,)h(we)f(use)g(the)f (notation)g Fs(F)41 b Fq(:=)28 b Fs(F)2503 4664 y Fm(\037)2551 4649 y Fq(\()p Fs(H)r(;)17 b(T)d Fq(\))p Fv(,)19 b Fs(Q)28 b Fq(:=)g Fs(Q)3182 4664 y Fm(\037)3230 4649 y Fq(\()p Fs(H)r(;)17 b(T)d Fq(\))p Fv(,)257 4769 y(and)25 b Fs(Q)503 4733 y Fn(#)594 4769 y Fq(:=)j Fs(Q)802 4733 y Fn(#)802 4794 y Fm(\037)865 4769 y Fq(\()p Fs(H)r(;)17 b(T)d Fq(\))p Fv(.)30 b(W)-8 b(e)25 b(observ)o(e)g(the)f(relations)416 4982 y Fs(H)g(\037)53 b Fq(=)f Fs(\037)17 b(H)922 4997 y Fm(\037)1020 4982 y Fq(+)p 1145 4927 62 4 v 49 w Fs(\037)1206 4940 y Fn(2)1262 4982 y Fs(W)d(\037)22 b(;)206 b Fv(and)189 b Fs(H)p 2122 4927 V 24 w(\037)53 b Fq(=)p 2364 4927 V 52 w Fs(\037)17 b(H)p 2523 4959 48 3 v 15 x Fm(\037)2620 4982 y Fq(+)50 b Fs(\037)2807 4940 y Fn(2)2863 4982 y Fs(W)p 2969 4927 62 4 v 14 w(\037)22 b(;)159 b Fv(\(II.15\))257 5185 y(which)25 b(follo)n(w)e(from)i Fs(\037)1088 5149 y Fn(2)1150 5185 y Fq(+)p 1248 5130 V 22 w Fs(\037)1309 5148 y Fn(2)1376 5185 y Fq(=)i Fp(1)p Fv(.)k(The)25 b(ne)o(xt)f(tw)o(o) g(identities,)1041 5388 y Fs(H)h(Q)52 b Fq(=)h Fs(\037)17 b(F)202 b Fv(and)189 b Fs(Q)2158 5347 y Fn(#)2238 5388 y Fs(H)60 b Fq(=)52 b Fs(F)31 b(\037)22 b(;)527 b Fv(\(II.16\))p eop %%Page: 13 14 13 13 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(13)257 573 y(are)27 b(of)g(k)o(e)o(y)e(importance)h(in)g(the)g (proof.)35 b(The)o(y)26 b(both)f(deri)n(v)o(e)g(from)i(a)f(simple)f (computation,)257 693 y(which)g(we)g(gi)n(v)o(e)e(only)h(for)h(the)g (\002rst)g(equality)f(in)g(\(II.16\),)733 913 y Fs(H)g(Q)84 b Fq(=)f Fs(H)8 b(\037)38 b Fr(\000)h Fs(H)p 1551 858 62 4 v 8 w(\037)17 b(H)1718 872 y Fl(\000)p Fn(1)p 1710 901 48 3 v 1710 938 a Fm(\037)p 1811 858 62 4 v 1811 913 a Fs(\037W)d(\037)999 1069 y Fq(=)83 b Fs(\037H)1300 1084 y Fm(\037)1386 1069 y Fq(+)p 1501 1014 V 39 w Fs(\037)1562 1028 y Fn(2)1601 1069 y Fs(W)14 b(\037)39 b Fr(\000)1923 989 y Fo(\000)p 1969 1014 V 80 x Fs(\037H)p 2111 1047 48 3 v 15 x Fm(\037)2181 1069 y Fq(+)22 b Fs(\037)2340 1028 y Fn(2)2379 1069 y Fs(W)p 2485 1014 62 4 v 14 w(\037)2546 989 y Fo(\001)2609 1069 y Fs(H)2698 1028 y Fl(\000)p Fn(1)p 2690 1057 48 3 v 2690 1094 a Fm(\037)p 2791 1014 62 4 v 2791 1069 a Fs(\037W)14 b(\037)999 1215 y Fq(=)83 b Fs(\037)17 b(F)35 b(:)1877 b Fv(\(II.17\))257 1435 y Ff(Pr)n(oof)34 b(of)g(\(i\).)58 b Fv(It)33 b(suf)n(\002ces)h(to)g (check)g(that)f(\(II.8\))h(holds)f(if)h Fs(F)47 b Fv(is)33 b(bounded)g(in)l(v)o(ertible)g(on)257 1555 y Fq(Ran)17 b Fs(\037)25 b Fv(and)g(that)f(\(II.9\))h(holds)f(if)h Fs(H)32 b Fv(is)25 b(bounded)f(in)l(v)o(ertible)f(on)h Fr(H)q Fv(.)404 1675 y(Thus,)h(suppose)h(\002rst)g(that)f(the)h (restriction)g(of)g Fs(F)40 b Fv(to)25 b Fq(Ran)17 b Fs(\037)26 b Fv(is)g(bounded)f(in)l(v)o(ertible)g(on)257 1796 y Fq(Ran)17 b Fs(\037)p Fv(,)25 b(and)f(de\002ne)1245 1916 y Fs(R)54 b Fq(:=)e Fs(Q)23 b(F)1705 1875 y Fl(\000)p Fn(1)1821 1916 y Fs(Q)1898 1875 y Fn(#)2011 1916 y Fq(+)p 2137 1861 V 50 w Fs(\037)16 b(H)2303 1875 y Fl(\000)p Fn(1)p 2295 1904 48 3 v 2295 1941 a Fm(\037)p 2397 1861 62 4 v 2397 1916 a Fs(\037)22 b(:)731 b Fv(\(II.18\))257 2090 y(Using)24 b(\(II.16\))h(and)g(\(II.15\),)g(we)g(obtain)849 2310 y Fs(H)g(R)84 b Fq(=)f Fs(H)24 b(Q)17 b(F)1548 2269 y Fl(\000)p Fn(1)1658 2310 y Fs(Q)1735 2269 y Fn(#)1848 2310 y Fq(+)1974 2230 y Fo(\000)p 2020 2256 V 80 x Fs(\037H)p 2162 2288 48 3 v 15 x Fm(\037)2232 2310 y Fq(+)22 b Fs(\037)2391 2269 y Fn(2)2430 2310 y Fs(W)p 2536 2256 62 4 v 14 w(\037)2597 2230 y Fo(\001)2659 2310 y Fs(H)2748 2269 y Fl(\000)p Fn(1)p 2740 2298 48 3 v 2740 2336 a Fm(\037)p 2842 2256 62 4 v 2842 2310 a Fs(\037)335 b Fv(\(II.19\))1113 2469 y Fq(=)83 b Fs(\037)17 b(Q)1427 2428 y Fn(#)1539 2469 y Fq(+)p 1665 2414 V 50 w Fs(\037)1726 2428 y Fn(2)1815 2469 y Fq(+)50 b Fs(\037)2002 2428 y Fn(2)2042 2469 y Fs(W)p 2148 2414 V 14 w(\037)16 b(H)2314 2428 y Fl(\000)p Fn(1)p 2306 2457 48 3 v 2306 2494 a Fm(\037)p 2408 2414 62 4 v 2408 2469 a Fs(\037)1113 2624 y Fq(=)83 b Fs(\037)1333 2583 y Fn(2)1422 2624 y Fq(+)p 1548 2569 V 50 w Fs(\037)1609 2583 y Fn(2)1701 2624 y Fq(=)52 b Fp(1)22 b Fs(;)257 2844 y Fv(and,)j(similarly)-6 b(,)23 b Fs(R)q(H)35 b Fq(=)27 b Fp(1)p Fv(.)k(Thus)24 b Fs(R)29 b Fq(=)f Fs(H)1775 2808 y Fl(\000)p Fn(1)1869 2844 y Fv(,)c(and)h(\(II.8\))g(holds)f (true.)404 2965 y(Con)l(v)o(ersely)-6 b(,)23 b(suppose)h(that)h Fs(H)32 b Fv(is)24 b(bounded)g(in)l(v)o(ertible,)g(and)g(de\002ne)1319 3159 y Fo(e)1301 3185 y Fs(R)54 b Fq(:=)e Fs(\037)17 b(H)1751 3143 y Fl(\000)p Fn(1)1862 3185 y Fs(\037)50 b Fq(+)p 2098 3130 V 49 w Fs(\037)17 b(T)2247 3143 y Fl(\000)p Fn(1)p 2341 3130 V 2341 3185 a Fs(\037)22 b(:)787 b Fv(\(II.20\))257 3405 y(Then,)25 b(using)f(again)g(\(II.15\))h(and)g (\(II.16\),)g(we)g(obtain)267 3625 y Fs(F)378 3599 y Fo(e)360 3625 y Fs(R)84 b Fq(=)f Fs(F)30 b(\037)17 b(H)937 3583 y Fl(\000)p Fn(1)1047 3625 y Fs(\037)50 b Fq(+)g Fs(F)p 1377 3570 V 30 w(\037)17 b(T)1526 3583 y Fl(\000)p Fn(1)p 1620 3570 V 1620 3625 a Fs(\037)1557 b Fv(\(II.21\))518 3770 y Fq(=)83 b Fs(Q)754 3729 y Fn(#)834 3770 y Fs(\037)50 b Fq(+)f Fs(H)1151 3785 y Fm(\037)p 1216 3715 V 1216 3770 a Fs(\037)16 b(T)1364 3729 y Fl(\000)p Fn(1)p 1459 3715 V 1459 3770 a Fs(\037)49 b Fr(\000)i Fs(\037W)p 1864 3715 V 14 w(\037)16 b(H)2030 3729 y Fl(\000)p Fn(1)p 2022 3757 48 3 v 2022 3795 a Fm(\037)p 2124 3715 62 4 v 2124 3770 a Fs(\037W)e(\037)p 2369 3715 V 17 w(\037)i(T)2517 3729 y Fl(\000)p Fn(1)p 2611 3715 V 2611 3770 a Fs(\037)518 3925 y Fq(=)83 b Fs(\037)738 3884 y Fn(2)822 3925 y Fr(\000)45 b Fs(\037W)p 1111 3870 V 14 w(\037)16 b(H)1277 3884 y Fl(\000)p Fn(1)p 1269 3913 48 3 v 1269 3950 a Fm(\037)p 1371 3870 62 4 v 1371 3925 a Fs(\037\037)44 b Fq(+)p 1657 3870 V 44 w Fs(\037)1719 3884 y Fn(2)1802 3925 y Fq(+)g Fs(W)2014 3940 y Fm(\037)p 2063 3870 V 2063 3925 a Fs(\037T)2195 3884 y Fl(\000)p Fn(1)p 2289 3870 V 2289 3925 a Fs(\037)50 b Fr(\000)g Fs(\037W)p 2694 3870 V 14 w(\037)16 b(H)2860 3884 y Fl(\000)p Fn(1)p 2852 3913 48 3 v 2852 3950 a Fm(\037)p 2954 3870 62 4 v 2954 3925 a Fs(\037W)e(\037)p 3182 3870 V(\037)j(T)3331 3884 y Fl(\000)p Fn(1)p 3425 3870 V 3425 3925 a Fs(\037)518 4081 y Fq(=)83 b Fp(1)44 b Fr(\000)h Fs(\037W)p 1066 4026 V 14 w(\037)16 b(H)1232 4039 y Fl(\000)p Fn(1)p 1224 4068 48 3 v 1224 4106 a Fm(\037)p 1326 4026 62 4 v 1326 4081 a Fs(\037\037)45 b Fq(+)f Fs(W)1705 4096 y Fm(\037)p 1753 4026 V 1753 4081 a Fs(\037T)1885 4039 y Fl(\000)p Fn(1)p 1979 4026 V 1979 4081 a Fs(\037)50 b Fr(\000)g Fs(\037W)p 2384 4026 V 14 w(\037)17 b(H)2551 4039 y Fl(\000)p Fn(1)p 2543 4068 48 3 v 2543 4106 a Fm(\037)p 2645 4026 62 4 v 2645 4081 a Fs(\037W)d(\037)p 2873 4026 V(\037)i(T)3021 4039 y Fl(\000)p Fn(1)p 3115 4026 V 3115 4081 a Fs(\037)518 4236 y Fq(=)83 b Fp(1)44 b Fr(\000)h Fs(\037W)p 1066 4182 V 14 w(\037)1127 4156 y Fo(\010)1185 4236 y Fs(H)1274 4195 y Fl(\000)p Fn(1)p 1266 4224 48 3 v 1266 4262 a Fm(\037)1407 4236 y Fr(\000)39 b Fs(T)1594 4195 y Fl(\000)p Fn(1)1727 4236 y Fq(+)f Fs(H)1930 4195 y Fl(\000)p Fn(1)p 1922 4224 V 1922 4262 a Fm(\037)p 2024 4182 62 4 v 2024 4236 a Fs(\037W)p 2191 4182 V 14 w(\037)17 b(T)2340 4195 y Fl(\000)p Fn(1)2434 4156 y Fo(\011)p 2492 4182 V 80 x Fs(\037\037)518 4382 y Fq(=)83 b Fp(1)22 b Fs(;)257 4602 y Fv(where)e(we)f(use)g(the)g (second)f(resolv)o(ent)g(identity)f(and)i(the)g(f)o(act)g(that)g Fs(T)14 b Fv(,)20 b Fs(\037)f Fv(and)p 2948 4547 V 19 w Fs(\037)g Fv(commute)f(to)257 4722 y(deri)n(v)o(e)26 b(the)g(last)g(equality)-6 b(.)33 b(Similarly)-6 b(,)25 b(one)i(checks)f(that)2294 4697 y Fo(e)2276 4722 y Fs(R)q(F)44 b Fq(=)30 b Fp(1)p Fv(.)36 b(Thus)25 b Fs(F)40 b Fv(is)26 b(\(globally\))257 4843 y(in)l(v)o(ertible)c(on)h Fr(H)i Fv(with)d(in)l(v)o(erse)h Fs(F)1472 4806 y Fl(\000)p Fn(1)1594 4843 y Fq(=)1715 4817 y Fo(e)1697 4843 y Fs(R)q Fv(.)31 b(Denoting)22 b(by)h Fs(P)37 b Fv(the)23 b(projection)f(onto)h Fq(Ran)16 b Fs(\037)p Fv(,)257 4963 y(we)25 b(observ)o(e)g(that)1341 5083 y Fs(F)66 b Fq(=)53 b Fs(P)30 b(F)g(P)63 b Fq(+)50 b Fs(P)2114 5042 y Fl(?)2189 5083 y Fs(T)30 b(P)2353 5042 y Fl(?)3238 5083 y Fv(\(II.22\))257 5258 y(is)d(block-diagonal)e (w)-6 b(.)26 b(r)-5 b(.)26 b(t.)h Fr(H)32 b Fq(=)f(Ran)16 b Fs(\037)24 b Fr(\010)g Fq(\(Ran)16 b Fs(\037)p Fq(\))2179 5222 y Fl(?)2238 5258 y Fv(.)36 b(Hence,)28 b(the)e(global)g(in)l(v)o (ertibility)257 5378 y(of)f Fs(F)39 b Fv(on)24 b Fr(H)i Fv(implies)e(the)g(in)l(v)o(ertibility)e(of)j(its)f(restriction)g(to)h Fq(Ran)16 b Fs(\037)p Fv(,)25 b(pro)o(ving)e(\(II.9\).)p eop %%Page: 14 15 14 14 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(14)257 573 y Ff(Pr)n(oof)25 b(of)g(\(ii\).)31 b Fv(If)25 b Fs( )32 b Fr(2)c(H)23 b(n)f(f)p Fq(0)p Fr(g)i Fv(solv)o(es)g Fs(H)8 b( )31 b Fq(=)d(0)c Fv(then)h(\(II.16\))g (implies)e(that)1341 793 y Fs(F)14 b Fq(\()p Fs(\037 )t Fq(\))53 b(=)f Fs(Q)1880 752 y Fn(#)1960 793 y Fs(H)24 b( )57 b Fq(=)52 b(0)22 b Fs(:)827 b Fv(\(II.23\))257 1013 y(Furthermore,)25 b(by)g(\(II.15\))1135 1233 y Fq(0)53 b(=)p 1365 1178 62 4 v 52 w Fs(\037)17 b(H)24 b( )56 b Fq(=)d Fs(H)p 1877 1210 48 3 v 15 x Fm(\037)p 1941 1178 62 4 v 1941 1233 a Fs(\037 )c Fq(+)p 2234 1178 V 44 w Fs(\037W)14 b(\037)2462 1192 y Fn(2)2501 1233 y Fs( )26 b(;)621 b Fv(\(II.24\))257 1453 y(and)25 b(hence)612 1673 y Fs(Q)17 b(\037 )57 b Fq(=)52 b Fs(\037)1076 1632 y Fn(2)1116 1673 y Fs( )43 b Fr(\000)p 1338 1618 V 39 w Fs(\037H)1488 1632 y Fl(\000)p Fn(1)p 1480 1660 48 3 v 1480 1698 a Fm(\037)p 1582 1618 62 4 v 1582 1673 a Fs(\037W)14 b(\037)1810 1632 y Fn(2)1849 1673 y Fs( )57 b Fq(=)52 b Fs(\037)2158 1632 y Fn(2)2198 1673 y Fs( )43 b Fq(+)p 2418 1618 V 38 w Fs(\037)2479 1632 y Fn(2)2519 1673 y Fs( )56 b Fq(=)d Fs( )26 b(:)355 b Fv(\(II.25\))257 1893 y(Therefore,)26 b Fs( )32 b Fr(6)p Fq(=)27 b(0)e Fv(implies)e Fs(\037 )32 b Fr(6)p Fq(=)27 b(0)p Fv(.)257 2013 y Ff(Pr)n(oof)e(of)g(\(iii\).)31 b Fv(If)25 b Fs(')i Fr(2)h Fq(Ran)17 b Fs(\037)22 b Fr(n)g(f)p Fq(0)p Fr(g)i Fv(solv)o(es)g Fs(F)14 b(')27 b Fq(=)h(0)c Fv(then)h(\(II.16\))g (implies)e(that)1414 2233 y Fs(H)h(Q')53 b Fq(=)f Fs(\037)17 b(F)d(')52 b Fq(=)h(0)22 b Fs(:)899 b Fv(\(II.26\))257 2453 y(Denote)30 b(by)p 705 2373 77 4 v 30 w Fs(P)43 b Fv(the)30 b(orthogonal)f(projection)g(onto)g Fq(Ran)p 2257 2398 62 4 v 16 w Fs(\037)q Fv(,)i(and)f(note,)h(in)e(passing,)h (that)g Fs(P)257 2574 y Fv(and)p 431 2494 77 4 v 30 w Fs(P)43 b Fv(commute,)30 b(b)n(ut)f Fs(P)p 1196 2494 V 14 w(P)50 b Fr(6)p Fq(=)37 b(0)p Fv(,)30 b(unless)f Fs(\037)h Fv(is)g(a)g(projection.)44 b(Since)31 b Fs(T)43 b Fv(is)29 b(in)l(v)o(ertible)g(on)257 2694 y Fq(Ran)p 449 2639 62 4 v 17 w Fs(\037)p Fv(,)c(the)f(tri)n(vial)g(identity)f Fs(F)41 b Fq(=)28 b Fs(T)36 b Fq(+)22 b Fs(\037W)14 b(Q)25 b Fv(implies)e(that)p 1047 2834 77 4 v 1047 2914 a Fs(P)66 b Fq(=)52 b Fs(T)1375 2873 y Fl(\000)p Fn(1)p 1486 2834 V 1486 2914 a Fs(P)30 b(T)66 b Fq(=)53 b Fs(T)1902 2873 y Fl(\000)p Fn(1)p 2012 2834 V 2012 2914 a Fs(P)30 b Fq(\()p Fs(F)53 b Fr(\000)39 b Fs(\037W)14 b(Q)p Fq(\))22 b Fs(;)532 b Fv(\(II.27\))257 3155 y(which,)25 b(together)f(with)p 1104 3075 V 24 w Fs(P)1181 3093 y Fl(?)1267 3155 y Fq(=)p 1371 3075 V 28 w Fs(P)1447 3093 y Fl(?)1506 3155 y Fs(Q)p Fv(,)h(gi)n(v)o(es)643 3392 y Fp(1)52 b Fq(=)p 880 3312 V 53 w Fs(P)g Fq(+)p 1109 3312 V 38 w Fs(P)1186 3330 y Fl(?)1298 3392 y Fq(=)g Fs(T)1497 3351 y Fl(\000)p Fn(1)p 1591 3312 V 1591 3392 a Fs(P)13 b(F)64 b Fq(+)1920 3311 y Fo(\000)p 1965 3312 V 1965 3392 a Fs(P)2042 3330 y Fl(?)2140 3392 y Fr(\000)39 b Fs(T)2327 3351 y Fl(\000)p Fn(1)p 2421 3312 V 2421 3392 a Fs(P)13 b(\037W)2664 3311 y Fo(\001)2726 3392 y Fs(Q)23 b(:)385 b Fv(\(II.28\))257 3612 y(Applying)24 b(\(II.28\))h(to)f Fs(')p Fv(,)h(we)g(obtain)f(that) 1255 3849 y Fs(')53 b Fq(=)1500 3769 y Fo(\000)p 1546 3769 V 80 x Fs(P)1622 3788 y Fl(?)1720 3849 y Fr(\000)39 b Fs(T)1907 3808 y Fl(\000)p Fn(1)p 2001 3769 V 2001 3849 a Fs(P)14 b(\037W)2245 3769 y Fo(\001)2307 3849 y Fs(Q')22 b(;)741 b Fv(\(II.29\))257 4069 y(which)25 b(implies)e(that)h Fs(Q')k Fr(6)p Fq(=)g(0)c Fv(pro)o(vided)g Fs(')k Fr(6)p Fq(=)f(0)p Fv(.)257 4190 y Ff(Pr)n(oof)k(of)f(\(i)o(v\).) 47 b Fv(By)31 b(\(i\),)g Fq(dim)15 b(Ker)p Fs(H)46 b Fq(=)37 b(0)30 b Fv(is)g(equi)n(v)n(alent)e(to)i Fq(dim)15 b(Ker)q Fs(F)51 b Fq(=)37 b(0)p Fv(,)32 b(assuming)257 4310 y(that)j Fq(\()p Fs(H)r(;)17 b(T)d Fq(\))34 b Fv(is)g(a)h (Feshbach)g(pair)-5 b(.)60 b(W)-8 b(e)35 b(may)g(therefore)g(assume)f (that)g Fq(Ker)p Fs(H)54 b Fr(6)p Fq(=)46 b(0)34 b Fv(and)257 4431 y Fq(Ker)p Fs(F)51 b Fr(6)p Fq(=)37 b(0)30 b Fv(are)g(both)f (nontri)n(vial.)44 b(Eq.)30 b(\(II.25\))g(sho)n(ws)f(that)g Fs(\037)37 b Fq(:)g(Ker)p Fs(H)45 b Fr(!)36 b Fq(Ker)q Fs(F)43 b Fv(is)30 b(in-)257 4551 y(jecti)n(v)o(e,)f(hence)g Fq(dim)15 b(Ker)p Fs(H)43 b Fr(\024)35 b Fq(dim)16 b(Ker)p Fs(F)e Fv(,)29 b(and)g(Eq.)f(\(II.29\))i(sho)n(ws)d(that)h Fs(Q)36 b Fq(:)f(Ker)p Fs(F)49 b Fr(!)257 4671 y Fq(Ker)p Fs(H)29 b Fv(is)21 b(injecti)n(v)o(e,)f(hence)i Fq(dim)15 b(Ker)q Fs(H)35 b Fr(\025)28 b Fq(dim)15 b(Ker)p Fs(F)f Fv(.)30 b(This)20 b(establishes)g(\(i)n(v\))h(and)g(more-)257 4792 y(o)o(v)o(er)j(that)h Fs(\037)i Fq(:)h(Ker)p Fs(H)36 b Fr(!)27 b Fq(Ker)p Fs(F)39 b Fv(and)24 b Fs(Q)k Fq(:)g(Ker)p Fs(F)42 b Fr(!)27 b Fq(Ker)p Fs(H)32 b Fv(are)26 b(actually)e (bijections.)257 4912 y Ff(Pr)n(oof)k(of)f(\(v\).)37 b Fv(By)28 b(assumption,)d Fs(H)p 1540 4889 48 3 v 15 x Fm(\037)1612 4912 y Fr(\000)f Fs(z)32 b Fv(is)27 b(bounded)f(in)l(v)o (ertible)f(on)i Fq(Ran)p 2975 4857 62 4 v 16 w Fs(\037)p Fv(,)h(for)f Fs(z)37 b Fq(=)32 b(0)p Fv(.)257 5032 y(Thus)21 b(the)g(\002rst)h(resolv)o(ent)e(identity)g(yields)h(the)g(e)o (xistence)g(of)h(an)f(open)g(neighborhood)f Fs(U)39 b Fr(\032)257 5153 y Fg(C)58 b Fv(of)31 b Fq(0)g Fv(and)g(a)h(constant)e Fs(C)47 b(<)39 b Fr(1)31 b Fv(such)g(that)g Fr(k)p 1977 5098 V Fs(\037)p Fq(\()p Fs(H)p 2157 5130 48 3 v 15 x Fm(\037)2231 5153 y Fr(\000)d Fs(z)t Fq(\))2423 5117 y Fl(\000)p Fn(1)p 2518 5098 62 4 v 2518 5153 a Fs(\037)p Fr(k)39 b(\024)h Fs(C)7 b Fv(,)33 b(for)e(all)g Fs(z)45 b Fr(2)40 b Fs(U)10 b Fv(.)257 5273 y(Therefore,)26 b Fq(\()p Fs(H)k Fr(\000)22 b Fs(z)t(;)17 b(T)36 b Fr(\000)23 b Fs(z)t Fq(\))j Fv(de\002nes)f(a)g(Feshbach)g(pair)l(,)g(for)g(all)f Fs(z)33 b Fr(2)28 b Fs(U)10 b Fv(.)p eop %%Page: 15 16 15 15 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(15)404 573 y(In)24 b(the)f(proof,)h(we)g(abbre)n(viate)g Fs(F)1561 588 y Fm(z)1629 573 y Fq(:=)j Fs(F)1822 588 y Fm(\037)1870 573 y Fq(\()p Fs(H)g Fr(\000)19 b Fs(z)t(;)e(T)33 b Fr(\000)19 b Fs(z)t Fq(\))p Fv(,)25 b Fs(Q)j Fq(:=)f Fs(Q)2840 588 y Fm(\037)2889 573 y Fq(\()p Fs(H)r(;)17 b(T)d Fq(\))p Fv(,)23 b(and)h(we)257 693 y(recall)32 b(that)f Fs(M)10 b Fv(,)34 b Fs(N)42 b Fv(are)33 b(de\002ned)f(as)f(in) g(\(II.10\)\226\(II.11\),)j(corresponding)d(to)g Fs(z)45 b Fq(=)40 b(0)p Fv(.)50 b(Our)257 814 y(\002rst)25 b(goal)g(is)f(to)h (sho)n(w)e(that,)h(for)i(an)o(y)e Fs( )32 b Fr(2)c(H)q Fv(,)475 993 y Fq(lim)474 1055 y Fm(")p Fl(&)p Fn(0)656 993 y Fq(Im)773 912 y Fo(\012)820 993 y Fs( )887 908 y Fo(\014)887 968 y(\014)942 993 y Fs(F)1019 951 y Fl(\000)p Fn(1)1005 1018 y Fm(i")1130 993 y Fs( )1197 912 y Fo(\013)1297 993 y Fq(=)53 b(lim)1425 1055 y Fm(")p Fl(&)p Fn(0)1608 993 y Fq(Im)1724 912 y Fo(\012)1771 993 y Fs(N)27 b( )1943 908 y Fo(\014)1943 968 y(\014)1999 912 y(\000)2044 993 y Fs(N)h(F)2213 1008 y Fn(0)2269 993 y Fs(N)49 b Fr(\000)39 b Fs(i")2591 912 y Fo(\001)2637 934 y Fl(\000)p Fn(1)2753 993 y Fs(N)27 b( )2925 912 y Fo(\013)2995 993 y Fs(:)216 b Fv(\(II.30\))257 1202 y(T)-8 b(o)25 b(this)f(end,)h(we)g(compute)f (the)g(dif)n(ference)h(between)g Fs(F)2249 1217 y Fm(i")2335 1202 y Fv(and)g Fs(F)2567 1217 y Fn(0)2629 1202 y Fr(\000)d Fs(i")p Fv(,)442 1371 y Fs(F)505 1386 y Fm(i")589 1371 y Fr(\000)g Fs(F)751 1386 y Fn(0)813 1371 y Fq(+)g Fs(i")2248 b Fv(\(II.31\))525 1516 y Fq(=)83 b Fr(\000)p Fs(\037)17 b(W)p 962 1461 62 4 v 31 w(\037)g Fq(\()p Fs(H)p 1159 1493 48 3 v 15 x Fm(\037)1228 1516 y Fr(\000)23 b Fs(i")p Fq(\))1445 1475 y Fl(\000)p Fn(1)p 1556 1461 62 4 v 1556 1516 a Fs(\037)17 b(W)30 b(\037)50 b Fq(+)f Fs(\037)17 b(W)p 2192 1461 V 30 w(\037)g(H)2359 1475 y Fl(\000)p Fn(1)p 2351 1504 48 3 v 2351 1541 a Fm(\037)p 2470 1461 62 4 v 2470 1516 a Fs(\037)f(W)31 b(\037)525 1671 y Fq(=)83 b Fr(\000)p Fs(i")17 b(\037)g(W)p 1057 1617 V 30 w(\037)g Fq(\()p Fs(H)p 1254 1649 48 3 v 15 x Fm(\037)1324 1671 y Fr(\000)23 b Fs(i")p Fq(\))1541 1630 y Fl(\000)p Fn(1)1651 1671 y Fs(H)1740 1630 y Fl(\000)p Fn(1)p 1732 1659 V 1732 1697 a Fm(\037)p 1851 1617 62 4 v 1851 1671 a Fs(\037)17 b(W)30 b(\037)525 1827 y Fq(=)83 b Fr(\000)p Fs(i")17 b(\037)g(W)p 1057 1772 V 30 w(\037)g(H)1224 1785 y Fl(\000)p Fn(2)p 1216 1814 48 3 v 1216 1852 a Fm(\037)p 1335 1772 62 4 v 1335 1827 a Fs(\037)f(W)30 b(\037)50 b Fq(+)g Fs(")1817 1786 y Fn(2)1873 1827 y Fs(\037)17 b(W)p 2073 1772 V 30 w(\037)f(H)2239 1785 y Fl(\000)p Fn(1)p 2231 1814 48 3 v 2231 1852 a Fm(\037)2333 1827 y Fq(\()p Fs(H)p 2452 1804 V 15 x Fm(\037)2522 1827 y Fr(\000)23 b Fs(i")p Fq(\))2739 1786 y Fl(\000)p Fn(1)2850 1827 y Fs(H)2939 1785 y Fl(\000)p Fn(1)p 2931 1814 V 2931 1852 a Fm(\037)p 3049 1772 62 4 v 3049 1827 a Fs(\037)17 b(W)30 b(\037)525 1982 y Fq(=)83 b Fr(\000)p Fs(i")17 b(M)961 1941 y Fl(\003)1018 1982 y Fs(M)61 b Fq(+)49 b Fs(")1344 1941 y Fn(2)1400 1982 y Fs(M)1504 1941 y Fl(\003)1561 1982 y Fq(\()p Fs(H)p 1680 1959 48 3 v 15 x Fm(\037)1750 1982 y Fr(\000)22 b Fs(i")p Fq(\))1966 1941 y Fl(\000)p Fn(1)2077 1982 y Fs(M)525 2127 y Fq(=)83 b Fr(\000)p Fs(i")17 b(M)961 2086 y Fl(\003)1018 2127 y Fs(M)61 b Fq(+)49 b Fs(")1344 2086 y Fn(2)1400 2127 y Fs(M)1504 2086 y Fl(\003)1561 2127 y Fs(H)1650 2086 y Fl(\000)p Fn(1)p 1642 2115 V 1642 2152 a Fm(\037)1760 2127 y Fs(M)61 b Fq(+)50 b Fs(i")2120 2086 y Fn(3)2176 2127 y Fs(M)2280 2086 y Fl(\003)2337 2127 y Fq(\()p Fs(H)p 2456 2105 V 15 x Fm(\037)2525 2127 y Fr(\000)23 b Fs(i")p Fq(\))2742 2086 y Fl(\000)p Fn(1)2853 2127 y Fs(H)2942 2086 y Fl(\000)p Fn(1)p 2934 2115 V 2934 2152 a Fm(\037)3052 2127 y Fs(M)33 b(:)257 2295 y Fv(T)-8 b(aking)25 b(into)e(account)i(that)g Fs(N)1334 2259 y Fl(\000)p Fn(2)1456 2295 y Fq(=)j Fp(1)22 b Fq(+)g Fs(M)1840 2259 y Fl(\003)1880 2295 y Fs(M)10 b Fv(,)26 b(this)d(is)i(equi)n(v)n(alent)e(to)338 2464 y Fs(F)401 2479 y Fm(i")484 2464 y Fr(\000)g Fs(F)647 2479 y Fn(0)709 2464 y Fq(+)f Fs(i"N)974 2423 y Fl(\000)p Fn(2)1121 2464 y Fq(=)52 b Fs(")1295 2423 y Fn(2)1351 2464 y Fs(M)1455 2423 y Fl(\003)1512 2464 y Fs(H)1601 2423 y Fl(\000)p Fn(1)p 1593 2451 V 1593 2489 a Fm(\037)1711 2464 y Fs(M)61 b Fq(+)50 b Fs(i")2071 2423 y Fn(3)2127 2464 y Fs(M)2231 2423 y Fl(\003)2287 2464 y Fq(\()p Fs(H)p 2406 2441 V 15 x Fm(\037)2476 2464 y Fr(\000)23 b Fs(i")p Fq(\))2693 2423 y Fl(\000)p Fn(1)2804 2464 y Fs(H)2893 2423 y Fl(\000)p Fn(1)p 2885 2451 V 2885 2489 a Fm(\037)3003 2464 y Fs(M)33 b(:)81 b Fv(\(II.32\))257 2642 y(Multiplying)27 b(\(II.32\))i(by)g Fs(F)1255 2601 y Fl(\000)p Fn(1)1241 2668 y Fm(i")1378 2642 y Fv(from)g(the)g(left)g(and)g(by)g Fq(\()p Fs(F)2317 2657 y Fn(0)2382 2642 y Fr(\000)d Fs(i"N)2652 2606 y Fl(\000)p Fn(2)2747 2642 y Fq(\))2785 2606 y Fl(\000)p Fn(1)2908 2642 y Fv(from)j(the)g(right,)257 2762 y(we)c(obtain)589 2931 y Fs(F)666 2889 y Fl(\000)p Fn(1)652 2956 y Fm(i")782 2931 y Fr(\000)882 2850 y Fo(\000)927 2931 y Fs(F)990 2946 y Fn(0)1052 2931 y Fr(\000)e Fs(i"N)1319 2890 y Fl(\000)p Fn(2)1413 2850 y Fo(\001)1459 2872 y Fl(\000)p Fn(1)1606 2931 y Fq(=)52 b Fr(\000)p Fs(")1857 2890 y Fn(2)1914 2931 y Fs(F)1991 2889 y Fl(\000)p Fn(1)1977 2956 y Fm(i")2101 2931 y Fs(M)2205 2890 y Fl(\003)2262 2931 y Fs(H)2351 2889 y Fl(\000)p Fn(1)p 2343 2918 V 2343 2956 a Fm(\037)2461 2931 y Fs(M)2583 2850 y Fo(\000)2628 2931 y Fs(F)2691 2946 y Fn(0)2753 2931 y Fr(\000)23 b Fs(i"N)3020 2890 y Fl(\000)p Fn(2)3115 2850 y Fo(\001)3160 2872 y Fl(\000)p Fn(1)783 3103 y Fq(+)k Fs(i")965 3062 y Fn(3)1021 3103 y Fs(F)1098 3062 y Fl(\000)p Fn(1)1084 3129 y Fm(i")1209 3103 y Fs(M)1313 3062 y Fl(\003)1369 3103 y Fq(\()p Fs(H)p 1488 3081 V 15 x Fm(\037)1558 3103 y Fr(\000)c Fs(i")p Fq(\))1775 3062 y Fl(\000)p Fn(1)1886 3103 y Fs(H)1975 3062 y Fl(\000)p Fn(1)p 1967 3091 V 1967 3128 a Fm(\037)2085 3103 y Fs(M)2207 3023 y Fo(\000)2252 3103 y Fs(F)2315 3118 y Fn(0)2377 3103 y Fr(\000)g Fs(i"N)2644 3062 y Fl(\000)p Fn(2)2738 3023 y Fo(\001)2784 3045 y Fl(\000)p Fn(1)2900 3103 y Fs(:)311 b Fv(\(II.33\))257 3282 y(Since)26 b Fr(k)p Fs(F)631 3240 y Fl(\000)p Fn(1)617 3307 y Fm(i")724 3282 y Fr(k)i Fq(=)g Fr(O)s Fq(\()p Fs(")1072 3245 y Fl(\000)p Fn(1)1165 3282 y Fq(\))d Fv(and)g Fr(k)p Fq(\()p Fs(F)1548 3297 y Fn(0)1610 3282 y Fr(\000)d Fs(i"N)1876 3245 y Fl(\000)p Fn(2)1971 3282 y Fq(\))2009 3245 y Fl(\000)p Fn(1)2103 3282 y Fr(k)28 b Fq(=)f Fr(O)s Fq(\()p Fs(")2450 3245 y Fl(\000)p Fn(1)2544 3282 y Fq(\))p Fv(,)e(as)g(well,)f(we)h(ha)n(v)o(e)g(that)955 3380 y Fo(\015)955 3440 y(\015)1010 3464 y Fs(F)1087 3423 y Fl(\000)p Fn(1)1073 3490 y Fm(i")1203 3464 y Fr(\000)1303 3384 y Fo(\000)1349 3464 y Fs(F)1412 3479 y Fn(0)1473 3464 y Fr(\000)e Fs(i"N)1740 3428 y Fl(\000)p Fn(2)1835 3384 y Fo(\001)1880 3406 y Fl(\000)p Fn(1)1975 3380 y Fo(\015)1975 3440 y(\015)2083 3464 y Fq(=)52 b Fr(O)s Fq(\(1\))236 b Fv(and)440 b(\(II.34\))608 3549 y Fo(\015)608 3609 y(\015)663 3634 y Fs(i")742 3598 y Fn(3)798 3634 y Fs(F)875 3592 y Fl(\000)p Fn(1)861 3659 y Fm(i")986 3634 y Fs(M)1090 3598 y Fl(\003)1147 3634 y Fq(\()p Fs(H)p 1266 3611 V 15 x Fm(\037)1335 3634 y Fr(\000)23 b Fs(i")p Fq(\))1552 3598 y Fl(\000)p Fn(1)1663 3634 y Fs(H)1752 3592 y Fl(\000)p Fn(1)p 1744 3621 V 1744 3659 a Fm(\037)1862 3634 y Fs(M)1984 3553 y Fo(\000)2029 3634 y Fs(F)2092 3649 y Fn(0)2154 3634 y Fr(\000)g Fs(i"N)2421 3598 y Fl(\000)p Fn(2)2516 3553 y Fo(\001)2561 3575 y Fl(\000)p Fn(1)2656 3549 y Fo(\015)2656 3609 y(\015)2763 3634 y Fq(=)53 b Fr(O)s Fq(\()p Fs(")p Fq(\))22 b Fs(:)93 b Fv(\(II.35\))257 3802 y(Eq.)37 b(\(II.35\))g(implies)e(that)h(the)h (second)f(term)g(on)h(the)f(right)g(side)g(of)h(\(II.33\))g(can)g(be)g (ne-)257 3922 y(glected,)25 b(as)g Fs(")i Fr(!)g Fq(0)p Fv(,)e(i.e.,)589 4115 y Fq(lim)588 4177 y Fm(")p Fl(&)p Fn(0)771 4115 y Fq(Im)887 4004 y Fo(D)948 4115 y Fs( )1015 4000 y Fo(\014)1015 4060 y(\014)1015 4120 y(\014)1070 4034 y(\010)1129 4115 y Fs(F)1206 4074 y Fl(\000)p Fn(1)1192 4140 y Fm(i")1344 4115 y Fr(\000)45 b Fq(\()p Fs(F)1567 4130 y Fn(0)1628 4115 y Fr(\000)23 b Fs(i"N)1895 4074 y Fl(\000)p Fn(2)1990 4115 y Fq(\))2028 4074 y Fl(\000)p Fn(1)2122 4034 y Fo(\011)2197 4115 y Fs( )2264 4004 y Fo(E)3238 4115 y Fv(\(II.36\))671 4348 y Fq(=)83 b Fr(\000)18 b Fq(lim)924 4411 y Fm(")p Fl(&)p Fn(0)1079 4238 y Fo(n)1145 4348 y Fs(")1191 4307 y Fn(2)1247 4348 y Fq(Im)1364 4238 y Fo(D)1424 4348 y Fs( )1491 4234 y Fo(\014)1491 4293 y(\014)1491 4353 y(\014)1547 4348 y Fs(F)1624 4307 y Fl(\000)p Fn(1)1610 4374 y Fm(i")1734 4348 y Fs(M)1838 4307 y Fl(\003)1895 4348 y Fs(H)1984 4307 y Fl(\000)p Fn(1)p 1976 4336 V 1976 4373 a Fm(\037)2095 4348 y Fs(M)2216 4267 y Fo(\000)2262 4348 y Fs(F)2325 4363 y Fn(0)2387 4348 y Fr(\000)k Fs(i"N)2653 4307 y Fl(\000)p Fn(2)2748 4267 y Fo(\001)2794 4290 y Fl(\000)p Fn(1)2904 4348 y Fs( )2971 4238 y Fo(E)17 b(o)3138 4348 y Fs(:)257 4569 y Fv(Thanks)32 b(to)f(\(II.34\))h(and)g(the)g(bound)f Fr(k)p Fq(\()p Fs(F)1743 4584 y Fn(0)1810 4569 y Fr(\000)d Fs(i"N)2082 4533 y Fl(\000)p Fn(2)2176 4569 y Fq(\))2214 4533 y Fl(\000)p Fn(1)2309 4569 y Fr(k)40 b Fq(=)h Fr(O)s Fq(\()p Fs(")2682 4533 y Fl(\000)p Fn(1)2776 4569 y Fq(\))p Fv(,)33 b(we)g(may)e(further)257 4689 y(replace)26 b Fs(F)647 4648 y Fl(\000)p Fn(1)633 4715 y Fm(i")766 4689 y Fv(by)e Fq(\()p Fs(F)991 4704 y Fn(0)1053 4689 y Fr(\000)f Fs(i"N)1320 4653 y Fl(\000)p Fn(2)1414 4689 y Fq(\))1452 4653 y Fl(\000)p Fn(1)1572 4689 y Fv(on)h(the)h(right)f(side)g(of)h (\(II.36\).)31 b(That)25 b(is,)364 4886 y Fq(lim)362 4948 y Fm(")p Fl(&)p Fn(0)545 4886 y Fq(Im)661 4775 y Fo(D)722 4886 y Fs( )789 4771 y Fo(\014)789 4831 y(\014)789 4891 y(\014)845 4805 y(\010)903 4886 y Fs(F)980 4844 y Fl(\000)p Fn(1)966 4911 y Fm(i")1118 4886 y Fr(\000)45 b Fq(\()p Fs(F)1341 4901 y Fn(0)1402 4886 y Fr(\000)23 b Fs(i"N)1669 4844 y Fl(\000)p Fn(2)1764 4886 y Fq(\))1802 4844 y Fl(\000)p Fn(1)1896 4805 y Fo(\011)1971 4886 y Fs( )2038 4775 y Fo(E)3238 4886 y Fv(\(II.37\))445 5119 y Fq(=)83 b Fr(\000)19 b Fq(lim)698 5182 y Fm(")p Fl(&)p Fn(0)853 5008 y Fo(n)920 5119 y Fs(")966 5078 y Fn(2)1021 5119 y Fq(Im)1138 5008 y Fo(D)1199 5119 y Fs( )1266 5004 y Fo(\014)1266 5064 y(\014)1266 5124 y(\014)1321 5038 y(\000)1367 5119 y Fs(F)1430 5134 y Fn(0)1491 5119 y Fr(\000)k Fs(i"N)1758 5078 y Fl(\000)p Fn(2)1853 5038 y Fo(\001)1899 5061 y Fl(\000)p Fn(1)2009 5119 y Fs(M)2113 5078 y Fl(\003)2170 5119 y Fs(H)2259 5078 y Fl(\000)p Fn(1)p 2251 5107 V 2251 5144 a Fm(\037)2370 5119 y Fs(M)2491 5038 y Fo(\000)2537 5119 y Fs(F)2600 5134 y Fn(0)2661 5119 y Fr(\000)g Fs(i"N)2928 5078 y Fl(\000)p Fn(2)3023 5038 y Fo(\001)3069 5061 y Fl(\000)p Fn(1)3179 5119 y Fs( )3246 5008 y Fo(E)17 b(o)445 5352 y Fq(=)85 b(lim)604 5415 y Fm(")p Fl(&)p Fn(0)787 5352 y Fq(Im)903 5242 y Fo(D)964 5352 y Fs(M)28 b(N)1417 5285 y(")p 1201 5329 479 4 v 1201 5421 a(N)10 b(F)1352 5436 y Fn(0)1392 5421 y Fs(N)32 b Fq(+)22 b Fs(i")1706 5352 y(N)27 b( )1878 5238 y Fo(\014)1878 5298 y(\014)1878 5357 y(\014)1933 5352 y Fs(H)2022 5311 y Fl(\000)p Fn(1)p 2014 5340 48 3 v 2014 5378 a Fm(\037)2133 5352 y Fs(M)g(N)2587 5285 y(")p 2369 5329 481 4 v 2369 5421 a(N)10 b(F)2520 5436 y Fn(0)2560 5421 y Fs(N)33 b Fr(\000)23 b Fs(i")2876 5352 y(N)k( )3048 5242 y Fo(E)3131 5352 y Fs(:)p eop %%Page: 16 17 16 16 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(16)257 573 y(Finally)-6 b(,)23 b(we)h(use)g(the)g(f)o(act)g(that)g Fs(")p Fq(\()p Fs(N)10 b(F)1606 588 y Fn(0)1645 573 y Fs(N)30 b Fr(\006)19 b Fs(i")p Fq(\))1966 537 y Fl(\000)p Fn(1)2084 573 y Fv(con)l(v)o(er)n(ges)24 b(strongly)e(to)i Fr(\007)p Fs(i)17 b(P)3147 588 y Fn(Ker)p Fm(N)7 b(F)3369 597 y Fh(0)3403 588 y Fm(N)3470 573 y Fv(,)257 693 y(where)38 b Fs(P)601 708 y Fn(Ker)p Fm(N)7 b(F)823 717 y Fh(0)857 708 y Fm(N)961 693 y Fv(denotes)37 b(the)g(projection)f(onto)g(the)h(k) o(ernel)g(of)h Fs(N)10 b(F)2833 708 y Fn(0)2873 693 y Fs(N)g Fv(.)68 b(Hence,)40 b(the)257 814 y(limit,)28 b(as)h Fs(")34 b Fr(&)h Fq(0)p Fv(,)29 b(of)g(the)f(matrix)g(element)g (on)g(the)h(right)f(side)g(of)h(\(II.37\))f(is)h(real,)h(and)e(we)257 934 y(arri)n(v)o(e)d(at)f(\(II.30\),)h(i.e.,)475 1146 y Fq(lim)474 1209 y Fm(")p Fl(&)p Fn(0)656 1146 y Fq(Im)773 1066 y Fo(\012)820 1146 y Fs( )887 1062 y Fo(\014)887 1121 y(\014)942 1146 y Fs(F)1019 1105 y Fl(\000)p Fn(1)1005 1172 y Fm(i")1130 1146 y Fs( )1197 1066 y Fo(\013)1297 1146 y Fq(=)53 b(lim)1425 1209 y Fm(")p Fl(&)p Fn(0)1608 1146 y Fq(Im)1724 1066 y Fo(\012)1771 1146 y Fs(N)27 b( )1943 1062 y Fo(\014)1943 1121 y(\014)1999 1066 y(\000)2044 1146 y Fs(N)h(F)2213 1161 y Fn(0)2269 1146 y Fs(N)49 b Fr(\000)39 b Fs(i")2591 1066 y Fo(\001)2637 1088 y Fl(\000)p Fn(1)2753 1146 y Fs(N)27 b( )2925 1066 y Fo(\013)2995 1146 y Fs(:)216 b Fv(\(II.38\))404 1400 y(In)23 b(order)g(to)g (establish)f(Eq.)h(\(II.13\),)h(we)f(combine)g(\(II.30\))g(\(or)h (\(II.38\)\))f(with)g(\(II.9\))g(and)257 1520 y(obtain)295 1733 y Fq(lim)293 1795 y Fm(")p Fl(&)p Fn(0)476 1733 y Fq(Im)592 1622 y Fo(D)653 1733 y Fs( )720 1618 y Fo(\014)720 1678 y(\014)720 1738 y(\014)776 1652 y(\000)821 1733 y Fs(N)k(F)989 1748 y Fn(0)1046 1733 y Fs(N)49 b Fr(\000)39 b Fs(i")1368 1652 y Fo(\001)1414 1674 y Fl(\000)p Fn(1)1525 1733 y Fs( )1592 1622 y Fo(E)1733 1733 y Fq(=)81 b(lim)1889 1795 y Fm(")p Fl(&)p Fn(0)2072 1733 y Fq(Im)2188 1652 y Fo(\012)2235 1733 y Fs(N)2323 1691 y Fl(\000)p Fn(1)2435 1733 y Fs( )2502 1648 y Fo(\014)2502 1708 y(\014)2557 1733 y Fs(F)2634 1691 y Fl(\000)p Fn(1)2620 1758 y Fm(i")2744 1733 y Fs(N)2832 1691 y Fl(\000)p Fn(1)2944 1733 y Fs( )3011 1652 y Fo(\013)376 1936 y Fq(=)k(lim)535 1999 y Fm(")p Fl(&)p Fn(0)718 1936 y Fq(Im)834 1855 y Fo(\012)881 1936 y Fs(\037)17 b(N)1047 1895 y Fl(\000)p Fn(1)1158 1936 y Fs( )1225 1851 y Fo(\014)1225 1911 y(\014)1281 1936 y Fq(\()p Fs(H)29 b Fr(\000)23 b Fs(i")p Fq(\))1646 1895 y Fl(\000)p Fn(1)1757 1936 y Fs(\037)17 b(N)1923 1895 y Fl(\000)p Fn(1)2034 1936 y Fs( )2101 1855 y Fo(\013)2198 1936 y Fq(+)50 b(Im)2440 1855 y Fo(\012)p 2487 1881 62 4 v 81 x Fs(\037)17 b(N)2653 1895 y Fl(\000)p Fn(1)2764 1936 y Fs( )2831 1851 y Fo(\014)2831 1911 y(\014)2887 1936 y Fs(T)2958 1895 y Fl(\000)p Fn(1)p 3068 1881 V 3068 1936 a Fs(\037)g(N)3234 1895 y Fl(\000)p Fn(1)3345 1936 y Fs( )3412 1855 y Fo(\013)376 2140 y Fq(=)85 b(lim)535 2202 y Fm(")p Fl(&)p Fn(0)718 2140 y Fq(Im)834 2059 y Fo(\012)881 2140 y Fs(\037)17 b(N)1047 2098 y Fl(\000)p Fn(1)1158 2140 y Fs( )1225 2055 y Fo(\014)1225 2115 y(\014)1281 2140 y Fq(\()p Fs(H)29 b Fr(\000)23 b Fs(i")p Fq(\))1646 2098 y Fl(\000)p Fn(1)1757 2140 y Fs(\037)17 b(N)1923 2098 y Fl(\000)p Fn(1)2034 2140 y Fs( )2101 2059 y Fo(\013)2170 2140 y Fs(:)1041 b Fv(\(II.39\))404 2393 y(The)30 b(proof)g(of)g(Eq.)g (\(II.12\))h(is)f(similar)-5 b(.)45 b(W)-8 b(e)30 b(use)h(again)e (\(II.30\))i(together)e(with)h(\(II.8\),)257 2514 y(which)25 b(yields)773 2726 y Fq(lim)772 2789 y Fm(")p Fl(&)p Fn(0)954 2726 y Fq(Im)1071 2645 y Fo(\012)1135 2726 y Fs( )1202 2641 y Fo(\014)1202 2701 y(\014)1257 2726 y Fq(\()p Fs(H)30 b Fr(\000)22 b Fs(i")p Fq(\))1622 2685 y Fl(\000)p Fn(1)1733 2726 y Fs( )1800 2645 y Fo(\013)855 2929 y Fq(=)84 b(lim)1014 2992 y Fm(")p Fl(&)p Fn(0)1196 2929 y Fq(Im)1313 2849 y Fo(\012)1360 2929 y Fs(Q)1437 2888 y Fl(\003)1493 2929 y Fs( )1560 2845 y Fo(\014)1560 2904 y(\014)1616 2929 y Fs(F)1693 2888 y Fl(\000)p Fn(1)1679 2955 y Fm(i")1803 2929 y Fs(Q)1880 2888 y Fl(\003)1936 2929 y Fs( )2003 2849 y Fo(\013)2100 2929 y Fq(+)50 b(Im)2343 2849 y Fo(\012)p 2390 2875 V 80 x Fs(\037)16 b( )2534 2845 y Fo(\014)2534 2904 y(\014)2590 2929 y Fs(H)2679 2888 y Fl(\000)p Fn(1)p 2671 2917 48 3 v 2671 2955 a Fm(\037)p 2789 2875 62 4 v 2789 2929 a Fs(\037)h( )2934 2849 y Fo(\013)855 3133 y Fq(=)84 b(lim)1014 3196 y Fm(")p Fl(&)p Fn(0)1196 3133 y Fq(Im)1313 3052 y Fo(\012)1360 3133 y Fs(Q)1437 3092 y Fl(\003)1493 3133 y Fs( )1560 3048 y Fo(\014)1560 3108 y(\014)1616 3133 y Fs(F)1693 3092 y Fl(\000)p Fn(1)1679 3158 y Fm(i")1803 3133 y Fs(Q)1880 3092 y Fl(\003)1936 3133 y Fs( )2003 3052 y Fo(\013)3238 3133 y Fv(\(II.40\))855 3353 y Fq(=)g(lim)1014 3416 y Fm(")p Fl(&)p Fn(0)1196 3353 y Fq(Im)1313 3272 y Fo(\012)1360 3353 y Fs(N)27 b(Q)1542 3312 y Fl(\003)1598 3353 y Fs( )1665 3268 y Fo(\014)1665 3328 y(\014)1721 3272 y(\000)1766 3353 y Fs(N)g(F)1934 3368 y Fn(0)1991 3353 y Fs(N)49 b Fr(\000)39 b Fs(i")2313 3272 y Fo(\001)2359 3295 y Fl(\000)p Fn(1)2475 3353 y Fs(N)27 b(Q)2657 3312 y Fl(\003)2714 3353 y Fs( )2781 3272 y Fo(\013)2850 3353 y Fs(:)3418 3606 y Fd(\003)257 3900 y Fe(II.2)119 b(Iterated)29 b(A)m(pplications)i(of)f(the)g(Smooth) g(F)m(eshbach)h(Map)257 4087 y Fv(In)k(this)f(section)g(we)h(study)e (iterated)i(applications)e(of)i(the)f(smooth)g(Feshbach)h(map.)60 b(T)-8 b(o)257 4218 y(moti)n(v)n(ate)27 b(our)h(study)-6 b(,)29 b(we)f(recall)i(from)e(\(II.14\))h(that)f(the)h(Feshbach)g(map)f Fs(F)2970 4167 y Fn(\(0\))2956 4245 y Fm(P)3064 4218 y Fq(\()p Fs(H)8 b Fq(\))28 b Fv(corre-)257 4354 y(sponding)j(to)h(a)g (projection)g Fs(P)45 b Fv(and)33 b(an)f(operator)g Fs(H)40 b Fv(is)32 b(gi)n(v)o(en)f(by)h Fs(F)2732 4303 y Fn(\(0\))2718 4381 y Fm(P)2826 4354 y Fq(\()p Fs(H)8 b Fq(\))40 b(=)i Fs(P)14 b(H)8 b(P)40 b Fr(\000)257 4475 y Fs(P)14 b(H)p 423 4394 77 4 v 8 w(P)f Fq(\()p 537 4394 V Fs(P)g(H)p 702 4394 V 8 w(P)g Fq(\))816 4438 y Fl(\000)p Fn(1)p 910 4394 V 910 4475 a Fs(P)h(H)8 b(P)14 b Fv(,)21 b(where)p 1464 4394 V 22 w Fs(P)41 b Fq(:=)28 b Fp(1)11 b Fr(\000)g Fs(P)j Fv(,)21 b(and)h(we)g(assume)f(that)p 2767 4394 V 21 w Fs(P)14 b(H)p 2933 4394 V 8 w(P)35 b Fv(is)21 b(in)l(v)o(ertible)257 4595 y(on)k Fq(Ran)p 557 4515 V Fs(P)13 b Fv(.)31 b(In)25 b(this)f(case,)h(Eq.)f(\(II.9\))i(is)e (equi)n(v)n(alent)f(to)1362 4817 y Fs(F)1439 4766 y Fn(\(0\))1425 4844 y Fm(P)1533 4817 y Fq(\()p Fs(H)8 b Fq(\))1698 4776 y Fl(\000)p Fn(1)1844 4817 y Fq(=)53 b Fs(P)30 b(H)2155 4776 y Fl(\000)p Fn(1)2265 4817 y Fs(P)36 b(;)847 b Fv(\(II.41\))257 5045 y(pro)o(vided)23 b Fs(H)31 b Fv(is)22 b(in)l(v)o(ertible,)h(where) g Fs(F)1604 4994 y Fn(\(0\))1590 5072 y Fm(P)1698 5045 y Fq(\()p Fs(H)8 b Fq(\))23 b Fv(is)g(de\002ned)h(in)f(\(II.14\).)31 b(This)22 b(identity)g(imme-)257 5166 y(diately)i(implies)g(the)g (follo)n(wing)f(composition)g(property)1438 5388 y Fs(F)1515 5337 y Fn(\(0\))1501 5415 y Fm(P)1648 5388 y Fr(\016)38 b Fs(F)1813 5337 y Fn(\(0\))1799 5415 y Fm(Q)1960 5388 y Fq(=)52 b Fs(F)2165 5337 y Fn(\(0\))2151 5415 y Fm(P)10 b(Q)2288 5388 y Fs(;)923 b Fv(\(II.42\))p eop %%Page: 17 18 17 17 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(17)257 573 y(for)27 b(commuting)d(projections)h Fs(P)39 b Fv(and)26 b Fs(Q)p Fv(,)h(assuming)e(that)g(the)h(Feshbach)h (maps)e(are)i(appli-)257 693 y(cable.)404 814 y(The)39 b(purpose)f(of)h(this)f(section)h(is)f(to)h(de)n(v)o(elop)f(a)h (generalization)f(of)i(\(II.42\))f(to)f(the)257 934 y(smooth)32 b(Feshbach)i(map.)56 b(T)-8 b(o)34 b(this)e(end,)k(we)d(consider)g(a)h (pair)g(of)f(commuting,)h(positi)n(v)o(e)257 1054 y(operators)27 b(bounded)f(by)h(unity)-6 b(,)26 b Fq(0)32 b Fr(\024)g Fs(\037)1654 1069 y Fn(1)1694 1054 y Fs(;)17 b(\037)1799 1069 y Fn(2)1870 1054 y Fr(\024)32 b Fq(1)p Fv(,)27 b(and)g(the)g (corresponding)f(complemen-)257 1175 y(tary)i(operators)p 839 1120 62 4 v 28 w Fs(\037)900 1198 y Fm(j)970 1175 y Fq(:=)33 b(\()p Fp(1)24 b Fr(\000)h Fs(\037)1387 1139 y Fn(2)1387 1199 y Fm(j)1427 1175 y Fq(\))1465 1139 y Fn(1)p Fm(=)p Fn(2)1575 1175 y Fv(.)39 b(Furthermore,)29 b(we)f(set)g Fs(\037)2535 1190 y Fn(12)2643 1175 y Fq(:=)33 b Fs(\037)2840 1190 y Fn(1)2879 1175 y Fs(\037)2940 1190 y Fn(2)3013 1175 y Fq(=)g Fs(\037)3183 1190 y Fn(2)3223 1175 y Fs(\037)3284 1190 y Fn(1)3351 1175 y Fv(and)p 257 1252 V 257 1307 a Fs(\037)318 1330 y Fn(12)421 1307 y Fq(:=)28 b(\()p Fp(1)22 b Fr(\000)g Fs(\037)828 1270 y Fn(2)828 1331 y(12)903 1307 y Fq(\))941 1270 y Fn(1)p Fm(=)p Fn(2)1051 1307 y Fv(.)31 b(Thus,)24 b(we)h(ha)n(v)o(e)1035 1513 y Fs(\037)1096 1472 y Fn(2)1096 1537 y(1)1158 1513 y Fq(+)p 1256 1458 V 22 w Fs(\037)1317 1472 y Fn(2)1317 1537 y(1)1409 1513 y Fq(=)52 b Fs(\037)1598 1472 y Fn(2)1598 1537 y(2)1660 1513 y Fq(+)p 1758 1458 V 22 w Fs(\037)1819 1472 y Fn(2)1819 1537 y(2)1911 1513 y Fq(=)h Fs(\037)2101 1472 y Fn(2)2101 1537 y(12)2197 1513 y Fq(+)p 2295 1458 V 22 w Fs(\037)2357 1472 y Fn(2)2357 1537 y(12)2484 1513 y Fq(=)f Fp(1)23 b Fs(:)520 b Fv(\(II.43\))257 1719 y(W)-8 b(e)22 b(suppose)f(to)g(be)h(gi)n(v)o(en)e(operators)i Fs(H)8 b Fv(,)21 b Fs(T)1791 1734 y Fn(1)1831 1719 y Fv(,)h(and)g Fs(T)2101 1734 y Fn(12)2176 1719 y Fv(,)g(where)g Fs(T)2545 1734 y Fn(1)2606 1719 y Fv(and)g Fs(T)2829 1734 y Fn(12)2926 1719 y Fv(commute)e(with)257 1839 y Fs(\037)318 1854 y Fn(1)358 1839 y Fs(;)d(\037)463 1854 y Fn(2)502 1839 y Fs(;)p 546 1784 V 17 w(\037)607 1863 y Fn(1)646 1839 y Fs(;)p 690 1784 V 17 w(\037)751 1863 y Fn(2)813 1839 y Fv(and)23 b(thus)f(also)g(with)g Fs(\037)1613 1854 y Fn(12)1688 1839 y Fs(;)p 1732 1784 V 17 w(\037)1793 1863 y Fn(12)1867 1839 y Fv(.)30 b(W)-8 b(e)23 b(assume)f(that)h Fs(H)8 b Fv(,)22 b Fs(T)2757 1854 y Fn(1)2797 1839 y Fv(,)h(and)g Fs(T)3069 1854 y Fn(12)3166 1839 y Fv(ha)n(v)o(e)f(the)257 1960 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Fm(\037)2785 5377 y Fh(2)2823 5273 y Fo(\000)2869 5353 y Fs(F)2932 5368 y Fn(1)2972 5353 y Fs(;)17 b(T)3073 5368 y Fn(2)3112 5273 y Fo(\001)3161 5353 y Fs(:)50 b Fv(\(II.51\))p eop %%Page: 18 19 18 18 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(18)257 574 y Ff(Remark)36 b(II.4.)46 b Ft(W)-9 b(e)35 b(observe)g(that)f Fs(F)1630 523 y Fn(\(0\))1616 601 y Fm(P)1724 574 y Fq(\()p Fs(H)8 b Fq(\))35 b Ft(is)f(determined)g(by)h (the)g(pr)l(ojection)e Fs(P)49 b Ft(and)257 694 y(the)32 b(oper)o(ator)d Fs(H)8 b Ft(,)33 b(wher)l(eas)f(the)f(second)g (\223input)g(par)o(ameter\224)g(of)g(a)g(F)-7 b(eshbac)o(h)30 b(pair)g(is)h(a)257 815 y Fv(deri)n(v)o(ed)c Ft(quantity)f(in)h(this)g (case)o(,)h Fs(T)46 b Fq(=)33 b Fs(P)14 b(H)8 b(P)37 b Fq(+)p 2018 735 77 4 v 24 w Fs(P)13 b(H)p 2183 735 V 8 w(P)g Ft(.)39 b(In)28 b(contr)o(ast,)e(ther)l(e)i(is,)f(a)h (priori,)257 935 y(no)j(pr)l(efer)l(ed)f(c)o(hoice)g(for)g Fs(T)44 b Ft(in)30 b(the)h(smooth)e(F)-7 b(eshbac)o(h)29 b(map)h(\226)g Fs(T)45 b Ft(is)30 b(mer)l(ely)g(r)l(equir)l(ed)h(to)257 1055 y(commute)e(with)g Fs(\037)g Ft(and)p 1115 1001 62 4 v 28 w Fs(\037)p Ft(.)43 b(While)29 b(this)f(yields)g(gr)l(eat)h (\003e)n(xibility)f(for)g(c)o(hoosing)f Fs(T)14 b Ft(,)30 b(the)f(r)l(e-)257 1176 y(quir)l(ement)c(\(II.47\))h(is)f(r)o(ather)f (r)l(estrictive)o(,)h(and)g(it)g(is)f(not)h(clear)-11 b(,)25 b(in)g(g)o(ener)o(al,)g(that)f(a)i(c)o(hoice)257 1296 y(for)j Fs(T)460 1311 y Fn(1)499 1296 y Ft(,)h Fs(T)611 1311 y Fn(12)686 1296 y Ft(,)g(and)f Fs(T)977 1311 y Fn(2)1046 1296 y Ft(ful\002lling)d(\(II.47\))k(e)n(xists,)f(at)g(all.) 42 b(In)29 b(this)f(sense)o(,)i(the)f(smooth)f(F)-7 b(esh-)257 1417 y(bac)o(h)24 b(map)h(does)f(not)h(possess)e(any)i(natur)o(al)e (composition)g(pr)l(operty)-5 b(.)257 1610 y(Pr)l(oof)o(.)29 b Fv(W)-8 b(e)23 b(may)g(assume)f(that)g Fs(H)31 b Fv(is)23 b(in)l(v)o(ertible;)e(otherwise,)i(we)g(pass)g(to)g Fs(H)2951 1625 y Fm(")3015 1610 y Fq(:=)k Fs(U)10 b Fq(\()p Fr(j)p Fs(H)e Fr(j)15 b Fq(+)257 1730 y 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b Fq(\()p Fs(H)r(;)17 b(T)d Fq(\))30 b Fv(be)g(a)h(Feshbach)g(pair)f(corresponding)g(to)f Fs(\037)p Fv(..)48 b(W)-8 b(e)31 b(assume)e(that)h Fs(H)r(;)17 b(T)52 b Fr(2)257 2756 y(L)p Fq(\()p Fr(D)s Fs(;)17 b Fr(H)q Fq(\))p Fv(,)32 b(where)g Fr(D)43 b Fq(:=)d(dom)o(\()p Fs(H)8 b Fq(\))39 b(=)h(dom)o(\()p Fs(T)14 b Fq(\))32 b Fv(and)f(that)g Fs(H)r(;)17 b(T)8 b(;)17 b(\037;)p 2770 2701 62 4 v 17 w(\037)31 b Fv(and)h(the)f(compo-)257 2877 y(sition)d(of)i(operators)f(in)g(the)g(de\002nition)f(of)i Fs(F)1866 2892 y Fm(\037)1914 2877 y Fq(\()p Fs(H)r(;)17 b(T)d Fq(\))29 b Fv(are)h(all)f(contained)f(in)h(the)g(domain,)257 2997 y Fq(dom\()p Fs(\016)t Fq(\))p Fv(,)41 b(of)d(a)g(gi)n(v)o(en)f (deri)n(v)n(ation)f Fs(\016)t Fv(.)71 b(This)37 b(implies,)j(in)d (particular)l(,)k(that)d Fs(F)3055 3012 y Fm(\037)3103 2997 y Fq(\()p Fs(H)r(;)17 b(T)d Fq(\))52 b Fr(2)257 3117 y Fq(dom\()p Fs(\016)t Fq(\))p Fv(,)24 b(and)h(we)g(ha)n(v)o(e)g (the)f(follo)n(wing)f(identities.)257 3302 y Ff(Theor)n(em)29 b(II.5.)42 b Ft(F)-10 b(or)27 b(a)g(given)g(F)-7 b(eshbac)o(h)26 b(pair)g Fq(\()p Fs(H)r(;)17 b(T)d Fq(\))27 b Ft(corr)l(esponding)f(to) h Fs(\037)g Ft(on)g(a)g(sepa-)257 3422 y(r)o(able)f(Hilbert)h(space)g Fr(H)h Ft(and)f(a)g(derivation)f Fs(\016)31 b Ft(with)c(the)g(pr)l (operties)e(speci\002ed)i(abo)o(ve)o(,)h(we)257 3543 y(have)d(that)569 3741 y Fs(\016)t Fq(\()p Fs(F)14 b Fq(\))83 b(=)g Fs(Q)1088 3700 y Fn(#)1168 3741 y Fs(\016)t Fq(\()p Fs(W)14 b Fq(\))j Fs(Q)43 b Fq(+)h Fs(Q)1731 3700 y Fn(#)1811 3741 y Fs(W)1933 3660 y Fo(\000)1979 3741 y Fs(\016)t Fq(\()p Fs(\037)p Fq(\))22 b Fr(\000)h Fs(\016)t Fq(\()p 2370 3686 V Fs(\037)p Fq(\))17 b Fs(M)2590 3660 y Fo(\001)3238 3741 y Fv(\(II.60\))1033 3896 y Fq(+)1131 3815 y Fo(\000)1177 3896 y Fs(\016)t Fq(\()p Fs(\037)p Fq(\))22 b Fr(\000)g Fs(M)1586 3855 y Fn(#)1667 3896 y Fs(\016)t Fq(\()p 1752 3841 V Fs(\037)p Fq(\))1851 3815 y Fo(\001)1913 3896 y Fs(W)30 b(Q)45 b Fq(+)f Fs(\016)t Fq(\()p Fs(T)14 b Fq(\))44 b(+)g Fs(M)2739 3855 y Fn(#)2819 3896 y Fs(\016)t Fq(\()p Fs(T)14 b Fq(\))j Fs(M)32 b(;)257 4094 y Ft(wher)l(e)27 b Fs(F)42 b Fq(:=)28 b Fs(F)822 4109 y Fm(\037)870 4094 y Fq(\()p Fs(H)r(;)17 b(T)d Fq(\))p Ft(,)25 b Fs(Q)k Fq(:=)g Fs(Q)1509 4109 y Fm(\037)1557 4094 y Fq(\()p Fs(H)r(;)17 b(T)d Fq(\))p Ft(,)25 b Fs(Q)1958 4058 y Fn(#)2049 4094 y Fq(:=)k Fs(Q)2258 4058 y Fn(#)2258 4119 y Fm(\037)2321 4094 y Fq(\()p Fs(H)r(;)17 b(T)d Fq(\))p Ft(,)25 b Fs(M)39 b Fq(:=)29 b Fs(H)2999 4053 y Fl(\000)p Fn(1)p 2991 4082 48 3 v 2991 4119 a Fm(\037)p 3093 4039 62 4 v 3093 4094 a Fs(\037W)14 b(\037)25 b Ft(and)257 4225 y Fs(M)361 4188 y Fn(#)453 4225 y Fq(:=)i Fs(\037W)p 750 4170 V 14 w(\037H)900 4183 y Fl(\000)p Fn(1)p 892 4212 48 3 v 892 4250 a Fm(\037)994 4225 y Ft(.)k(In)25 b(particular)-11 b(,)23 b(if)h Fs(\016)t Fq(\()p Fs(\037)p Fq(\))k(=)f Fs(\016)t Fq(\()p Fs(T)14 b Fq(\))27 b(=)h(0)d Ft(then)1430 4436 y Fs(\016)t Fq(\()p Fs(F)14 b Fq(\))52 b(=)h Fs(Q)1888 4395 y Fn(#)1968 4436 y Fs(\016)t Fq(\()p Fs(H)8 b Fq(\))17 b Fs(Q)k(:)916 b Fv(\(II.61\))257 4634 y Ft(Pr)l(oof)o(.)51 b Fv(T)-8 b(o)32 b(v)o(erify)f(\(II.60\),)k(we)d(observ)o(e)g(that)f Fs(Q)42 b Fq(=)f Fs(\037)27 b Fr(\000)p 2375 4579 62 4 v 28 w Fs(\037M)43 b Fv(and)32 b Fs(Q)2826 4598 y Fn(#)2931 4634 y Fq(=)41 b Fs(\037)27 b Fr(\000)h Fs(M)3345 4598 y Fn(#)p 3409 4579 V 3409 4634 a Fs(\037)p Fv(.)257 4754 y(Applying)c(Leibniz')g(rule,)h(we)g(obtain)f(that)576 4952 y Fs(\016)t Fq(\()p Fs(F)14 b Fq(\))83 b(=)g Fs(\016)t Fq(\()p Fs(T)14 b Fq(\))44 b(+)g Fs(\037)17 b(\016)t Fq(\()p Fs(W)d Fq(\))j Fs(\037)43 b Fq(+)h Fs(\016)t Fq(\()p Fs(\037)p Fq(\))17 b Fs(W)30 b(\037)44 b Fq(+)g Fs(\037)17 b(W)30 b(\016)t Fq(\()p Fs(\037)p Fq(\))1040 5098 y Fr(\000)22 b Fs(\016)t Fq(\()p Fs(\037)p Fq(\))17 b Fs(W)p 1462 5043 V 30 w(\037)g(M)55 b Fr(\000)45 b Fs(\037)17 b(\016)t Fq(\()p Fs(W)d Fq(\))p 2135 5043 V 17 w Fs(\037)h(M)55 b Fr(\000)45 b Fs(\037)17 b(W)30 b(\016)t Fq(\()p 2767 5043 V Fs(\037)p Fq(\))17 b Fs(M)1040 5243 y Fr(\000)22 b Fs(M)1243 5202 y Fn(#)1324 5243 y Fs(\016)t Fq(\()p 1409 5188 V Fs(\037)p Fq(\))17 b Fs(W)30 b(\037)44 b Fr(\000)h Fs(M)1978 5202 y Fn(#)p 2058 5188 V 2058 5243 a Fs(\037)17 b(\016)t Fq(\()p Fs(W)d Fq(\))j Fs(\037)43 b Fr(\000)i Fs(M)2712 5202 y Fn(#)p 2792 5188 V 2792 5243 a Fs(\037)17 b(W)30 b(\016)t Fq(\()p Fs(\037)p Fq(\))1040 5388 y Fr(\000)22 b Fs(\037)17 b(W)p 1339 5333 V 30 w(\037)g(\016)1464 5307 y Fo(\000)1510 5388 y Fs(H)1599 5347 y Fl(\000)p Fn(1)p 1591 5376 48 3 v 1591 5413 a Fm(\037)1693 5307 y Fo(\001)p 1755 5333 62 4 v 1755 5388 a Fs(\037)g(W)30 b(\037)22 b(:)1173 b Fv(\(II.62\))p eop %%Page: 20 21 20 20 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(20)257 573 y(Application)24 b(of)h(\(II.59\))g(to)f(the)h(last)f (line)g(of)h(\(II.62\))g(gi)n(v)o(es)414 793 y Fr(\000)17 b Fs(\037)g(W)p 708 738 62 4 v 30 w(\037)f(\016)832 712 y Fo(\000)878 793 y Fs(H)967 752 y Fl(\000)p Fn(1)p 959 780 48 3 v 959 818 a Fm(\037)1061 712 y Fo(\001)p 1123 738 62 4 v 1123 793 a Fs(\037)h(W)30 b(\037)83 b Fq(=)g Fs(M)1730 752 y Fn(#)1810 793 y Fs(\016)t Fq(\()p Fs(T)14 b Fq(\))j Fs(M)55 b Fq(+)44 b Fs(M)2394 752 y Fn(#)p 2474 738 V 2474 793 a Fs(\037)17 b(\016)t Fq(\()p Fs(W)d Fq(\))p 2798 738 V 17 w Fs(\037)i(M)269 b Fv(\(II.63\))1648 951 y Fq(+)22 b Fs(M)1850 910 y Fn(#)1931 951 y Fs(\016)t Fq(\()p 2016 897 V Fs(\037)o Fq(\))17 b Fs(W)p 2253 897 V 30 w(\037)g(M)55 b Fq(+)44 b Fs(M)2704 910 y Fn(#)p 2785 897 V 2785 951 a Fs(\037)16 b(W)31 b(\016)t Fq(\()p 3070 897 V Fs(\037)p Fq(\))17 b Fs(M)32 b(:)257 1171 y Fv(Inserting)24 b(\(II.63\))i(into)e(\(II.62\),)h(we)g(obtain)f(that) 569 1391 y Fs(\016)t Fq(\()p Fs(F)14 b Fq(\))83 b(=)g(\()p Fs(\037)22 b Fr(\000)h Fs(M)1336 1350 y Fn(#)p 1399 1337 V 1399 1391 a Fs(\037)p Fq(\))17 b Fs(\016)t Fq(\()p Fs(W)d Fq(\))j(\()p Fs(\037)k Fr(\000)p 1981 1337 V 23 w Fs(\037M)10 b Fq(\))45 b(+)f(\()p Fs(\037)22 b Fr(\000)h Fs(M)2674 1350 y Fn(#)p 2738 1337 V 2738 1391 a Fs(\037)p Fq(\))17 b Fs(W)30 b(\016)t Fq(\()p Fs(\037)p Fq(\))1033 1537 y(+)22 b Fs(\016)t Fq(\()p Fs(\037)p Fq(\))17 b Fs(W)30 b Fq(\()p Fs(\037)22 b Fr(\000)p 1675 1482 V 23 w Fs(\037M)10 b Fq(\))45 b Fr(\000)f Fq(\()p Fs(\037)23 b Fr(\000)f Fs(M)2369 1496 y Fn(#)p 2433 1482 V 2433 1537 a Fs(\037)p Fq(\))17 b Fs(W)30 b(\016)t Fq(\()p 2756 1482 V Fs(\037)p Fq(\))17 b Fs(M)1033 1682 y Fr(\000)23 b Fs(M)1237 1641 y Fn(#)1317 1682 y Fs(\016)t Fq(\()p 1402 1627 V Fs(\037)p Fq(\))17 b Fs(W)30 b Fq(\()p Fs(\037)22 b Fr(\000)p 1860 1627 V 22 w Fs(\037)q(M)10 b Fq(\))45 b(+)f Fs(\016)t Fq(\()p Fs(T)14 b Fq(\))44 b(+)g Fs(M)2691 1641 y Fn(#)2771 1682 y Fs(\016)t Fq(\()p Fs(T)14 b Fq(\))j Fs(M)852 1827 y Fq(=)83 b Fs(Q)1088 1786 y Fn(#)1168 1827 y Fs(\016)t Fq(\()p Fs(W)14 b Fq(\))j Fs(Q)43 b Fq(+)h Fs(Q)1731 1786 y Fn(#)1811 1827 y Fs(W)1933 1747 y Fo(\000)1979 1827 y Fs(\016)t Fq(\()p Fs(\037)p Fq(\))22 b Fr(\000)h Fs(\016)t Fq(\()p 2370 1772 V Fs(\037)p Fq(\))17 b Fs(M)2590 1747 y Fo(\001)3238 1827 y Fv(\(II.64\))1033 1983 y Fq(+)1131 1902 y 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b(and)f(deri)n(v)n(ati)n(v)o(es)e(with)h(respect)i(to) f(parameters.)30 b(In)24 b(particular)l(,)f(if)g Fs(z)33 b Fr(7!)27 b Fs(H)8 b Fq(\()p Fs(z)t Fq(\))257 2693 y Fv(is)28 b(an)h(analytic)e(f)o(amily)-6 b(,)28 b(and)g Fs(T)42 b Fv(and)29 b Fs(\037)f Fv(are)h(independent)e(of)i Fs(z)t Fv(,)h(then)d(\(II.61\))i(implies)e(that)257 2813 y Fs(F)320 2828 y Fm(\037)368 2813 y Fq(\()p Fs(H)8 b Fq(\()p Fs(z)t Fq(\))p Fs(;)17 b(T)d Fq(\))25 b Fv(is)f(analytic,)g (too,)h(because)462 3033 y Fs(@)517 3048 y Fn(\026)-39 b Fm(z)553 3033 y Fs(F)616 3048 y Fm(\037)664 3033 y Fq(\()p Fs(H)8 b Fq(\()p Fs(z)t Fq(\))p Fs(;)17 b(T)d Fq(\))52 b(=)h Fs(Q)1327 2992 y Fn(#)1327 3058 y Fm(\037)1390 3033 y Fq(\()p Fs(H)8 b Fq(\()p Fs(z)t Fq(\))p Fs(;)17 b(T)d Fq(\))1812 2953 y Fo(\000)1857 3033 y Fs(@)1912 3048 y Fn(\026)-39 b Fm(z)1948 3033 y Fs(H)8 b Fq(\()p Fs(z)t Fq(\))2162 2953 y Fo(\001)2225 3033 y Fs(Q)2302 3048 y Fm(\037)2350 3033 y Fq(\()p Fs(H)g Fq(\()p Fs(z)t Fq(\))p Fs(;)17 b(T)d Fq(\))52 b(=)g(0)22 b Fs(;)205 b Fv(\(II.65\))257 3253 y(in)32 b(this)g(case.)54 b(F)o(or)32 b(e)o(xamples)g(of)g(applications)f(of)i(deri)n(v)n(ations,)f(we)h (refer)g(the)f(reader)i(to)257 3374 y([6].)d(In)24 b(a)h(forthcoming)d (publication,)h(deri)n(v)n(ations)f(play)i(a)g(crucial)g(role)g(in)g (the)g(analysis)f(of)257 3494 y(the)i Ft(Nelson)g(model)f Fv(at)h(total)f(momentum)f Fs(P)1815 3509 y Fn(tot)1937 3494 y Fq(=)k(0)p Fv(.)p eop %%Page: 21 22 21 21 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(21)257 573 y Fu(III)143 b(The)35 b(Renormalization)d(Gr)m(oup)j (based)g(on)568 756 y(the)g(Smooth)f(F)l(eshbach)h(Map)257 979 y Fv(In)20 b(this)f(section)h(we)g(de)n(v)o(elop)e(an)i(operator)n (-theoretic)g(renormalization)f(group)h(\(RG\))g(trans-)257 1099 y(formation)k(based)h(on)g(the)g(smooth)f(Feshbach)h(map)g(and)g (compare)g(it)f(to)h(the)g(one)g(in)f([2,)h(3],)257 1220 y(which)c(w)o(as)h(based)f(on)g(the)g(Feshbach)h(\(projection\))f(map.) 29 b(It)21 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y Fs(H)60 b Fq(=)53 b Fs(T)14 b Fq([)p Fs(H)1708 3822 y Fm(f)1753 3807 y Fq(])50 b(+)f Fs(W)64 b Fr(\000)50 b Fs(E)28 b Fr(\001)22 b Fp(1)g Fs(;)762 b Fv(\(III.2\))257 4027 y(where)31 b Fs(E)45 b Fr(2)38 b Fg(C)57 b Fv(is)30 b(a)h(comple)o(x)e(number)l(,)i Fs(T)52 b Fr(2)39 b Fs(C)2047 3991 y Fn(1)2086 4027 y Fq(\()p Fs(I)8 b Fq(\))31 b Fv(is)f(a)h(continuously)d(dif)n (ferentiable)257 4148 y(function)33 b(on)h Fs(I)52 b Fq(:=)44 b([0)p Fs(;)17 b Fq(1])33 b Fv(with)g Fs(T)14 b Fq([0])44 b(=)g(0)p Fv(,)35 b(and)f Fs(T)14 b Fq([)p Fs(H)2243 4163 y Fm(f)2288 4148 y Fq(])44 b Fr(2)h(B)s Fq([)p Fr(H)2649 4163 y Fn(red)2752 4148 y Fq(])34 b Fv(is)f(de\002ned)h(by)f(the)257 4268 y(functional)24 b(calculus.)31 b(Moreo)o(v)o(er)l(,)23 b Fs(W)42 b Fq(=)1741 4193 y Fo(P)1846 4297 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)2117 4268 y Fs(W)2209 4283 y Fm(m;n)2338 4268 y Fv(,)25 b(with)349 4488 y Fs(W)441 4503 y Fm(m;n)665 4488 y Fr(\021)97 b Fs(W)931 4503 y Fm(m;n)1060 4488 y Fq([)p Fs(w)1157 4503 y Fm(m;n)1286 4488 y Fq(])1942 b Fv(\(III.3\))653 4702 y Fq(:=)83 b Fs(P)902 4717 y Fn(red)1020 4567 y Fo(Z)1076 4792 y Fm(B)1132 4762 y Fk(m)p Fh(+)p Fk(n)1129 4813 y Fh(1)1364 4635 y Fs(dK)1505 4599 y Fn(\()p Fm(m;n)p Fn(\))p 1307 4679 440 4 v 1307 4771 a Fr(j)p Fs(K)1425 4742 y Fn(\()p Fm(m;n)p Fn(\))1608 4771 y Fr(j)1636 4742 y Fn(1)p Fm(=)p Fn(2)1784 4702 y Fs(a)1835 4661 y Fl(\003)1874 4702 y Fq(\()p Fs(k)1966 4661 y Fn(\()p Fm(m)p Fn(\))2088 4702 y Fq(\))17 b Fs(w)2213 4717 y Fm(m;n)2341 4622 y Fo(\002)2382 4702 y Fs(H)2463 4717 y Fm(f)2509 4702 y Fq(;)g Fs(K)2643 4661 y Fn(\()p Fm(m;n)p Fn(\))2826 4622 y Fo(\003)2884 4702 y Fs(a)p Fq(\()2976 4676 y(~)2973 4702 y Fs(k)3027 4661 y Fn(\()p Fm(n)p Fn(\))3130 4702 y Fq(\))22 b Fs(P)3253 4717 y Fn(red)3377 4702 y Fs(;)p eop %%Page: 22 23 22 22 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(22)257 573 y(where)28 b Fs(B)602 588 y Fn(1)669 573 y Fv(denotes)f(the)h(unit)e(ball)h(in)g Fg(R)1681 537 y Fm(d)1728 573 y Fv(,)h(and)f Fs(P)2015 588 y Fn(red)2145 573 y Fv(is)g(a)h(shorthand)e(for)i Fp(1)p Fq([)p Fs(H)3039 588 y Fm(f)3117 573 y Fs(<)k Fq(1])p Fv(.)39 b(W)-8 b(e)257 693 y(use)25 b(the)g(notation)568 899 y Fs(k)622 862 y Fn(\()p Fm(m)p Fn(\))794 899 y Fq(:=)49 b(\()p Fs(k)1035 914 y Fn(1)1074 899 y Fs(;)17 b(:)g(:)g(:)f(;)h(k)1344 914 y Fm(m)1410 899 y Fq(\))50 b Fr(2)g Fg(R)1680 862 y Fm(d)q(m)1811 899 y Fs(;)1976 872 y Fq(~)1973 899 y Fs(k)2027 862 y Fn(\()p Fm(n)p Fn(\))2179 899 y Fq(:=)g(\()2373 872 y(~)2370 899 y Fs(k)2421 914 y Fn(1)2460 899 y Fs(;)17 b(:)g(:)g(:)f(;)2681 872 y Fq(~)2679 899 y Fs(k)2730 914 y Fm(n)2776 899 y Fq(\))50 b Fr(2)g Fg(R)3046 862 y Fm(dn)3157 899 y Fs(;)71 b Fv(\(III.4\))611 1044 y Fs(K)701 1008 y Fn(\()p Fm(m;n)p Fn(\))935 1044 y Fq(:=)50 b(\()p Fs(k)1180 1008 y Fn(\()p Fm(m)p Fn(\))1301 1044 y Fs(;)1347 1018 y Fq(~)1345 1044 y Fs(k)1399 1008 y Fn(\()p Fm(n)p Fn(\))1501 1044 y Fq(\))22 b Fs(;)135 b(dK)1864 1008 y Fn(\()p Fm(m;n)p Fn(\))2097 1044 y Fq(:=)2250 969 y Fo(Q)2344 995 y Fm(m)2344 1073 y(i)p Fn(=1)2479 1044 y Fs(d)2530 1008 y Fm(d)2570 1044 y Fs(k)2621 1059 y Fm(i)2693 969 y Fo(Q)2787 995 y Fm(n)2787 1073 y(i)p Fn(=1)2922 1044 y Fs(d)2973 1008 y Fm(d)3016 1018 y Fq(~)3013 1044 y Fs(k)3064 1059 y Fm(i)3115 1044 y Fs(;)113 b Fv(\(III.5\))790 1199 y Fs(a)841 1162 y Fl(\003)880 1199 y Fq(\()p Fs(k)972 1162 y Fn(\()p Fm(m)p Fn(\))1094 1199 y Fq(\))50 b(:=)1334 1124 y Fo(Q)1429 1150 y Fm(m)1429 1228 y(i)p Fn(=1)1564 1199 y Fs(a)1615 1162 y Fl(\003)1654 1199 y Fq(\()p Fs(k)1743 1214 y Fm(i)1771 1199 y Fq(\))22 b Fs(;)135 b(a)p Fq(\()2085 1172 y(~)2082 1199 y Fs(k)2136 1162 y Fn(\()p Fm(n)p Fn(\))2238 1199 y Fq(\))50 b(:=)2479 1124 y Fo(Q)2573 1150 y Fm(n)2573 1228 y(i)p Fn(=1)2708 1199 y Fs(a)p Fq(\()2800 1172 y(~)2797 1199 y Fs(k)2848 1214 y Fm(i)2876 1199 y Fq(\))22 b Fs(;)292 b Fv(\(III.6\))821 1353 y Fr(j)p Fs(K)939 1317 y Fn(\()p Fm(m;n)p Fn(\))1123 1353 y Fr(j)43 b Fq(:=)i Fr(j)p Fs(k)1424 1317 y Fn(\()p Fm(m)p Fn(\))1545 1353 y Fr(j)22 b(\001)g(j)1676 1327 y Fq(~)1673 1353 y Fs(k)1727 1317 y Fn(\()p Fm(n)p Fn(\))1828 1353 y Fr(j)g Fs(;)88 b Fr(j)p Fs(k)2075 1317 y Fn(\()p Fm(m)p Fn(\))2196 1353 y Fr(j)44 b Fq(:=)g Fr(j)p Fs(k)2494 1368 y Fn(1)2533 1353 y Fr(j)17 b(\001)g(\001)g(\001)d(j)p Fs(k)2789 1368 y Fm(m)2855 1353 y Fr(j)22 b Fs(;)323 b Fv(\(III.7\))1269 1498 y Fq(\006[)p Fs(k)1420 1462 y Fn(\()p Fm(m)p Fn(\))1542 1498 y Fq(])45 b(:=)f Fr(j)p Fs(k)1840 1513 y Fn(1)1879 1498 y Fr(j)22 b Fq(+)g Fs(:)17 b(:)g(:)k Fq(+)h Fr(j)p Fs(k)2340 1513 y Fm(m)2406 1498 y Fr(j)g Fs(:)772 b Fv(\(III.8\))257 1704 y(The)27 b(function)g Fs(w)869 1719 y Fm(m;n)997 1704 y Fq([)17 b Fr(\001)g Fs(;)g(K)1220 1668 y Fn(\()p Fm(m;n)p Fn(\))1403 1704 y Fq(])32 b Fr(2)g Fs(C)1637 1668 y Fn(1)1676 1704 y Fq(\()p Fs(I)8 b Fq(\))27 b Fv(is)g(continuously)e(dif)n(ferentiable)h (on)h Fs(I)8 b Fv(,)27 b(point-)257 1841 y(wise,)g(for)h(almost)d(e)n (v)o(ery)i Fs(K)1263 1805 y Fn(\()p Fm(m;n)p Fn(\))1478 1841 y Fr(2)32 b Fs(B)1655 1790 y Fn(\()p Fm(m)p Fn(+)p Fm(n)p Fn(\))1650 1865 y(1)1875 1841 y Fv(,)27 b(its)f(deri)n(v)n(ati)n (v)o(e)f(being)h(denoted)g(by)h Fs(@)3233 1856 y Fm(r)3272 1841 y Fs(w)3342 1856 y Fm(m;n)3470 1841 y Fv(.)257 1961 y(As)42 b(a)g(function)e(of)i Fs(K)1084 1925 y Fn(\()p Fm(m;n)p Fn(\))1268 1961 y Fv(,)k(it)41 b(is)g(totally)f(symmetric)g(w) -6 b(.)41 b(r)-5 b(.)42 b(t.)f(the)g(v)n(ariables)g Fs(k)3239 1925 y Fn(\()p Fm(m)p Fn(\))3419 1961 y Fq(=)257 2081 y(\()p Fs(k)346 2096 y Fn(1)385 2081 y Fs(;)17 b(:)g(:)g(:)f(;)h(k)655 2096 y Fm(m)721 2081 y Fq(\))25 b Fv(and)956 2055 y Fq(~)953 2081 y Fs(k)1007 2045 y Fn(\()p Fm(n)p Fn(\))1137 2081 y Fq(=)i(\()1281 2055 y(~)1278 2081 y Fs(k)1329 2096 y Fn(1)1368 2081 y Fs(;)17 b(:)g(:)g(:)f(;)1589 2055 y Fq(~)1587 2081 y Fs(k)1638 2096 y Fm(n)1684 2081 y Fq(\))p Fv(,)25 b(and)g(obe)o(ys)f(the)g Fs(L)2409 2045 y Fn(2)2449 2081 y Fv(-norm)h(bound)949 2287 y Fr(k)p Fs(w)1069 2302 y Fm(m;n)1198 2287 y Fr(k)1248 2246 y Fn(#)1248 2311 y Fm(\026)1363 2287 y Fq(:=)53 b Fr(k)p Fs(w)1639 2302 y Fm(m;n)1767 2287 y Fr(k)1817 2302 y Fm(\026)1913 2287 y Fq(+)d Fr(k)p Fs(@)2140 2302 y Fm(r)2178 2287 y Fs(w)2248 2302 y Fm(m;n)2377 2287 y Fr(k)2427 2302 y Fm(\026)2526 2287 y Fs(<)j Fr(1)21 b Fs(;)452 b Fv(\(III.9\))257 2492 y(where)426 2755 y Fr(k)p Fs(w)546 2770 y Fm(m;n)675 2755 y Fr(k)725 2770 y Fm(\026)824 2755 y Fq(:=)979 2614 y Fo(\022)1069 2619 y(Z)1125 2845 y Fm(B)1181 2815 y Fk(m)p Fh(+)p Fk(n)1178 2866 y Fh(1)1449 2687 y Fs(dK)1590 2651 y Fn(\()p Fm(m;n)p Fn(\))p 1355 2732 513 4 v 1355 2760 a Fo(\014)1355 2820 y(\014)1389 2845 y Fs(K)1479 2816 y Fn(\()p Fm(m;n)p Fn(\))1663 2760 y Fo(\014)1663 2820 y(\014)1696 2787 y Fn(3+2)p Fm(\026)1917 2755 y Fq(sup)1932 2836 y Fm(r)r Fl(2)p Fm(I)2080 2670 y Fo(\014)2080 2730 y(\014)2113 2755 y Fs(w)2183 2770 y Fm(m;n)2312 2755 y Fq([)p Fs(r)s Fq(;)17 b Fs(K)2520 2714 y Fn(\()p Fm(m;n)p Fn(\))2704 2755 y Fq(])2731 2670 y Fo(\014)2731 2730 y(\014)2764 2696 y Fn(2)2825 2614 y Fo(\023)2899 2637 y Fn(1)p Fm(=)p Fn(2)3009 2755 y Fs(;)169 b Fv(\(III.10\))257 3055 y(for)25 b(some)f Fs(\026)j(>)h Fq(0)p Fv(.)i(The)25 b(Banach)g(space)g(of)f(these)h(functions)e(is)h (denoted)g(by)g Fr(W)3069 3019 y Fn(#)3061 3080 y Fm(m;n)3191 3055 y Fv(.)30 b(More-)257 3176 y(o)o(v)o(er)l(,)24 b Fs(W)569 3191 y Fm(m;n)698 3176 y Fq([)p Fs(w)795 3191 y Fm(m;n)924 3176 y Fq(])h Fv(stresses)f(the)h(dependence)g(of)g Fs(W)2147 3191 y Fm(m;n)2301 3176 y Fv(on)g Fs(w)2496 3191 y Fm(m;n)2624 3176 y Fv(.)404 3296 y(The)20 b(follo)n(wing)e (basic)h(bound)h(links)e(the)i Fs(L)1899 3260 y Fn(2)1939 3296 y Fv(-norm)g(de\002ned)g(in)f(\(III.10\))i(to)e(the)h(operator)257 3417 y(norm)25 b(on)f Fr(B)s Fq([)p Fr(H)796 3432 y Fn(red)899 3417 y Fq(])h Fv(and)g(hence)g(gi)n(v)o(es)e(an)i(unambiguous)e (meaning)h(to)g Fs(H)8 b Fv(.)257 3607 y Ff(Theor)n(em)32 b(III.1.)44 b Ft(F)l(ix)30 b Fs(\026)37 b(>)h Fq(0)30 b Ft(and)g Fs(m;)17 b(n)39 b Fr(2)f Fg(N)1958 3622 y Fn(0)2034 3607 y Ft(suc)o(h)29 b(that)h Fs(m)d Fq(+)f Fs(n)38 b Fr(\025)g Fq(1)p Ft(.)48 b(Suppose)29 b(that)257 3728 y Fs(w)327 3743 y Fm(m;n)484 3728 y Fr(2)f(W)676 3743 y Fm(m;n)805 3728 y Ft(,)c(and)f(let)g Fs(W)1242 3743 y Fm(m;n)1398 3728 y Fq(:=)28 b Fs(W)1621 3743 y Fm(m;n)1750 3728 y Fq([)p Fs(w)1847 3743 y Fm(m;n)1976 3728 y Fq(])23 b Ft(be)h(as)f(de\002ned)g(in)f(\(III.3\).)31 b(Then)24 b(on)f Fr(H)3393 3743 y Fn(red)356 3896 y Fo(\015)356 3956 y(\015)411 3980 y Fs(W)503 3995 y Fm(m;n)632 3896 y Fo(\015)632 3956 y(\015)688 4019 y Fn(op)819 3980 y Fr(\024)949 3896 y Fo(\015)949 3956 y(\015)1004 3980 y Fq(\()p Fs(P)1119 3939 y Fl(?)1105 4005 y Fn(\012)1178 3980 y Fs(H)1259 3995 y Fm(f)1304 3980 y Fq(\))1342 3939 y Fl(\000)p Fm(m=)p Fn(2)1550 3980 y Fs(W)1642 3995 y Fm(m;n)1788 3980 y Fq(\()p Fs(P)1903 3939 y Fl(?)1889 4005 y Fn(\012)1962 3980 y Fs(H)2043 3995 y Fm(f)2088 3980 y Fq(\))2126 3939 y Fl(\000)p Fm(n=)p Fn(2)2298 3896 y Fo(\015)2298 3956 y(\015)2353 4019 y Fn(op)2485 3980 y Fr(\024)2779 3913 y Fq(1)p 2625 3958 357 4 v 2625 3977 a Fr(p)p 2708 3977 274 4 v 79 x Fs(m)2793 4028 y Fm(m)2876 4056 y Fs(n)2934 4028 y Fm(n)3008 3980 y Fr(k)p Fs(w)3128 3995 y Fm(m;n)3257 3980 y Fr(k)3307 3995 y Fm(\026)3370 3980 y Fs(;)3205 4149 y Fv(\(III.11\))257 4270 y Ft(wher)l(e)i Fr(k)36 b(\001)g(k)721 4285 y Fn(op)824 4270 y Ft(denotes)23 b(the)h(oper)o(ator)f(norm)g(on)h Fr(B)s Fq([)p Fr(H)2207 4285 y Fn(red)2310 4270 y Fq(])p Ft(,)g Fs(P)2449 4285 y Fn(\012)2532 4270 y Fq(:=)k Fr(j)p Fq(\012)p Fr(ih)p Fq(\012)p Fr(j)c Ft(is)f(the)h(ortho)o(g-)257 4390 y(onal)g(pr)l(ojection)f(onto)h(the)h(vacuum)g(vector)g(in)f Fr(F)10 b Ft(,)24 b(and)h Fs(P)2332 4354 y Fl(?)2318 4415 y Fn(\012)2418 4390 y Fq(:=)j Fp(1)22 b Fr(\000)g Fs(P)2789 4405 y Fn(\012)2844 4390 y Ft(.)404 4581 y Fv(Before)30 b(turning)e(to)g(the)h(proof)g(of)g(Theorem)f(III.1)i(we)f (recall)g(a)g(v)o(ery)g(useful)f(identity)-6 b(,)257 4701 y(the)25 b Ff(Pull-Thr)n(ough)h(F)n(ormula)f Fv(\(and)g(its)f (adjoint)f(relation\),)337 4907 y Fs(a)p Fq(\()p Fs(k)s Fq(\))17 b Fs(F)d Fq([)p Fs(H)720 4922 y Fm(f)765 4907 y Fq(])53 b(=)f Fs(F)14 b Fq([)p Fs(H)1158 4922 y Fm(f)1225 4907 y Fq(+)22 b Fr(j)p Fs(k)s Fr(j)p Fq(])17 b Fs(a)p Fq(\()p Fs(k)s Fq(\))94 b Fv(and)71 b Fs(F)14 b Fq([)p Fs(H)2152 4922 y Fm(f)2197 4907 y Fq(])j Fs(a)2292 4865 y Fl(\003)2331 4907 y Fq(\()p Fs(k)s Fq(\))53 b(=)f Fs(a)2693 4865 y Fl(\003)2733 4907 y Fq(\()p Fs(k)s Fq(\))17 b Fs(F)d Fq([)p Fs(H)3065 4922 y Fm(f)3132 4907 y Fq(+)22 b Fr(j)p Fs(k)s Fr(j)p Fq(])f Fs(;)3205 5027 y Fv(\(III.12\))257 5147 y(which)i(holds)g(on)g Fr(H)971 5162 y Fn(red)1096 5147 y Fv(in)g(the)g(sense)h(of)f(operator)n(-v)n(alued)g(distrib)n (utions,)e(for)i(e)n(v)o(ery)g(boun-)257 5268 y(ded,)f(measurable)g (function)e Fs(F)14 b Fv(.)29 b(The)22 b(Pull-Through)e(F)o(ormula)h (\(III.12\))h(has)f(the)g(follo)n(wing)257 5388 y(consequence.)p eop %%Page: 23 24 23 23 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(23)257 573 y Ff(Lemma)26 b(III.2.)40 b Ft(F)-10 b(or)24 b Fs( )32 b Fr(2)c(H)1303 588 y Fn(red)1405 573 y Ft(,)d Fs(m)j Fr(\025)g Fq(1)p Ft(,)d Fs(\013)j(>)g Fq(0)p Ft(,)c(and)h Fs(k)2293 537 y Fn(\()p Fm(m)p Fn(\))2414 573 y Fs(inB)2584 537 y Fm(m)2579 597 y Fn(1)2651 573 y Ft(,)937 712 y Fo(\015)937 772 y(\015)992 796 y Fs(a)p Fq(\()p Fs(k)1135 755 y Fn(\()p Fm(m)p Fn(\))1257 796 y Fq(\))17 b Fs( )1379 712 y Fo(\015)1379 772 y(\015)1486 796 y Fr(\024)53 b Fs(m)1701 755 y Fl(\000)p Fm(\013m)1896 712 y Fo(\014)1896 772 y(\014)1929 796 y Fs(k)1983 755 y Fn(\()p Fm(m)p Fn(\))2105 712 y Fo(\014)2105 772 y(\014)2138 738 y Fl(\000)p Fm(\013)2270 712 y Fo(\015)2270 772 y(\015)2325 796 y Fs(a)p Fq(\()p Fs(k)2468 755 y Fn(\()p Fm(m)p Fn(\))2590 796 y Fq(\))17 b Fs( )2712 712 y Fo(\015)2712 772 y(\015)2789 796 y Fs(:)389 b Fv(\(III.13\))257 1009 y Ft(Pr)l(oof)o(.)35 b Fv(F)o(or)27 b(arbitrary)g Fs( )35 b Fq(=)d Fp(1)p Fq([)p Fs(H)1444 1024 y Fm(f)1520 1009 y Fs(<)g Fq(1])p Fs( )j Fq(=)c Fp(1)p Fq([)p Fs(H)2073 1024 y Fm(f)2150 1009 y Fs(<)g Fq(1])2333 973 y Fn(2)2373 1009 y Fs( )g Fv(in)26 b Fr(H)2655 1024 y Fn(red)2757 1009 y Fv(,)i(the)e (Pull-Through)257 1129 y(F)o(ormula)e(\(III.12\))i(yields)409 1342 y Fs(a)p Fq(\()p Fs(k)552 1301 y Fn(\()p Fm(m)p Fn(\))673 1342 y Fq(\))p Fs( )87 b Fq(=)c Fs(a)p Fq(\()p Fs(k)1163 1301 y Fn(\()p Fm(m)p Fn(\))1285 1342 y Fq(\))17 b Fp(1)p Fq([)p Fs(H)1504 1357 y Fm(f)1576 1342 y Fs(<)28 b Fq(1])17 b Fs( )84 b Fq(=)c Fp(1)2132 1261 y Fo(\002)2173 1342 y Fs(H)2254 1357 y Fm(f)2322 1342 y Fq(+)22 b(\006[)p Fs(k)2571 1301 y Fn(\()p Fm(m)p Fn(\))2693 1342 y Fq(])27 b Fs(<)h Fq(1)2900 1261 y Fo(\003)2958 1342 y Fs(a)p Fq(\()p Fs(k)3101 1301 y Fn(\()p Fm(m)p Fn(\))3223 1342 y Fq(\))17 b Fs( )861 1501 y Fq(=)83 b Fp(1)1076 1420 y Fo(\002)1118 1501 y Fq(\006[)p Fs(k)1269 1460 y Fn(\()p Fm(m)p Fn(\))1391 1501 y Fq(])27 b Fs(<)h Fq(1)1598 1420 y Fo(\003)1656 1501 y Fs(a)p Fq(\()p Fs(k)1799 1460 y Fn(\()p Fm(m)p Fn(\))1921 1501 y Fq(\))17 b Fs( )25 b(;)1114 b Fv(\(III.14\))257 1722 y(where)38 b Fq(\006[)p Fs(k)689 1686 y Fn(\()p Fm(m)p Fn(\))811 1722 y Fq(])f Fv(is)g(de\002ned)h(in)e (\(III.8\).)69 b(Thus)36 b(\(III.13\))i(follo)n(ws)d(by)i(sho)n(wing)e (that,)40 b(for)257 1843 y Fs(a)308 1858 y Fn(1)348 1843 y Fs(;)17 b(:)g(:)g(:)f(;)h(a)618 1858 y Fm(m)713 1843 y Fr(2)29 b Fq(\(0)p Fs(;)17 b Fq(1])25 b Fv(with)f Fs(a)1293 1858 y Fn(1)1356 1843 y Fq(+)e Fs(:)17 b(:)g(:)22 b Fq(+)h Fs(a)1741 1858 y Fm(m)1836 1843 y Fr(\024)29 b Fq(1)p Fv(,)c(we)h(ha)n(v)o(e)f(that)g Fs(a)2619 1858 y Fn(1)2675 1843 y Fr(\001)17 b(\001)g(\001)e Fs(a)2859 1858 y Fm(m)2955 1843 y Fr(\024)29 b Fs(m)3146 1807 y Fl(\000)p Fm(m)3268 1843 y Fv(,)c(with)257 1963 y Fs(a)308 1978 y Fm(i)377 1963 y Fq(=)39 b Fr(j)p Fs(k)571 1978 y Fm(i)599 1963 y Fr(j)p Fv(.)50 b(Indeed,)34 b(this)c(inequality)g(is)h(equi)n(v)n (alent)f(to)h Fq(ln)o(\(1)p Fs(=a)2554 1978 y Fn(1)2593 1963 y Fq(\))c(+)g Fs(:)17 b(:)g(:)26 b Fq(+)h(ln\(1)p Fs(=a)3274 1978 y Fm(m)3340 1963 y Fq(\))40 b Fr(\025)257 2084 y Fs(m)34 b Fq(ln)o(\()p Fs(m)p Fq(\))p Fv(,)29 b(and)f(we)h(may)f(assume)f(that)h Fs(a)1739 2099 y Fn(1)1803 2084 y Fq(+)d Fs(:)17 b(:)g(:)24 b Fq(+)g Fs(a)2194 2099 y Fm(m)2295 2084 y Fq(=)33 b(1)p Fv(.)41 b(This)27 b(inequality)g(is)h (a)g(con-)257 2204 y(sequence)g(of)g(Jensen')-5 b(s)26 b(inequality)h(applied)f(to)i(the)f(con)l(v)o(e)o(x)f(function)h Fs(F)14 b Fq(\()p Fs(x)p Fq(\))33 b(:=)f Fs(x)i Fq(ln)o(\()p Fs(x)p Fq(\))p Fv(;)257 2324 y(more)25 b(e)o(xplicitly)390 2485 y Fm(m)349 2515 y Fo(X)360 2725 y Fm(j)t Fn(=1)510 2609 y Fq(ln)608 2529 y Fo(\000)654 2609 y Fs(a)705 2568 y Fl(\000)p Fn(1)705 2635 y Fm(j)799 2529 y Fo(\001)897 2609 y Fq(=)1067 2485 y Fm(m)1026 2515 y Fo(X)1036 2725 y Fm(j)t Fn(=1)1186 2609 y Fs(a)1237 2624 y Fm(j)1274 2609 y Fs(F)1351 2529 y Fo(\000)1396 2609 y Fs(a)1447 2568 y Fl(\000)p Fn(1)1447 2635 y Fm(j)1542 2529 y Fo(\001)1640 2609 y Fr(\025)53 b Fs(F)1847 2499 y Fo(\020)1964 2485 y Fm(m)1923 2515 y Fo(X)1934 2725 y Fm(j)t Fn(=1)2084 2609 y Fs(a)2135 2624 y Fm(j)2188 2609 y Fs(a)2239 2568 y Fl(\000)p Fn(1)2239 2635 y Fm(j)2333 2499 y Fo(\021)2445 2609 y Fq(=)g Fs(F)14 b Fq(\()p Fs(m)p Fq(\))52 b(=)h Fs(m)33 b Fq(ln\()p Fs(m)p Fq(\))22 b Fs(:)3205 2824 y Fv(\(III.15\))3418 2944 y Fd(\003)257 3065 y Ft(Pr)l(oof)c(of)g (Theor)l(em)i(III.1:)28 b Fv(F)o(or)19 b Fs( )t(;)e(')27 b Fr(2)h(H)1736 3080 y Fn(red)1838 3065 y Fv(,)20 b(we)g(estimate)e (the)g(matrix)g(element)h Fr(h)p Fs( )t Fr(j)e Fs(W)3367 3080 y Fm(m;n)3495 3065 y Fs(')p Fr(i)257 3185 y Fv(by)i(means)h(of)f (the)g(Cauchy-Schw)o(arz)i(inequality)d(and)h(Lemma)g(III.2)h(with)e Fs(\013)29 b Fq(:=)e(\(1)p Fs(=)p Fq(2\))r(+)r Fs(\026)p Fv(,)257 3464 y Fr(jh)p Fs( )t Fr(j)g Fs(W)538 3479 y Fm(m;n)684 3464 y Fs(')p Fr(ij)52 b Fq(=)995 3319 y Fo(\014)995 3379 y(\014)995 3439 y(\014)995 3499 y(\014)1045 3328 y(Z)1100 3554 y Fm(B)1156 3524 y Fk(m)p Fh(+)p Fk(n)1153 3575 y Fh(1)1388 3396 y Fs(dK)1529 3360 y Fn(\()p Fm(m;n)p Fn(\))p 1331 3441 440 4 v 1331 3532 a Fr(j)p Fs(K)1449 3503 y Fn(\()p Fm(m;n)p Fn(\))1633 3532 y Fr(j)1661 3503 y Fn(1)p Fm(=)p Fn(2)1808 3383 y Fo(\012)1855 3464 y Fs(a)p Fq(\()p Fs(k)1998 3423 y Fn(\()p Fm(m)p Fn(\))2120 3464 y Fq(\))17 b Fs( )2258 3379 y Fo(\014)2258 3439 y(\014)2316 3464 y Fs(w)2386 3479 y Fm(m;n)2515 3383 y Fo(\002)2556 3464 y Fs(H)2637 3479 y Fm(f)2683 3464 y Fq(;)g Fs(K)2817 3423 y Fn(\()p Fm(m;n)p Fn(\))3000 3383 y Fo(\003)3067 3464 y Fs(a)p Fq(\()3159 3437 y(~)3156 3464 y Fs(k)3210 3423 y Fn(\()p Fm(n)p Fn(\))3312 3464 y Fq(\))g Fs(')3431 3383 y Fo(\013)3477 3319 y(\014)3477 3379 y(\014)3477 3439 y(\014)3477 3499 y(\014)340 3761 y Fr(\024)501 3626 y Fo(Z)556 3851 y Fm(B)612 3822 y Fk(m)p Fh(+)p Fk(n)609 3873 y Fh(1)844 3694 y Fs(dK)985 3658 y Fn(\()p Fm(m;n)p Fn(\))p 787 3739 V 787 3830 a Fr(j)p Fs(K)905 3801 y Fn(\()p Fm(m;n)p Fn(\))1089 3830 y Fr(j)1117 3801 y Fn(1)p Fm(=)p Fn(2)1278 3761 y Fq(sup)1293 3843 y Fm(r)r Fl(2)p Fm(I)1441 3677 y Fo(\014)1441 3737 y(\014)1475 3761 y Fs(w)1545 3776 y Fm(m;n)1673 3761 y Fq([)p Fs(r)s Fq(;)g Fs(K)1881 3720 y Fn(\()p Fm(m;n)p Fn(\))2065 3761 y Fq(])2092 3677 y Fo(\014)2092 3737 y(\014)2150 3677 y(\015)2150 3737 y(\015)2205 3761 y Fs(a)p Fq(\()p Fs(k)2348 3720 y Fn(\()p Fm(m)p Fn(\))2470 3761 y Fq(\))g Fs( )2592 3677 y Fo(\015)2592 3737 y(\015)2675 3677 y(\015)2675 3737 y(\015)2730 3761 y Fs(a)p Fq(\()2822 3735 y(~)2819 3761 y Fs(k)2873 3720 y Fn(\()p Fm(n)p Fn(\))2975 3761 y Fq(\))g Fs(')3094 3677 y Fo(\015)3094 3737 y(\015)3258 3761 y Fv(\(III.16\))340 4066 y Fr(\024)665 3998 y Fq(1)p 511 4043 357 4 v 511 4063 a Fr(p)p 594 4063 274 4 v 79 x Fs(m)679 4113 y Fm(m)762 4142 y Fs(n)820 4113 y Fm(n)911 3930 y Fo(Z)966 4155 y Fm(B)1022 4126 y Fk(m)p Fh(+)p Fk(n)1019 4177 y Fh(1)1187 4066 y Fs(dK)1328 4024 y Fn(\()p Fm(m;n)p Fn(\))1547 3992 y Fq(sup)1693 4015 y Fm(r)r Fl(2)p Fm(I)1831 3907 y Fo(\014)1831 3967 y(\014)1864 3992 y Fs(w)1934 4007 y Fm(m;n)2063 3992 y Fq([)p Fs(r)s Fq(;)g Fs(K)2271 3956 y Fn(\()p Fm(m;n)p Fn(\))2454 3992 y Fq(])2481 3907 y Fo(\014)2481 3967 y(\014)p 1547 4043 968 4 v 1797 4134 a Fr(j)p Fs(K)1915 4105 y Fn(\()p Fm(m;n)p Fn(\))2099 4134 y Fr(j)2127 4105 y Fn(1+)p Fm(\026)2549 3981 y Fo(\015)2549 4041 y(\015)2605 4066 y Fs(a)p Fq(\()p Fs(k)2748 4024 y Fn(\()p Fm(m)p Fn(\))2869 4066 y Fq(\))g Fs( )2991 3981 y Fo(\015)2991 4041 y(\015)3074 3981 y(\015)3074 4041 y(\015)3129 4066 y Fs(a)p Fq(\()3221 4039 y(~)3218 4066 y Fs(k)3272 4024 y Fn(\()p Fm(n)p Fn(\))3374 4066 y Fq(\))g Fs(')3493 3981 y Fo(\015)3493 4041 y(\015)340 4378 y Fr(\024)511 4310 y Fs(D)592 4325 y Fm(m)658 4310 y Fq(\()p Fs( )t Fq(\))801 4274 y Fn(1)p Fm(=)p Fn(2)928 4310 y Fs(D)1009 4325 y Fm(n)1056 4310 y Fq(\()p Fs(')p Fq(\))1196 4274 y Fn(1)p Fm(=)p Fn(2)p 511 4355 795 4 v 730 4375 a Fr(p)p 813 4375 274 4 v 79 x Fs(m)898 4425 y Fm(m)981 4454 y Fs(n)1039 4425 y Fm(n)1332 4237 y Fo(\022)1422 4242 y(Z)1478 4468 y Fm(B)1534 4438 y Fk(m)p Fh(+)p Fk(n)1531 4489 y Fh(1)1802 4310 y Fs(dK)1943 4274 y Fn(\()p Fm(m;n)p Fn(\))p 1708 4355 513 4 v 1708 4383 a Fo(\014)1708 4443 y(\014)1741 4468 y Fs(K)1831 4439 y Fn(\()p Fm(m;n)p Fn(\))2016 4383 y Fo(\014)2016 4443 y(\014)2049 4410 y Fn(3+2)p Fm(\026)2269 4378 y Fq(sup)2285 4459 y Fm(r)r Fl(2)p Fm(I)2433 4293 y Fo(\014)2433 4353 y(\014)2466 4378 y Fs(w)2536 4393 y Fm(m;n)2665 4378 y Fq([)p Fs(r)s Fq(;)g Fs(K)2873 4337 y Fn(\()p Fm(m;n)p Fn(\))3056 4378 y Fq(])3083 4293 y Fo(\014)3083 4353 y(\014)3117 4319 y Fn(2)3156 4237 y Fo(\023)3229 4260 y Fn(1)p Fm(=)p Fn(2)3362 4378 y Fs(;)257 4671 y Fv(where)977 4817 y Fs(D)1058 4832 y Fm(m)1124 4817 y Fq(\()p Fs( )t Fq(\))53 b(:=)1475 4681 y Fo(Z)1531 4907 y Fm(B)1587 4884 y Fk(m)1584 4928 y Fh(1)1667 4817 y Fs(dk)1772 4776 y Fn(\()p Fm(m)p Fn(\))1921 4732 y Fo(\014)1921 4792 y(\014)1954 4817 y Fs(k)2008 4776 y Fn(\()p Fm(m)p Fn(\))2129 4732 y Fo(\014)2129 4792 y(\014)2190 4732 y(\015)2190 4792 y(\015)2246 4817 y Fs(a)p Fq(\()p Fs(k)2389 4776 y Fn(\()p Fm(m)p Fn(\))2510 4817 y Fq(\))17 b Fs( )2632 4732 y Fo(\015)2632 4792 y(\015)2687 4758 y Fn(2)2749 4817 y Fs(:)429 b Fv(\(III.17\))257 5066 y(Clearly)-6 b(,)25 b Fs(D)675 5081 y Fn(0)714 5066 y Fq(\()p Fs( )t Fq(\))j(=)g Fr(k)p Fs( )t Fr(k)1156 5030 y Fn(2)1195 5066 y Fv(,)c(and)h(we)g(observ)o(e)f(that)482 5327 y Fs(D)563 5342 y Fm(m)629 5327 y Fq(\()p Fs( )t Fq(\))53 b(:=)980 5191 y Fo(Z)1036 5417 y Fm(B)1092 5387 y Fk(m)p Fi(\000)p Fh(1)1089 5438 y(1)1250 5327 y Fs(dk)1355 5286 y Fn(\()p Fm(m)p Fl(\000)p Fn(1\))1594 5242 y Fo(\014)1594 5302 y(\014)1627 5327 y Fs(k)1681 5286 y Fn(\()p Fm(m)p Fl(\000)p Fn(1\))1893 5242 y Fo(\014)1893 5302 y(\014)1954 5242 y(\015)1954 5302 y(\015)2009 5327 y Fs(H)2098 5276 y Fn(1)p Fm(=)p Fn(2)2090 5355 y Fm(f)2224 5327 y Fs(a)p Fq(\()p Fs(k)2367 5286 y Fn(\()p Fm(m)p Fl(\000)p Fn(1\))2579 5327 y Fq(\))17 b Fs(P)2711 5286 y Fl(?)2697 5351 y Fn(\012)2769 5327 y Fs( )2836 5242 y Fo(\015)2836 5302 y(\015)2892 5268 y Fn(2)2953 5327 y Fs(;)225 b Fv(\(III.18\))p eop %%Page: 24 25 24 24 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(24)257 573 y(for)25 b Fs(m)j Fr(\025)h Fq(1)p Fv(.)h(The)25 b(Pull-Through)f(F)o(ormula)g(\(III.12\))h(implies)e(that)289 708 y Fo(\015)289 768 y(\015)344 793 y Fs(H)433 742 y Fn(1)p Fm(=)p Fn(2)425 820 y Fm(f)559 793 y Fs(a)p Fq(\()p Fs(k)702 752 y Fn(\()p Fm(m)p Fl(\000)p Fn(1\))914 793 y Fq(\))17 b Fs(P)1046 752 y Fl(?)1032 817 y Fn(\012)1104 793 y Fs( )1171 708 y Fo(\015)1171 768 y(\015)1306 793 y Fq(=)1463 708 y Fo(\015)1463 768 y(\015)1518 793 y Fs(H)1607 742 y Fn(1)p Fm(=)p Fn(2)1599 820 y Fm(f)1733 793 y Fs(a)p Fq(\()p Fs(k)1876 752 y Fn(\()p Fm(m)p Fl(\000)p Fn(1\))2088 793 y Fq(\))g Fs(H)2232 742 y Fl(\000)p Fn(1)p Fm(=)p Fn(2)2224 820 y Fm(f)2412 793 y Fs(H)2501 742 y Fn(1)p Fm(=)p Fn(2)2493 820 y Fm(f)2628 793 y Fs(P)2705 752 y Fl(?)2691 817 y Fn(\012)2763 793 y Fs( )2830 708 y Fo(\015)2830 768 y(\015)3205 793 y Fv(\(III.19\))275 1011 y Fq(=)455 867 y Fo(\015)455 926 y(\015)455 986 y(\015)455 1046 y(\015)510 900 y(\020)835 944 y Fs(H)916 959 y Fm(f)p 579 988 637 4 v 579 1079 a Fs(H)660 1094 y Fm(f)728 1079 y Fq(+)22 b(\006[)p Fs(k)977 1051 y Fn(\()p Fm(m)p Fl(\000)p Fn(1\))1189 1079 y Fq(])1226 900 y Fo(\021)1286 923 y Fn(1)p Fm(=)p Fn(2)1412 1011 y Fs(a)p Fq(\()p Fs(k)1555 970 y Fn(\()p Fm(m)p Fl(\000)p Fn(1\))1767 1011 y Fq(\))17 b Fs(H)1911 960 y Fn(1)p Fm(=)p Fn(2)1903 1039 y Fm(f)2037 1011 y Fs(P)2114 970 y Fl(?)2100 1036 y Fn(\012)2172 1011 y Fs( )2239 867 y Fo(\015)2239 926 y(\015)2239 986 y(\015)2239 1046 y(\015)2347 1011 y Fr(\024)2477 926 y Fo(\015)2477 986 y(\015)2532 1011 y Fs(a)p Fq(\()p Fs(k)2675 970 y Fn(\()p Fm(m)p Fl(\000)p Fn(1\))2887 1011 y Fq(\))g Fs(H)3031 960 y Fn(1)p Fm(=)p Fn(2)3023 1039 y Fm(f)3157 1011 y Fs(P)3234 970 y Fl(?)3220 1036 y Fn(\012)3292 1011 y Fs( )3359 926 y Fo(\015)3359 986 y(\015)3437 1011 y Fs(:)257 1273 y Fv(Hence,)26 b(for)f Fs(m)j Fr(\025)g Fq(1)p Fv(,)c(we)i(ha)n(v)o(e)268 1482 y Fs(D)349 1497 y Fm(m)416 1482 y Fq(\()p Fs( )t Fq(\))52 b Fr(\024)i Fs(D)823 1497 y Fm(m)p Fl(\000)p Fn(1)979 1482 y Fq(\()p Fs(H)1106 1431 y Fn(1)p Fm(=)p Fn(2)1098 1510 y Fm(f)1216 1482 y Fs(P)1293 1441 y Fl(?)1279 1507 y Fn(\012)1351 1482 y Fs( )t Fq(\))f Fr(\024)g(\001)17 b(\001)g(\001)51 b(\024)i Fs(D)2019 1497 y Fn(0)2058 1482 y Fq(\()p Fs(H)2185 1431 y Fm(m=)p Fn(2)2177 1510 y Fm(f)2322 1482 y Fs(P)2399 1441 y Fl(?)2385 1507 y Fn(\012)2457 1482 y Fs( )t Fq(\))g(=)2743 1398 y Fo(\015)2743 1457 y(\015)2798 1482 y Fq(\()p Fs(H)2917 1497 y Fm(f)2963 1482 y Fs(P)3040 1441 y Fl(?)3026 1507 y Fn(\012)3098 1482 y Fq(\))3136 1441 y Fm(m=)p Fn(2)3273 1482 y Fs( )3340 1398 y Fo(\015)3340 1457 y(\015)3395 1424 y Fn(2)3457 1482 y Fs(;)3205 1603 y Fv(\(III.20\))257 1723 y(where)30 b(the)f(last)g(e)o(xpression)f(is)h(an)h(upper)f(bound)g(when)g Fs(m)36 b Fq(=)g(0)29 b Fv(and)h Fs(m)36 b Fr(\025)h Fq(1)p Fv(.)44 b(Inserting)257 1843 y(this)24 b(into)g(\(III.16\),)h (we)g(obtain)367 2053 y Fr(jh)p Fs( )t Fr(j)i Fs(W)648 2068 y Fm(m;n)794 2053 y Fs(')p Fr(ij)52 b(\024)h Fs(m)1192 2012 y Fl(\000)p Fm(m=)p Fn(2)1401 2053 y Fs(n)1459 2012 y Fl(\000)p Fm(n=)p Fn(2)1654 2053 y Fr(k)p Fs(w)1774 2068 y Fm(m;n)1902 2053 y Fr(k)1952 2068 y Fm(\026)2026 1968 y Fo(\015)2026 2028 y(\015)2082 2053 y Fq(\()p Fs(H)2201 2068 y Fm(f)2246 2053 y Fs(P)2323 2012 y Fl(?)2309 2078 y Fn(\012)2381 2053 y Fq(\))2419 2012 y Fm(m=)p Fn(2)2556 2053 y Fs( )2623 1968 y Fo(\015)2623 2028 y(\015)2706 1968 y(\015)2706 2028 y(\015)2762 2053 y Fq(\()p Fs(H)2881 2068 y Fm(f)2926 2053 y Fs(P)3003 2012 y Fl(?)2989 2078 y Fn(\012)3061 2053 y Fq(\))3099 2012 y Fm(n=)p Fn(2)3217 2053 y Fs(')3281 1968 y Fo(\015)3281 2028 y(\015)3358 2053 y Fs(;)3205 2173 y Fv(\(III.21\))257 2294 y(from)24 b(which)g(the)g(right)f(inequality)g(in)h(\(III.11\))g(follo)n(ws)f (immediately)-6 b(.)28 b(Using,)c(moreo)o(v)o(er)l(,)257 2414 y(that)c Fq(0)27 b Fr(\024)h Fs(H)689 2429 y Fm(f)762 2414 y Fr(\024)g Fq(1)20 b Fv(on)g Fr(H)1140 2429 y Fn(red)1242 2414 y Fv(,)h(the)f(left)f(inequality)g(in)g(\(III.11\))h(is)g(also)f (a)h(direct)g(consequence)257 2535 y(of)25 b(\(III.21\).)2738 b Fd(\003)404 2655 y Fv(Theorem)27 b(III.1)h(says)f(that)g(the)g (\002niteness)g(of)h Fr(k)p Fs(w)2153 2670 y Fm(m;n)2282 2655 y Fr(k)2332 2670 y Fm(\026)2406 2655 y Fv(insures)e(that)h Fs(W)2984 2670 y Fm(m;n)3141 2655 y Fv(de\002nes)h(a)257 2775 y(bounded)e(operator)g(on)g Fr(B)s Fq([)p Fr(H)1290 2790 y Fn(red)1393 2775 y Fq(])p Fv(.)34 b(In)27 b(order)f(to)g(be)g (able)g(to)g(add)g(up)g(all)f(components)g Fs(W)3366 2790 y Fm(m;n)257 2896 y Fv(of)g Fs(W)14 b Fv(,)25 b(we)g(\002x)g(tw)o (o)f(numbers)g Fs(\026)k(>)f Fq(0)p Fv(,)e Fq(0)i Fs(<)h(\030)k(<)27 b Fq(1)p Fv(,)e(and)g(de\002ne)g(the)g(Banach)g(space)1404 3119 y Fr(W)1510 3072 y Fn(#)1502 3145 y Fl(\025)p Fn(1)1649 3119 y Fq(:=)1855 3025 y Fo(M)1805 3235 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)2072 3119 y Fr(W)2178 3078 y Fn(#)2170 3144 y Fm(m;n)2321 3119 y Fs(;)857 b Fv(\(III.22\))257 3429 y(to)25 b(consist)e(of)i(all)g(sequences)g Fs(w)p 1323 3445 73 4 v 30 w Fq(:=)i(\()p Fs(w)1661 3444 y Fm(m;n)1790 3429 y Fq(\))1828 3444 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)2107 3429 y Fv(obe)o(ying)983 3659 y Fr(k)p Fs(w)p 1033 3675 V 2 w Fr(k)1155 3611 y Fn(#)1155 3686 y Fm(\026;\030)1307 3659 y Fq(:=)1516 3564 y Fo(X)1463 3774 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)1730 3659 y Fs(\030)1778 3617 y Fl(\000)p Fn(\()p Fm(m)p Fn(+)p Fm(n)p Fn(\))2079 3659 y Fr(k)p Fs(w)2199 3674 y Fm(m;n)2327 3659 y Fr(k)2377 3617 y Fn(#)2377 3683 y Fm(\026)2493 3659 y Fs(<)52 b Fr(1)22 b Fs(:)435 b Fv(\(III.23\))257 3986 y(Note)25 b(that)f Fr(W)756 3939 y Fn(#)748 4012 y Fl(\025)p Fn(1)868 3986 y Fv(depends)g(on)h(the)f(tw)o(o)h(parameters)g Fs(\026)i(>)h Fq(0)c Fv(and)h Fq(0)j Fs(<)f(\030)32 b(<)c Fq(1)p Fv(.)404 4107 y(Ne)o(xt)c(we)h(de\002ne)g(the)g(Banach)h(spaces) 482 4340 y Fr(T)108 b Fq(:=)830 4230 y Fo(n)897 4340 y Fs(f)38 b Fr(2)28 b Fs(C)1154 4299 y Fn(1)1193 4340 y Fq(\()p Fs(I)8 b Fq(\))1345 4226 y Fo(\014)1345 4286 y(\014)1345 4345 y(\014)1403 4340 y Fs(f)j Fq(\(0\))27 b(=)h(0)22 b Fs(;)41 b Fr(k)p Fs(f)11 b Fr(k)2016 4355 y Fl(T)2104 4340 y Fq(:=)27 b(sup)2249 4422 y Fm(r)r Fl(2)p Fm(I)2398 4340 y Fr(j)p Fs(f)2485 4299 y Fl(0)2507 4340 y Fq(\()p Fs(r)s Fq(\))p Fr(j)g Fs(<)h Fr(1)2889 4230 y Fo(o)2977 4340 y Fs(;)201 b Fv(\(III.24\))368 4584 y Fr(W)474 4537 y Fn(#)466 4608 y(0)p Fm(;)p Fn(0)644 4584 y Fq(:=)830 4473 y Fo(n)897 4584 y Fs(w)967 4599 y Fn(0)p Fm(;)p Fn(0)1088 4584 y Fr(2)28 b Fs(C)1259 4543 y Fn(1)1299 4584 y Fq(\()p Fs(I)8 b Fq(\))1450 4469 y Fo(\014)1450 4529 y(\014)1450 4589 y(\014)1508 4584 y Fr(k)p Fs(w)1628 4599 y Fn(0)p Fm(;)p Fn(0)1722 4584 y Fr(k)1772 4543 y Fn(#)1772 4608 y Fm(\026)1863 4584 y Fq(:=)27 b Fr(k)p Fs(w)2113 4599 y Fn(0)p Fm(;)p Fn(0)2207 4584 y Fr(k)2257 4599 y Fm(\026)2326 4584 y Fq(+)22 b Fr(k)p Fs(@)2525 4599 y Fm(r)2563 4584 y Fs(w)2633 4599 y Fn(0)p Fm(;)p Fn(0)2727 4584 y Fr(k)2777 4599 y Fm(\026)2851 4584 y Fs(<)28 b Fr(1)3055 4473 y Fo(o)3121 4584 y Fs(;)57 b Fv(\(III.25\))257 4821 y(where)26 b Fr(k)p Fs(w)646 4836 y Fn(0)p Fm(;)p Fn(0)740 4821 y Fr(k)790 4836 y Fm(\026)864 4821 y Fq(:=)h(sup)1141 4844 y Fm(r)r Fl(2)p Fm(I)1279 4821 y Fr(j)p Fs(w)1377 4836 y Fn(0)p Fm(;)p Fn(0)1470 4821 y Fq([)p Fs(r)s Fq(])p Fr(j)p Fv(.)k(W)-8 b(e)25 b(observ)o(e)f(that)g(there)h(is)g(a)g(natural)g(bijection)527 5040 y Fr(W)633 4992 y Fn(#)625 5064 y(0)p Fm(;)p Fn(0)772 5040 y Fr(!)52 b Fg(C)c Fr(\010)23 b(T)48 b Fs(;)253 b(w)1570 5055 y Fn(0)p Fm(;)p Fn(0)1716 5040 y Fr(7!)53 b Fs(w)1939 5055 y Fn(0)p Fm(;)p Fn(0)2033 5040 y Fq([0])22 b Fr(\010)g Fq(\()p Fs(w)2365 5055 y Fn(0)p Fm(;)p Fn(0)2482 5040 y Fr(\000)g Fs(w)2651 5055 y Fn(0)p Fm(;)p Fn(0)2745 5040 y Fq([0]\))g Fs(:)270 b Fv(\(III.26\))257 5268 y(W)-8 b(e)20 b(shall)e(henceforth)h(not)f(distinguish)e(between)j Fr(W)2118 5221 y Fn(#)2110 5292 y(0)p Fm(;)p Fn(0)2224 5268 y Fv(and)g Fg(C)g Fr(\010)q(T)32 b Fv(.)c(W)-8 b(e)20 b(write)e Fs(W)3134 5283 y Fn(0)p Fm(;)p Fn(0)3229 5268 y Fq([)p Fs(w)3326 5283 y Fn(0)p Fm(;)p Fn(0)3420 5268 y Fq(])28 b(:=)257 5388 y Fs(w)327 5403 y Fn(0)p Fm(;)p Fn(0)421 5388 y Fq([)p Fs(H)529 5403 y Fm(f)575 5388 y Fq(])f Fr(2)i(B)s Fq([)p Fr(H)903 5403 y Fn(red)1006 5388 y Fq(])p Fv(.)p eop %%Page: 25 26 25 25 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(25)404 573 y(De\002ning)668 807 y Fr(W)774 760 y Fn(#)766 833 y Fl(\025)p Fn(0)914 807 y Fq(:=)1119 712 y Fo(M)1069 923 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(0)1336 807 y Fr(W)1442 766 y Fn(#)1434 832 y Fm(m;n)1616 807 y Fq(=)53 b Fg(C)75 b Fr(\010)51 b(T)75 b(\010)2300 712 y Fo(M)2250 923 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)2517 807 y Fr(W)2623 766 y Fn(#)2615 832 y Fm(m;n)2767 807 y Fs(;)411 b Fv(\(III.27\))257 1120 y(with)24 b(the)h(norm)977 1255 y Fo(\015)977 1315 y(\015)1032 1340 y Fs(w)p 1032 1356 73 4 v 1105 1255 a Fo(\015)1105 1315 y(\015)1160 1281 y Fn(#)1160 1379 y Fm(\026;\030)1313 1340 y Fq(:=)1521 1245 y Fo(X)1468 1456 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(0)1735 1340 y Fs(\030)1783 1299 y Fl(\000)p Fn(\()p Fm(m)p Fn(+)p Fm(n)p Fn(\))2084 1340 y Fr(k)p Fs(w)2204 1355 y Fm(m;n)2333 1340 y Fr(k)2383 1299 y Fn(#)2383 1364 y Fm(\026)2498 1340 y Fs(<)53 b Fr(1)21 b Fs(;)430 b Fv(\(III.28\))257 1678 y(we)30 b(observ)o(e)e(that)h(an)o(y)f (sequence)h Fs(w)p 1484 1694 V 38 w Fq(:=)35 b(\()p Fs(w)1838 1693 y Fm(m;n)1967 1678 y Fq(\))2005 1693 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(0)2288 1678 y Fv(in)29 b Fr(W)2501 1631 y Fn(#)2493 1704 y Fl(\025)p Fn(0)2617 1678 y Fv(de\002nes)g(an)g (operator)g(of)257 1798 y(the)c(form)g(\(III.2\))g(on)g Fr(H)1097 1813 y Fn(red)1224 1798 y Fv(by)f(setting)506 2033 y Fs(H)8 b Fq(\()p Fs(w)p 633 2048 V 2 w Fq(\))83 b(:=)1065 1938 y Fo(X)1012 2148 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(0)1279 2033 y Fs(W)1371 2048 y Fm(m;n)1500 2033 y Fq([)p Fs(w)p 1527 2048 V 3 w Fq(])1578 b Fv(\(III.29\))840 2307 y Fq(=)1012 2227 y Fo(\000)1058 2307 y Fs(w)1128 2322 y Fn(0)p Fm(;)p Fn(0)1222 2307 y Fq([)p Fs(H)1330 2322 y Fm(f)1375 2307 y Fq(])22 b Fr(\000)h Fs(w)1594 2322 y Fn(0)p Fm(;)p Fn(0)1688 2307 y Fq([0])1791 2227 y Fo(\001)1886 2307 y Fq(+)50 b Fs(w)2082 2322 y Fn(0)p Fm(;)p Fn(0)2176 2307 y Fq([0])22 b Fr(\001)g Fp(1)50 b Fq(+)2636 2213 y Fo(X)2583 2423 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)2849 2307 y Fs(W)2941 2322 y Fm(m;n)3071 2307 y Fq([)p Fs(w)p 3098 2323 V 2 w Fq(])22 b Fs(;)257 2627 y Fv(where)33 b(we)f(write)f Fs(W)1009 2642 y Fm(m;n)1138 2627 y Fq([)p Fs(w)p 1165 2643 V 3 w Fq(])40 b(:=)h Fs(W)1541 2642 y Fm(m;n)1670 2627 y Fq([)p Fs(w)1767 2642 y Fm(m;n)1896 2627 y Fq(])p Fv(.)51 b(Ne)o(xt,)33 b(we)f(deri)n(v)o(e)f(a)h (corollary)f(of)h(Theo-)257 2756 y(rem)22 b(III.1)h(which)e(gi)n(v)o (es)f(meaning)i(to)f(\(III.29\))h(and)g(sho)n(ws)f(that)g Fs(H)36 b Fq(:)27 b Fr(W)2784 2709 y Fn(#)2776 2782 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(0)3059 2756 y Fr(!)g(B)s Fq([)p Fr(H)3365 2771 y Fn(red)3468 2756 y Fq(])257 2877 y Fv(is)e(an)g(embedding.)k(Furthermore,)c(we)g(demonstrate)f(that)g (this)g(embedding)g(is)g(injecti)n(v)o(e.)257 3080 y Ff(Theor)n(em)33 b(III.3.)44 b Ft(F)-10 b(or)30 b(any)h Fs(\026)39 b(>)g Fq(0)31 b Ft(and)g Fq(0)39 b Fs(<)g(\030)44 b(<)39 b Fq(1)p Ft(,)32 b(the)f(map)g Fs(H)47 b Fq(:)39 b Fr(W)2921 3033 y Fn(#)2913 3106 y Fl(\025)p Fn(0)3047 3080 y Fr(!)g(B)s Fq([)p Fr(H)3365 3095 y Fn(red)3468 3080 y Fq(])257 3219 y Ft(de\002nes)29 b(an)g(injective)g(embedding)o (,)g(the)g(subspace)g Fs(H)8 b Fq(\()p Fr(W)2329 3172 y Fn(#)2321 3245 y Fl(\025)p Fn(0)2415 3219 y Fq(\))36 b Fr(\022)g(B)s Fq([)p Fr(H)2781 3234 y Fn(red)2884 3219 y Fq(])29 b Ft(is)g(closed,)h(and)257 3340 y(we)c(have)f(the)g(oper)o (ator)e(norm)h(estimates)1373 3488 y Fo(\015)1373 3548 y(\015)1445 3573 y Fs(H)8 b Fq(\()p Fs(w)p 1572 3588 V 1 w Fq(\))1698 3488 y Fo(\015)1698 3548 y(\015)1753 3612 y Fn(op)1885 3573 y Fr(\024)2015 3488 y Fo(\015)2015 3548 y(\015)2087 3573 y Fs(w)p 2087 3588 V 2176 3488 a Fo(\015)2176 3548 y(\015)2231 3514 y Fn(#)2231 3612 y Fm(\026;\030)2353 3573 y Fs(;)825 b Fv(\(III.30\))257 3826 y Ft(for)25 b(any)f Fs(w)p 567 3842 V 30 w Fq(=)k(\()p Fs(w)879 3841 y Fm(m;n)1008 3826 y Fq(\))1046 3841 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(0)1328 3826 y Fr(2)g(W)1528 3779 y Fn(#)1520 3852 y Fl(\025)p Fn(0)1614 3826 y Ft(,)d(and)1341 3991 y Fo(\015)1341 4051 y(\015)1413 4075 y Fs(H)8 b Fq(\()p Fs(w)p 1540 4091 V 1 w Fq(\))1666 3991 y Fo(\015)1666 4051 y(\015)1721 4114 y Fn(op)1853 4075 y Fr(\024)53 b Fs(\030)2047 3991 y Fo(\015)2047 4051 y(\015)2119 4075 y Fs(w)p 2119 4091 V 2208 3991 a Fo(\015)2208 4051 y(\015)2263 4017 y Fn(#)2263 4114 y Fm(\026;\030)2385 4075 y Fs(;)793 b Fv(\(III.31\))257 4329 y Ft(for)25 b(any)f Fs(w)p 567 4345 V 30 w Fq(=)k(\()p Fs(w)879 4344 y Fm(m;n)1008 4329 y Fq(\))1046 4344 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)1328 4329 y Fr(2)g(W)1528 4281 y Fn(#)1520 4355 y Fl(\025)p Fn(1)1614 4329 y Ft(.)257 4532 y(Pr)l(oof)o(.)35 b Fv(The)27 b(estimates)e(\(III.30\))i(and)g(\(III.31\))g(follo)n(w)f(directly)g (from)h(Theorem)f(III.1,)h(us-)257 4652 y(ing)21 b(that)f Fq(0)27 b Fs(<)h(\030)k(<)c Fq(1)p Fv(.)h(Eq.)20 b(\(III.30\))i (immediately)d(implies)g(the)i(closedness)f(of)h Fs(H)8 b Fq(\()p Fr(W)3266 4605 y Fn(#)3258 4678 y Fl(\025)p Fn(0)3352 4652 y Fq(\))28 b Fr(\022)257 4773 y(B)s Fq([)p Fr(H)436 4788 y Fn(red)539 4773 y Fq(])p Fv(,)404 4893 y(The)d(injecti)n(vity)d(of)j Fs(H)33 b Fv(is)24 b(more)h(subtle.)30 b(W)-8 b(e)25 b(observ)o(e)g(that)f(by)h(linearity)-6 b(,)23 b(it)i(suf)n(\002ces)g(to)257 5014 y(sho)n(w)f(that)g(if)h Fs(H)8 b Fq(\()p Fs(w)p 877 5030 V 2 w Fq(\))28 b(=)f(0)e Fv(then)f Fs(w)p 1388 5030 V 30 w Fq(=)k(0)p Fv(.)404 5134 y(W)-8 b(e)23 b(\002rst)h(pro)o(v)o(e)e(that)h(if)g Fs(H)8 b Fq(\()p Fs(w)p 1367 5150 V 2 w Fq(\))28 b(=)f(0)d Fv(then)f Fs(w)1946 5149 y Fn(0)p Fm(;)p Fn(0)2067 5134 y Fq(=)28 b(0)p Fv(.)i(T)-8 b(o)23 b(this)f(end,)i(we)g(pick)f(a)g (function)257 5254 y Fs(f)42 b Fr(2)32 b Fs(C)522 5218 y Fl(1)515 5279 y Fn(0)596 5254 y Fq([)p Fs(B)697 5269 y Fn(1)737 5254 y Fq(;)17 b Fg(R)847 5218 y Fn(+)912 5254 y Fq(])p Fv(,)27 b(with)1195 5174 y Fo(R)1278 5254 y Fr(j)p Fs(f)11 b Fq(\()p Fs(x)p Fq(\))p Fr(j)1524 5218 y Fn(2)1579 5254 y Fs(d)1630 5218 y Fm(d)1670 5254 y Fs(x)32 b Fq(=)f(1)p Fv(,)c(and)f(de\002ne)i Fs(f)2454 5269 y Fm(";k)2548 5254 y Fq(\()p Fs(x)p Fq(\))k(:=)f Fs(")2891 5218 y Fl(\000)p Fm(d=)p Fn(2)3057 5254 y Fs(f)3116 5174 y Fo(\000)3161 5254 y Fs(")3207 5218 y Fl(\000)p Fn(1)3301 5254 y Fq(\()p Fs(x)24 b Fr(\000)p eop %%Page: 26 27 26 26 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(26)257 573 y Fs(k)s Fq(\))349 492 y Fo(\001)395 573 y Fv(.)31 b(A)25 b(simple)e(computation)g(gi)n(v)o(es)362 684 y Fo(\012)409 765 y Fs(a)460 724 y Fl(\003)499 765 y Fq(\()p Fs(f)585 780 y Fm(";k)680 765 y Fq(\)\012)788 680 y Fo(\014)788 740 y(\014)849 765 y Fs(H)8 b Fq(\()p Fs(w)p 976 781 73 4 v 2 w Fq(\))28 b Fs(a)1165 724 y Fl(\003)1205 765 y Fq(\()p Fs(f)1291 780 y Fm(";k)1385 765 y Fq(\)\012)1493 684 y Fo(\013)445 910 y Fq(=)604 829 y Fo(\012)651 910 y Fs(a)702 869 y Fl(\003)741 910 y Fq(\()p Fs(f)827 925 y Fm(";k)922 910 y Fq(\)\012)1030 825 y Fo(\014)1030 885 y(\014)1086 910 y Fs(w)1156 925 y Fn(0)p Fm(;)p Fn(0)1250 910 y Fq([)p Fs(H)1358 925 y Fm(f)1403 910 y Fq(])22 b Fs(a)1503 869 y Fl(\003)1543 910 y Fq(\()p Fs(f)1629 925 y Fm(";k)1724 910 y Fq(\)\012)1832 829 y Fo(\013)1929 910 y Fq(+)2055 829 y Fo(\012)2102 910 y Fs(a)2153 869 y Fl(\003)2193 910 y Fq(\()p Fs(f)2279 925 y Fm(";k)2373 910 y Fq(\)\012)2481 825 y Fo(\014)2481 885 y(\014)2537 910 y Fs(W)2629 925 y Fn(1)p Fm(;)p Fn(1)2724 910 y Fq([)p Fs(w)2821 925 y Fn(1)p Fm(;)p Fn(1)2915 910 y Fq(])g Fs(a)3015 869 y Fl(\003)3055 910 y Fq(\()p Fs(f)3141 925 y Fm(";k)3236 910 y Fq(\)\012)3344 829 y Fo(\013)445 1114 y Fq(=)604 978 y Fo(Z)659 1204 y Fm(B)712 1213 y Fh(1)767 1114 y Fs(d)818 1073 y Fm(d)859 1114 y Fs(x)p Fr(j)p Fs(f)990 1129 y Fm(";k)1085 1114 y Fq(\()p Fs(x)p Fq(\))p Fr(j)1244 1073 y Fn(2)1305 1114 y Fs(w)1375 1129 y Fn(0)p Fm(;)p Fn(0)1469 1114 y Fq([)p Fs(!)t Fq(\()p Fs(x)p Fq(\)])1486 b Fv(\(III.32\))631 1396 y Fq(+)751 1260 y Fo(Z)807 1486 y Fm(B)863 1463 y Fh(2)860 1507 y(1)1003 1328 y Fs(d)1054 1292 y Fm(d)1094 1328 y Fs(x)17 b(d)1217 1292 y Fm(d)1257 1328 y Fs(x)1312 1292 y Fl(0)p 929 1373 482 4 v 929 1464 a Fr(j)p Fs(x)p Fr(j)1040 1435 y Fn(1)p Fm(=)p Fn(2)1166 1464 y Fr(j)p Fs(x)1249 1435 y Fl(0)1272 1464 y Fr(j)1300 1435 y Fn(1)p Fm(=)p Fn(2)p 1442 1309 275 4 v 1442 1396 a Fs(f)1490 1411 y Fm(";k)1585 1396 y Fq(\()p Fs(x)p Fq(\))g Fs(f)1781 1411 y Fm(";k)1876 1396 y Fq(\()p Fs(x)1969 1355 y Fl(0)1993 1396 y Fq(\))22 b Fs(w)2123 1411 y Fn(1)p Fm(;)p Fn(1)2217 1396 y Fq([0;)17 b Fs(x;)g(x)2491 1355 y Fl(0)2514 1396 y Fq(])22 b Fs(:)257 1656 y Fv(The)31 b(second)f(term)h(in)f(the)g(last)g(line)g(of)h (\(III.32\))g(tends)f(to)g Fq(0)p Fv(,)i(because)f Fs(f)2857 1671 y Fm(";k)2990 1656 y Fr(!)38 b Fq(0)31 b Fv(weakly)257 1776 y(in)e Fs(L)430 1740 y Fn(2)470 1776 y Fq(\()p Fs(B)582 1791 y Fn(1)621 1776 y Fq(\))p Fv(,)i(and)e(the)g(\002rst)g(term)g (tends)f(to)h Fs(w)1851 1791 y Fn(0)p Fm(;)p Fn(0)1945 1776 y Fq([)p Fs(!)t Fq(\()p Fs(k)s Fq(\)])g Fv(since)g Fr(j)p Fs(f)2533 1791 y Fm(";k)2627 1776 y Fq(\()p Fs(x)p Fq(\))p Fr(j)2786 1740 y Fn(2)2861 1776 y Fr(!)36 b Fs(\016)t Fq(\()p Fs(x)25 b Fr(\000)h Fs(k)s Fq(\))p Fv(,)k(as)257 1897 y Fs(")e Fr(!)f Fq(0)p Fv(.)k(By)25 b(assumption,)d Fs(H)8 b Fq(\()p Fs(w)p 1334 1913 73 4 v 2 w Fq(\))28 b(=)f(0)p Fv(,)e(and)g(we)g(obtain)628 2089 y Fs(w)698 2104 y Fn(0)p Fm(;)p Fn(0)792 2089 y Fq([)p Fs(!)t Fq(\()p Fs(k)s Fq(\)])52 b(=)i(lim)1221 2148 y Fm(")p Fl(!)p Fn(0)1376 2008 y Fo(\012)1423 2089 y Fs(a)1474 2047 y Fl(\003)1514 2089 y Fq(\()p Fs(f)1600 2104 y Fm(";k)1695 2089 y Fq(\)\012)1803 2004 y Fo(\014)1803 2064 y(\014)1864 2089 y Fs(H)8 b Fq(\()p Fs(w)p 1991 2105 V 2 w Fq(\))28 b Fs(a)2180 2047 y Fl(\003)2219 2089 y Fq(\()p Fs(f)2305 2104 y Fm(";k)2400 2089 y Fq(\)\012)2508 2008 y Fo(\013)2608 2089 y Fq(=)53 b(0)21 b Fs(;)371 b Fv(\(III.33\))257 2307 y(for)25 b(an)o(y)g Fs(k)30 b Fr(2)f Fs(B)816 2322 y Fn(1)855 2307 y Fv(.)404 2428 y(T)-8 b(o)34 b(proceed)h(in)f(the)h (proof,)i(we)e(use)f(induction)f(in)i Fs(J)55 b Fq(:=)45 b Fs(m)30 b Fq(+)f Fs(n)46 b Fr(\025)h Fq(1)p Fv(.)60 b(Although)257 2548 y(technically)37 b(some)n(what)g(more)g(in)l(v)n (olv)o(ed,)i(the)f(idea)g(for)g(the)f(proof)h(is)f(the)h(same)f(as)h (for)257 2668 y Fs(m)50 b Fq(=)f Fs(n)h Fq(=)f(0)p Fv(.)66 b(W)-8 b(e)37 b(pro)o(v)o(e)e(that)h Fs(H)8 b Fq(\()p Fs(w)p 1630 2684 V 2 w Fq(\))49 b(=)g(0)37 b Fv(and)f Fs(w)2250 2683 y Fm(m;n)2428 2668 y Fq(=)49 b(0)p Fv(,)40 b(for)c(all)h Fq(\()p Fs(m;)17 b(n)p Fq(\))49 b Fr(2)h Fg(N)3450 2632 y Fn(2)3450 2693 y(0)257 2789 y Fv(with)32 b Fs(m)27 b Fq(+)h Fs(n)41 b Fr(\024)g Fs(J)c Fr(\000)28 b Fq(1)p Fv(,)33 b(imply)e Fs(w)1538 2804 y Fm(m;n)1707 2789 y Fq(=)41 b(0)p Fv(,)33 b(for)g(all)e Fq(\()p Fs(m;)17 b(n)p Fq(\))41 b Fr(2)h Fg(N)2689 2753 y Fn(2)2689 2813 y(0)2766 2789 y Fv(with)31 b Fs(m)d Fq(+)f Fs(n)41 b Fq(=)g Fs(J)9 b Fv(.)257 2909 y(Thus,)24 b(\002x)g Fq(\()p Fs(m;)17 b(n)p Fq(\))24 b Fv(with)f Fs(m)c Fq(+)g Fs(n)28 b Fq(=)g Fs(J)9 b Fv(,)24 b(pick)f Fs(g)1866 2924 y Fn(1)1905 2909 y Fs(;)17 b(:)g(:)g(:)f(;)h(g)2171 2924 y Fm(m)2237 2909 y Fs(;)g(h)2337 2924 y Fn(1)2377 2909 y Fs(;)g(:)g(:)g(:)e(;)i(h) 2651 2924 y Fm(n)2726 2909 y Fr(2)28 b Fs(L)2886 2873 y Fn(2)2926 2909 y Fq(\()p Fs(B)3038 2924 y Fn(1)3077 2909 y Fq(\))c Fv(such)g(that)257 2949 y Fo(R)340 3030 y Fr(j)p Fs(g)415 3045 y Fm(i)443 3030 y Fq(\()p Fs(k)s Fq(\))p Fr(j)601 2993 y Fn(2)657 3030 y Fr(j)p Fs(k)s Fr(j)767 2993 y Fl(\000)p Fn(3)p Fl(\000)p Fn(2)p Fm(\026)1010 3030 y Fs(d)1061 2993 y Fm(d)1101 3030 y Fs(k)49 b Fq(=)1323 2949 y Fo(R)1406 3030 y Fr(j)p Fs(h)1490 3045 y Fm(j)1526 3030 y Fq(\()p Fs(k)s Fq(\))p Fr(j)1684 2993 y Fn(2)1740 3030 y Fr(j)p Fs(k)s Fr(j)1850 2993 y Fl(\000)p Fn(3)p Fl(\000)p Fn(2)p Fm(\026)2093 3030 y Fs(d)2144 2993 y Fm(d)2184 3030 y Fs(k)g Fq(=)d(1)p Fv(.)60 b(W)-8 b(e)35 b(set)g Fs(g)p 2851 3065 51 4 v 49 w Fq(:=)46 b Fs(g)3143 3045 y Fn(1)3182 3030 y Fs(;)17 b(:)g(:)g(:)f(g)3404 3045 y Fm(m)3470 3030 y Fv(,)257 3150 y Fs(h)p 257 3166 57 4 v 28 w Fq(:=)28 b Fs(h)528 3165 y Fn(1)567 3150 y Fs(;)17 b(:)g(:)g(:)f(h)798 3165 y Fm(n)870 3150 y Fv(and)25 b(furthermore)816 3342 y Fq(\011)892 3357 y Fm(g)p 892 3382 36 3 v 2 w(;";k)1125 3342 y Fq(:=)83 b Fs(a)1362 3301 y Fl(\003)1402 3342 y Fq(\()p Fs(g)1487 3357 y Fn(1)1526 3342 y Fq(\))17 b Fr(\001)g(\001)g(\001)d Fs(a)1764 3301 y Fl(\003)1804 3342 y Fq(\()p Fs(g)1889 3357 y Fm(m)1955 3342 y Fq(\))j Fs(a)2061 3301 y Fl(\003)2100 3342 y Fq(\()p Fs(f)2186 3357 y Fm(";k)2281 3342 y Fq(\))g(\012)53 b Fr(2)g(F)2660 3301 y Fn(\()p Fm(m)p Fn(+1\))2893 3342 y Fs(;)285 b Fv(\(III.34\))817 3509 y Fq(\010)887 3524 y Fm(h)p 887 3536 41 3 v(;";k)1125 3509 y Fq(:=)83 b Fs(a)1362 3468 y Fl(\003)1402 3509 y Fq(\()p Fs(h)1496 3524 y Fn(1)1535 3509 y Fq(\))17 b Fr(\001)g(\001)g(\001)e Fs(a)1774 3468 y Fl(\003)1813 3509 y Fq(\()p Fs(h)1907 3524 y Fm(n)1954 3509 y Fq(\))i Fs(a)2060 3468 y Fl(\003)2100 3509 y Fq(\()p Fs(f)2186 3524 y Fm(";k)2280 3509 y Fq(\))g(\012)56 b Fr(2)g(F)2665 3468 y Fn(\()p Fm(n)p Fn(+1\))2878 3509 y Fs(:)300 b Fv(\(III.35\))257 3701 y(Then)26 b(we)f(observ)o(e)g(that) g Fr(h)p Fq(\011)1253 3716 y Fm(g)p 1253 3741 36 3 v 2 w(;";k)1420 3701 y Fr(j)k Fs(W)1569 3716 y Fm(m)p Fn(+)p Fm(i;n)p Fn(+)p Fm(j)1864 3701 y Fq([)p Fs(w)p 1891 3717 73 4 v 2 w Fq(])17 b(\010)2077 3716 y Fm(h)p 2077 3728 41 3 v(;";k)2233 3701 y Fr(i)25 b Fv(v)n(anishes,)f(unless)h(the)g (follo)n(wing)257 3821 y(three)g(conditions)e(are)j(ful\002lled,)1149 4013 y Fs(i)44 b Fr(\024)h Fq(1)22 b Fs(;)206 b(j)50 b Fr(\024)45 b Fq(1)21 b Fs(;)112 b Fv(and)94 b Fs(i)45 b Fq(=)f Fs(j)28 b(:)601 b Fv(\(III.36\))257 4205 y(Moreo)o(v)o(er)l(,) 24 b Fs(w)763 4220 y Fm(m)p Fn(+)p Fm(i;n)p Fn(+)p Fm(j)1086 4205 y Fq(=)j(0)p Fv(,)e(unless)f Fs(i)e Fq(+)g Fs(j)34 b Fr(\025)28 b Fq(0)p Fv(,)d(by)f(the)h(induction)e(hypothesis.)28 b(Thus)565 4376 y Fo(\012)612 4457 y Fq(\011)688 4472 y Fm(g)p 688 4497 36 3 v 2 w(;";k)855 4372 y Fo(\014)855 4432 y(\014)916 4457 y Fs(H)8 b Fq(\()p Fs(w)p 1043 4473 73 4 v 2 w Fq(\))17 b(\010)1240 4472 y Fm(h)p 1240 4484 41 3 v -1 w(;";k)1395 4376 y Fo(\013)1522 4457 y Fq(=)1678 4317 y Fo(\034)1753 4457 y Fq(\011)1829 4472 y Fm(g)p 1829 4497 36 3 v 2 w(;";k)1996 4313 y Fo(\014)1996 4372 y(\014)1996 4432 y(\014)1996 4492 y(\014)2102 4362 y(X)2073 4573 y Fm(i)p Fn(+)p Fm(j)t Fl(\025)p Fn(0)2291 4457 y Fs(W)2383 4472 y Fm(m)p Fn(+)p Fm(i;n)p Fn(+)p Fm(j)2679 4457 y Fq([)p Fs(w)p 2706 4473 73 4 v 2 w Fq(])g(\010)2892 4472 y Fm(h)p 2892 4484 41 3 v(;";k)3047 4317 y Fo(\035)3205 4457 y Fv(\(III.37\))648 4714 y Fq(=)807 4633 y Fo(\012)854 4714 y Fq(\011)930 4729 y Fm(g)p 930 4753 36 3 v 2 w(;";k)1097 4629 y Fo(\014)1097 4689 y(\014)1158 4714 y Fs(W)1250 4729 y Fm(m)p Fn(+1)p Fm(;n)p Fn(+1)1559 4714 y Fq([)p Fs(w)p 1586 4730 73 4 v 3 w Fq(])g(\010)1773 4729 y Fm(h)p 1773 4741 41 3 v(;";k)1928 4633 y Fo(\013)2025 4714 y Fq(+)2151 4633 y Fo(\012)2198 4714 y Fq(\011)2274 4729 y Fm(g)p 2274 4753 36 3 v 1 w(;";k)2441 4629 y Fo(\014)2441 4689 y(\014)2501 4714 y Fs(W)2593 4729 y Fm(m;n)2722 4714 y Fq([)p Fs(w)p 2749 4730 73 4 v 3 w Fq(])g(\010)2936 4729 y Fm(h)p 2936 4741 41 3 v(;";k)3091 4633 y Fo(\013)3160 4714 y Fs(:)257 4906 y Fv(T)-8 b(o)25 b(analyse)g(the)f(\002rst)h(term) g(in)f(the)h(sum)f(in)g(the)h(last)f(line)g(of)h(\(III.37\),)h(we)f (note)f(that)257 5017 y Fo(\012)304 5097 y Fq(\011)380 5112 y Fm(g)p 380 5137 36 3 v 2 w(;";k)547 5013 y Fo(\014)547 5073 y(\014)608 5097 y Fs(W)700 5112 y Fm(m)p Fn(+1)p Fm(;n)p Fn(+1)1010 5097 y Fq([)p Fs(w)p 1037 5113 73 4 v 2 w Fq(])17 b(\010)1223 5112 y Fm(h)p 1223 5124 41 3 v(;";k)1378 5017 y Fo(\013)1478 5097 y Fq(=)52 b(\()p Fs(m)23 b Fq(+)f(1\)!)17 b(\()p Fs(n)22 b Fq(+)g(1\)!)938 b Fv(\(III.38\))282 5191 y Fo(Z)337 5417 y Fm(B)393 5387 y Fk(m)p Fh(+)p Fk(n)p Fh(+2)390 5438 y(1)702 5259 y Fs(dX)842 5223 y Fn(\()p Fm(m)p Fn(+1)p Fm(;n)p Fn(+1\))p 645 5304 619 4 v 645 5395 a Fr(j)p Fs(X)762 5366 y Fn(\()p Fm(m)p Fn(+1)p Fm(;n)p Fn(+1\))1126 5395 y Fr(j)1154 5366 y Fn(1)p Fm(=)p Fn(2)1306 5327 y Fs(w)1376 5342 y Fm(m)p Fn(+1)p Fm(;n)p Fn(+1)1685 5327 y Fq([0;)34 b Fs(X)1911 5286 y Fn(\()p Fm(m)p Fn(+1)p Fm(;n)p Fn(+1\))2275 5327 y Fq(])2319 5246 y Fo(\000)2364 5327 y Fq(\011)2440 5342 y Fm(g)p 2440 5367 36 3 v 2 w(;";k)2612 5327 y Fr(\012)23 b Fq(\010)2782 5342 y Fm(h)p 2782 5354 41 3 v(;";k)2938 5246 y Fo(\001)2983 5327 y Fq([)p Fs(X)3099 5286 y Fn(\()p Fm(m)p Fn(+1)p Fm(;n)p Fn(+1\))3463 5327 y Fq(])g Fs(;)p eop %%Page: 27 28 27 27 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(27)257 573 y(where)612 692 y Fo(\000)658 773 y Fq(\011)734 788 y Fm(g)p 734 813 36 3 v 2 w(;";k)907 773 y Fr(\012)22 b Fq(\010)1076 788 y Fm(h)p 1076 800 41 3 v(;";k)1232 692 y Fo(\001)1277 773 y Fq([)p Fs(X)1393 732 y Fn(\()p Fm(m)p Fn(+1)p Fm(;n)p Fn(+1\))1758 773 y Fq(])52 b(:=)1265 b Fv(\(III.39\))779 918 y Fs(g)826 933 y Fn(1)865 918 y Fq(\()p Fs(x)958 933 y Fn(1)997 918 y Fq(\))17 b Fr(\001)g(\001)g (\001)e Fs(g)1232 933 y Fm(m)1298 918 y Fq(\()p Fs(x)1391 933 y Fm(m)1458 918 y Fq(\))i Fs(f)1561 933 y Fm(";k)1656 918 y Fq(\()p Fs(x)1749 933 y Fm(m)p Fn(+1)1906 918 y Fq(\))22 b Fs(h)2022 933 y Fn(1)2061 918 y Fq(\()6 b(~)-55 b Fs(x)2154 933 y Fn(1)2194 918 y Fq(\))17 b Fr(\001)g(\001)g(\001)e Fs(h)2438 933 y Fm(n)2485 918 y Fq(\()6 b(~)-55 b Fs(x)2578 933 y Fm(n)2625 918 y Fq(\))17 b Fs(f)2728 933 y Fm(";k)2822 918 y Fq(\()6 b(~)-55 b Fs(x)2915 933 y Fm(n)p Fn(+1)3053 918 y Fq(\))22 b Fs(:)257 1118 y Fv(Since)k Fq(\011)580 1133 y Fm(g)p 580 1158 36 3 v 1 w(;";k)752 1118 y Fr(\012)d Fq(\010)922 1133 y Fm(h)p 922 1145 41 3 v(;";k)1105 1118 y Fr(!)k Fq(0)e Fv(weakly)g(in)f Fs(L)1787 1082 y Fn(2)1827 1118 y Fq(\()p Fs(B)1944 1077 y Fm(m)p Fn(+)p Fm(n)p Fn(+2)1939 1143 y(1)2198 1118 y Fq(\))p Fv(,)h(as)g Fs(")i Fr(!)h Fq(0)p Fv(,)c(we)h(ha)n(v)o(e)g(that)1077 1334 y Fq(lim)1075 1393 y Fm(")p Fl(!)p Fn(0)1230 1253 y Fo(\012)1277 1334 y Fq(\011)1353 1349 y Fm(g)p 1353 1373 36 3 v 2 w(;";k)1520 1249 y Fo(\014)1520 1309 y(\014)1581 1334 y Fs(W)1673 1349 y Fm(m)p Fn(+1)p Fm(;n)p Fn(+1)1983 1334 y Fq([)p Fs(w)p 2010 1350 73 4 v 2 w Fq(])17 b(\010)2196 1349 y Fm(h)p 2196 1361 41 3 v(;";k)2351 1253 y Fo(\013)2451 1334 y Fq(=)52 b(0)22 b Fs(:)528 b Fv(\(III.40\))257 1559 y(The)25 b(second)g(term)f(in)h(the)f(sum)g(in)h(the)f(last)h (line)f(of)h(\(III.37\))g(equals)473 1751 y Fo(\012)520 1832 y Fq(\011)596 1847 y Fm(g)p 596 1872 36 3 v 2 w(;";k)763 1747 y Fo(\014)763 1807 y(\014)824 1832 y Fs(W)916 1847 y Fm(m;n)1045 1832 y Fq([)p Fs(w)p 1072 1848 73 4 v 3 w Fq(])17 b(\010)1259 1847 y Fm(h)p 1259 1859 41 3 v(;";k)1414 1751 y Fo(\013)1514 1832 y Fq(=)52 b Fs(m)p Fq(!)17 b Fs(n)p Fq(!)1873 1691 y Fo(\032)1948 1832 y Fs(A)p Fq(\()p Fs(")p Fq(\))22 b(+)2303 1707 y Fm(m)2263 1737 y Fo(X)2268 1946 y Fm(\026)p Fn(=1)2464 1707 y Fm(m)2423 1737 y Fo(X)2430 1946 y Fm(\027)t Fn(=1)2584 1832 y Fs(B)2658 1847 y Fm(\026\027)2743 1832 y Fq(\()p Fs(")p Fq(\))2865 1691 y Fo(\033)2962 1832 y Fs(;)216 b Fv(\(III.41\))257 2125 y(where)493 2368 y Fs(A)p Fq(\()p Fs(")p Fq(\))83 b(:=)957 2232 y Fo(Z)1012 2458 y Fm(B)1068 2428 y Fk(m)p Fh(+)p Fk(n)p Fh(+1)1065 2479 y(1)1378 2300 y Fs(dX)1518 2264 y Fn(\()p Fm(m;n)p Fn(\))p 1320 2345 439 4 v 1320 2436 a Fr(j)p Fs(X)1437 2407 y Fn(\()p Fm(m;n)p Fn(\))1621 2436 y Fr(j)1649 2407 y Fn(1)p Fm(=)p Fn(2)1785 2368 y Fs(dy)30 b Fr(j)p Fs(f)1990 2383 y Fm(";k)2085 2368 y Fq(\()p Fs(y)t Fq(\))p Fr(j)2241 2327 y Fn(2)3205 2368 y Fv(\(III.42\))p 957 2520 718 4 v 957 2607 a Fs(g)1004 2622 y Fn(1)1043 2607 y Fq(\()p Fs(x)1136 2622 y Fn(1)1176 2607 y Fq(\))17 b Fr(\001)g(\001)g(\001)d Fs(g)1410 2622 y Fm(m)1477 2607 y Fq(\()p Fs(x)1570 2622 y Fm(m)1637 2607 y Fq(\))i Fs(h)1747 2622 y Fn(1)1787 2607 y Fq(\()6 b(~)-55 b Fs(x)1880 2622 y Fn(1)1920 2607 y Fq(\))17 b Fr(\001)g(\001)g(\001)d Fs(h)2163 2622 y Fm(n)2210 2607 y Fq(\()6 b(~)-55 b Fs(x)2303 2622 y Fm(n)2350 2607 y Fq(\))45 b Fs(w)2503 2622 y Fm(m;n)2631 2607 y Fq([)p Fs(!)t Fq(\()p Fs(y)t Fq(\);)32 b Fs(X)2999 2566 y 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Fq(4\))p Fv(,)j(is)f(a)i(smooth)d (characteristic)501 4731 y(function)24 b(of)h Fq([0)p Fs(;)17 b(\032)p Fq(])25 b Fv(\(so)g(that)p 1509 4676 62 4 v 24 w Fs(\037)1570 4754 y Fm(\032)1610 4731 y Fq([)p Fs(H)1718 4746 y Fm(f)1764 4731 y Fq(])i(:=)1949 4647 y Fo(p)p 2049 4647 492 4 v 84 x Fq(1)22 b Fr(\000)g Fs(\037)2280 4746 y Fm(\032)2321 4731 y Fq([)p Fs(H)2429 4746 y Fm(f)2474 4731 y Fq(])2501 4702 y Fn(2)2565 4731 y Fv(is)j(smooth,)e(as)i (well\);)331 4934 y(\(S\))49 b(a)30 b(unitary)e(scaling)h (transformation)f Fs(S)1874 4949 y Fm(\032)1944 4934 y Fv(which)h(maps)f Fp(1)p Fq([)p Fs(H)2620 4949 y Fm(f)2702 4934 y Fs(<)35 b(\032)p Fq(])i Fr(7!)e Fp(1)p Fq([)p Fs(H)3226 4949 y Fm(f)3308 4934 y Fs(<)g Fq(1])501 5055 y Fv(and)25 b Fs(\037)731 5070 y Fm(\032)771 5055 y Fq([)p Fs(H)879 5070 y Fm(f)925 5055 y Fq(])i Fr(7!)h Fs(\037)1168 5070 y Fn(1)1207 5055 y Fq([)p Fs(H)1315 5070 y Fm(f)1361 5055 y Fq(])p Fv(,)d(follo)n(wed)e(by)i(multiplication)d(by)i Fs(\032)2697 5018 y Fl(\000)p Fn(1)2792 5055 y Fv(;)325 5258 y(\(E\))49 b(an)25 b(analytic)g(transformation)e Fs(E)1644 5273 y Fm(\032)1709 5258 y Fv(of)i(the)g(spectral)g (parameter)g Fs(z)32 b Fr(2)c Fs(D)2974 5273 y Fn(1)p Fm(=)p Fn(2)3109 5258 y Fv(in)d Fs(w)p 3212 5274 73 4 v 2 w Fq([)p Fs(z)t Fq(])p Fv(.)p eop %%Page: 29 30 29 29 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(29)257 573 y Ff(\(S\))39 b Fv(In)f(Subsect.)g(III.3)h(we)f(sho)n (w)f(that)h(a)g(small)f(ball)h(in)f Fr(W)2449 588 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(0)2742 573 y Fv(centered)i(at)f Fs(w)p 3228 589 73 4 v 3301 597 a Fn(0)3392 573 y Fq(:=)257 693 y(0)24 b Fr(\010)g Fs(r)j Fr(\010)d Fq(\(0\))728 708 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)1009 693 y Fv(is)j(contained)f(in)h Fq(dom)16 b Fr(R)1907 708 y Fm(\032)1975 693 y Fv(and)27 b(is)f(mapped)h(into)f(itself.)37 b(T)-8 b(o)26 b(this)g(end,)257 814 y(we)g(\002rst)g(introduce)g(the)f Ff(scaling)h(transf)n(ormation)g Fs(S)2181 829 y Fm(\032)2250 814 y Fq(:)k Fr(B)s Fq([)p Fr(F)10 b Fq(])30 b Fr(!)f(B)s Fq([)p Fr(F)10 b Fq(])p Fv(,)26 b(for)g Fs(A)k Fr(2)g(B)s Fq([)p Fr(F)10 b Fq(])p Fv(,)257 934 y(by)1409 1095 y Fs(S)1469 1110 y Fm(\032)1509 1095 y Fq(\()p Fs(A)p Fq(\))52 b(:=)1877 1028 y(1)p 1876 1073 51 4 v 1876 1164 a Fs(\032)1964 1095 y Fq(\000)2025 1110 y Fm(\032)2093 1095 y Fs(A)28 b Fq(\000)2255 1054 y Fl(\003)2255 1120 y Fm(\032)2317 1095 y Fs(;)861 b Fv(\(III.48\))257 1314 y(where)26 b Fq(\000)587 1329 y Fm(\032)652 1314 y Fv(is)e(the)h(unitary)f (dilatation)f(on)i Fr(F)34 b Fv(de\002ned)25 b(by)g Fq(\000)2333 1329 y Fm(\032)2373 1314 y Fq(\012)j(:=)g(\012)23 b Fs(;)i Fv(and)391 1534 y Fq(\000)452 1549 y Fm(\032)509 1534 y Fs(a)560 1492 y Fl(\003)600 1534 y Fq(\()p Fs(k)s Fq(\))17 b(\000)808 1492 y Fl(\003)808 1558 y Fm(\032)900 1534 y Fq(:=)53 b Fs(\032)1106 1492 y Fl(\000)p Fm(d=)p Fn(2)1289 1534 y Fs(a)1340 1492 y Fl(\003)1379 1534 y Fq(\()p Fs(\032)1467 1492 y Fl(\000)p Fn(1)1562 1534 y Fs(k)s Fq(\))22 b Fs(;)135 b Fq(\000)1899 1549 y Fm(\032)1956 1534 y Fs(a)p Fq(\()p Fs(k)s Fq(\))17 b(\000)2215 1492 y Fl(\003)2215 1558 y Fm(\032)2307 1534 y Fq(:=)53 b Fs(\032)2513 1492 y Fl(\000)p Fm(d=)p Fn(2)2696 1534 y Fs(a)p Fq(\()p Fs(\032)2835 1492 y Fl(\000)p Fn(1)2929 1534 y Fs(k)s Fq(\))23 b Fs(;)134 b Fv(\(III.49\))257 1754 y(for)25 b(all)g Fs(k)31 b Fr(2)d Fg(R)765 1717 y Fm(d)811 1754 y Fv(.)j(It)25 b(is)f(easy)h(to)g(check)g (that)f Fq(\000)1841 1769 y Fm(\032)1881 1754 y Fs(H)1962 1769 y Fm(f)2008 1754 y Fq(\000)2069 1717 y Fl(\003)2069 1778 y Fm(\032)2136 1754 y Fq(=)k Fs(\032H)2371 1769 y Fm(f)2416 1754 y Fv(,)d(and)g(hence)463 1984 y Fq(\000)524 1999 y Fm(\032)581 1984 y Fs(\037)642 1999 y Fm(\032)683 1984 y Fq([)p Fs(H)791 1999 y Fm(f)836 1984 y Fq(])17 b(\000)941 1943 y Fl(\003)941 2008 y Fm(\032)1033 1984 y Fq(=)53 b Fs(\037)1223 1999 y Fn(1)1262 1984 y Fq([)p Fs(H)1370 1999 y Fm(f)1416 1984 y Fq(])118 b Fv(and)142 b Fs(S)1907 1999 y Fm(\032)1947 1903 y Fo(\000)1992 1984 y Fs(T)14 b Fq([)p Fs(H)2171 1999 y Fm(f)2217 1984 y Fq(])2244 1903 y Fo(\001)2342 1984 y Fq(=)52 b Fs(\032)2520 1943 y Fl(\000)p Fn(1)2632 1984 y Fs(T)14 b Fq([)p Fs(\032)j(H)2878 1999 y Fm(f)2922 1984 y Fq(])23 b Fs(;)206 b Fv(\(III.50\))257 2204 y(for)25 b(an)o(y)e Fs(T)41 b Fr(2)28 b(T)e Fv(.)31 b(F)o(or)23 b Fs(H)8 b Fq(\()p Fs(w)p 1179 2220 73 4 v 2 w Fq(\))28 b(=)g Fs(T)14 b Fq(\()p Fs(H)1611 2219 y Fm(f)1655 2204 y Fq(\))19 b(+)g Fs(E)25 b Fr(\001)19 b Fp(1)g Fq(+)f Fs(W)c Fv(,)24 b(smoothly)e(projected)i(by)g Fs(\037)3250 2219 y Fm(\032)3290 2204 y Fq([)p Fs(H)3398 2219 y Fm(f)3443 2204 y Fq(])p Fv(,)257 2324 y(we)h(\002nd)g(that)635 2544 y Fs(S)695 2559 y Fm(\032)736 2463 y Fo(\000)781 2544 y Fs(\037)842 2559 y Fm(\032)883 2544 y Fq([)p Fs(H)991 2559 y Fm(f)1036 2544 y Fq(])17 b Fs(H)8 b Fq(\()p Fs(w)p 1207 2560 V 2 w Fq(\))p Fs(\037)1378 2559 y Fm(\032)1418 2544 y Fq([)p Fs(H)1526 2559 y Fm(f)1571 2544 y Fq(])1598 2463 y Fo(\001)3205 2544 y Fv(\(III.51\))718 2727 y Fq(=)83 b Fs(\037)938 2742 y Fn(1)978 2727 y Fq([)p Fs(H)1086 2742 y Fm(f)1131 2727 y Fq(])1175 2616 y Fo(n)1241 2727 y Fs(\032)1291 2686 y Fl(\000)p Fn(1)1386 2727 y Fs(T)14 b Fq([)p Fs(\032)j(H)1632 2742 y Fm(f)1677 2727 y Fq(])50 b(+)f Fs(\032)1929 2686 y Fl(\000)p Fn(1)2041 2727 y Fs(E)28 b Fr(\001)22 b Fp(1)50 b Fq(+)f Fs(S)2482 2742 y Fm(\032)2522 2727 y Fq(\()p Fs(W)14 b Fq(\))2704 2616 y Fo(o)2787 2727 y Fs(\037)2848 2742 y Fn(1)2888 2727 y Fq([)p Fs(H)2996 2742 y Fm(f)3041 2727 y Fq(])22 b Fs(:)257 2974 y Fv(Note)j(that)f Fs(S)710 2989 y Fm(\032)750 2974 y Fq(\()p Fs(H)869 2989 y Fm(f)914 2974 y Fq(\))k(=)f Fs(H)1164 2989 y Fm(f)1209 2974 y Fv(,)e(which)f(means)g(that)g(the)g (operator)h Fs(H)2564 2989 y Fm(f)2637 2974 y Fq(=)i Fs(H)8 b Fq([)p Fs(w)p 2856 2990 V 2928 2998 a Fn(0)2968 2974 y Fq(])24 b Fv(correspond-)257 3095 y(ing)36 b(to)g(the)g(center)h (of)g(the)f(small)f(ball)h(in)g Fr(W)1901 3110 y Fl(\025)p Fn(0)2032 3095 y Fv(mentioned)g(abo)o(v)o(e)f(is)h(a)h Ft(\002xed)f(point)h Fv(of)257 3215 y Fs(S)317 3230 y Fm(\032)358 3215 y Fv(.)k(Further)28 b(note)g(that)g Fs(S)1183 3230 y Fm(\032)1223 3215 y Fq(\()p Fs(E)j Fr(\001)24 b Fp(1)p Fq(\))34 b(=)g Fs(\032)1704 3179 y Fl(\000)p Fn(1)1798 3215 y Fs(E)h Ft(e)n(xpands)28 b Fv(under)g(the)g(scaling)f (map)h(at)g(a)h(rate)257 3335 y Fs(\032)307 3299 y Fl(\000)p Fn(1)402 3335 y Fv(.)38 b(\(T)-8 b(o)27 b(control)g(this)f(e)o (xpansion)g(it)h(is)g(necessary)g(to)g(suitably)f(project)h(the)g (spectral)g(pa-)257 3456 y(rameter)32 b(by)g(means)f(of)h(the)f (analytic)g(transformation)f Fs(E)2316 3471 y Fm(\032)2389 3456 y Fv(alluded)h(to)g(abo)o(v)o(e\).)50 b(The)32 b(k)o(e)o(y)257 3576 y(idea)21 b(of)g(the)f(renormalization)g(group)g(is)h(that)f(the)g (interaction)g(contracts)h(under)f(the)h(scaling)257 3696 y(transformation,)j(and)h(it)f(is)g(instructi)n(v)o(e)f(to)h (check)h(that)g Fs(S)2270 3711 y Fm(\032)2335 3696 y Fv(contracts)f Fs(W)14 b Fv(.)404 3826 y(T)-8 b(o)20 b(this)g(end)h(we)g(remark)g(that)g(the)g(scaling)f(map)g Fs(S)2166 3841 y Fm(\032)2228 3826 y Fv(restricted)g(to)h Fs(H)8 b Fq([)p Fr(W)2940 3779 y Fn(#)2932 3852 y Fl(\025)p Fn(0)3026 3826 y Fq(])28 b Fr(\022)g(B)s Fq([)p Fr(H)3365 3841 y Fn(red)3468 3826 y Fq(])257 3946 y Fv(induces)d(a)g(scaling)f (map)g Fs(s)1205 3961 y Fm(\032)1271 3946 y Fv(on)g Fr(W)1493 3961 y Fl(\025)p Fn(0)1613 3946 y Fv(by)641 4166 y Fs(S)701 4181 y Fm(\032)742 4086 y Fo(\000)787 4166 y Fs(H)8 b Fq(\()p Fs(w)p 914 4182 V 2 w Fq(\))1024 4086 y Fo(\001)1122 4166 y Fq(=:)53 b Fs(H)1367 4086 y Fo(\000)1412 4166 y Fs(s)1458 4181 y Fm(\032)1498 4166 y Fq(\()p Fs(w)p 1536 4182 V 3 w Fq(\))1647 4086 y Fo(\001)1745 4166 y Fq(=:)f Fs(H)1989 4086 y Fo(\000)2035 4166 y Fq(\()p Fs(s)2119 4181 y Fm(\032)2159 4166 y Fq(\()p Fs(w)2267 4181 y Fm(m;n)2396 4166 y Fq(\)\))2472 4181 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(0)2726 4086 y Fo(\001)2794 4166 y Fs(;)384 b Fv(\(III.52\))257 4386 y(and)25 b(it)f(is)h(easy)g (to)f(v)o(erify)g(that,)h(for)g(all)f Fq(\()p Fs(m;)17 b(n)p Fq(\))28 b Fr(2)g Fg(N)2079 4350 y Fn(2)2079 4411 y(0)2125 4386 y Fv(,)541 4623 y Fs(s)587 4638 y Fm(\032)627 4623 y Fq(\()p Fs(w)735 4638 y Fm(m;n)864 4623 y Fq(\))902 4542 y Fo(\002)943 4623 y Fs(r)m(;)17 b(K)1118 4582 y Fn(\()p Fm(m;n)p Fn(\))1302 4542 y Fo(\003)1396 4623 y Fq(=)53 b Fs(\032)1585 4554 y Fk(d)p Fi(\000)p Fh(1)p 1585 4567 111 3 v 1625 4608 a(2)1705 4582 y Fn(\()p Fm(m)p Fn(+)p Fm(n)p Fn(\))p Fl(\000)p Fn(1)2037 4623 y Fs(w)2107 4638 y Fm(m;n)2236 4542 y Fo(\002)2277 4623 y Fs(\032)17 b(r)30 b(;)45 b(\032)17 b(K)2647 4582 y Fn(\()p Fm(m;n)p Fn(\))2831 4542 y Fo(\003)2894 4623 y Fs(:)284 b Fv(\(III.53\))257 4843 y(Inserting)24 b(this)g(into)g(\(III.51\),)h(we)h(obtain)310 5063 y Fs(S)370 5078 y Fm(\032)410 4982 y Fo(\000)455 5063 y Fs(\037)516 5078 y Fm(\032)557 5063 y Fq([)p Fs(H)665 5078 y Fm(f)710 5063 y Fq(])17 b Fs(H)8 b Fq(\()p Fs(w)p 881 5079 73 4 v 2 w Fq(\))p Fs(\037)1052 5078 y Fm(\032)1092 5063 y Fq([)p Fs(H)1200 5078 y Fm(f)1245 5063 y Fq(])1272 4982 y Fo(\001)3205 5063 y Fv(\(III.54\))393 5246 y Fq(=)82 b Fs(\037)612 5261 y Fn(1)652 5246 y Fq([)p Fs(H)760 5261 y Fm(f)805 5246 y Fq(])849 5135 y Fo(n)915 5246 y Fs(\032)965 5204 y Fl(\000)p Fn(1)1060 5246 y Fs(T)14 b Fq([)p Fs(\032)j(H)1306 5261 y Fm(f)1351 5246 y Fq(])50 b Fr(\000)g Fs(\032)1605 5204 y Fl(\000)p Fn(1)1716 5246 y Fs(E)29 b Fr(\001)21 b Fp(1)50 b Fq(+)2151 5151 y Fo(X)2098 5361 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)2365 5246 y Fs(W)2457 5261 y Fm(m;n)2586 5165 y Fo(\002)2627 5246 y Fs(s)2673 5261 y Fm(\032)2714 5246 y Fq(\()p Fs(w)2822 5261 y Fm(m;n)2950 5246 y Fq(\))2988 5165 y Fo(\003)3030 5135 y(o)3113 5246 y Fs(\037)3174 5261 y Fn(1)3213 5246 y Fq([)p Fs(H)3321 5261 y Fm(f)3367 5246 y Fq(])22 b Fs(;)p eop %%Page: 30 31 30 30 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(30)257 573 y(and)27 b(we)f(note)g(that)g(by)g(Theorem)g(III.1,)h (the)f(operator)g(norm)g(of)g Fs(W)2626 588 y Fm(m;n)2755 492 y Fo(\002)2797 573 y Fs(s)2843 588 y Fm(\032)2883 573 y Fq(\()p Fs(w)2991 588 y Fm(m;n)3120 573 y Fq(\))3158 492 y Fo(\003)3225 573 y Fv(is)g(con-)257 693 y(trolled)e(by)411 913 y Fr(k)p Fs(s)507 928 y Fm(\032)547 913 y Fq(\()p Fs(w)655 928 y Fm(m;n)784 913 y Fq(\))p Fr(k)872 928 y Fm(\026)3205 913 y Fv(\(III.55\))495 1142 y Fq(=)84 b Fs(\032)715 1073 y Fk(d)p Fi(\000)p Fh(1)p 715 1086 111 3 v 755 1127 a(2)835 1101 y Fn(\()p Fm(m)p Fn(+)p Fm(n)p Fn(\))p Fl(\000)p Fn(1)1161 1002 y Fo(\022)1251 1007 y(Z)1307 1232 y Fm(B)1363 1202 y Fk(m)p Fh(+)p Fk(n)1360 1253 y Fh(1)1631 1075 y Fs(dK)1772 1039 y Fn(\()p Fm(m;n)p Fn(\))p 1538 1119 513 4 v 1538 1148 a Fo(\014)1538 1208 y(\014)1571 1233 y Fs(K)1661 1204 y Fn(\()p Fm(m;n)p Fn(\))1845 1148 y Fo(\014)1845 1208 y(\014)1878 1174 y Fn(3+2)p Fm(\026)2104 1142 y Fq(sup)2119 1224 y Fm(r)r Fl(2)p Fm(I)2268 1058 y Fo(\014)2268 1117 y(\014)2301 1142 y Fs(w)2371 1157 y Fm(m;n)2500 1142 y Fq([)p Fs(\032)17 b(r)i Fq(;)33 b Fs(\032)17 b(K)2874 1101 y Fn(\()p Fm(m;n)p Fn(\))3058 1142 y Fq(])3085 1058 y Fo(\014)3085 1117 y(\014)3119 1084 y Fn(2)3158 1002 y Fo(\023)3231 1024 y Fn(1)p Fm(=)p Fn(2)495 1469 y Fq(=)84 b Fs(\032)705 1427 y Fn(\(1+)p Fm(\026)p Fn(\)\()p Fm(m)p Fn(+)p Fm(n)p Fn(\))p Fl(\000)p Fn(1)1219 1328 y Fo(\022)1309 1333 y(Z)1364 1559 y Fm(B)1420 1529 y Fk(m)p Fh(+)p Fk(n)1417 1569 y(\032)1689 1401 y Fs(dK)1830 1365 y Fn(\()p Fm(m;n)p Fn(\))p 1595 1446 V 1595 1474 a Fo(\014)1595 1534 y(\014)1628 1559 y Fs(K)1718 1530 y Fn(\()p Fm(m;n)p Fn(\))1902 1474 y Fo(\014)1902 1534 y(\014)1935 1500 y Fn(3+2)p Fm(\026)2194 1469 y Fq(sup)2161 1554 y Fm(r)r Fl(2)p Fn([0)p Fm(;\032)p Fn(])2389 1384 y Fo(\014)2389 1444 y(\014)2422 1469 y Fs(w)2492 1484 y Fm(m;n)2621 1469 y Fq([)p Fs(r)s Fq(;)17 b Fs(K)2829 1427 y Fn(\()p Fm(m;n)p Fn(\))3013 1469 y Fq(])3040 1384 y Fo(\014)3040 1444 y(\014)3073 1410 y Fn(2)3112 1328 y Fo(\023)3186 1350 y Fn(1)p Fm(=)p Fn(2)494 1718 y Fr(\024)84 b Fs(\032)705 1677 y Fn(\(1+)p Fm(\026)p Fn(\)\()p Fm(m)p Fn(+)p Fm(n)p Fn(\))p Fl(\000)p Fn(1)1219 1718 y Fr(k)p Fs(w)1339 1733 y Fm(m;n)1467 1718 y Fr(k)1517 1733 y Fm(\026)1644 1718 y Fr(\024)d Fs(\032)1852 1677 y Fm(\026)p Fn(\()p Fm(m)p Fn(+)p Fm(n)p Fn(\))2130 1718 y Fr(k)p Fs(w)2250 1733 y Fm(m;n)2379 1718 y Fr(k)2429 1733 y Fm(\026)2497 1718 y Fs(;)257 1938 y Fv(for)30 b Fs(m)d Fq(+)e Fs(n)37 b Fr(\025)h Fq(1)p Fv(.)45 b(Since)30 b Fs(\026)37 b(>)g Fq(0)p Fv(,)31 b(this)d(estimate)h(sho)n(ws)g(that)g Fs(S)2561 1953 y Fm(\032)2631 1938 y Fv(contracts)h Fr(k)p Fs(w)3141 1953 y Fm(m;n)3269 1938 y Fr(k)3319 1953 y Fm(\026)3396 1938 y Fv(by)257 2058 y(at)k(least)g(a)h(f)o(actor)f(of)g Fs(\032)1091 2022 y Fm(\026)1183 2058 y Fs(<)45 b Fq(1)p Fv(.)58 b(W)-8 b(e)35 b(sho)n(w)e(in)g(Subsect.)h(III.3)h(belo)n(w)e (that)h(this)f(contrac-)257 2179 y(tion)25 b(actually)f(dominates)g (the)h(modi\002cations)f(the)h(other)g(tw)o(o)g(ingredients)f(of)h(the) g(smooth)257 2299 y(Feshbach)h(map)e(impose)g(on)g(the)h(ar)n(gument)f Fs(w)p 1838 2315 73 4 v 28 w Fv(of)h(the)f(renormalization)g(map)g Fr(R)3126 2314 y Fm(\032)3167 2299 y Fv(.)257 2444 y Ff(\(E\))d Fv(The)f(e)o(xpansion)f(of)h(multiples)e(of)i(the)g (identity)f(operator)h(under)g Fs(S)2731 2459 y Fm(\032)2791 2444 y Fv(suggests)f(to)g Ft(r)l(enor)n(-)257 2565 y(malize)35 b(the)g(spectr)o(al)g(par)o(ameter)h Fv(of)g(the)f(problem.)62 b(Gi)n(v)o(en)34 b Fs(w)p 2488 2581 V 50 w Fr(2)48 b(W)2820 2580 y Fl(\025)p Fn(0)2950 2565 y Fv(with)35 b Fs(E)6 b Fq([)p Fs(z)t Fq(])48 b(:=)257 2685 y Fr(\000)p Fs(w)404 2700 y Fn(0)p Fm(;)p Fn(0)499 2685 y Fq([)p Fs(z)t Fq(;)17 b(0])p Fv(,)25 b(this)f(is)g(accomplished)g(by)h(means)f(of)h(an)g Ff(analytic)f(map)i Fv(de\002ned)f(by)1068 2905 y Fs(E)1140 2920 y Fm(\032)1233 2905 y Fq(:)53 b Fr(U)10 b Fq([)p Fs(w)p 1412 2921 V 3 w Fq(])50 b Fr(!)f Fs(D)1792 2921 y Fn(1)p Fm(=)p Fn(2)1924 2905 y Fs(;)92 b(z)54 b Fr(7!)c Fs(\032)2342 2864 y Fl(\000)p Fn(1)2453 2905 y Fs(E)6 b Fq([)p Fs(z)t Fq(])23 b Fs(;)521 b Fv(\(III.56\))257 3125 y(where)1128 3245 y Fr(U)10 b Fq([)p Fs(w)p 1227 3261 V 2 w Fq(])53 b(:=)1535 3135 y Fo(n)1601 3245 y Fs(z)33 b Fr(2)28 b Fs(D)1854 3261 y Fn(1)p Fm(=)p Fn(2)1989 3131 y Fo(\014)1989 3191 y(\014)1989 3250 y(\014)2047 3245 y Fr(j)p Fs(E)6 b Fq([)p Fs(z)t Fq(])p Fr(j)28 b(\024)2427 3178 y Fs(\032)p 2427 3223 51 4 v 2428 3314 a Fq(2)2509 3135 y Fo(o)2598 3245 y Fs(:)580 b Fv(\(III.57\))257 3451 y(W)-8 b(e)29 b(no)n(w)e(sho)n(w)g(that)h Fs(E)1099 3466 y Fm(\032)1168 3451 y Fv(is)g(a)g(bijection)f(and)h(that)g Fr(U)10 b Fq([)p Fs(w)p 2161 3467 73 4 v 3 w Fq(])28 b Fv(is)g(close)g(to)g(the)g(disc)g(of)g(radius)g Fs(\032)257 3571 y Fv(centered)e(at)f Fq(0)p Fv(,)f(pro)o(vided)g Fs(w)p 1193 3587 V 27 w Fv(is)g(close)h(to)f Fs(w)p 1713 3587 V 1786 3595 a Fn(0)1825 3571 y Fv(.)31 b(T)-8 b(o)25 b(this)f(end,)g(we)h(de\002ne)h(a)f(polydisc)401 3818 y Fr(D)s Fq(\()p Fs(";)17 b(\016)t Fq(\))81 b(:=)961 3707 y Fo(n)1028 3818 y Fs(w)p 1028 3834 V 30 w Fr(2)28 b(W)1320 3833 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(0)1600 3703 y Fo(\014)1600 3763 y(\014)1600 3823 y(\014)1720 3818 y Fq(sup)1675 3899 y Fm(z)s Fl(2)p Fm(D)1816 3913 y Fh(1)p Fk(=)p Fh(2)1929 3733 y Fo(\015)1929 3793 y(\015)1984 3818 y Fs(T)14 b Fq([)p Fs(z)t Fq(;)j Fs(r)s Fq(])22 b Fr(\000)h Fs(r)s Fr(k)2468 3833 y Fl(T)2555 3818 y Fr(\024)28 b Fs(")22 b(;)450 b Fv(\(III.58\))1007 4084 y Fq(sup)961 4165 y Fm(z)s Fl(2)p Fm(D)1102 4179 y Fh(1)p Fk(=)p Fh(2)1216 4084 y Fr(j)p Fs(E)6 b Fq([)p Fs(z)t Fq(])22 b Fr(\000)h Fs(z)t Fr(j)28 b(\024)g Fs(\016)e(;)157 b Fq(sup)1964 4165 y Fm(z)s Fl(2)p Fm(D)2105 4179 y Fh(1)p Fk(=)p Fh(2)2219 3999 y Fo(\015)2219 4059 y(\015)2274 4084 y Fq(\()p Fs(w)2382 4099 y Fm(m;n)2511 4084 y Fq([)p Fs(z)t Fq(]\))2652 4099 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)2907 4084 y Fr(k)2957 4037 y Fn(#)2957 4112 y Fm(\026;\030)3084 4084 y Fr(\024)28 b Fs(\016)3236 3973 y Fo(o)3325 4084 y Fs(;)257 4379 y Fv(in)d Fr(W)458 4394 y Fl(\025)p Fn(0)553 4379 y Fv(,)f(for)h(all)g Fs(";)17 b(\016)31 b(>)c Fq(0)p Fv(.)k(Note)25 b(that)719 4517 y Fo(n)785 4628 y Fs(w)p 785 4644 V 30 w Fr(2)j(W)1077 4643 y Fl(\025)p Fn(0)1200 4513 y Fo(\014)1200 4573 y(\014)1200 4633 y(\014)1233 4628 y Fr(k)p Fs(w)p 1283 4644 V 24 w Fr(\000)23 b Fs(w)p 1477 4644 V 1550 4652 a Fn(0)1589 4628 y Fr(k)1639 4643 y Fm(\026;\030)1766 4628 y Fr(\024)28 b Fs(")1917 4517 y Fo(o)2036 4628 y Fr(\022)53 b(D)s Fq(\()p Fs(";)17 b(")p Fq(\))747 b Fv(\(III.59\))1511 4840 y Fr(\022)1671 4730 y Fo(n)1738 4840 y Fs(w)p 1738 4856 V 30 w Fr(2)28 b(W)2030 4855 y Fl(\025)p Fn(0)2152 4726 y Fo(\014)2152 4786 y(\014)2152 4845 y(\014)2185 4840 y Fr(k)p Fs(w)p 2235 4856 V 25 w Fr(\000)22 b Fs(w)p 2429 4856 V 2502 4864 a Fn(0)2541 4840 y Fr(k)2591 4855 y Fm(\026;\030)2718 4840 y Fr(\024)29 b Fq(3)p Fs(")2919 4730 y Fo(o)3007 4840 y Fs(;)257 5088 y Fv(is)19 b(comparable)h(to)f(the)g(ball)h(of)f (radius)g Fs(")h Fv(about)f Fs(w)p 1908 5104 V 1980 5112 a Fn(0)2020 5088 y Fq([)p Fs(z)t Fq(])28 b(:=)g Fr(\000)p Fs(z)7 b Fr(\010)s Fs(r)e Fr(\010)s Fq(\(0\))2745 5103 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)3020 5088 y Fv(in)19 b Fr(W)3215 5103 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(0)3470 5088 y Fv(,)257 5208 y(i.e.,)25 b Fs(H)8 b Fq(\()p Fs(w)p 556 5224 V 2 w Fq(\))25 b Fv(is)f(close)h(to)1331 5329 y Fs(H)8 b Fq(\()p Fs(w)p 1458 5344 V 1529 5353 a Fn(0)1569 5329 y Fq([)p Fs(z)t Fq(]\))53 b(=)g Fs(H)1973 5344 y Fm(f)2068 5329 y Fr(\000)d Fs(z)27 b Fr(\001)22 b Fp(1)g Fs(:)783 b Fv(\(III.60\))p eop %%Page: 31 32 31 31 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(31)257 573 y Ff(Lemma)26 b(III.4.)40 b Ft(F)l(ix)24 b Fq(0)k Fs(<)f(\032)h Fr(\024)h Fq(1)p Fs(=)p Fq(2)p Ft(,)24 b Fs(\026)j(>)h Fq(0)p Ft(,)c(and)g Fq(0)k Fs(<)f(\030)33 b(<)27 b Fq(1)p Ft(.)k(Then)1342 772 y Fs(D)1423 788 y Fn(3)p Fm(\032=)p Fn(8)1622 772 y Fr(\022)53 b(U)10 b Fq([)p Fs(w)p 1851 788 73 4 v 3 w Fq(])53 b Fr(\022)g Fs(D)2215 788 y Fn(5)p Fm(\032=)p Fn(8)2383 772 y Fs(;)795 b Fv(\(III.61\))257 972 y Ft(for)25 b(all)f Fs(w)p 529 988 V 30 w Fr(2)k(D)s Fq(\()p Fs(\032=)p Fq(8)p Fs(;)17 b(\032=)p Fq(8\))p Ft(,)24 b(and)1381 1087 y Fo(\014)1381 1147 y(\014)1415 1171 y Fs(\032)17 b(@)1533 1186 y Fm(z)1573 1171 y Fs(E)1645 1186 y Fm(\032)1685 1171 y Fq([)p Fs(z)t Fq(])23 b Fr(\000)g Fq(1)1960 1087 y Fo(\014)1960 1147 y(\014)2046 1171 y Fr(\024)53 b Fq(1)p Fs(=)p Fq(5)21 b Fs(;)834 b Fv(\(III.62\))257 1371 y Ft(for)33 b(all)f Fs(z)48 b Fr(2)c(U)10 b Fq([)p Fs(w)p 847 1387 V 3 w Fq(])p Ft(.)56 b(In)33 b(particular)-11 b(,)34 b Fs(E)1670 1386 y Fm(\032)1753 1371 y Fq(:)44 b Fr(U)10 b Fq([)p Fs(w)p 1923 1387 V 3 w Fq(])43 b Fr(!)g Fs(D)2290 1386 y Fn(1)p Fm(=)p Fn(2)2433 1371 y Ft(is)33 b(a)g(bijection,)h(for)f(all) f Fs(w)p 3313 1387 V 46 w Fr(2)257 1491 y(D)s Fq(\()p Fs(\032=)p Fq(8)p Fs(;)17 b(\032=)p Fq(8\))p Ft(.)257 1677 y(Pr)l(oof)o(.)55 b Fv(If)34 b Fs(w)p 660 1693 V 46 w Fr(2)44 b(D)s Fq(\()p Fs(";)17 b(")p Fq(\))32 b Fv(then)h Fq(sup)1562 1701 y Fm(z)s Fl(2)p Fm(D)1703 1715 y Fh(1)p Fk(=)p Fh(2)1820 1677 y Fr(j)p Fs(E)6 b Fq([)p Fs(z)t Fq(])29 b Fr(\000)g Fs(z)t Fr(j)44 b(\024)g Fs(")p Fv(.)57 b(Thus,)35 b(if)e Fs(z)48 b Fr(2)c(U)10 b Fq([)p Fs(w)p 3190 1693 V 3 w Fq(])34 b Fv(then)257 1811 y Fr(j)p Fs(z)t Fr(j)g(\024)517 1767 y Fm(\032)p 517 1788 37 4 v 517 1845 a Fn(2)588 1811 y Fq(+)24 b Fr(j)p Fs(E)6 b Fq([)p Fs(z)t Fq(])25 b Fr(\000)g Fs(z)t Fr(j)34 b(\024)1257 1767 y Fm(\032)p 1257 1788 V 1257 1845 a Fn(2)1327 1811 y Fq(+)25 b Fs(")p Fv(,)j(and)g Fr(j)p Fs(z)t Fr(j)34 b(2)g Fs(D)2019 1826 y Fn(\()p Fm(\032=)p Fn(2\)+)p Fm(")2272 1811 y Fv(.)41 b(Con)l(v)o(ersely)-6 b(,)27 b(if)h Fr(j)p Fs(z)t Fr(j)34 b(2)g Fs(D)3242 1826 y Fn(\()p Fm(\032=)p Fn(2\))p Fl(\000)p Fm(")257 1931 y Fv(then)c Fr(j)p Fs(E)6 b Fq([)p Fs(z)t Fq(])p Fr(j)39 b(\024)f(j)p Fs(E)6 b Fq([)p Fs(z)t Fq(])27 b Fr(\000)g Fs(z)t Fr(j)g Fq(+)f Fr(j)p Fs(z)t Fr(j)38 b(\024)g Fs(\032=)p Fq(2)p Fv(,)32 b(and)e Fs(z)43 b Fr(2)38 b(U)10 b Fq([)p Fs(w)p 2324 1947 73 4 v 3 w Fq(])p Fv(.)48 b(This)29 b(pro)o(v)o(es)g(\(III.61\))i(with)257 2051 y Fs(")d Fq(:=)f Fs(\032=)p Fq(8)p Fv(.)404 2172 y(T)-8 b(o)38 b(sho)n(w)f(that)h Fs(E)1051 2187 y Fm(\032)1130 2172 y Fv(is)g(a)g(bijection,)i(we)f(use)f(Cauchy')-5 b(s)38 b(estimate)f(with)h(contour)g(on)257 2292 y Fs(@)5 b(D)394 2308 y Fn(1)p Fm(=)p Fn(2)533 2292 y Fq(=)28 b Fr(fj)p Fs(z)t Fr(j)f Fq(=)h(1)p Fs(=)p Fq(2)p Fr(g)p Fv(.)g(Since)19 b Fr(U)10 b Fq([)p Fs(w)p 1512 2308 V 3 w Fq(])27 b Fr(\022)i Fs(D)1826 2308 y Fn(\()p Fm(\032=)p Fn(2\)+)p Fm(")2079 2292 y Fv(,)20 b(we)f(ha)n(v)o(e)f(that)h Fq(dist)2789 2211 y Fo(\000)2834 2292 y Fr(U)10 b Fq([)p Fs(w)p 2933 2308 V 3 w Fq(])17 b Fs(;)33 b(@)5 b(D)3247 2308 y Fn(1)p Fm(=)p Fn(2)3358 2211 y Fo(\001)3431 2292 y Fr(\025)267 2373 y Fn(1)p 267 2390 36 4 v 267 2447 a(2)313 2413 y Fq(\(1)24 b Fr(\000)i Fs(\032)p Fq(\))f Fr(\000)g Fs(")p Fv(.)42 b(Thus,)29 b(we)g(obtain)f(the)g(follo)n(wing)f(estimate)h(on)g (the)g(comple)o(x)g(deri)n(v)n(ati)n(v)o(e)257 2533 y(of)d Fs(E)437 2548 y Fm(\032)478 2533 y Fv(,)620 2695 y Fo(\014)620 2755 y(\014)654 2780 y Fs(\032)17 b(@)772 2795 y Fm(z)812 2780 y Fs(E)884 2795 y Fm(\032)924 2780 y Fq([)p Fs(z)t Fq(])23 b Fr(\000)g Fq(1)1199 2695 y Fo(\014)1199 2755 y(\014)1285 2780 y Fq(=)1413 2695 y Fo(\014)1413 2755 y(\014)1446 2780 y Fs(@)1497 2795 y Fm(z)1537 2780 y Fs(E)6 b Fq([)p Fs(z)t Fq(])24 b Fr(\000)e Fq(1)1890 2695 y Fo(\014)1890 2755 y(\014)1976 2780 y Fr(\024)2279 2712 y Fq(2)17 b Fs(")p 2116 2757 438 4 v 2116 2848 a Fq(1)22 b Fr(\000)g Fs(\032)h Fr(\000)f Fq(2)p Fs(")2615 2780 y(<)53 b Fq(1)22 b Fs(;)363 b Fv(\(III.63\))257 3030 y(pro)o(vided)18 b Fs(")28 b Fq(:=)f Fs(\032=)p Fq(8)19 b Fv(and)g Fs(\032)28 b(<)g Fq(2)p Fs(=)p Fq(3)p Fv(,)20 b(which)e(pro)o(v)o(es)g(\(III.62\))i(and)f(hence)g(that)g Fs(E)3046 3045 y Fm(\032)3114 3030 y Fq(:)28 b Fr(U)10 b Fq([)p Fs(w)p 3268 3046 73 4 v 3 w Fq(])28 b Fr(!)257 3150 y Fs(D)338 3166 y Fn(1)p Fm(=)p Fn(2)473 3150 y Fv(is)d(in)l(v)o(ertible.)2450 b Fd(\003)257 3292 y Ff(\(F\))37 b Fv(W)-8 b(e)37 b(no)n(w)f(describe)h(the)g(construction)e(of)i(the)f (third)g(element)h(of)f Fr(R)2910 3307 y Fm(\032)2951 3292 y Fv(,)k(the)c Ft(decima-)257 3413 y(tion)d(of)h(de)l(gr)l(ees)g (of)g(fr)l(eedom)g Fv(by)f(the)h(smooth)f(Feshbach)h(map.)58 b(In)34 b(order)g(to)g(apply)f(the)257 3533 y(smooth)c(Feshbach)i(map)e (as)i(indicated)e(abo)o(v)o(e,)h(we)h(need)f(to)g(v)o(erify)-6 b(,)30 b(for)h(a)f(gi)n(v)o(en)f Fs(w)p 3217 3549 V 2 w Fv(,)j(that)257 3654 y Fq(\()p Fs(H)8 b Fq(\()p Fs(w)p 422 3670 V 2 w Fq([)p Fs(z)t Fq(]\))p Fs(;)17 b(W)771 3669 y Fn(0)p Fm(;)p Fn(0)866 3654 y Fq([)p Fs(w)p 893 3670 V 3 w Fq([)p Fs(z)t Fq(]]\))37 b Fv(de\002nes)g(a)g(Feshbach)g (pair)g(corresponding)e(to)h Fs(\037)2957 3669 y Fm(\032)2998 3654 y Fq([)p Fs(H)3106 3669 y Fm(f)3151 3654 y Fq(])p Fv(,)k(for)d(all)257 3774 y Fs(z)c Fr(2)28 b(U)10 b Fq([)p Fs(w)p 528 3790 V 3 w Fq(])p Fv(.)257 3960 y Ff(Lemma)33 b(III.5.)44 b Ft(F)l(ix)32 b Fq(0)41 b Fs(<)f(\032)i(<)f Fq(1)p Fs(=)p Fq(2)p Ft(,)33 b Fs(\026)40 b(>)h Fq(0)p Ft(,)34 b(and)d Fq(0)41 b Fs(<)g(\030)k(<)c Fq(1)p Ft(.)52 b(Then)33 b(it)e(follows)g(that)257 4080 y Fq(\()p Fs(H)8 b Fq(\()p Fs(w)p 422 4096 V 2 w Fq([)p Fs(z)t Fq(]\))p Fs(;)17 b(W)771 4095 y Fn(0)p Fm(;)p Fn(0)866 4080 y Fq([)p Fs(w)p 893 4096 V 3 w Fq([)p Fs(z)t Fq(]]\))34 b Ft(de\002nes)g(a)g(F)-7 b(eshbac)o(h)32 b(pair)h(corr)l(esponding)f (to)h Fs(\037)2958 4095 y Fm(\032)2999 4080 y Fq([)p Fs(H)3107 4095 y Fm(f)3152 4080 y Fq(])p Ft(,)j(for)d(all)257 4201 y Fs(w)p 257 4217 V 30 w Fr(2)c(D)s Fq(\()p Fs(\032=)p Fq(8)p Fs(;)17 b(\032=)p Fq(8\))23 b Ft(and)i(all)f Fs(z)32 b Fr(2)c(U)10 b Fq([)p Fs(w)p 1546 4217 V 3 w Fq(])p Ft(.)257 4386 y(Pr)l(oof)o(.)28 b Fv(W)-8 b(e)24 b(remark)f(that)f Fs(H)8 b Fq(\()p Fs(w)p 1291 4402 V 2 w Fq([)p Fs(z)t Fq(]\))24 b Fv(and)f Fs(W)1787 4401 y Fn(0)p Fm(;)p Fn(0)1881 4386 y Fq([)p Fs(w)p 1908 4402 V 3 w Fq([)p Fs(z)t Fq(]])h Fv(de\002ne)f(bounded)g(operators)f(on)h Fr(H)3368 4401 y Fn(red)3470 4386 y Fv(,)257 4507 y(and)31 b(we)g(only)f(need)g(to)h (check)g(the)f(in)l(v)o(ertibility)e(of)i Fs(T)14 b Fq([)p Fs(z)t Fq(;)j Fs(H)2410 4522 y Fm(f)2456 4507 y Fq(])26 b Fr(\000)h Fs(E)6 b Fq([)p Fs(z)t Fq(])40 b(=)e Fs(W)3040 4522 y Fn(0)p Fm(;)p Fn(0)3134 4507 y Fq([)p Fs(w)p 3161 4523 V 3 w Fq([)p Fs(z)t Fq(]])32 b Fv(on)257 4627 y Fq(Ran)p 449 4572 62 4 v 17 w Fs(\037)510 4651 y Fm(\032)550 4627 y Fq([)p Fs(H)658 4642 y Fm(f)703 4627 y Fq(])p Fv(,)26 b(which)e(follo)n(ws)f(if)i(we)h(can)f(sho)n(w)f(that)g Fs(T)14 b Fq([)p Fs(z)t Fq(;)j Fs(r)s Fq(])23 b Fr(\000)f Fs(E)6 b Fq([)p Fs(z)t Fq(])26 b Fv(is)f(a)o(w)o(ay)g(from)f Fq(0)h Fv(for)257 4748 y(all)g Fs(r)30 b Fr(2)e Fq([3)p Fs(\032=)p Fq(4)p Fs(;)17 b Fq(1])p Fv(.)30 b(Indeed,)25 b(we)g(ha)n(v)o(e)610 4947 y Fr(j)p Fs(T)14 b Fq([)p Fs(z)t Fq(;)j Fs(r)s Fq(])22 b Fr(\000)h Fs(E)6 b Fq([)p Fs(z)t Fq(])p Fr(j)83 b(\025)h Fs(r)41 b Fr(\000)f(j)p Fs(T)14 b Fq([)p Fs(z)t Fq(;)j Fs(r)s Fq(])k Fr(\000)i Fs(r)s Fr(j)38 b(\000)h(j)p Fs(E)6 b Fq([)p Fs(z)t Fq(])p Fr(j)1317 5092 y(\025)84 b Fs(r)1525 5012 y Fo(\000)1570 5092 y Fq(1)39 b Fr(\000)g Fq(sup)1803 5172 y Fm(z)s(;r)1937 5092 y Fr(j)p Fs(T)2036 5051 y Fl(0)2059 5092 y Fq([)p Fs(z)t Fq(;)17 b Fs(r)s Fq(])23 b Fr(\000)f Fq(1)p Fr(j)2452 5012 y Fo(\001)2542 5092 y Fr(\000)83 b Fq(sup)2663 5178 y Fm(z)s Fl(2U)7 b Fn([)p Fm(w)p 2817 5190 53 3 v 1 w Fn(])2905 5092 y Fr(j)p Fs(E)f Fq([)p Fs(z)t Fq(])p Fr(j)1317 5370 y(\025)1488 5302 y Fq(3)17 b Fs(\032)p 1488 5347 116 4 v 1521 5438 a Fq(4)1614 5370 y(\(1)k Fr(\000)i Fs(\032=)p Fq(8\))44 b Fr(\000)2184 5302 y Fs(\032)p 2184 5347 51 4 v 2185 5438 a Fq(2)2297 5370 y Fr(\025)2437 5302 y Fq(5)17 b Fs(\032)p 2437 5347 116 4 v 2446 5438 a Fq(32)2615 5370 y Fr(\025)2755 5302 y Fs(\032)p 2755 5347 51 4 v 2756 5438 a Fq(8)2837 5370 y Fs(:)341 b Fv(\(III.64\))p eop %%Page: 32 33 32 32 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(32)3418 573 y Fd(\003)404 712 y Fv(Ha)n(ving)18 b(described)i(the)f(smooth)e(Feshbach)j(map)f Fs(F)2219 728 y Fm(\037)2263 736 y Fk(\032)2299 728 y Fn([)p Fm(H)2377 740 y Fk(f)2416 728 y Fn(])2440 712 y Fq(\()p Fs(H)8 b Fq(\()p Fs(w)p 2605 728 73 4 v 2 w Fq([)p Fs(z)t Fq(]\))p Fs(;)17 b(W)2954 727 y Fn(0)p Fm(;)p Fn(0)3049 712 y Fq([)p Fs(w)p 3076 728 V 2 w Fq([)p Fs(z)t Fq(]]\))k Fv(cor)n(-)257 832 y(responding)g(to)f(the)i(Feshbach)f(pair)h Fq(\()p Fs(H)8 b Fq(\()p Fs(w)p 1703 848 V 1 w Fq([)p Fs(z)t Fq(]\))p Fs(;)17 b(W)2051 847 y Fn(0)p Fm(;)p Fn(0)2146 832 y Fq([)p Fs(w)p 2173 848 V 3 w Fq([)p Fs(z)t Fq(]]\))p Fv(,)23 b(the)e(scaling)g(transformation)257 953 y Fs(S)317 968 y Fm(\032)358 953 y Fv(,)26 b(and)g(the)f(analytic)h (transformation)e Fs(E)1752 968 y Fm(\032)1819 953 y Fv(of)i(the)g(spectral)f(parameter)l(,)i(we)f(are)h(prepared)257 1073 y(to)e(de\002ne)g(the)g Ff(r)n(enormalization)g(transf)n(ormation) g Fr(R)2238 1088 y Fm(\032)2279 1073 y Fv(.)404 1194 y(F)o(or)f Fs(w)p 565 1210 V 31 w Fr(2)k(D)s Fq(\()p Fs(\032=)p Fq(8)p Fs(;)17 b(\032=)p Fq(8\))23 b Fv(and)i Fs(\020)35 b Fr(2)28 b Fs(D)1701 1209 y Fn(1)p Fm(=)p Fn(2)1811 1194 y Fv(,)d(we)g(set)425 1405 y Fr(R)509 1364 y Fm(H)509 1430 y(\032)576 1405 y Fq(\()p Fs(w)p 614 1421 V 3 w Fq(\)[)p Fs(\020)8 b Fq(])51 b(:=)i Fs(S)1097 1420 y Fm(\032)1137 1295 y Fo(\020)1197 1405 y Fs(F)1260 1421 y Fm(\037)1304 1429 y Fk(\032)1340 1421 y Fn([)p Fm(H)1418 1433 y Fk(f)1457 1421 y Fn(])1481 1325 y Fo(\000)1543 1405 y Fs(H)8 b Fq(\()p Fs(w)p 1670 1421 V 2 w Fq([)p Fs(E)1847 1364 y Fl(\000)p Fn(1)1841 1430 y Fm(\032)1941 1405 y Fq(\()p Fs(\020)g Fq(\)]\))p Fs(;)41 b(W)2293 1420 y Fn(0)p Fm(;)p Fn(0)2387 1405 y Fq([)p Fs(w)p 2414 1421 V 3 w Fq([)p Fs(E)2592 1364 y Fl(\000)p Fn(1)2586 1430 y Fm(\032)2686 1405 y Fq(\()p Fs(\020)8 b Fq(\)]])2884 1325 y Fo(\001)2929 1295 y(\021)3011 1405 y Fs(:)167 b Fv(\(III.65\))257 1610 y(Eq.)19 b(\(III.65\))g(requires)g(some)f (comment.)28 b(W)-8 b(e)19 b(assume)f(that)h Fs(\020)35 b Fr(2)28 b Fs(D)2598 1626 y Fn(1)p Fm(=)p Fn(2)2727 1610 y Fv(and)18 b Fs(w)p 2889 1626 V 30 w Fr(2)29 b(D)s Fq(\()p Fs(\032=)p Fq(8)p Fs(;)17 b(\032=)p Fq(8\))257 1730 y Fv(which,)j(by)e(Lemma)h(III.4,)h(implies)d(that)i Fs(z)32 b Fq(:=)c Fs(E)1974 1694 y Fl(\000)p Fn(1)1968 1755 y Fm(\032)2068 1730 y Fq(\()p Fs(\020)8 b Fq(\))27 b Fr(2)h(U)10 b Fq([)p Fs(w)p 2415 1746 V 3 w Fq(])p Fv(.)29 b(Furthermore,)20 b(Lemma)e(III.5)257 1861 y(then)24 b(insures)g(that)934 1780 y Fo(\000)979 1861 y Fs(H)8 b Fq(\()p Fs(w)p 1106 1877 V 2 w Fq([)p Fs(z)t Fq(]\))21 b Fr(\000)g Fs(z)t(;)c(W)1623 1876 y Fn(0)p Fm(;)p Fn(0)1718 1861 y Fq([)p Fs(w)p 1745 1877 V 2 w Fq([)p Fs(z)t Fq(]])k Fr(\000)g Fs(z)2115 1780 y Fo(\001)2186 1861 y Fv(is)j(a)g(Feshbach)h (pair)f(corresponding)257 1981 y(to)h Fs(\037)421 1996 y Fm(\032)461 1981 y Fq([)p Fs(H)569 1996 y Fm(f)615 1981 y Fq(])p Fv(,)g(and)g(hence)g(the)g(smooth)f(Feshbach)h(map)g Fs(F)2245 1997 y Fm(\037)2289 2005 y Fk(\032)2324 1997 y Fn([)p Fm(H)2402 2009 y Fk(f)2442 1997 y Fn(])2466 1981 y Fq(\()p Fs(H)8 b Fq(\()p Fs(w)p 2631 1997 V 1 w Fq([)p Fs(z)t Fq(]\))p Fs(;)17 b(W)2979 1996 y Fn(0)p Fm(;)p Fn(0)3074 1981 y Fq([)p Fs(w)p 3101 1997 V 3 w Fq([)p Fs(z)t Fq(]]\))26 b Fv(de-)257 2102 y(\002nes)c(an)f(operator)g (on)g Fq(Ran)16 b Fs(\037)1308 2117 y Fm(\032)1349 2102 y Fq([)p Fs(H)1457 2117 y Fm(f)1502 2102 y Fq(])28 b Fr(\032)g Fq(Ran)16 b Fp(1)p Fq([)p Fs(H)2017 2117 y Fm(f)2090 2102 y Fs(<)28 b(\032)p Fq(])p Fv(.)i(The)21 b(application)f(of)h(the)g(scaling)257 2222 y(map)31 b Fs(S)520 2237 y Fm(\032)592 2222 y Fv(to)g Fs(F)764 2238 y Fm(\037)808 2246 y Fk(\032)843 2238 y Fn([)p Fm(H)921 2250 y Fk(f)961 2238 y Fn(])985 2222 y Fq(\()p Fs(H)8 b Fq(\()p Fs(w)p 1150 2238 V 1 w Fq([)p Fs(z)t Fq(]\))p Fs(;)17 b(W)1498 2237 y Fn(0)p Fm(;)p Fn(0)1593 2222 y Fq([)p Fs(w)p 1620 2238 V 3 w Fq([)p Fs(z)t Fq(]]\))32 b Fv(yields)e(a)i(bounded)e(operator)i Fr(R)3060 2186 y Fm(H)3060 2247 y(\032)3127 2222 y Fq(\()p Fs(w)p 3165 2238 V 3 w Fq(\)[)p Fs(\020)8 b Fq(])38 b(=)257 2343 y Fs(S)317 2358 y Fm(\032)358 2343 y Fq(\()p Fs(F)459 2358 y Fm(\037)503 2366 y Fk(\032)539 2358 y Fn([)p Fm(H)617 2370 y Fk(f)656 2358 y Fn(])680 2343 y Fq(\()p Fs(H)8 b Fq(\()p Fs(w)p 845 2358 V 1 w Fq([)p Fs(z)t Fq(]\))p Fs(;)17 b(W)1193 2358 y Fn(0)p Fm(;)p Fn(0)1288 2343 y Fq([)p Fs(w)p 1315 2358 V 3 w Fq([)p Fs(z)t Fq(]]\)\))26 b Fv(on)e Fr(H)1828 2358 y Fn(red)1931 2343 y Fv(.)404 2463 y(In)e(the)h(ne)o(xt)f(subsection)f(we)i(demonstrate)f(that)g Fs(S)2171 2478 y Fm(\032)2234 2463 y Fv(actually)g(maps)g Fs(F)2868 2478 y Fm(\037)2912 2486 y Fk(\032)2948 2478 y Fn([)p Fm(H)3026 2490 y Fk(f)3066 2478 y Fn(])3089 2463 y Fq(\()p Fs(H)8 b Fq(\()p Fs(w)p 3254 2479 V 2 w Fq([)p Fs(z)t Fq(]\))p Fs(;)257 2601 y(W)349 2616 y Fn(0)p Fm(;)p Fn(0)444 2601 y Fq([)p Fs(w)p 471 2617 V 2 w Fq([)p Fs(z)t Fq(]]\))31 b Fv(into)e(the)g(subspace)h Fs(H)8 b Fq([)p Fr(W)1690 2554 y Fn(#)1682 2627 y Fl(\025)p Fn(0)1776 2601 y Fq(])37 b Fr(\022)g(B)s Fq([)p Fr(H)2133 2616 y Fn(red)2236 2601 y Fq(])30 b Fv(by)f(computing)48 b Fq(^)-69 b Fs(w)p 2878 2617 V 39 w Fr(2)37 b(W)3188 2616 y Fl(\025)p Fn(0)3313 2601 y Fv(such)257 2722 y(that)31 b Fs(H)8 b Fq(\()20 b(^)-69 b Fs(w)p 565 2738 V 2 w Fq([)p Fs(\020)8 b Fq(]\))40 b(=)g Fr(R)1020 2686 y Fm(H)1020 2746 y(\032)1087 2722 y Fq(\()p Fs(w)p 1125 2738 V 3 w Fq(\)[)p Fs(\020)8 b Fq(])p Fv(.)50 b(By)32 b(Theorem)f(III.3,)54 b Fq(^)-69 b Fs(w)p 2188 2738 V 34 w Fv(is)31 b(the)g(unique)g(element) g(in)g Fr(W)3400 2737 y Fl(\025)p Fn(0)257 2842 y Fv(with)24 b(this)g(property)-6 b(,)24 b(and)h(we)g(may)f(thus)g(set)h Fr(R)1926 2857 y Fm(\032)1967 2842 y Fq(\()p Fs(w)p 2005 2858 V 2 w Fq(\))j(:=)47 b(^)-69 b Fs(w)p 2273 2858 V 3 w Fv(.)404 2963 y(Our)21 b(ne)o(xt)f(goal)g(is)h(to)g(determine)40 b Fq(^)-69 b Fs(w)p 1569 2979 V 30 w Fq(=)28 b Fr(R)1857 2978 y Fm(\032)1897 2963 y Fq(\()p Fs(w)p 1935 2979 V 3 w Fq(\))p Fv(.)h(T)-8 b(o)21 b(this)f(end,)i(we)f(write)g(out)f Fr(R)3175 2926 y Fm(H)3175 2987 y(\032)3243 2963 y Fq(\()p Fs(w)p 3281 2979 V 2 w Fq(\)[)p Fs(\020)8 b Fq(])257 3083 y Fv(e)o(xplicitly)-6 b(.)27 b(According)21 b(to)g(the)g (de\002nition)f(\(II.5\))i(of)f(the)g(smooth)e(Feshbach)j(map,)f(we)h (ha)n(v)o(e)257 3203 y(that)257 3381 y Fs(F)320 3396 y Fm(\037)364 3404 y Fk(\032)400 3396 y Fn([)p Fm(H)478 3408 y Fk(f)518 3396 y Fn(])541 3300 y Fo(\000)587 3381 y Fs(H)8 b Fq(\()p Fs(w)p 714 3397 V 2 w Fq([)p Fs(z)t Fq(]\))p Fs(;)34 b(W)1080 3396 y Fn(0)p Fm(;)p Fn(0)1175 3381 y Fq([)p Fs(w)p 1202 3397 V 2 w Fq([)p Fs(z)t Fq(]])1404 3300 y Fo(\001)1503 3381 y Fq(=)53 b Fs(T)14 b Fq([)p Fs(z)t Fq(;)j Fs(H)1904 3396 y Fm(f)1949 3381 y Fq(])23 b Fr(\000)f Fs(E)6 b Fq([)p Fs(z)t Fq(])51 b(+)f Fs(W)2548 3396 y Fm(\037)2592 3404 y Fk(\032)2628 3396 y Fn([)p Fm(H)2706 3408 y Fk(f)2745 3396 y Fn(])2769 3381 y Fq([)p Fs(z)t Fq(])389 b Fv(\(III.66\))380 3552 y Fr(\000)28 b Fs(\037)546 3567 y Fm(\032)587 3552 y Fq([)p Fs(H)695 3567 y Fm(f)740 3552 y Fq(])17 b Fs(W)d Fq([)p Fs(z)t Fq(])p 1010 3497 62 4 v 17 w Fs(\037)1071 3575 y Fm(\032)1111 3552 y Fq([)p Fs(H)1219 3567 y Fm(f)1265 3552 y Fq(])1292 3471 y Fo(\000)1337 3552 y Fs(T)g Fq([)p Fs(z)t Fq(;)j Fs(H)1609 3567 y Fm(f)1655 3552 y Fq(])22 b Fr(\000)h Fs(E)6 b Fq([)p Fs(z)t Fq(])40 b(+)e Fs(W)p 2231 3530 44 3 v 15 x Fm(\037)2275 3584 y Fk(\032)2311 3567 y Fn([)p Fm(H)2389 3579 y Fk(f)2428 3567 y Fn(])2452 3552 y Fq([)p Fs(z)t Fq(])2555 3471 y Fo(\001)2602 3494 y Fl(\000)p Fn(1)p 2696 3497 62 4 v 2696 3552 a Fs(\037)2757 3575 y Fm(\032)2797 3552 y Fq([)p Fs(H)2905 3567 y Fm(f)2951 3552 y Fq(])17 b Fs(W)d Fq([)p Fs(z)t Fq(])j Fs(\037)3282 3567 y Fm(\032)3322 3552 y Fq([)p Fs(H)3430 3567 y Fm(f)3475 3552 y Fq(])23 b Fs(;)257 3729 y Fv(where)310 3906 y Fs(T)14 b Fq([)p Fs(z)t Fq(;)j Fs(r)s Fq(])22 b Fr(\000)h Fs(E)6 b Fq([)p Fs(z)t Fq(])53 b(:=)g Fs(W)1179 3921 y Fn(0)p Fm(;)p Fn(0)1273 3826 y Fo(\002)1315 3906 y Fs(w)1385 3921 y Fn(0)p Fm(;)p Fn(0)1479 3906 y Fq([)p Fs(z)t Fq(;)17 b Fs(r)s Fq(])1673 3826 y Fo(\003)1715 3906 y Fs(;)64 b(W)14 b Fq([)p Fs(z)t Fq(])53 b(=)2196 3832 y Fo(P)2301 3935 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)2573 3906 y Fs(W)2665 3921 y Fm(m;n)2794 3826 y Fo(\002)2835 3906 y Fs(w)2905 3921 y Fm(m;n)3034 3906 y Fq([)p Fs(z)t Fq(])3137 3826 y Fo(\003)3179 3906 y Fs(;)o Fv(\(III.67\))328 4052 y Fs(W)420 4067 y Fm(\037)464 4075 y Fk(\032)500 4067 y Fn([)p Fm(H)578 4079 y Fk(f)617 4067 y Fn(])641 4052 y Fq([)p Fs(z)t Fq(])h(=)e Fs(\037)987 4067 y Fm(\032)1027 4052 y Fq([)p Fs(H)1135 4067 y Fm(f)1181 4052 y Fq(])p Fs(W)14 b Fq([)p Fs(z)t Fq(])p Fs(\037)1478 4067 y Fm(\032)1519 4052 y Fq([)p Fs(H)1627 4067 y Fm(f)1672 4052 y Fq(])p Fs(;)64 b(W)p 1882 4030 44 3 v 15 x Fm(\037)1926 4084 y Fk(\032)1962 4067 y Fn([)p Fm(H)2040 4079 y Fk(f)2079 4067 y Fn(])2103 4052 y Fq([)p Fs(z)t Fq(])54 b(=)p 2388 3997 62 4 v 52 w Fs(\037)2449 4075 y Fm(\032)2490 4052 y Fq([)p Fs(H)2598 4067 y Fm(f)2643 4052 y Fq(])p Fs(W)14 b Fq([)p Fs(z)t Fq(])p 2879 3997 V Fs(\037)2941 4075 y Fm(\032)2981 4052 y Fq([)p Fs(H)3089 4067 y Fm(f)3134 4052 y Fq(])p Fs(:)j Fv(\(III.68\))257 4229 y(Note)25 b(that)462 4347 y Fo(\015)462 4407 y(\015)p 517 4377 V 25 x Fs(\037)578 4455 y Fm(\032)619 4432 y Fq([)p Fs(H)727 4447 y Fm(f)772 4432 y Fq(])799 4391 y Fn(2)839 4432 y Fr(j)p Fs(T)14 b Fq([)p Fs(z)t Fq(;)j Fs(H)1139 4447 y Fm(f)1184 4432 y Fq(])22 b Fr(\000)h Fs(E)6 b Fq([)p Fs(z)t Fq(])p Fr(j)1542 4391 y Fl(\000)p Fn(1)1637 4347 y Fo(\015)1637 4407 y(\015)1744 4432 y Fr(\024)1885 4364 y Fq(8)p 1885 4409 51 4 v 1885 4500 a Fs(\032)2063 4432 y Fv(and)118 b Fr(k)p Fs(W)14 b Fq([)p Fs(z)t Fq(])p Fr(k)53 b(\024)2827 4364 y Fs(\030)21 b(\032)p 2827 4409 115 4 v 2859 4500 a Fq(8)2973 4432 y Fs(;)205 b Fv(\(III.69\))257 4660 y(because)32 b(of)f Fs(w)p 718 4675 73 4 v 43 w Fr(2)40 b(D)s Fq(\()p Fs(\032=)p Fq(8)p Fs(;)17 b(\032=)p Fq(8\))p Fv(,)32 b(\(III.31\),)h(and)e(\(III.64\).)51 b(Eq.)31 b(\(III.69\))h(implies)e(that)g(the)257 4780 y(Neumann)f(series)h(e)o(xpansion)e(in)h Fs(W)p 1562 4757 48 3 v 15 x Fm(\037)1640 4780 y Fv(of)g(the)g(resolv)o(ent)g(in)g (\(III.66\))h(is)f(norm)g(con)l(v)o(er)n(gent)257 4900 y(and)c(yields)458 5078 y Fs(F)521 5093 y Fm(\037)565 5101 y Fk(\032)601 5093 y Fn([)p Fm(H)679 5105 y Fk(f)719 5093 y Fn(])742 4997 y Fo(\000)788 5078 y Fs(H)8 b Fq(\()p Fs(w)p 915 5093 73 4 v 2 w Fq([)p Fs(z)t Fq(]\))p Fs(;)34 b(W)1281 5093 y Fn(0)p Fm(;)p Fn(0)1376 5078 y Fq([)p Fs(w)1473 5093 y Fn(0)p Fm(;)p Fn(0)1567 5078 y Fq([)p Fs(z)t Fq(]])1697 4997 y Fo(\001)1796 5078 y Fq(=)53 b Fs(T)14 b Fq([)p Fs(z)t Fq(;)j Fs(H)2197 5093 y Fm(f)2242 5078 y Fq(])22 b Fr(\000)h Fs(E)6 b Fq([)p Fs(z)t Fq(])633 b Fv(\(III.70\))652 5311 y Fq(+)809 5187 y Fl(1)772 5217 y Fo(X)775 5428 y Fm(L)p Fn(=1)916 5311 y Fq(\()p Fr(\000)p Fq(1\))1118 5270 y Fm(L)p Fl(\000)p Fn(1)1277 5311 y Fs(\037)1338 5326 y Fm(\032)1379 5311 y Fq([)p Fs(H)1487 5326 y Fm(f)1532 5311 y Fq(])p Fs(W)14 b Fq([)p Fs(z)t Fq(])1768 5171 y Fo(\022)p 2015 5181 62 4 v 2015 5236 a Fs(\037)2077 5259 y Fm(\032)2117 5236 y Fq([)p Fs(H)2225 5251 y Fm(f)2270 5236 y Fq(])2297 5200 y Fn(2)p 1852 5288 649 4 v 1852 5380 a Fs(T)g Fq([)p Fs(z)t Fq(;)j Fs(H)2124 5395 y Fm(f)2169 5380 y Fq(])23 b Fr(\000)f Fs(E)6 b Fq([)p Fs(z)t Fq(])2538 5311 y Fs(W)14 b Fq([)p Fs(z)t Fq(])2747 5171 y Fo(\023)2821 5193 y Fm(L)p Fl(\000)p Fn(1)2963 5311 y Fs(\037)3024 5326 y Fm(\032)3065 5311 y Fq([)p Fs(H)3173 5326 y Fm(f)3218 5311 y Fq(])22 b Fs(:)p eop %%Page: 33 34 33 33 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(33)257 573 y(The)38 b(Neumann)e(series)h(on)g(the)g(right)g(side)f (of)i(\(III.70\))f(is)g(not)g(yet)g(normal-ordered,)j(in)257 693 y(the)26 b(sense)g(that)f(all)h(annihilation)e(operators)i(are)g (mo)o(v)o(ed)f(to)g(the)h(right)f(of)h(functions)f(of)h Fs(H)3450 708 y Fm(f)257 814 y Fv(and)37 b(all)g(creation)g(operators)g (are)h(mo)o(v)o(ed)d(to)i(the)f(left)h(of)g(functions)f(of)h Fs(H)2977 829 y Fm(f)3023 814 y Fv(.)67 b(The)37 b(right)257 934 y(side)25 b(\(III.70\))h(can)f(be)g(normal)g(ordered)g(by)g (systematically)e(using)h(the)h(Pull-Through)f(for)n(-)257 1054 y(mula)29 b(\(III.12\))i(and)e(W)l(ick')-5 b(s)30 b(theorem;)h(see)f(Theorem)g(III.6.)46 b(T)-8 b(o)29 b(formulate)g(it,)i(we)f(intro-)257 1175 y(duce)293 1367 y Fs(W)399 1326 y Fm(m;n)385 1391 y(p;q)528 1286 y Fo(\002)569 1367 y Fs(w)p 569 1383 73 4 v 642 1282 a Fo(\014)642 1342 y(\014)675 1367 y Fs(r)s Fq(;)j Fs(K)872 1326 y Fn(\()p Fm(m;n)p Fn(\))1056 1286 y Fo(\003)1150 1367 y Fq(:=)53 b Fs(P)1369 1382 y Fn(red)1471 1367 y Fr(\001)1706 b Fv(\(III.71\))294 1455 y Fo(Z)349 1681 y Fm(B)405 1648 y Fk(p)p Fh(+)p Fk(q)402 1702 y Fh(1)607 1523 y Fs(dX)747 1487 y Fn(\()p Fm(p;q)r Fn(\))p 549 1568 403 4 v 549 1659 a Fr(j)p Fs(X)666 1630 y Fn(\()p Fm(p;q)r Fn(\))814 1659 y Fr(j)842 1630 y Fn(1)p Fm(=)p Fn(2)989 1591 y Fs(a)1040 1550 y Fl(\003)1080 1591 y Fq(\()p Fs(x)1173 1550 y Fn(\()p Fm(p)p Fn(\))1268 1591 y Fq(\))17 b Fs(w)1393 1606 y Fm(m)p Fn(+)p Fm(p;n)p Fn(+)p Fm(q)1700 1510 y Fo(\002)1742 1591 y Fs(H)1823 1606 y Fm(f)1890 1591 y Fq(+)22 b Fs(r)s Fq(;)33 b Fs(k)2149 1550 y Fn(\()p Fm(m)p Fn(\))2271 1591 y Fs(;)17 b(x)2370 1550 y Fn(\()p Fm(p)p Fn(\))2481 1591 y Fs(;)2544 1565 y Fq(~)2541 1591 y Fs(k)2595 1550 y Fn(\()p Fm(n)p Fn(\))2697 1591 y Fs(;)23 b Fq(~)-55 b Fs(x)2796 1550 y Fn(\()p Fm(q)r Fn(\))2889 1510 y Fo(\003)2947 1591 y Fs(a)p Fq(\()6 b(~)-55 b Fs(x)3091 1550 y Fn(\()p Fm(q)r Fn(\))3185 1591 y Fq(\))22 b Fs(P)3308 1606 y Fn(red)3432 1591 y Fs(;)257 1868 y Fv(for)k Fs(m)d Fq(+)f Fs(n)29 b Fr(\025)f Fq(1)p Fv(,)e(which)e(de\002nes)i(an)f(operator)h (for)f(each)h Fs(K)2389 1832 y Fn(\()p Fm(m;n)p Fn(\))2601 1868 y Fr(2)j Fs(B)2775 1827 y Fm(m)p Fn(+)p Fm(n)2770 1892 y Fn(1)2939 1868 y Fv(,)d(a.e.)32 b(Here)26 b(we)257 1988 y(use)d(notation)e(for)h Fs(X)989 1952 y Fn(\()p Fm(p;q)r Fn(\))1137 1988 y Fv(,)h Fs(x)1240 1952 y Fn(\()p Fm(p)p Fn(\))1335 1988 y Fv(,)29 b Fq(~)-55 b Fs(x)1438 1952 y Fn(\()p Fm(q)r Fn(\))1531 1988 y Fv(,)23 b(etc.)30 b(similar)21 b(to)h(the)g(one)h(introduced)e(in)h(Eqs.)g(\(III.4\)\226) 257 2109 y(\(III.8\).)32 b(F)o(or)25 b Fs(m)j Fq(=)f Fs(n)h Fq(=)g(0)p Fv(,)c(we)h(set)437 2301 y Fs(W)529 2316 y Fm(p;q)622 2220 y Fo(\002)663 2301 y Fs(w)p 663 2317 73 4 v 736 2216 a Fo(\014)736 2276 y(\014)769 2301 y Fs(r)816 2220 y Fo(\003)910 2301 y Fq(:=)2192 b Fv(\(III.72\))603 2522 y Fs(P)666 2537 y Fn(red)784 2386 y Fo(Z)840 2612 y Fm(B)896 2579 y Fk(p)p Fh(+)p Fk(q)893 2633 y Fh(1)1097 2455 y Fs(dX)1237 2419 y Fn(\()p Fm(p;q)r Fn(\))p 1040 2499 403 4 v 1040 2590 a Fr(j)p Fs(X)1157 2562 y Fn(\()p Fm(p;q)r Fn(\))1304 2590 y Fr(j)1332 2562 y Fn(1)p Fm(=)p Fn(2)1480 2522 y Fs(a)1531 2481 y Fl(\003)1570 2522 y Fq(\()p Fs(x)1663 2481 y Fn(\()p Fm(p)p Fn(\))1758 2522 y Fq(\))17 b Fs(w)1883 2537 y Fm(p;q)1976 2441 y Fo(\002)2017 2522 y Fs(H)2098 2537 y Fm(f)2166 2522 y Fq(+)22 b Fs(r)s Fq(;)33 b Fs(x)2426 2481 y Fn(\()p Fm(p)p Fn(\))2537 2522 y Fs(;)40 b Fq(~)-55 b Fs(x)2653 2481 y Fn(\()p Fm(q)r Fn(\))2746 2441 y Fo(\003)2804 2522 y Fs(a)p Fq(\()6 b(~)-55 b Fs(x)2948 2481 y Fn(\()p Fm(q)r Fn(\))3042 2522 y Fq(\))22 b Fs(P)3165 2537 y Fn(red)3289 2522 y Fs(:)257 2806 y Ff(Theor)n(em)27 b(III.6)d(\(W)n(ick)h(Ordering\).)42 b Ft(Let)25 b Fs(w)p 1817 2822 73 4 v 30 w Fq(=)j(\()p Fs(w)2129 2821 y Fm(m;n)2257 2806 y Fq(\))2295 2821 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)2577 2806 y Fr(2)g(W)2777 2759 y Fn(#)2769 2833 y Fl(\025)p Fn(1)2864 2806 y Ft(,)d(write)f Fs(W)3235 2821 y Fm(m;n)3392 2806 y Fq(:=)257 2927 y Fs(W)349 2942 y Fm(m;n)478 2927 y Fq([)p Fs(w)575 2942 y Fm(m;n)704 2927 y Fq(])p Ft(,)g Fs(W)41 b Fq(=)1016 2852 y Fo(P)1122 2956 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)1393 2927 y Fs(W)1485 2942 y Fm(m;n)1614 2927 y Ft(,)23 b(and)f Fs(F)1897 2942 y Fn(0)1937 2927 y Fs(;)17 b(:)g(:)g(:)f(;)h(F)2219 2942 y Fm(L)2299 2927 y Fr(2)28 b(W)2491 2942 y Fn(0)p Fm(;)p Fn(0)2585 2927 y Ft(,)c(and)e(denote)h (by)g Fs(S)3272 2942 y Fm(M)3373 2927 y Ft(the)257 3047 y(gr)l(oup)h(of)g(permutations)f(of)h Fs(M)36 b Ft(elements.)30 b(Then)1055 3239 y Fs(F)1118 3254 y Fn(0)1174 3239 y Fs(W)g(F)1359 3254 y Fn(1)1415 3239 y Fs(W)h Fr(\001)17 b(\001)g(\001)e Fs(W)30 b(F)1856 3254 y Fm(L)p Fl(\000)p Fn(1)2015 3239 y Fs(W)g(F)2200 3254 y Fm(L)2305 3239 y Fq(=)52 b Fs(H)8 b Fq([)20 b(~)-69 b Fs(w)p 2549 3255 V 2 w Fq(])23 b Fs(;)507 b Fv(\(III.73\))257 3445 y Ft(wher)l(e)54 b Fq(~)-69 b Fs(w)p 530 3461 V 45 w Fr(2)43 b Fq(\()16 b Fo(e)-71 b Fs(w)865 3394 y Fn(\(sym)n(\))862 3472 y Fm(M)s(;N)1047 3445 y Fq(\))1085 3460 y Fm(M)7 b Fn(+)p Fm(N)g Fl(\025)p Fn(0)1414 3445 y Fr(2)43 b(W)1629 3398 y Fn(#)1621 3471 y Fl(\025)p Fn(0)1749 3445 y Ft(is)32 b(given)g(by)h(the)g(symmetrization)e(w)-7 b(.)32 b(r)-11 b(.)33 b(t.)f Fs(k)3361 3409 y Fn(\()p Fm(M)7 b Fn(\))257 3585 y Ft(and)434 3559 y Fq(~)432 3585 y Fs(k)486 3549 y Fn(\()p Fm(N)g Fn(\))608 3585 y Ft(,)502 3777 y Fo(e)-71 b Fs(w)559 3727 y Fn(\(sym)n(\))556 3804 y Fm(M)s(;N)741 3777 y Fq([)p Fs(r)s Fq(;)33 b Fs(K)965 3736 y Fn(\()p Fm(M)s(;N)7 b Fn(\))1178 3777 y Fq(])2000 b Fv(\(III.74\))569 3982 y Fq(:=)872 3915 y(1)p 765 3959 265 4 v 765 4051 a Fs(M)10 b Fq(!)17 b Fs(N)10 b Fq(!)1083 3888 y Fo(X)1055 4099 y Fm(\031)r Fl(2)p Fm(S)1188 4110 y Fk(M)1295 3888 y Fo(X)1276 4099 y Fn(~)-39 b Fm(\031)r Fl(2)p Fm(S)1405 4110 y Fk(N)1496 3982 y Fo(e)-72 b Fs(w)1549 3997 y Fm(M)s(;N)1707 3982 y Fq([)p Fs(r)s Fq(;)33 b Fs(k)1892 3998 y Fm(\031)r Fn(\(1\))2029 3982 y Fs(;)17 b(:)g(:)g(:)f(;)h(k)2299 3998 y Fm(\031)r Fn(\()p Fm(M)7 b Fn(\))2491 3982 y Fq(;)2554 3956 y(~)2552 3982 y Fs(k)2607 3998 y Fn(~)-39 b Fm(\031)q Fn(\(1\))2740 3982 y Fs(;)17 b(:)g(:)g(:)f(;)2961 3956 y Fq(~)2959 3982 y Fs(k)3014 3998 y Fn(~)-39 b Fm(\031)q Fn(\()p Fm(N)7 b Fn(\))3191 3982 y Fq(])22 b Fs(;)257 4268 y Ft(of)278 4460 y Fo(e)-72 b Fs(w)331 4475 y Fm(M)s(;N)489 4460 y Fq([)p Fs(r)s Fq(;)33 b Fs(K)713 4419 y Fn(\()p Fm(M)s(;N)7 b Fn(\))926 4460 y Fq(])52 b(=)2124 b Fv(\(III.75\))598 4606 y Fo(X)437 4807 y Fk(m)491 4822 y Fh(1)526 4807 y(+)p Fk(:::)p Fh(+)p Fk(m)731 4823 y(L)777 4807 y Fh(=)p Fk(M)r(;)466 4866 y(n)504 4881 y Fh(1)539 4866 y(+)p Fk(:::)o Fh(+)p Fk(n)727 4883 y(L)774 4866 y Fh(=)p Fk(N)1113 4606 y Fo(X)969 4794 y Fk(p)1001 4809 y Fh(1)1035 4794 y Fk(;q)1082 4809 y Fh(1)1117 4794 y Fk(;:::;p)1244 4811 y(L)1290 4794 y(;q)1337 4811 y(L)1384 4794 y Fh(:)939 4855 y Fk(m)993 4872 y(`)1024 4855 y Fh(+)p Fk(p)1103 4872 y(`)1132 4855 y Fh(+)p Fk(n)1217 4872 y(`)1247 4855 y Fh(+)p Fk(q)1322 4872 y(`)1352 4855 y Fi(\025)p Fh(1)1497 4577 y Fm(L)1457 4606 y Fo(Y)1461 4819 y Fm(`)p Fn(=1)1601 4561 y Fo(\032\022)1749 4634 y Fs(m)1834 4649 y Fm(`)1889 4634 y Fq(+)22 b Fs(p)2036 4649 y Fm(`)1868 4769 y Fs(p)1917 4784 y Fm(`)2069 4561 y Fo(\023)q(\022)2216 4634 y Fs(n)2274 4649 y Fm(`)2329 4634 y Fq(+)g Fs(q)2470 4649 y Fm(`)2322 4769 y Fs(q)2365 4784 y Fm(`)2504 4561 y Fo(\023\033)427 5071 y Fs(F)490 5086 y Fn(0)530 5071 y Fq([)p Fs(r)j Fq(+)h(~)-53 b Fs(r)768 5086 y Fn(0)807 5071 y Fq(])851 4931 y Fo(\034)926 5071 y Fq(\012)996 4927 y Fo(\014)996 4986 y(\014)996 5046 y(\014)996 5106 y(\014)1049 5046 y(f)1046 5071 y Fs(W)1138 5086 y Fn(1)1178 4990 y Fo(\002)1219 5071 y Fs(r)25 b Fq(+)d Fs(r)1430 5086 y Fn(1)1469 5071 y Fq(;)34 b Fs(K)1620 5020 y Fn(\()p Fm(m)1709 5029 y Fh(1)1744 5020 y Fm(;n)1807 5029 y Fh(1)1841 5020 y Fn(\))1613 5095 y(1)1873 4990 y Fo(\003)1942 5071 y Fs(F)2005 5086 y Fn(1)2045 5071 y Fq([)p Fs(H)2153 5086 y Fm(f)2220 5071 y Fq(+)22 b Fs(r)j Fq(+)h(~)-53 b Fs(r)2529 5086 y Fn(1)2568 5071 y Fq(])2626 5046 y Fo(f)2623 5071 y Fs(W)2715 5086 y Fn(2)2754 4990 y Fo(\002)2796 5071 y Fs(r)25 b Fq(+)d Fs(r)3007 5086 y Fn(2)3046 5071 y Fq(;)34 b Fs(K)3197 5020 y Fn(\()p Fm(m)3286 5029 y Fh(2)3321 5020 y Fm(;n)3384 5029 y Fh(2)3418 5020 y Fn(\))3190 5095 y(2)3450 4990 y Fo(\003)900 5343 y Fr(\001)17 b(\001)g(\001)e Fs(F)1096 5358 y Fm(L)p Fl(\000)p Fn(1)1238 5343 y Fq([)p Fs(H)1346 5358 y Fm(f)1414 5343 y Fq(+)22 b Fs(r)i Fq(+)j(~)-54 b Fs(r)1722 5358 y Fm(L)p Fl(\000)p Fn(1)1865 5343 y Fq(])1922 5318 y Fo(f)1919 5343 y Fs(W)2011 5358 y Fm(L)2064 5263 y Fo(\002)2105 5343 y Fs(r)25 b Fq(+)d Fs(r)2316 5358 y Fm(L)2368 5343 y Fq(;)33 b Fs(K)2518 5292 y Fn(\()p Fm(m)2607 5303 y Fk(L)2655 5292 y Fm(;n)2718 5303 y Fk(L)2763 5292 y Fn(\))2511 5370 y Fm(L)2795 5263 y Fo(\003)2864 5343 y Fq(\012)2934 5203 y Fo(\035)3009 5343 y Fs(F)3072 5358 y Fm(L)3125 5343 y Fq([)p Fs(r)25 b Fq(+)h(~)-53 b Fs(r)3363 5358 y Fm(L)3415 5343 y Fq(])22 b Fs(;)p eop %%Page: 34 35 34 34 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(34)257 573 y Ft(wher)l(e)740 759 y Fo(f)737 784 y Fs(W)829 799 y Fm(`)862 703 y Fo(\002)903 784 y Fs(r)25 b Fq(+)d Fs(r)1114 799 y Fm(`)1147 784 y Fq(;)33 b Fs(K)1297 733 y Fn(\()p Fm(m)1386 745 y Fk(`)1417 733 y Fm(;n)1480 745 y Fk(`)1510 733 y Fn(\))1290 812 y Fm(`)1541 703 y Fo(\003)1666 784 y Fq(:=)83 b Fs(W)1958 743 y Fm(m)2020 755 y Fk(`)2050 743 y Fm(;n)2113 755 y Fk(`)1944 809 y Fm(p)1980 821 y Fk(`)2010 809 y Fm(;q)2062 821 y Fk(`)2147 784 y Fq([)p Fs(w)p 2174 800 73 4 v 2246 699 a Fo(\014)2246 759 y(\014)2301 784 y Fs(r)25 b Fq(+)d Fs(r)2512 799 y Fm(`)2545 784 y Fq(;)33 b Fs(K)2695 733 y Fn(\()p Fm(m)2784 745 y Fk(`)2815 733 y Fm(;n)2878 745 y Fk(`)2908 733 y Fn(\))2688 812 y Fm(`)2939 784 y Fq(])23 b Fs(;)189 b Fv(\(III.76\))279 1112 y Fs(K)369 1071 y Fn(\()p Fm(M)s(;N)7 b Fn(\))678 1112 y Fq(=)97 b(\()p Fs(K)979 1061 y Fn(\()p Fm(m)1068 1070 y Fh(1)1103 1061 y Fm(;n)1166 1070 y Fh(1)1200 1061 y Fn(\))972 1136 y(1)1232 1112 y Fs(;)17 b(:)g(:)g(:)f(;)h(K)1541 1061 y Fn(\()p Fm(m)1630 1072 y Fk(L)1677 1061 y Fm(;n)1740 1072 y Fk(L)1785 1061 y Fn(\))1534 1139 y Fm(L)1817 1112 y Fq(\))22 b Fs(;)135 b(K)2129 1061 y Fn(\()p Fm(m)2218 1073 y Fk(`)2249 1061 y Fm(;n)2312 1073 y Fk(`)2341 1061 y Fn(\))2122 1140 y Fm(`)2453 1112 y Fq(=)80 b(\()p Fs(k)2701 1061 y Fn(\()p Fm(m)2790 1073 y Fk(`)2821 1061 y Fn(\))2698 1140 y Fm(`)2853 1112 y Fs(;)2899 1086 y Fq(~)2897 1112 y Fs(k)2951 1061 y Fn(\()p Fm(n)3021 1073 y Fk(`)3051 1061 y Fn(\))2948 1140 y Fm(`)3082 1112 y Fq(\))22 b Fs(;)36 b Fv(\(III.77\))505 1276 y Fs(r)549 1291 y Fm(`)665 1276 y Fq(:=)83 b(\006[)951 1249 y(~)948 1276 y Fs(k)1002 1225 y Fn(\()p Fm(n)1072 1234 y Fh(1)1107 1225 y Fn(\))999 1300 y(1)1139 1276 y Fq(])22 b(+)g Fs(:)17 b(:)g(:)22 b Fq(+)g(\006[)1621 1249 y(~)1618 1276 y Fs(k)1672 1223 y Fn(\()p Fm(n)1742 1235 y Fk(`)p Fi(\000)p Fh(1)1851 1223 y Fn(\))1669 1303 y Fm(`)p Fl(\000)p Fn(1)1883 1276 y Fq(])38 b(+)h(\006[)p Fs(k)2214 1223 y Fn(\()p Fm(m)2303 1235 y Fk(`)p Fh(+1)2412 1223 y Fn(\))2211 1303 y Fm(`)p Fn(+1)2443 1276 y Fq(])22 b(+)g Fs(:)17 b(:)g(:)22 b Fq(+)g(\006[)p Fs(k)2976 1225 y Fn(\()p Fm(m)3065 1236 y Fk(L)3113 1225 y Fn(\))2973 1303 y Fm(L)3144 1276 y Fq(])h Fs(;)-16 b Fv(\(III.78\))509 1446 y Fq(~)-53 b Fs(r)549 1461 y Fm(`)665 1446 y Fq(:=)83 b(\006[)951 1420 y(~)948 1446 y Fs(k)1002 1395 y Fn(\()p Fm(n)1072 1404 y Fh(1)1107 1395 y Fn(\))999 1471 y(1)1139 1446 y Fq(])22 b(+)g Fs(:)17 b(:)g(:)22 b Fq(+)g(\006[)1621 1420 y(~)1618 1446 y Fs(k)1672 1395 y Fn(\()p Fm(n)1742 1407 y Fk(`)1773 1395 y Fn(\))1669 1474 y Fm(`)1804 1446 y Fq(])39 b(+)g(\006[)p Fs(k)2136 1393 y Fn(\()p Fm(m)2225 1405 y Fk(`)p Fh(+1)2333 1393 y Fn(\))2133 1474 y Fm(`)p Fn(+1)2365 1446 y Fq(])22 b(+)g Fs(:)17 b(:)g(:)22 b Fq(+)g(\006[)p Fs(k)2898 1395 y Fn(\()p Fm(m)2987 1406 y Fk(L)3034 1395 y Fn(\))2895 1473 y Fm(L)3066 1446 y Fq(])g Fs(:)63 b Fv(\(III.79\))257 1657 y Ft(Pr)l(oof)o(.)29 b Fv(See)d([3,)f(Theorem)f(A.4].)1993 b Fd(\003)404 1778 y Fv(As)18 b(a)h(direct)g(consequence)g(of)g(Theorem)g(III.6)g(and)g (Eqs.)f(\(III.52\)\226\(III.53\))h(and)g(\(III.71\)\226)257 1898 y(\(III.72\),)26 b(we)f(\002nd)g(a)g(sequence)45 b Fq(^)-69 b Fs(w)p 1378 1914 V 27 w Fv(such)25 b(that)f Fs(H)8 b Fq(\()19 b(^)-68 b Fs(w)p 1984 1914 V 2 w Fq([)p Fs(\020)8 b Fq(]\))27 b(=)g Fr(R)2413 1862 y Fm(H)2413 1923 y(\032)2481 1898 y Fq(\()p Fs(w)p 2519 1914 V 2 w Fq(\)[)p Fs(\020)8 b Fq(])257 2018 y(=)28 b Fs(S)421 2033 y Fm(\032)461 2018 y Fq(\()p Fs(F)562 2034 y Fm(\037)606 2042 y Fk(\032)642 2034 y Fn([)p Fm(H)720 2046 y Fk(f)759 2034 y Fn(])783 2018 y Fr(f)17 b Fs(H)8 b Fq(\()p Fs(w)p 977 2034 V 1 w Fq([)p Fs(E)1153 1982 y Fl(\000)p Fn(1)1147 2043 y Fm(\032)1248 2018 y Fq(\()p Fs(\020)g Fq(\)]\))p Fs(;)41 b(W)1600 2033 y Fn(0)p Fm(;)p Fn(0)1694 2018 y Fq([)p Fs(w)p 1721 2034 V 3 w Fq([)p Fs(E)1899 1982 y Fl(\000)p Fn(1)1893 2043 y Fm(\032)1993 2018 y Fq(\()p Fs(\020)8 b Fq(\)]])17 b(\))p Fr(g)23 b Fv(as)i(follo)n(ws.)257 2214 y Ff(Theor)n(em)32 b(III.7.)43 b Ft(Let)30 b Fs(\020)43 b Fr(2)37 b Fs(D)1360 2230 y Fn(1)p Fm(=)p Fn(2)1470 2214 y Ft(,)31 b Fs(w)p 1526 2230 V 39 w Fr(2)37 b(D)s Fq(\()p Fs(\032=)p Fq(8)p Fs(;)17 b(\032=)p Fq(8\))p Ft(,)29 b(and)h(set)f Fs(z)41 b Fq(:=)c Fs(E)2912 2178 y Fl(\000)p Fn(1)2906 2239 y Fm(\032)3006 2214 y Fq(\()p Fs(\020)8 b Fq(\))36 b Fr(2)g(U)10 b Fq([)p Fs(w)p 3370 2230 V 3 w Fq(])p Ft(.)257 2361 y(Then)42 b Fs(H)8 b Fq(\()19 b(^)-68 b Fs(w)p 625 2377 V 2 w Fq([)p Fs(\020)8 b Fq(]\))58 b(=)g Fr(R)1116 2325 y Fm(H)1116 2385 y(\032)1183 2361 y Fq(\()p Fs(w)p 1221 2377 V 3 w Fq(\)[)p Fs(\020)8 b Fq(])41 b Ft(determines)60 b Fq(^)-69 b Fs(w)p 1956 2377 V 61 w Fq(=)58 b(\()20 b(^)-69 b Fs(w)2332 2310 y Fn(\()p Fm(sy)r(m)p Fn(\))2329 2388 y Fm(M)s(;N)2523 2361 y Fq(\))2561 2376 y Fm(M)7 b Fn(+)p Fm(N)g Fl(\025)p Fn(0)2889 2361 y Ft(to)41 b(be)h(the)f(sym-)257 2501 y(metrization)23 b(w)-7 b(.)25 b(r)-11 b(.)25 b(t.)f Fs(k)1054 2465 y Fn(\()p Fm(M)7 b Fn(\))1213 2501 y Ft(and)1390 2475 y Fq(~)1387 2501 y Fs(k)1441 2465 y Fn(\()p Fm(N)g Fn(\))1588 2501 y Ft(\(as)25 b(in)f(Eq.)h(\(III.74\)\))g(of)277 2784 y Fq(^)-69 b Fs(w)327 2799 y Fm(M)s(;N)485 2784 y Fq([)p Fs(\020)8 b Fq(;)32 b Fs(r)s Fq(;)h Fs(K)819 2743 y Fn(\()p Fm(M)s(;N)7 b Fn(\))1032 2784 y Fq(])53 b(=)f Fs(\032)1300 2715 y Fk(d)p Fi(\000)p Fh(1)p 1301 2728 111 3 v 1341 2769 a(2)1421 2743 y Fn(\()p Fm(M)7 b Fn(+)p Fm(N)g Fn(\))p Fl(\000)p Fn(1)1838 2660 y Fl(1)1802 2689 y Fo(X)1805 2901 y Fm(L)p Fn(=1)1946 2784 y Fq(\()p Fr(\000)p Fq(1\))2148 2743 y Fm(L)p Fl(\000)p Fn(1)3224 2784 y Fv(\(III.80\))594 3029 y Fo(X)433 3229 y Fk(m)487 3244 y Fh(1)522 3229 y(+)p Fk(:::)p Fh(+)p Fk(m)727 3246 y(L)773 3229 y Fh(=)p Fk(M)r(;)462 3289 y(n)500 3304 y Fh(1)535 3289 y(+)p Fk(:::)o Fh(+)p Fk(n)723 3306 y(L)770 3289 y Fh(=)p Fk(N)1109 3029 y Fo(X)965 3217 y Fk(p)997 3232 y Fh(1)1031 3217 y Fk(;q)1078 3232 y Fh(1)1113 3217 y Fk(;:::;p)1240 3234 y(L)1286 3217 y(;q)1333 3234 y(L)1380 3217 y Fh(:)935 3278 y Fk(m)989 3295 y(`)1020 3278 y Fh(+)p Fk(p)1099 3295 y(`)1128 3278 y Fh(+)p Fk(n)1213 3295 y(`)1243 3278 y Fh(+)p Fk(q)1318 3295 y(`)1348 3278 y Fi(\025)p Fh(1)1493 2999 y Fm(L)1453 3029 y Fo(Y)1457 3241 y Fm(`)p Fn(=1)1597 2983 y Fo(\032\022)1745 3057 y Fs(m)1830 3072 y Fm(`)1885 3057 y Fq(+)22 b Fs(p)2032 3072 y Fm(`)1864 3192 y Fs(p)1913 3207 y Fm(`)2065 2983 y Fo(\023)q(\022)2212 3057 y Fs(n)2270 3072 y Fm(`)2325 3057 y Fq(+)g Fs(q)2466 3072 y Fm(`)2318 3192 y Fs(q)2361 3207 y Fm(`)2500 2983 y Fo(\023\033)2648 3124 y Fs(\037)2709 3139 y Fn(1)2748 3124 y Fq([)p Fs(r)j Fq(+)i(~)-54 b Fs(r)2986 3139 y Fn(0)3026 3124 y Fq(])17 b Fs(\037)3131 3139 y Fn(1)3170 3124 y Fq([)p Fs(r)25 b Fq(+)h(~)-53 b Fs(r)3408 3139 y Fm(L)3460 3124 y Fq(])423 3356 y Fo(\034)498 3496 y Fq(\012)568 3352 y Fo(\014)568 3411 y(\014)568 3471 y(\014)568 3531 y(\014)621 3471 y(f)618 3496 y Fs(W)710 3511 y Fn(1)750 3415 y Fo(\002)791 3496 y Fs(z)t Fq(;)35 b Fs(\032)p Fq(\()p Fs(r)25 b Fq(+)d Fs(r)1201 3511 y Fn(1)1240 3496 y Fq(\);)33 b Fs(\032K)1478 3445 y Fn(\()p Fm(m)1567 3454 y Fh(1)1603 3445 y Fm(;n)1666 3454 y Fh(1)1700 3445 y Fn(\))1471 3521 y(1)1732 3415 y Fo(\003)1790 3356 y(\022)p 2128 3374 62 4 v 2128 3429 a Fs(\037)2189 3392 y Fn(2)2189 3453 y(1)2229 3429 y Fq([)p Fs(H)2337 3444 y Fm(f)2404 3429 y Fq(+)22 b Fs(r)j Fq(+)h(~)-53 b Fs(r)2713 3444 y Fn(1)2752 3429 y Fq(])p 1873 3473 1162 4 v 1873 3564 a Fs(T)14 b Fq([)p Fs(z)t Fq(;)34 b Fs(\032)p Fq(\()p Fs(H)2250 3579 y Fm(f)2318 3564 y Fq(+)22 b Fs(r)i Fq(+)j(~)-54 b Fs(r)2626 3579 y Fn(1)2666 3564 y Fq(\)])22 b Fr(\000)h Fs(E)6 b Fq([)p Fs(z)t Fq(])3045 3356 y Fo(\023)545 3747 y(f)542 3772 y Fs(W)634 3787 y Fn(2)673 3692 y Fo(\002)715 3772 y Fs(z)t Fq(;)34 b Fs(\032)p Fq(\()p Fs(r)25 b Fq(+)d Fs(r)1124 3787 y Fn(2)1163 3772 y Fq(\);)33 b Fs(\032K)1401 3721 y Fn(\()p Fm(m)1490 3730 y Fh(2)1526 3721 y Fm(;n)1589 3730 y Fh(2)1623 3721 y Fn(\))1394 3797 y(2)1655 3692 y Fo(\003)1713 3632 y(\022)p 2051 3650 62 4 v 2051 3705 a Fs(\037)2112 3668 y Fn(2)2112 3729 y(1)2152 3705 y Fq([)p Fs(H)2260 3720 y Fm(f)2327 3705 y Fq(+)22 b Fs(r)j Fq(+)i(~)-54 b Fs(r)2636 3720 y Fn(2)2676 3705 y Fq(])p 1796 3749 1162 4 v 1796 3841 a Fs(T)14 b Fq([)p Fs(z)t Fq(;)34 b Fs(\032)p Fq(\()p Fs(H)2173 3856 y Fm(f)2241 3841 y Fq(+)22 b Fs(r)j Fq(+)h(~)-53 b Fs(r)2550 3856 y Fn(2)2589 3841 y Fq(\)])22 b Fr(\000)h Fs(E)6 b Fq([)p Fs(z)t Fq(])2968 3632 y Fo(\023)3058 3772 y Fr(\001)17 b(\001)g(\001)542 4048 y(\001)g(\001)g(\001)674 3908 y Fo(\022)p 1013 3926 62 4 v 1013 3981 a Fs(\037)1074 3944 y Fn(2)1074 4006 y(1)1113 3981 y Fq([)p Fs(H)1221 3996 y Fm(f)1289 3981 y Fq(+)22 b Fs(r)j Fq(+)h(~)-53 b Fs(r)1598 3996 y Fm(L)p Fl(\000)p Fn(1)1740 3981 y Fq(])p 758 4025 1265 4 v 758 4117 a Fs(T)14 b Fq([)p Fs(z)t Fq(;)34 b Fs(\032)p Fq(\()p Fs(H)1135 4132 y Fm(f)1202 4117 y Fq(+)22 b Fs(r)j Fq(+)h(~)-53 b Fs(r)1511 4132 y Fm(L)p Fl(\000)p Fn(1)1653 4117 y Fq(\)])22 b Fr(\000)h Fs(E)6 b Fq([)p Fs(z)t Fq(])2032 3908 y Fo(\023)2125 4023 y(f)2122 4048 y Fs(W)2214 4063 y Fm(L)2266 3968 y Fo(\002)2308 4048 y Fs(z)t Fq(;)34 b Fs(\032)p Fq(\()p Fs(r)25 b Fq(+)d Fs(r)2717 4063 y Fm(L)2769 4048 y Fq(\);)33 b Fs(\032K)3007 3997 y Fn(\()p Fm(m)3096 4008 y Fk(L)3144 3997 y Fm(;n)3207 4008 y Fk(L)3253 3997 y Fn(\))3000 4075 y Fm(L)3284 3968 y Fo(\003)3342 4048 y Fq(\012)3412 3908 y Fo(\035)3487 4048 y Fs(;)257 4312 y Ft(for)25 b Fs(M)32 b Fq(+)22 b Fs(N)39 b Fr(\025)28 b Fq(1)p Ft(,)c(and)360 4523 y Fq(^)-69 b Fs(w)410 4538 y Fn(0)p Fm(;)p Fn(0)505 4523 y Fq([)p Fs(\020)8 b Fq(;)32 b Fs(r)s Fq(])44 b Fr(\000)h Fs(\032)932 4482 y Fl(\000)p Fn(1)1043 4523 y Fs(w)1113 4538 y Fn(0)p Fm(;)p Fn(0)1207 4523 y Fq([)p Fs(z)t Fq(;)34 b Fs(\032)17 b(r)s Fq(])1720 b Fv(\(III.81\))423 4747 y Fq(=)83 b Fs(\032)632 4706 y Fl(\000)p Fn(1)802 4622 y Fl(1)766 4652 y Fo(X)769 4864 y Fm(L)p Fn(=2)910 4747 y Fq(\()p Fr(\000)p Fq(1\))1112 4706 y Fm(L)p Fl(\000)p Fn(1)1292 4652 y Fo(X)1223 4863 y Fm(p)1259 4872 y Fh(1)1294 4863 y Fn(+)p Fm(q)1381 4872 y Fh(1)1415 4863 y Fl(\025)p Fn(1)1521 4747 y Fr(\001)17 b(\001)g(\001)1735 4652 y Fo(X)1654 4863 y Fm(p)1690 4874 y Fk(L)1736 4863 y Fn(+)p Fm(q)1823 4874 y Fk(L)1869 4863 y Fl(\025)p Fn(1)1976 4747 y Fs(\037)2037 4762 y Fn(1)2076 4747 y Fq([)p Fs(r)s Fq(])582 4925 y Fo(\034)657 5066 y Fq(\012)727 4921 y Fo(\014)727 4981 y(\014)727 5041 y(\014)727 5100 y(\014)777 5066 y Fs(W)869 5081 y Fm(p)905 5090 y Fh(1)940 5081 y Fm(;q)992 5090 y Fh(1)1029 4985 y Fo(\002)1071 5066 y Fs(w)p 1071 5082 73 4 v 2 w Fq([)p Fs(z)t Fq(])1246 4981 y Fo(\014)1246 5041 y(\014)1303 5066 y Fs(\032r)1400 4985 y Fo(\003)1458 4925 y(\022)p 1796 4943 62 4 v 1796 4998 a Fs(\037)1857 4962 y Fn(2)1857 5023 y(1)1897 4998 y Fq([)p Fs(H)2005 5013 y Fm(f)2072 4998 y Fq(+)22 b Fs(r)s Fq(])p 1541 5043 958 4 v 1541 5134 a Fs(T)14 b Fq([)p Fs(z)t Fq(;)34 b Fs(\032)p Fq(\()p Fs(H)1918 5149 y Fm(f)1986 5134 y Fq(+)22 b Fs(r)s Fq(\)])f Fr(\000)i Fs(E)6 b Fq([)p Fs(z)t Fq(])2509 4925 y Fo(\023)2582 5066 y Fs(W)2674 5081 y Fm(p)2710 5090 y Fh(2)2745 5081 y Fm(;q)2797 5090 y Fh(2)2834 4985 y Fo(\002)2876 5066 y Fs(w)p 2876 5082 73 4 v 2 w Fq([)p Fs(z)t Fq(])3051 4981 y Fo(\014)3051 5041 y(\014)3108 5066 y Fs(\032r)3205 4985 y Fo(\003)3279 5066 y Fr(\001)17 b(\001)g(\001)819 5342 y(\001)g(\001)g(\001)951 5201 y Fo(\022)p 1290 5219 62 4 v 1290 5274 a Fs(\037)1351 5238 y Fn(2)1351 5299 y(1)1390 5274 y Fq([)p Fs(H)1498 5289 y Fm(f)1566 5274 y Fq(+)22 b Fs(r)s Fq(])p 1035 5319 958 4 v 1035 5410 a Fs(T)14 b Fq([)p Fs(z)t Fq(;)34 b Fs(\032)p Fq(\()p Fs(H)1412 5425 y Fm(f)1479 5410 y Fq(+)22 b Fs(r)s Fq(\)])g Fr(\000)h Fs(E)6 b Fq([)p Fs(z)t Fq(])2003 5201 y Fo(\023)2093 5342 y Fs(W)2185 5357 y Fm(p)2221 5368 y Fk(L)2267 5357 y Fm(;q)2319 5368 y Fk(L)2368 5261 y Fo(\002)2410 5342 y Fs(w)p 2410 5358 73 4 v 2 w Fq([)p Fs(z)t Fq(])2585 5257 y Fo(\014)2585 5317 y(\014)2641 5342 y Fs(\032r)2738 5261 y Fo(\003)2824 5342 y Fq(\012)2894 5201 y Fo(\035)2986 5342 y Fs(\037)3047 5357 y Fn(1)3086 5342 y Fq([)p Fs(r)s Fq(])22 b Fs(;)p eop %%Page: 35 36 35 35 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(35)257 573 y Ft(wher)l(e)44 b(the)f(notation)e(intr)l(oduced)h(in) g(Eqs.)h(\(III.71\)\226\(III.72\))g(and)g(\(III.76\)\226\(III.79\))h (is)257 693 y(used.)31 b(In)25 b(particular)-11 b(,)23 b(the)h(summation)f(on)i(the)f(right)g(side)h(of)f(\(III.81\))h(starts) f(with)g Fs(L)k Fq(=)g(2)p Ft(.)404 855 y Fv(W)-8 b(e)27 b(remark)f(that)g(Theorem)h(III.7)g(determines)45 b Fq(^)-69 b Fs(w)p 2096 871 73 4 v 29 w Fv(from)27 b Fs(w)p 2416 871 V 33 w Fr(2)k(D)s Fq(\()p Fs(\032=)p Fq(8)p Fs(;)17 b(\032=)p Fq(8\))26 b Fv(only)f(as)i(a)257 975 y(sequence)k(of)g(inte)o (gral)e(k)o(ernels)h(that)g(de\002ne)h(an)g(operator)f(in)g Fr(B)s Fq([)p Fr(H)2645 990 y Fn(red)2748 975 y Fq(])p Fv(.)48 b(In)31 b(order)g(to)f(ha)n(v)o(e)277 1095 y 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5310 y Fs(;)p eop %%Page: 36 37 36 36 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(36)257 573 y(where)26 b Fs(F)589 588 y Fn(0)628 573 y Fq([)p Fs(r)s Fq(])i(:=)f Fs(\037)948 588 y Fn(1)988 573 y Fq([)p Fs(r)e Fq(+)h(~)-53 b Fs(r)1226 588 y Fn(0)1265 573 y Fq(])p Fv(,)25 b Fs(F)1405 588 y Fm(L)1457 573 y Fq([)p Fs(r)s Fq(])j(:=)g Fs(\037)1778 588 y Fn(1)1817 573 y Fq([)p Fs(r)d Fq(+)i(~)-54 b Fs(r)2055 588 y Fm(L)2107 573 y Fq(])25 b Fv(and)1292 852 y Fs(F)1355 867 y Fm(`)1388 852 y Fq([)p Fs(r)s Fq(])52 b(:=)p 1835 730 62 4 v 1835 785 a Fs(\037)1896 808 y Fn(1)1936 785 y Fq([)p Fs(r)25 b Fq(+)h(~)-53 b Fs(r)2174 800 y Fm(`)2207 785 y Fq(])2234 748 y Fn(2)p 1707 829 695 4 v 1707 920 a Fs(T)14 b Fq([)p Fs(\032)p Fq(\()p Fs(r)25 b Fq(+)h(~)-53 b Fs(r)2104 935 y Fm(`)2137 920 y Fq(\)])22 b Fr(\000)h Fs(E)2434 852 y(;)744 b Fv(\(III.84\))257 1122 y(for)25 b Fs(`)j Fq(=)g(1)p Fs(;)17 b(:)g(:)g(:)e(;)i(L)22 b Fr(\000)h Fq(1)p Fv(.)31 b(W)-8 b(e)25 b(note)f(that)292 1413 y Fq(^)-69 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1906 y Fi(\025)p Fh(1)1507 1628 y Fm(L)1467 1658 y Fo(Y)1472 1870 y Fm(`)p Fn(=1)1611 1612 y Fo(\032\022)1759 1685 y Fs(m)1844 1700 y Fm(`)1900 1685 y Fq(+)22 b Fs(p)2047 1700 y Fm(`)1879 1821 y Fs(p)1928 1836 y Fm(`)2080 1612 y Fo(\023\022)2226 1685 y Fs(n)2284 1700 y Fm(`)2340 1685 y Fq(+)g Fs(q)2481 1700 y Fm(`)2332 1821 y Fs(q)2375 1836 y Fm(`)2514 1612 y Fo(\023\033)2690 1753 y Fs(V)2747 1768 y Fm(m;p;n;q)p 2747 1792 238 3 v 2984 1753 a Fq([)p Fs(r)s Fq(;)17 b Fs(K)3192 1711 y Fn(\()p Fm(M)s(;N)7 b Fn(\))3405 1753 y Fq(])22 b Fs(:)257 2122 y Fv(T)-8 b(o)25 b(estimate)44 b Fq(^)-69 b Fs(w)p 742 2138 73 4 v 2 w Fv(,)25 b(we)g(start)g(with)f(the)g(follo)n(wing)f(preparatory) i(lemma.)257 2325 y Ff(Lemma)20 b(III.10.)33 b Ft(F)-10 b(or)18 b(\002xed)i Fs(L)28 b Fr(2)g Fg(N)34 b Ft(and)19 b Fs(m;)e(p;)g(n;)g(q)p 1717 2361 371 4 v 31 w Fr(2)28 b Fg(N)2275 2289 y Fn(4)q Fm(L)2275 2350 y Fn(0)2369 2325 y Ft(,)20 b(the)f(k)o(ernel)h Fs(V)2880 2340 y Fm(m;p;n;q)p 2880 2365 238 3 v 3145 2325 a Fr(2)28 b(W)3345 2278 y Fn(#)3337 2352 y Fm(M)s(;N)257 2446 y Ft(and)609 2666 y Fs(\032)669 2597 y Fk(d)p Fi(\000)p Fh(1)p 670 2610 111 3 v 710 2651 a(2)790 2625 y Fn(\()p Fm(M)7 b Fn(+)p Fm(N)g Fn(\))p Fl(\000)p Fn(1)1154 2581 y Fo(\015)1154 2641 y(\015)1210 2666 y Fs(V)1267 2681 y Fm(m;p;n;q)p 1267 2706 238 3 v 1504 2666 a Fr(k)1554 2625 y Fn(#)1554 2690 y Fm(\026)3205 2666 y Fv(\(III.86\))692 2937 y Fr(\024)84 b Fq(2\()p Fs(L)22 b Fq(+)g(1\))17 b Fs(C)1307 2896 y Fn(\()p Fm(L)p Fn(+1\))1520 2937 y Fs(\032)1570 2896 y Fn(\(1+)p Fm(\026)p Fn(\)\()p Fm(M)7 b Fn(+)p Fm(N)g Fn(\))p Fl(\000)p Fm(L)2197 2812 y(L)2157 2842 y Fo(Y)2161 3055 y Fm(`)p Fn(=1)2311 2762 y Fo(\015)2311 2821 y(\015)2366 2846 y Fs(w)2436 2861 y Fm(m)2498 2873 y Fk(`)2528 2861 y Fn(+)p Fm(p)2619 2873 y Fk(`)2649 2861 y Fm(;n)2712 2873 y Fk(`)2741 2861 y Fn(+)p Fm(q)2828 2873 y Fk(`)2862 2846 y Fq([)p Fs(z)t Fq(])2965 2762 y Fo(\015)2965 2821 y(\015)3021 2788 y Fn(#)3021 2885 y Fm(\026)p 2311 2914 774 4 v 2524 2934 a Fo(p)p 2623 2934 248 4 v 2623 3018 a Fs(p)2672 2971 y Fm(p)2708 2983 y Fk(`)2672 3046 y Fm(`)2759 3018 y Fs(q)2806 2971 y Fm(q)2838 2983 y Fk(`)2802 3046 y Fm(`)3116 2937 y Fs(;)257 3241 y Ft(with)25 b(the)g(con)l (vention)f(that)g Fs(p)1288 3204 y Fm(p)1355 3241 y Fq(:=)k(1)p Ft(,)c(for)g Fs(p)k Fq(=)f(0)p Ft(.)257 3444 y(Pr)l(oof)o(.)70 b Fv(W)-8 b(e)39 b(\002rst)f(note)h(that)f Fr(j)p Fs(F)1427 3459 y Fn(0)1466 3444 y Fq([)p Fs(r)s Fq(])p Fr(j)52 b(\024)i Fs(C)1848 3459 y Fn(\002)1907 3444 y Fv(,)41 b Fr(j)p Fs(@)2052 3459 y Fm(r)2091 3444 y Fs(F)2154 3459 y Fn(0)2193 3444 y Fq([)p Fs(r)s Fq(])p Fr(j)53 b(\024)g Fs(C)2575 3459 y Fn(\002)2634 3444 y Fv(,)42 b Fr(j)p Fs(F)2792 3459 y Fm(L)2844 3444 y Fq([)p Fs(r)s Fq(])p Fr(j)52 b(\024)i Fs(C)3226 3459 y Fn(\002)3284 3444 y Fv(,)42 b(and)257 3564 y Fr(j)p Fs(@)336 3579 y Fm(r)374 3564 y Fs(F)437 3579 y Fm(L)490 3564 y Fq([)p Fs(r)s Fq(])p Fr(j)27 b(\024)h Fs(C)821 3579 y Fn(\002)880 3564 y Fv(,)d(and)g(that,)f(for)h Fs(`)i Fq(=)h(1)p Fs(;)17 b(:)g(:)g(:)f(;)h(L)22 b Fr(\000)g Fq(1)p Fv(,)400 3844 y Fr(j)p Fs(F)491 3859 y Fm(`)524 3844 y Fq([)p Fs(r)s Fq(])p Fr(j)g Fq(+)g Fr(j)p Fs(@)852 3859 y Fm(r)890 3844 y Fs(F)953 3859 y Fm(`)986 3844 y Fq([)p Fs(r)s Fq(])p Fr(j)82 b(\024)1358 3700 y Fo(\014)1358 3759 y(\014)1358 3819 y(\014)1358 3879 y(\014)p 1549 3722 62 4 v 1549 3777 a Fs(\037)1610 3740 y Fn(2)1610 3801 y(1)1650 3777 y Fq([)p Fs(r)25 b Fq(+)h(~)-53 b Fs(r)1888 3792 y Fm(`)1921 3777 y Fq(])p 1401 3821 695 4 v 1401 3912 a Fs(T)14 b Fq([)p Fs(\032)p Fq(\()p Fs(r)25 b Fq(+)h(~)-53 b Fs(r)1798 3927 y Fm(`)1831 3912 y Fq(\)])22 b Fr(\000)h Fs(E)2106 3700 y Fo(\014)2106 3759 y(\014)2106 3819 y(\014)2106 3879 y(\014)2189 3844 y Fq(+)2314 3700 y Fo(\014)2314 3759 y(\014)2314 3819 y(\014)2314 3879 y(\014)2358 3777 y Fq(2)p 2407 3722 62 4 v Fs(\037)2467 3800 y Fn(1)2507 3777 y Fq([)p Fs(r)i Fq(+)h(~)-53 b Fs(r)2745 3792 y Fm(`)2778 3777 y Fq(])17 b Fs(@)2873 3792 y Fm(r)p 2911 3722 V 2911 3777 a Fs(\037)2972 3800 y Fn(1)3011 3777 y Fq([)p Fs(r)25 b Fq(+)i(~)-54 b Fs(r)3249 3792 y Fm(`)3282 3777 y Fq(])p 2358 3821 952 4 v 2486 3912 a Fs(T)14 b Fq([)p Fs(\032)p Fq(\()p Fs(r)25 b Fq(+)h(~)-53 b Fs(r)2883 3927 y Fm(`)2916 3912 y Fq(\)])22 b Fr(\000)h Fs(E)3319 3700 y Fo(\014)3319 3759 y(\014)3319 3819 y(\014)3319 3879 y(\014)1386 4119 y Fq(+)1489 3974 y Fo(\014)1489 4034 y(\014)1489 4094 y(\014)1489 4153 y(\014)p 1532 3996 62 4 v 1532 4051 a Fs(\037)1593 4015 y Fn(2)1593 4076 y(1)1633 4051 y Fq([)p Fs(r)i Fq(+)h(~)-53 b Fs(r)1871 4066 y Fm(`)1904 4051 y Fq(])27 b Fs(\032)17 b(@)2076 4066 y Fm(r)2115 4051 y Fs(T)d Fq([)p Fs(z)t Fq(;)j Fs(\032)p Fq(\()p Fs(r)25 b Fq(+)i(~)-54 b Fs(r)2605 4066 y Fm(`)2638 4051 y Fq(\)])p 1532 4096 1171 4 v 1713 4187 a(\()p Fs(T)14 b Fq([)p Fs(\032)p Fq(\()p Fs(r)25 b Fq(+)h(~)-53 b Fs(r)2148 4202 y Fm(`)2180 4187 y Fq(\)])23 b Fr(\000)f Fs(E)6 b Fq(\))2483 4158 y Fn(2)2713 3974 y Fo(\014)2713 4034 y(\014)2713 4094 y(\014)2713 4153 y(\014)1197 4382 y Fr(\024)1368 4314 y Fs(C)1438 4329 y Fn(\002)p 1368 4359 129 4 v 1407 4450 a Fs(\032)1529 4382 y(;)1649 b Fv(\(III.87\))257 4646 y(for)24 b(some)g(constant)f Fq(1)k Fr(\024)h Fs(C)1239 4661 y Fn(\002)1326 4646 y Fs(<)f Fr(1)d Fv(only)f(depending)g(on)g (the)h(smooth)e(cutof)n(f)h(function)g Fq(\002)p Fv(.)257 4849 y Ff(Remark)43 b(III.11.)50 b Ft(Estimate)41 b(\(III.87\))h(holds) g(uniformly)e(in)i Fs(r)62 b Fr(2)e Fs(I)8 b Ft(,)47 b(as)41 b(follows)h(fr)l(om)257 4969 y(Lemma)34 b(III.5,)i(whic)o(h)d (insur)l(es)f(that)h Fr(j)p Fs(T)14 b Fq([)p Fs(\032)p Fq(\()p Fs(r)31 b Fq(+)h(~)-53 b Fs(r)2032 4984 y Fm(`)2065 4969 y Fq(\)])28 b Fr(\000)h Fs(E)6 b Fr(j)43 b(\025)h Fs(\032=)p Fq(8)p Ft(.)57 b(Note)33 b(that)g(it)g(is)f(Es-)257 5090 y(timate)f(\(III.87\))g(that)f(simpli\002es)g(the)h(analysis)e (consider)o(ably)-5 b(,)31 b(as)g(compar)l(ed)f(to)h([3],)h(be-)257 5210 y(cause)25 b(the)g(functions)e Fs(F)1109 5225 y Fm(`)1142 5210 y Fq([)p Fs(r)s Fq(])i Ft(ar)l(e)g(continuously)e(dif)n (fer)l(entiable)g(in)i Fs(r)s Ft(.)p eop %%Page: 37 38 37 37 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(37)404 573 y(Ne)o(xt)42 b(we)i(estimate)e Fr(j)p Fs(V)1259 588 y Fm(m;p;n;q)p 1259 613 238 3 v 1496 573 a Fr(j)p Fv(,)48 b(using)42 b(that)h Fr(jh)p Fq(\012)p Fr(j)p Fs(A)p Fq(\012)p Fr(ij)61 b(\024)h(k)p Fs(A)p Fr(k)2797 588 y Fn(op)2876 573 y Fv(,)48 b(for)43 b(an)o(y)g Fs(A)62 b Fr(2)257 693 y(B)s Fq([)p Fr(H)436 708 y Fn(red)539 693 y Fq(])p Fv(.)31 b(W)-8 b(e)25 b(ha)n(v)o(e)g(that)587 809 y Fo(\014)587 869 y(\014)620 894 y Fs(V)677 909 y Fm(m;p;n;q)p 677 934 V 914 894 a Fq([)p Fs(r)s Fq(;)17 b Fs(K)1122 853 y Fn(\()p Fm(M)s(;N)7 b Fn(\))1334 894 y Fq(])1361 809 y Fo(\014)1361 869 y(\014)3205 894 y Fv(\(III.88\))670 1150 y Fr(\024)870 1026 y Fm(L)830 1055 y Fo(Y)834 1268 y Fm(`)p Fn(=0)974 1065 y Fo(\015)974 1125 y(\015)1029 1150 y Fs(F)1092 1165 y Fm(`)1125 1150 y Fq([)p Fs(H)1233 1165 y Fm(f)1301 1150 y Fq(+)22 b Fs(r)s Fq(])1473 1065 y Fo(\015)1473 1125 y(\015)1528 1189 y Fn(op)1691 1026 y Fm(L)1651 1055 y Fo(Y)1655 1268 y Fm(`)p Fn(=1)1795 1035 y Fo(\015)1795 1095 y(\015)1795 1155 y(\015)1853 1125 y(f)1850 1150 y Fs(W)1942 1165 y Fm(`)1975 1069 y Fo(\002)2017 1150 y Fs(z)t Fq(;)34 b Fs(\032)p Fq(\()p Fs(r)25 b Fq(+)d Fs(r)2426 1165 y Fm(`)2459 1150 y Fq(\);)33 b Fs(\032K)2697 1099 y Fn(\()p Fm(m)2786 1111 y Fk(`)2817 1099 y Fm(;n)2880 1111 y Fk(`)2910 1099 y Fn(\))2690 1178 y Fm(`)2941 1069 y Fo(\003)2983 1035 y(\015)2983 1095 y(\015)2983 1155 y(\015)3038 1219 y Fn(op)3139 1150 y Fs(:)257 1436 y Fv(W)-8 b(e)26 b(emplo)o(y)d (Leibniz')i(rule)g(to)f(compute)g Fs(@)1799 1451 y Fm(r)1838 1436 y Fs(V)1895 1451 y Fm(m;p;n;q)p 1895 1475 V 2132 1436 a Fv(,)h(with)f(the)g(result)371 1664 y Fs(@)422 1679 y Fm(r)460 1664 y Fs(V)517 1679 y Fm(m;p;n;q)p 517 1704 V 755 1664 a Fq([)p Fs(r)s Fq(;)17 b Fs(K)963 1623 y Fn(\()p Fm(M)s(;N)7 b Fn(\))1175 1664 y Fq(])53 b(=)1395 1639 y Fo(e)1383 1664 y Fs(V)1462 1623 y Fn(\()p Fm(i)p Fn(\))1440 1689 y Fm(m;p;n;q)p 1440 1714 V 1678 1664 a Fq([)p Fs(r)s Fq(;)17 b Fs(K)1886 1623 y Fn(\()p Fm(M)s(;N)7 b Fn(\))2098 1664 y Fq(])50 b(+)2312 1639 y Fo(e)2300 1664 y Fs(V)2379 1623 y Fn(\()p Fm(ii)p Fn(\))2357 1689 y Fm(m;p;n;q)p 2357 1714 V 2595 1664 a Fq([)p Fs(r)s Fq(;)17 b Fs(K)2803 1623 y Fn(\()p Fm(M)s(;N)7 b Fn(\))3015 1664 y Fq(])22 b Fs(;)114 b Fv(\(III.89\))257 1874 y(where)359 2049 y Fo(e)348 2075 y Fs(V)426 2033 y Fn(\()p Fm(i)p Fn(\))405 2099 y Fm(m;p;n;q)p 405 2124 V 642 2075 a Fq([)p Fs(r)s Fq(;)17 b Fs(K)850 2033 y Fn(\()p Fm(M)s(;N)7 b Fn(\))1063 2075 y Fq(])52 b(:=)1960 b Fv(\(III.90\))444 2218 y Fm(L)396 2248 y Fo(X)406 2458 y Fm(j)t Fn(=0)556 2202 y Fo(\034)631 2343 y Fq(\012)701 2198 y Fo(\014)701 2258 y(\014)701 2318 y(\014)701 2378 y(\014)754 2214 y Fm(j)t Fl(\000)p Fn(1)751 2248 y Fo(Y)755 2460 y Fm(`)p Fn(=1)895 2232 y Fo(n)962 2343 y Fs(F)1025 2358 y Fm(`)p Fl(\000)p Fn(1)1148 2343 y Fq([)p Fs(H)1256 2358 y Fm(f)1323 2343 y Fq(+)22 b Fs(r)s Fq(])1526 2318 y Fo(f)1523 2343 y Fs(W)1615 2358 y Fm(`)1648 2262 y Fo(\002)1689 2343 y Fs(z)t Fq(;)34 b Fs(\032)p Fq(\()p Fs(r)25 b Fq(+)d Fs(r)2098 2358 y Fm(`)2131 2343 y Fq(\);)34 b Fs(\032K)2370 2292 y Fn(\()p Fm(m)2459 2304 y Fk(`)2490 2292 y Fm(;n)2553 2304 y Fk(`)2582 2292 y Fn(\))2363 2371 y Fm(`)2614 2262 y Fo(\003)2655 2232 y(o)2750 2262 y(\000)2795 2343 y Fs(@)2846 2358 y Fm(r)2885 2343 y Fs(F)2948 2358 y Fm(j)2984 2343 y Fq([)p Fs(H)3092 2358 y Fm(f)3160 2343 y Fq(+)22 b Fs(r)s Fq(])3332 2262 y Fo(\001)947 2570 y Fm(L)908 2599 y Fo(Y)868 2812 y Fm(`)p Fn(=)p Fm(j)t Fn(+1)1091 2583 y Fo(n)1160 2669 y(f)1157 2694 y Fs(W)1249 2709 y Fm(`)1282 2613 y Fo(\002)1324 2694 y Fs(z)t Fq(;)34 b Fs(\032)p Fq(\()p Fs(r)25 b Fq(+)d Fs(r)1733 2709 y Fm(`)1766 2694 y Fq(\);)33 b Fs(\032K)2004 2643 y Fn(\()p Fm(m)2093 2655 y Fk(`)2125 2643 y Fm(;n)2188 2655 y Fk(`)2217 2643 y Fn(\))1997 2722 y Fm(`)2249 2613 y Fo(\003)2318 2694 y Fs(F)2381 2709 y Fm(`)2414 2694 y Fq([)p Fs(H)2522 2709 y Fm(f)2589 2694 y Fq(+)22 b Fs(r)s Fq(])2761 2583 y Fo(o)2855 2694 y Fq(\012)2925 2554 y Fo(\035)3023 2694 y Fs(;)257 2991 y Fv(and)365 3167 y Fo(e)353 3192 y Fs(V)432 3151 y Fn(\()p Fm(ii)p Fn(\))410 3217 y Fm(m;p;n;q)p 410 3242 V 648 3192 a Fq([)p Fs(r)s Fq(;)17 b Fs(K)856 3151 y Fn(\()p Fm(M)s(;N)7 b Fn(\))1068 3192 y Fq(])53 b(:=)1954 b Fv(\(III.91\))449 3336 y Fm(L)401 3366 y Fo(X)412 3576 y Fm(j)t Fn(=1)562 3320 y Fo(\034)636 3460 y Fq(\012)706 3316 y Fo(\014)706 3376 y(\014)706 3435 y(\014)706 3495 y(\014)740 3460 y Fs(F)803 3475 y Fn(0)843 3460 y Fq([)p Fs(H)951 3475 y Fm(f)1018 3460 y Fq(+)22 b Fs(r)s Fq(])1226 3332 y Fm(j)t Fl(\000)p Fn(1)1223 3366 y Fo(Y)1227 3578 y Fm(`)p Fn(=1)1367 3350 y Fo(n)1436 3435 y(f)1433 3460 y Fs(W)1525 3475 y Fm(`)1559 3380 y Fo(\002)1600 3460 y Fs(z)t Fq(;)34 b Fs(\032)p Fq(\()p Fs(r)25 b Fq(+)d Fs(r)2009 3475 y Fm(`)2042 3460 y Fq(\);)33 b Fs(\032K)2280 3409 y Fn(\()p Fm(m)2369 3421 y Fk(`)2401 3409 y Fm(;n)2464 3421 y Fk(`)2493 3409 y Fn(\))2273 3488 y Fm(`)2525 3380 y Fo(\003)2594 3460 y Fs(F)2657 3475 y Fm(`)2690 3460 y Fq([)p Fs(H)2798 3475 y Fm(f)2865 3460 y Fq(+)22 b Fs(r)s Fq(])3037 3350 y Fo(o)874 3736 y Fs(\032)941 3655 y Fo(\000)986 3736 y Fs(@)1037 3751 y Fm(r)1079 3710 y Fo(f)1076 3736 y Fs(W)1168 3751 y Fm(j)1204 3655 y Fo(\002)1246 3736 y Fs(z)t Fq(;)34 b Fs(\032)p Fq(\()p Fs(r)25 b Fq(+)d Fs(r)1655 3751 y Fm(j)1691 3736 y Fq(\);)34 b Fs(\032K)1930 3682 y Fn(\()p Fm(m)2019 3692 y Fk(j)2053 3682 y Fm(;n)2116 3692 y Fk(j)2147 3682 y Fn(\))1923 3761 y Fm(j)2179 3655 y Fo(\003\001)717 3863 y Fm(L)677 3893 y Fo(Y)637 4105 y Fm(`)p Fn(=)p Fm(j)t Fn(+1)860 3877 y Fo(n)927 3987 y Fs(F)990 4002 y Fm(`)p Fl(\000)p Fn(1)1113 3987 y Fq([)p Fs(H)1221 4002 y Fm(f)1288 3987 y Fq(+)22 b Fs(r)s Fq(])1491 3962 y Fo(f)1488 3987 y Fs(W)1580 4002 y Fm(`)1613 3907 y Fo(\002)1654 3987 y Fs(z)t Fq(;)35 b Fs(\032)p Fq(\()p Fs(r)25 b Fq(+)d Fs(r)2064 4002 y Fm(`)2097 3987 y Fq(\);)33 b Fs(\032K)2335 3936 y Fn(\()p Fm(m)2424 3948 y Fk(`)2455 3936 y Fm(;n)2518 3948 y Fk(`)2548 3936 y Fn(\))2328 4015 y Fm(`)2579 3907 y Fo(\003)2621 3877 y(o)2715 3987 y Fs(F)2778 4002 y Fm(L)2830 3987 y Fq([)p Fs(H)2938 4002 y Fm(f)3005 3987 y Fq(+)22 b Fs(r)s Fq(])28 b(\012)3275 3847 y Fo(\035)3372 3987 y Fs(:)257 4284 y Fv(Proceeding)d(in)g(the)g (same)f(w)o(ay)h(as)g(in)f(Eq.)h(\(III.88\))g(and)g(using)f (\(III.87\),)h(we)g(obtain)594 4490 y Fo(\014)594 4550 y(\014)12 b(e)627 4575 y Fs(V)706 4534 y Fn(\()p Fm(i)p Fn(\))684 4599 y Fm(m;p;n;q)p 684 4624 V 922 4575 a Fq([)p Fs(r)s Fq(;)17 b Fs(K)1130 4534 y Fn(\()p Fm(M)s(;N)7 b Fn(\))1342 4575 y Fq(])1369 4490 y Fo(\014)1369 4550 y(\014)1455 4575 y Fr(\024)1585 4434 y Fo(\032)1724 4450 y Fm(L)1676 4480 y Fo(X)1687 4690 y Fm(j)t Fn(=0)1837 4490 y Fo(\015)1837 4550 y(\015)1892 4575 y Fs(@)1943 4590 y Fm(r)1981 4575 y Fs(F)2044 4590 y Fm(j)2081 4575 y Fq([)p Fs(H)2189 4590 y Fm(f)2256 4575 y Fq(+)22 b Fs(r)s Fq(])2428 4490 y Fo(\015)2428 4550 y(\015)2484 4614 y Fn(op)2626 4450 y Fm(L)2586 4480 y Fo(Y)2589 4681 y Fk(`)p Fh(=0)p Fk(;)2599 4737 y(`)p Fi(6)p Fh(=)p Fk(j)2737 4490 y Fo(\015)2737 4550 y(\015)2793 4575 y Fs(F)2856 4590 y Fm(`)2889 4575 y Fq([)p Fs(H)2997 4590 y Fm(f)3064 4575 y Fq(+)g Fs(r)s Fq(])3236 4490 y Fo(\015)3236 4550 y(\015)3291 4614 y Fn(op)3370 4434 y Fo(\033)1349 4845 y Fm(L)1310 4875 y Fo(Y)1314 5087 y Fm(`)p Fn(=1)1454 4855 y Fo(\015)1454 4915 y(\015)1454 4975 y(\015)1512 4945 y(f)1509 4970 y Fs(W)1601 4985 y Fm(`)1634 4889 y Fo(\002)1676 4970 y Fs(z)t Fq(;)34 b Fs(\032)p Fq(\()p Fs(r)25 b Fq(+)d Fs(r)2085 4985 y Fm(`)2118 4970 y Fq(\);)33 b Fs(\032K)2356 4919 y Fn(\()p Fm(m)2445 4931 y Fk(`)2476 4919 y Fm(;n)2539 4931 y Fk(`)2569 4919 y Fn(\))2349 4998 y Fm(`)2600 4889 y Fo(\003)2642 4855 y(\015)2642 4915 y(\015)2642 4975 y(\015)2697 5039 y Fn(op)3205 4970 y Fv(\(III.92\))677 5310 y Fr(\024)83 b Fq(\()p Fs(L)23 b Fq(+)f(1\))17 b Fs(C)1243 5269 y Fm(L)p Fn(+1)1236 5337 y(\002)1401 5310 y Fs(\032)1451 5269 y Fl(\000)p Fm(L)p Fn(+1)1732 5186 y Fm(L)1693 5216 y Fo(Y)1697 5428 y Fm(`)p Fn(=1)1837 5196 y Fo(\015)1837 5256 y(\015)1837 5315 y(\015)1895 5285 y(f)1892 5310 y Fs(W)1984 5325 y Fm(`)2017 5230 y Fo(\002)2059 5310 y Fs(z)t Fq(;)34 b Fs(\032)p Fq(\()p Fs(r)25 b Fq(+)d Fs(r)2468 5325 y Fm(`)2501 5310 y Fq(\);)33 b Fs(\032K)2739 5260 y Fn(\()p Fm(m)2828 5272 y Fk(`)2859 5260 y Fm(;n)2922 5272 y Fk(`)2952 5260 y Fn(\))2732 5338 y Fm(`)2983 5230 y Fo(\003)3025 5196 y(\015)3025 5256 y(\015)3025 5315 y(\015)3080 5379 y Fn(op)p eop %%Page: 38 39 38 38 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(38)257 573 y(and)515 654 y Fo(\014)515 714 y(\014)559 713 y(e)548 739 y Fs(V)626 697 y Fn(\()p Fm(ii)p Fn(\))605 763 y Fm(m;p;n;q)p 605 788 238 3 v 842 739 a Fq([)p Fs(r)s Fq(;)17 b Fs(K)1050 697 y Fn(\()p Fm(M)s(;N)7 b Fn(\))1263 739 y Fq(])1290 654 y Fo(\014)1290 714 y(\014)598 1004 y Fr(\024)798 880 y Fm(L)758 910 y Fo(Y)762 1122 y Fm(`)p Fn(=0)902 919 y Fo(\015)902 979 y(\015)957 1004 y Fs(F)1020 1019 y Fm(`)1054 1004 y Fq([)p Fs(H)1162 1019 y Fm(f)1229 1004 y Fq(+)22 b Fs(r)s Fq(])1401 919 y Fo(\015)1401 979 y(\015)1456 1043 y Fn(op)1562 864 y Fo(\032)1702 880 y Fm(L)1654 910 y Fo(X)1664 1119 y Fm(j)t Fn(=1)1814 919 y Fo(\015)1814 979 y(\015)1870 1004 y Fs(@)1921 1019 y Fm(r)1962 979 y Fo(f)1959 1004 y Fs(W)2051 1019 y Fm(j)2088 923 y Fo(\002)2129 1004 y Fs(z)t Fq(;)34 b Fs(\032)p Fq(\()p Fs(r)25 b Fq(+)d Fs(r)2538 1019 y Fm(j)2575 1004 y Fq(\);)33 b Fs(\032K)2813 950 y Fn(\()p Fm(m)2902 960 y Fk(j)2936 950 y Fm(;n)2999 960 y Fk(j)3031 950 y Fn(\))2806 1030 y Fm(j)3062 923 y Fo(\003)3104 919 y(\015)3104 979 y(\015)3159 1043 y Fn(op)1277 1231 y Fm(L)1238 1261 y Fo(Y)1241 1461 y Fk(`)p Fh(=1)p Fk(;)1251 1518 y(`)p Fi(6)p Fh(=)p Fk(j)1389 1241 y Fo(\015)1389 1301 y(\015)1389 1360 y(\015)1447 1330 y(f)1444 1355 y Fs(W)1536 1370 y Fm(`)1569 1275 y Fo(\002)1611 1355 y Fs(z)t Fq(;)h Fs(\032)p Fq(\()p Fs(r)25 b Fq(+)d Fs(r)2020 1370 y Fm(`)2053 1355 y Fq(\);)33 b Fs(\032K)2291 1305 y Fn(\()p Fm(m)2380 1317 y Fk(`)2411 1305 y Fm(;n)2474 1317 y Fk(`)2504 1305 y Fn(\))2284 1383 y Fm(`)2535 1275 y Fo(\003)2577 1241 y(\015)2577 1301 y(\015)2577 1360 y(\015)2632 1424 y Fn(op)2711 1215 y Fo(\033)598 1751 y Fr(\024)83 b Fs(C)835 1709 y Fm(L)p Fn(+1)828 1778 y(\002)994 1751 y Fs(\032)1044 1709 y Fl(\000)p Fm(L)p Fn(+2)1269 1610 y Fo(\032)1409 1626 y Fm(L)1361 1656 y Fo(X)1371 1866 y Fm(j)t Fn(=1)1521 1666 y Fo(\015)1521 1726 y(\015)1579 1725 y(f)1576 1751 y Fs(W)1682 1709 y Fl(0)1668 1775 y Fm(j)1705 1670 y Fo(\002)1747 1751 y Fs(z)t Fq(;)34 b Fs(\032)p Fq(\()p Fs(r)25 b Fq(+)d Fs(r)2156 1766 y Fm(j)2192 1751 y Fq(\);)34 b Fs(\032K)2431 1696 y Fn(\()p Fm(m)2520 1706 y Fk(j)2554 1696 y Fm(;n)2617 1706 y Fk(j)2649 1696 y Fn(\))2424 1776 y Fm(j)2680 1670 y Fo(\003)2722 1666 y(\015)2722 1726 y(\015)2777 1790 y Fn(op)1277 1977 y Fm(L)1238 2007 y Fo(Y)1241 2208 y Fk(`)p Fh(=1)p Fk(;)1251 2264 y(`)p Fi(6)p Fh(=)p Fk(j)1389 1987 y Fo(\015)1389 2047 y(\015)1389 2107 y(\015)1447 2077 y(f)1444 2102 y Fs(W)1536 2117 y Fm(`)1569 2021 y Fo(\002)1611 2102 y Fs(z)t Fq(;)g Fs(\032)p Fq(\()p Fs(r)25 b Fq(+)d Fs(r)2020 2117 y Fm(`)2053 2102 y Fq(\);)33 b Fs(\032K)2291 2051 y Fn(\()p Fm(m)2380 2063 y Fk(`)2411 2051 y Fm(;n)2474 2063 y Fk(`)2504 2051 y Fn(\))2284 2130 y Fm(`)2535 2021 y Fo(\003)2577 1987 y(\015)2577 2047 y(\015)2577 2107 y(\015)2632 2171 y Fn(op)2711 1961 y Fo(\033)2808 2102 y Fs(;)370 b Fv(\(III.93\))257 2406 y(where)460 2546 y Fo(f)457 2571 y Fs(W)563 2530 y Fl(0)549 2596 y Fm(j)586 2491 y Fo(\002)628 2571 y Fs(z)t Fq(;)34 b Fs(r)25 b Fq(+)d Fs(r)949 2586 y Fm(j)985 2571 y Fq(;)33 b Fs(K)1135 2517 y Fn(\()p Fm(m)1224 2527 y Fk(j)1258 2517 y Fm(;n)1321 2527 y Fk(j)1353 2517 y Fn(\))1128 2597 y Fm(j)1384 2491 y Fo(\003)1478 2571 y Fq(:=)53 b Fs(W)1740 2530 y Fm(m)1802 2540 y Fk(j)1835 2530 y Fm(;n)1898 2540 y Fk(j)1726 2596 y Fm(p)1762 2606 y Fk(j)1794 2596 y Fm(;q)1846 2606 y Fk(j)1934 2571 y Fq([)p Fs(@)2012 2586 y Fm(r)2050 2571 y Fs(w)p 2050 2587 73 4 v 3 w Fq([)p Fs(z)t Fq(])2226 2487 y Fo(\014)2226 2546 y(\014)2282 2571 y Fs(r)25 b Fq(+)d Fs(r)2493 2586 y Fm(j)2529 2571 y Fq(;)34 b Fs(K)2680 2517 y Fn(\()p Fm(m)2769 2527 y Fk(j)2802 2517 y Fm(;n)2865 2527 y Fk(j)2897 2517 y Fn(\))2673 2597 y Fm(j)2929 2571 y Fq(])22 b Fs(;)200 b Fv(\(III.94\))257 2737 y(with)31 b Fs(@)517 2752 y Fm(r)556 2737 y Fs(w)p 556 2753 V 42 w Fq(:=)40 b(\()p Fs(@)900 2752 y Fm(r)939 2737 y Fs(w)1009 2752 y Fm(m;n)1137 2737 y Fq(\))1175 2752 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(0)1430 2737 y Fv(,)33 b(similar)d(to)h(\(III.71\))h(and)g (\(III.76\).)51 b(Inserting)31 b(\(III.92\)\226)257 2857 y(\(III.93\))26 b(into)e(\(III.89\))h(and)g(adding)f(the)g(result)h(to) f(\(III.88\),)h(we)g(obtain)320 3034 y Fq(sup)335 3115 y Fm(r)r Fl(2)p Fm(I)483 2949 y Fo(\014)483 3009 y(\014)517 3034 y Fs(V)574 3049 y Fm(m;p;n;q)p 574 3074 238 3 v 811 3034 a Fq([)p Fs(r)s Fq(;)17 b Fs(K)1019 2993 y Fn(\()p Fm(M)s(;N)7 b Fn(\))1231 3034 y Fq(])1258 2949 y Fo(\014)1258 3009 y(\014)1292 2975 y Fn(2)1375 3034 y Fq(+)50 b(sup)1516 3115 y Fm(r)r Fl(2)p Fm(I)1665 2949 y Fo(\014)1665 3009 y(\014)1698 3034 y Fs(@)1749 3049 y Fm(r)1787 3034 y Fs(V)1844 3049 y Fm(m;p;n;q)p 1844 3074 V 2082 3034 a Fq([)p Fs(r)s Fq(;)17 b Fs(K)2290 2993 y Fn(\()p Fm(M)s(;N)7 b Fn(\))2502 3034 y Fq(])2529 2949 y Fo(\014)2529 3009 y(\014)2562 2975 y Fn(2)3205 3034 y Fv(\(III.95\))317 3263 y Fr(\024)92 b Fq(\()p Fs(L)22 b Fq(+)g(1\))797 3222 y Fn(2)853 3263 y Fs(C)930 3212 y Fn(2\()p Fm(L)p Fn(+1\))923 3290 y(\002)1179 3263 y Fs(\032)1229 3222 y Fl(\000)p Fn(2)p Fm(L)p Fn(+2)526 3379 y Fm(L)486 3409 y Fo(Y)490 3621 y Fm(`)p Fn(=1)630 3363 y Fo(\032)721 3503 y Fq(sup)736 3585 y Fm(r)r Fl(2)p Fm(I)885 3389 y Fo(\015)885 3449 y(\015)885 3508 y(\015)943 3478 y(f)940 3503 y Fs(W)1032 3518 y Fm(`)1065 3423 y Fo(\002)1107 3503 y Fs(z)t Fq(;)34 b Fs(r)s Fq(;)f Fs(\032K)1464 3452 y Fn(\()p Fm(m)1553 3464 y Fk(`)1584 3452 y Fm(;n)1647 3464 y Fk(`)1676 3452 y Fn(\))1457 3531 y Fm(`)1708 3423 y Fo(\003)1750 3389 y(\015)1750 3449 y(\015)1750 3508 y(\015)1805 3415 y Fn(2)1805 3572 y(op)1933 3503 y Fq(+)50 b Fs(\032)34 b Fq(sup)2158 3585 y Fm(r)r Fl(2)p Fm(I)2306 3389 y Fo(\015)2306 3449 y(\015)2306 3508 y(\015)2364 3478 y(f)2361 3503 y Fs(W)2467 3462 y Fl(0)2453 3528 y Fm(`)2490 3423 y Fo(\002)2532 3503 y Fs(z)t Fq(;)g Fs(r)s Fq(;)f Fs(\032K)2889 3452 y Fn(\()p Fm(m)2978 3464 y Fk(`)3009 3452 y Fm(;n)3072 3464 y Fk(`)3102 3452 y Fn(\))2882 3531 y Fm(`)3133 3423 y Fo(\003)3175 3389 y(\015)3175 3449 y(\015)3175 3508 y(\015)3230 3415 y Fn(2)3230 3572 y(op)3309 3363 y Fo(\033)3406 3503 y Fs(;)257 3770 y Fv(and)j(inte)o(grating)e(this)h(inequality)f(against)h(the)h (measure)f Fr(j)p Fs(K)2478 3734 y Fn(\()p Fm(M)s(;N)7 b Fn(\))2691 3770 y Fr(j)2719 3734 y Fl(\000)p Fn(3)p Fl(\000)p Fn(2)p Fm(\026)2962 3770 y Fs(dK)3103 3734 y Fn(\()p Fm(M)s(;N)g Fn(\))3316 3770 y Fv(,)38 b(we)257 3891 y(arri)n(v)o(e)25 b(at)269 4015 y Fo(\015)269 4074 y(\015)325 4099 y Fs(V)382 4114 y Fm(m;p;n;q)p 382 4139 V 619 4099 a Fr(k)669 4058 y Fn(2)669 4124 y Fm(M)s(;N)907 4099 y Fq(=)1063 3964 y Fo(Z)1119 4189 y Fm(B)1175 4159 y Fk(M)5 b Fh(+)p Fk(N)1172 4210 y Fh(1)1462 4032 y Fs(dK)1603 3996 y Fn(\()p Fm(M)s(;N)i Fn(\))p 1373 4076 531 4 v 1373 4168 a Fr(j)p Fs(K)1491 4139 y Fn(\()p Fm(M)s(;N)g Fn(\))1704 4168 y Fr(j)1732 4139 y Fn(3+2)p Fm(\026)1958 4099 y Fq(sup)1973 4181 y Fm(r)r Fl(2)p Fm(I)2121 4015 y Fo(\014)2121 4074 y(\014)2154 4099 y Fs(V)2211 4114 y Fm(m;p;n;q)p 2211 4139 238 3 v 2449 4099 a Fq([)p Fs(r)s Fq(;)17 b Fs(K)2657 4058 y Fn(\()p Fm(M)s(;N)7 b Fn(\))2869 4099 y Fq(])2896 4015 y Fo(\014)2896 4074 y(\014)2929 4041 y Fn(2)3205 4099 y Fv(\(III.96\))255 4424 y Fr(\024)103 b Fq(\()p Fs(L)23 b Fq(+)f(1\))747 4382 y Fn(2)803 4424 y Fs(C)880 4373 y Fn(2\()p Fm(L)p Fn(+1\))873 4451 y(\002)1129 4424 y Fs(\032)1179 4382 y Fl(\000)p Fn(2)p Fm(L)p Fn(+2)1495 4299 y Fm(L)1456 4329 y Fo(Y)1460 4541 y Fm(`)p Fn(=1)1600 4283 y Fo(\032)1691 4288 y(Z)1746 4513 y Fm(B)1802 4473 y Fk(m)1856 4491 y(`)1887 4473 y Fh(+)p Fk(n)1972 4491 y(`)1799 4535 y Fh(1)2125 4356 y Fs(dK)2266 4320 y Fn(\()p Fm(m)2355 4332 y Fk(`)2386 4320 y Fm(;n)2449 4332 y Fk(`)2479 4320 y Fn(\))p 2037 4401 562 4 v 2037 4492 a Fr(j)p Fs(K)2155 4463 y Fn(\()p Fm(m)2244 4475 y Fk(`)2275 4463 y Fm(;n)2338 4475 y Fk(`)2367 4463 y Fn(\))2399 4492 y Fr(j)2427 4463 y Fn(3+2)p Fm(\026)672 4618 y Fo(n)755 4729 y Fq(sup)770 4810 y Fm(r)r Fl(2)p Fm(I)918 4614 y Fo(\015)918 4674 y(\015)918 4734 y(\015)977 4703 y(f)974 4729 y Fs(W)1066 4744 y Fm(`)1099 4648 y Fo(\002)1140 4729 y Fs(z)t Fq(;)34 b Fs(r)s Fq(;)f Fs(\032K)1497 4678 y Fn(\()p Fm(m)1586 4690 y Fk(`)1617 4678 y Fm(;n)1680 4690 y Fk(`)1710 4678 y Fn(\))1490 4756 y Fm(`)1741 4648 y Fo(\003)1783 4614 y(\015)1783 4674 y(\015)1783 4734 y(\015)1838 4640 y Fn(2)1838 4798 y(op)1967 4729 y Fq(+)49 b Fs(\032)34 b Fq(sup)2191 4810 y Fm(r)r Fl(2)p Fm(I)2339 4614 y Fo(\015)2339 4674 y(\015)2339 4734 y(\015)2398 4703 y(f)2395 4729 y Fs(W)2501 4688 y Fl(0)2487 4753 y Fm(`)2524 4648 y Fo(\002)2565 4729 y Fs(z)t Fq(;)g Fs(r)s Fq(;)f Fs(\032K)2922 4678 y Fn(\()p Fm(m)3011 4690 y Fk(`)3043 4678 y Fm(;n)3106 4690 y Fk(`)3135 4678 y Fn(\))2915 4756 y Fm(`)3167 4648 y Fo(\003)3208 4614 y(\015)3208 4674 y(\015)3208 4734 y(\015)3263 4640 y Fn(2)3263 4798 y(op)3342 4618 y Fo(o)3408 4588 y(\033)255 5037 y Fq(=)104 b(\()p Fs(L)23 b Fq(+)f(1\))747 4996 y Fn(2)803 5037 y Fs(C)880 4986 y Fn(2\()p Fm(L)p Fn(+1\))873 5064 y(\002)1129 5037 y Fs(\032)1179 4996 y Fn(\(3+2)p Fm(\026)p Fl(\000)p Fm(d)p Fn(\)\()p Fm(M)7 b Fn(+)p Fm(N)g Fn(\))p Fl(\000)p Fn(2)p Fm(L)p Fn(+2)2057 4913 y Fm(L)2017 4943 y Fo(Y)2022 5155 y Fm(`)p Fn(=1)2161 4897 y Fo(\032)2253 4902 y(Z)2308 5127 y Fm(B)2364 5087 y Fk(m)2418 5104 y(`)2449 5087 y Fh(+)p Fk(n)2534 5104 y(`)2361 5138 y(\032)2687 4970 y Fs(dK)2828 4934 y Fn(\()p Fm(m)2917 4946 y Fk(`)2948 4934 y Fm(;n)3011 4946 y Fk(`)3040 4934 y Fn(\))p 2599 5014 V 2599 5106 a Fr(j)p Fs(K)2717 5077 y Fn(\()p Fm(m)2806 5089 y Fk(`)2836 5077 y Fm(;n)2899 5089 y Fk(`)2929 5077 y Fn(\))2960 5106 y Fr(j)2988 5077 y Fn(3+2)p Fm(\026)672 5232 y Fo(n)755 5342 y Fq(sup)770 5424 y Fm(r)r Fl(2)p Fm(I)918 5228 y Fo(\015)918 5288 y(\015)918 5347 y(\015)977 5317 y(f)974 5342 y Fs(W)1066 5357 y Fm(`)1099 5262 y Fo(\002)1140 5342 y Fs(z)t Fq(;)34 b Fs(r)s Fq(;)f Fs(K)1447 5292 y Fn(\()p Fm(m)1536 5304 y Fk(`)1567 5292 y Fm(;n)1630 5304 y Fk(`)1660 5292 y Fn(\))1440 5370 y Fm(`)1691 5262 y Fo(\003)1733 5228 y(\015)1733 5288 y(\015)1733 5347 y(\015)1788 5254 y Fn(2)1788 5411 y(op)1916 5342 y Fq(+)50 b Fs(\032)34 b Fq(sup)2141 5424 y Fm(r)r Fl(2)p Fm(I)2289 5228 y Fo(\015)2289 5288 y(\015)2289 5347 y(\015)2347 5317 y(f)2344 5342 y Fs(W)2450 5301 y Fl(0)2436 5367 y Fm(`)2473 5262 y Fo(\002)2515 5342 y Fs(z)t Fq(;)g Fs(r)s Fq(;)f Fs(K)2822 5292 y Fn(\()p Fm(m)2911 5304 y Fk(`)2942 5292 y Fm(;n)3005 5304 y Fk(`)3034 5292 y Fn(\))2815 5370 y Fm(`)3066 5262 y Fo(\003)3107 5228 y(\015)3107 5288 y(\015)3107 5347 y(\015)3163 5254 y Fn(2)3163 5411 y(op)3241 5232 y Fo(o)3308 5202 y(\033)3405 5342 y Fs(:)p eop %%Page: 39 40 39 39 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(39)257 573 y(As)24 b(a)f(k)o(e)o(y)g(step,)g(we)h(no)n(w)f(con)l (v)o(ert)f(the)i(operator)f(norms)g(on)g(the)g(right)g(side)g(of)g (\(III.96\))h(into)257 693 y Fs(L)323 657 y Fn(2)363 693 y Fv(-norms)h(by)h(means)f(of)h(Inequality)f(\(III.11\))h(in)g (Theorem)f(III.1.)35 b(Indeed,)26 b(inserting)e(this)257 824 y(estimate)h(into)f(the)h Fs(`)983 788 y Fm(th)1079 824 y Fv(f)o(actor)h(on)f(the)g(right)g(side)g(of)g(\(III.96\),)h (pointwise)e(in)h Fs(K)3063 773 y Fn(\()p Fm(m)3152 785 y Fk(`)3183 773 y Fm(;n)3246 785 y Fk(`)3275 773 y Fn(\))3056 852 y Fm(`)3332 824 y Fv(a.e.,)257 944 y(and)g(inte)o(grating)e(o)o(v)o (er)h(the)h(v)n(ariables)f Fs(X)1698 908 y Fn(\()p Fm(p)1761 920 y Fk(`)1791 908 y Fm(;q)1843 920 y Fk(`)1872 908 y Fn(\))1903 944 y Fv(,)h(we)g(obtain)297 1091 y Fo(Z)352 1317 y Fm(B)408 1277 y Fk(m)462 1294 y(`)493 1277 y Fh(+)p Fk(n)578 1294 y(`)405 1338 y Fh(1)732 1160 y Fs(dK)873 1124 y Fn(\()p Fm(m)962 1136 y Fk(`)992 1124 y Fm(;n)1055 1136 y Fk(`)1085 1124 y Fn(\))p 643 1204 562 4 v 643 1295 a Fr(j)p Fs(K)761 1267 y Fn(\()p Fm(m)850 1279 y Fk(`)881 1267 y Fm(;n)944 1279 y Fk(`)973 1267 y Fn(\))1005 1295 y Fr(j)1033 1267 y Fn(3+2)p Fm(\026)1231 1227 y Fq(sup)1246 1308 y Fm(r)r Fl(2)p Fm(I)1395 1113 y Fo(\015)1395 1172 y(\015)1395 1232 y(\015)1453 1202 y(f)1450 1227 y Fs(W)1542 1242 y Fm(`)1575 1146 y Fo(\002)1617 1227 y Fs(z)t Fq(;)34 b Fs(r)s Fq(;)f Fs(\032K)1974 1176 y Fn(\()p Fm(m)2063 1188 y Fk(`)2094 1176 y Fm(;n)2157 1188 y Fk(`)2186 1176 y Fn(\))1967 1255 y Fm(`)2218 1146 y Fo(\003)2259 1113 y(\015)2259 1172 y(\015)2259 1232 y(\015)2315 1139 y Fn(2)2315 1296 y(op)3205 1227 y Fv(\(III.97\))380 1525 y Fr(\024)700 1457 y Fq(1)p 551 1502 348 4 v 551 1522 a Fo(p)p 650 1522 248 4 v 650 1606 a Fs(p)699 1559 y Fm(p)735 1571 y Fk(`)699 1634 y Fm(`)785 1606 y Fs(q)832 1559 y Fm(q)864 1571 y Fk(`)828 1634 y Fm(`)941 1389 y Fo(Z)996 1615 y Fm(B)1052 1574 y Fk(m)1106 1592 y(`)1137 1574 y Fh(+)p Fk(n)1222 1592 y(`)1049 1636 y Fh(1)1376 1457 y Fs(dK)1517 1421 y Fn(\()p Fm(m)1606 1433 y Fk(`)1636 1421 y Fm(;n)1699 1433 y Fk(`)1729 1421 y Fn(\))p 1287 1502 562 4 v 1287 1593 a Fr(j)p Fs(K)1405 1564 y Fn(\()p Fm(m)1494 1576 y Fk(`)1525 1564 y Fm(;n)1588 1576 y Fk(`)1617 1564 y Fn(\))1649 1593 y Fr(j)1677 1564 y Fn(3+2)p Fm(\026)1859 1410 y Fo(\015)1859 1470 y(\015)1859 1530 y(\015)1914 1525 y Fs(w)1984 1540 y Fm(m)2046 1552 y Fk(`)2076 1540 y Fn(+)p Fm(p)2167 1552 y Fk(`)2196 1540 y Fm(;n)2259 1552 y Fk(`)2289 1540 y Fn(+)p Fm(q)2376 1552 y Fk(`)2409 1525 y Fq([)p Fs(z)t Fq(;)i Fr(\001)17 b Fq(;)32 b Fs(k)2705 1484 y Fn(\()p Fm(m)2794 1496 y Fk(`)2825 1484 y Fn(\))2857 1525 y Fs(;)h Fr(\001)17 b Fq(;)3024 1498 y(~)3022 1525 y Fs(k)3076 1484 y Fn(\()p Fm(n)3146 1496 y Fk(`)3176 1484 y Fn(\))3207 1525 y Fs(;)34 b Fr(\001)3313 1444 y Fo(\003)3354 1410 y(\015)3354 1470 y(\015)3354 1530 y(\015)3409 1436 y Fn(2)3409 1594 y Fm(\026)381 1802 y Fq(=)700 1735 y(1)p 551 1779 348 4 v 551 1799 a Fo(p)p 650 1799 248 4 v 650 1883 a Fs(p)699 1836 y Fm(p)735 1848 y Fk(`)699 1911 y Fm(`)785 1883 y Fs(q)832 1836 y Fm(q)864 1848 y Fk(`)828 1911 y Fm(`)925 1688 y Fo(\015)925 1747 y(\015)925 1807 y(\015)980 1802 y Fs(w)1050 1817 y Fm(m)1112 1829 y Fk(`)1142 1817 y Fn(+)p Fm(p)1233 1829 y Fk(`)1262 1817 y Fm(;n)1325 1829 y Fk(`)1355 1817 y Fn(+)p Fm(q)1442 1829 y Fk(`)1475 1802 y Fq([)p Fs(z)t Fq(])1578 1688 y Fo(\015)1578 1747 y(\015)1578 1807 y(\015)1635 1714 y Fn(2)1635 1871 y Fm(\026)1703 1802 y Fs(:)257 2091 y Fv(This)24 b(estimate,)g(inserted)h(into)f(\(III.96\),)h(yields) 648 2324 y Fs(\032)708 2255 y Fk(d)p Fi(\000)p Fh(1)p 708 2268 111 3 v 748 2309 a(2)829 2283 y Fn(\()p Fm(M)7 b Fn(+)p Fm(N)g Fn(\))p Fl(\000)p Fn(1)1193 2239 y Fo(\015)1193 2299 y(\015)1248 2324 y Fs(V)1305 2339 y Fm(m;p;n;q)p 1305 2364 238 3 v 1543 2324 a Fr(k)1593 2283 y Fn(#)1593 2348 y Fm(\026)1708 2324 y Fr(\024)1420 b Fv(\(III.98\))814 2595 y Fq(2\()p Fs(L)22 b Fq(+)g(1\))17 b Fs(C)1268 2544 y Fn(\()p Fm(L)p Fn(+1\))1261 2622 y(\002)1481 2595 y Fs(\032)1531 2554 y Fn(\(1+)p Fm(\026)p Fn(\)\()p Fm(M)7 b Fn(+)p Fm(N)g Fn(\))p Fl(\000)p Fm(L)2158 2471 y(L)2118 2501 y Fo(Y)2122 2713 y Fm(`)p Fn(=1)2272 2420 y Fo(\015)2272 2479 y(\015)2328 2504 y Fs(w)2398 2519 y Fm(m)2460 2531 y Fk(`)2490 2519 y Fn(+)p Fm(p)2581 2531 y Fk(`)2610 2519 y Fm(;n)2673 2531 y Fk(`)2702 2519 y Fn(+)p Fm(q)2789 2531 y Fk(`)2823 2504 y Fq([)p Fs(z)t Fq(])2926 2420 y Fo(\015)2926 2479 y(\015)2982 2446 y Fn(#)2982 2543 y Fm(\026)p 2272 2572 774 4 v 2485 2592 a Fo(p)p 2585 2592 248 4 v 84 x Fs(p)2634 2629 y Fm(p)2670 2641 y Fk(`)2634 2704 y Fm(`)2720 2676 y Fs(q)2767 2629 y Fm(q)2799 2641 y Fk(`)2763 2704 y Fm(`)3077 2595 y Fs(:)3418 2898 y Fd(\003)404 3018 y Fv(W)-8 b(e)31 b(are)h(no)n(w)e(prepared)i(for)f (the)g(proof)g(of)h(the)e Ft(main)h(theor)l(em)g Fv(about)f(the)h (renormal-)257 3139 y(ization)24 b(transformation)g Fr(R)1249 3154 y Fm(\032)1290 3139 y Fv(,)g(namely)-6 b(,)24 b(the)h (codimension-1)e(contracti)n(vity)-6 b(.)257 3259 y Ff(Pr)n(oof)33 b(of)g(Theor)n(em)h(III.8.)54 b Fv(First,)34 b(we)f(apply)f(Lemma)g (III.10)h(to)f(\(III.85\))i(and)e(use)h(that)257 3299 y Fo(\000)303 3335 y Fm(m)p Fn(+)p Fm(p)362 3414 y(p)456 3299 y Fo(\001)529 3379 y Fr(\024)28 b Fq(2)683 3343 y Fm(m)p Fn(+)p Fm(p)840 3379 y Fv(.)j(This)24 b(yields)451 3613 y Fo(\015)451 3673 y(\015)527 3698 y Fq(^)-69 b Fs(w)577 3713 y Fm(M)s(;N)734 3698 y Fq([)p Fs(\020)8 b Fq(])839 3613 y Fo(\015)839 3673 y(\015)894 3639 y Fn(#)894 3737 y Fm(\026)1009 3698 y Fr(\024)1176 3573 y Fl(1)1139 3603 y Fo(X)1142 3814 y Fm(L)p Fn(=1)1300 3698 y Fq(2)17 b Fs(C)1436 3713 y Fn(\002)1511 3698 y Fq(\()p Fs(L)22 b Fq(+)g(1\))1839 3587 y Fo(\020)1908 3630 y Fs(C)1978 3645 y Fn(\002)p 1908 3675 129 4 v 1947 3766 a Fs(\032)2047 3587 y Fo(\021)2106 3609 y Fm(L)2175 3617 y Fo(\000)2221 3698 y Fq(2)17 b Fs(\032)2337 3657 y Fn(\(1+)p Fm(\026)p Fn(\))2528 3617 y Fo(\001)2574 3639 y Fm(M)7 b Fn(+)p Fm(N)3205 3698 y Fv(\(III.99\))623 3943 y Fo(X)462 4143 y Fk(m)516 4158 y Fh(1)551 4143 y(+)p Fk(:::)o Fh(+)p Fk(m)755 4160 y(L)802 4143 y Fh(=)p Fk(M)r(;)491 4203 y(n)529 4218 y Fh(1)564 4203 y(+)p Fk(:::)o Fh(+)p Fk(n)752 4220 y(L)798 4203 y Fh(=)p Fk(N)1121 3943 y Fo(X)976 4130 y Fk(p)1008 4145 y Fh(1)1043 4130 y Fk(;q)1090 4145 y Fh(1)1125 4130 y Fk(;:::;p)1252 4147 y(L)1298 4130 y(;q)1345 4147 y(L)1391 4130 y Fh(:)947 4191 y Fk(m)1001 4209 y(`)1032 4191 y Fh(+)p Fk(p)1111 4209 y(`)1140 4191 y Fh(+)p Fk(n)1225 4209 y(`)1255 4191 y Fh(+)p Fk(q)1330 4209 y(`)1360 4191 y Fi(\025)p Fh(1)1505 3913 y Fm(L)1465 3943 y Fo(Y)1469 4155 y Fm(`)p Fn(=1)1609 3897 y Fo(\032)1684 3927 y(\020)1811 3970 y Fq(2)p 1753 4015 165 4 v 1753 4044 a Fr(p)p 1836 4044 82 4 v 62 x Fs(p)1885 4121 y Fm(`)1928 3927 y Fo(\021)1987 3949 y Fm(p)2023 3961 y Fk(`)2041 3927 y Fo(\020)2165 3970 y Fq(2)p 2110 4015 160 4 v 2110 4044 a Fr(p)p 2193 4044 77 4 v 62 x Fs(q)2236 4121 y Fm(`)2279 3927 y Fo(\021)2339 3949 y Fm(q)2371 3961 y Fk(`)2404 3953 y Fo(\015)2404 4013 y(\015)2460 4037 y Fs(w)2530 4052 y Fm(m)2592 4064 y Fk(`)2622 4052 y Fn(+)p Fm(p)2713 4064 y Fk(`)2742 4052 y Fm(;n)2805 4064 y Fk(`)2835 4052 y Fn(+)p Fm(q)2922 4064 y Fk(`)2955 4037 y Fq([)p Fs(z)t Fq(])3058 3953 y Fo(\015)3058 4013 y(\015)3114 3979 y Fn(#)3114 4077 y Fm(\026)3177 3897 y Fo(\033)3274 4037 y Fs(:)257 4406 y Fv(Summing)24 b(o)o(v)o(er)f(the)i(indices)e Fs(m;)17 b(p;)g(n;)g(q)p 1321 4441 371 4 v 4 w Fv(,)24 b(as)h(required)g(in)f (\(III.85\),)h(we)f(deri)n(v)o(e)g(the)g(follo)n(w-)p eop %%Page: 40 41 40 40 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(40)257 573 y(ing)25 b(bound,)476 712 y Fo(\015)476 772 y(\015)531 797 y Fq(\()20 b(^)-69 b Fs(w)639 812 y Fm(M)s(;N)797 797 y Fq([)p Fs(\020)8 b Fq(]\))940 812 y Fm(M)f Fn(+)p Fm(N)g Fl(\025)p Fn(1)1226 797 y Fr(k)1276 750 y Fn(#)1276 825 y Fm(\026;\030)1428 797 y Fq(=)1626 702 y Fo(X)1557 914 y Fm(M)g Fn(+)p Fm(N)g Fl(\025)p Fn(1)1856 797 y Fs(\030)1904 756 y Fl(\000)p Fn(\()p Fm(M)g Fn(+)p Fm(N)g Fn(\))2227 712 y Fo(\015)2227 772 y(\015)2303 797 y Fq(^)-69 b Fs(w)2353 812 y Fm(M)s(;N)2510 797 y Fq([)p Fs(\020)8 b Fq(])p Fr(k)2665 756 y Fn(#)2665 822 y Fm(\026)3155 797 y Fv(\(III.100\))461 1132 y Fr(\024)104 b Fq(4)17 b Fs(C)778 1147 y Fn(\002)853 1132 y Fs(\032)903 1091 y Fn(\(1+)p Fm(\026)p Fn(\))1148 1007 y Fl(1)1111 1037 y Fo(X)1114 1249 y Fm(L)p Fn(=1)1255 1132 y Fq(\()p Fs(L)23 b Fq(+)f(1\))1584 991 y Fo(\022)1666 1064 y Fs(C)1736 1079 y Fn(\002)p 1666 1109 129 4 v 1706 1200 a Fs(\032)1805 991 y Fo(\023)1879 1014 y Fm(L)2017 1037 y Fo(X)1947 1249 y Fm(M)7 b Fn(+)p Fm(N)g Fl(\025)p Fn(1)2418 1037 y Fo(X)2257 1237 y Fk(m)2311 1252 y Fh(1)2346 1237 y(+)p Fk(:::)o Fh(+)p Fk(m)2550 1254 y(L)2597 1237 y Fh(=)p Fk(M)r(;)2286 1297 y(n)2324 1312 y Fh(1)2359 1297 y(+)p Fk(:::)o Fh(+)p Fk(n)2547 1314 y(L)2593 1297 y Fh(=)p Fk(N)2932 1037 y Fo(X)2788 1225 y Fk(p)2820 1240 y Fh(1)2855 1225 y Fk(;q)2902 1240 y Fh(1)2937 1225 y Fk(;:::)o(;p)3063 1241 y(L)3109 1225 y(;q)3156 1241 y(L)3203 1225 y Fh(:)2759 1285 y Fk(m)2813 1303 y(`)2843 1285 y Fh(+)p Fk(p)2922 1303 y(`)2952 1285 y Fh(+)p Fk(n)3037 1303 y(`)3067 1285 y Fh(+)p Fk(q)3142 1303 y(`)3172 1285 y Fi(\025)p Fh(1)682 1412 y Fm(L)642 1442 y Fo(Y)646 1655 y Fm(`)p Fn(=1)786 1397 y Fo(\032)861 1426 y(\020)956 1470 y Fq(2)17 b Fs(\030)p 930 1514 165 4 v 930 1543 a Fr(p)p 1013 1543 82 4 v 62 x Fs(p)1062 1620 y Fm(`)1105 1426 y Fo(\021)1165 1449 y Fm(p)1201 1461 y Fk(`)1251 1426 y Fo(\020)1344 1470 y Fq(2)g Fs(\030)p 1320 1514 160 4 v 1320 1543 a Fr(p)p 1403 1543 77 4 v 62 x Fs(q)1446 1620 y Fm(`)1490 1426 y Fo(\021)1549 1449 y Fm(q)1581 1461 y Fk(`)1642 1537 y Fs(\030)1690 1496 y Fl(\000)p Fn(\()p Fm(m)1834 1508 y Fk(`)1864 1496 y Fn(+)p Fm(p)1955 1508 y Fk(`)1985 1496 y Fn(+)p Fm(n)2083 1508 y Fk(`)2113 1496 y Fn(+)p Fm(q)2200 1508 y Fk(`)2229 1496 y Fn(\))2283 1452 y Fo(\015)2283 1512 y(\015)2338 1537 y Fs(w)2408 1552 y Fm(m)2470 1564 y Fk(`)2500 1552 y Fn(+)p Fm(p)2591 1564 y Fk(`)2621 1552 y Fm(;n)2684 1564 y Fk(`)2713 1552 y Fn(+)p Fm(q)2800 1564 y Fk(`)2833 1537 y Fq([)p Fs(z)t Fq(])2936 1452 y Fo(\015)2936 1512 y(\015)2993 1479 y Fn(#)2993 1576 y Fm(\026)3056 1397 y Fo(\033)461 1861 y Fr(\024)104 b Fq(4)17 b Fs(C)778 1876 y Fn(\002)853 1861 y Fs(\032)903 1820 y Fn(\(1+)p Fm(\026)p Fn(\))1148 1737 y Fl(1)1111 1767 y Fo(X)1114 1978 y Fm(L)p Fn(=1)1255 1861 y Fq(\()p Fs(L)23 b Fq(+)f(1\))1584 1721 y Fo(\022)1666 1794 y Fs(C)1736 1809 y Fn(\002)p 1666 1838 129 4 v 1706 1930 a Fs(\032)1805 1721 y Fo(\023)1879 1743 y Fm(L)642 2044 y Fo(\032)787 2090 y(X)733 2301 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)1000 2044 y Fo(\022)1131 2060 y Fm(m)1090 2090 y Fo(X)1099 2299 y Fm(p)p Fn(=0)1251 2074 y Fo(\020)1330 2117 y Fq(2)17 b Fs(\030)p 1320 2162 132 4 v 1320 2191 a Fr(p)p 1403 2191 49 4 v 62 x Fs(p)1462 2074 y Fo(\021)1522 2096 y Fm(p)1561 2044 y Fo(\023)g(\022)1792 2060 y Fm(n)1741 2090 y Fo(X)1751 2299 y Fm(q)r Fn(=0)1902 2074 y Fo(\020)1980 2117 y Fq(2)g Fs(\030)p 1971 2162 130 4 v 1971 2191 a Fr(p)p 2054 2191 47 4 v 62 x Fs(q)2111 2074 y Fo(\021)2170 2096 y Fm(q)2209 2044 y Fo(\023)2298 2185 y Fs(\030)2346 2144 y Fl(\000)p Fn(\()p Fm(m)p Fn(+)p Fm(n)p Fn(\))2642 2185 y Fr(k)p Fs(w)2762 2200 y Fm(m;n)2891 2185 y Fq([)p Fs(z)t Fq(])p Fr(k)3044 2144 y Fn(#)3044 2209 y Fm(\026)3107 2044 y Fo(\033)3182 2067 y Fm(L)461 2519 y Fr(\024)104 b Fq(4)17 b Fs(C)778 2534 y Fn(\002)853 2519 y Fs(\032)903 2478 y Fn(\(1+)p Fm(\026)p Fn(\))1148 2394 y Fl(1)1111 2424 y Fo(X)1114 2636 y Fm(L)p Fn(=1)1255 2519 y Fq(\()p Fs(L)23 b Fq(+)f(1\))1584 2378 y Fo(\022)1666 2452 y Fs(C)1736 2467 y Fn(\002)p 1666 2496 129 4 v 1706 2587 a Fs(\032)1805 2378 y Fo(\023)1879 2401 y Fm(L)1947 2519 y Fs(A)2020 2478 y Fn(2)p Fm(L)2124 2408 y Fo(\020)2184 2434 y(\015)2184 2494 y(\015)2239 2519 y Fq(\()p Fs(w)2347 2534 y Fm(m;n)2476 2519 y Fq([)p Fs(z)t Fq(]\))2617 2534 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)2872 2434 y Fo(\015)2872 2494 y(\015)2928 2558 y Fm(\026;\030)3027 2408 y Fo(\021)3087 2431 y Fm(L)3161 2519 y Fs(;)257 2812 y Fv(where)748 2984 y Fs(A)52 b Fq(:=)1066 2860 y Fl(1)1029 2890 y Fo(X)1038 3099 y Fm(p)p Fn(=0)1189 2874 y Fo(\020)1268 2917 y Fq(2)17 b Fs(\030)p 1259 2962 132 4 v 1259 2991 a Fr(p)p 1342 2991 49 4 v 62 x Fs(p)1401 2874 y Fo(\021)1460 2896 y Fm(p)1553 2984 y Fr(\024)1719 2860 y Fl(1)1683 2890 y Fo(X)1692 3099 y Fm(p)p Fn(=0)1843 2904 y Fo(\000)1889 2984 y Fq(2)g Fs(\030)2003 2904 y Fo(\001)2047 2926 y Fm(p)2140 2984 y Fq(=)2417 2917 y(1)p 2278 2962 328 4 v 2278 3053 a(1)44 b Fr(\000)h Fq(2)17 b Fs(\030)2638 2984 y(;)490 b Fv(\(III.101\))257 3255 y(since)38 b Fs(\030)56 b Fq(=)727 3193 y Fr(p)p 810 3193 51 4 v 62 x Fs(\032)q(=)p Fq(\(4)p Fs(C)1067 3270 y Fn(\002)1125 3255 y Fq(\))c Fr(\024)g Fq(1)p Fs(=)p Fq(4)p Fv(,)41 b(by)c(assumption.)69 b(Note)37 b(that)h(in)f(\(III.101\))h(we)h(ha)n(v)o(e)257 3375 y(dropped)33 b(the)h(f)o(actor)f Fs(p)1086 3339 y Fl(\000)p Fm(p=)p Fn(2)1285 3375 y Fv(gained)f(in)h(Theorem)h(III.1.) 56 b(Our)34 b(assumption)d(also)i(insures)257 3495 y(that)515 3731 y Fs(B)58 b Fq(:=)972 3664 y Fs(C)1042 3679 y Fn(\002)p 812 3708 449 4 v 812 3799 a Fs(\032)17 b Fq(\(1)22 b Fr(\000)g Fq(2)p Fs(\030)5 b Fq(\))1222 3771 y Fn(2)1293 3646 y Fo(\015)1293 3706 y(\015)1348 3731 y Fq(\()p Fs(w)1456 3746 y Fm(m;n)1585 3731 y Fq([)p Fs(z)t Fq(]\))1726 3746 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)1981 3646 y Fo(\015)1981 3706 y(\015)2037 3770 y Fm(\026;\030)2189 3731 y Fr(\024)2329 3664 y Fq(4)17 b Fs(C)2465 3679 y Fn(\002)2540 3664 y Fs(\016)p 2329 3708 258 4 v 2433 3799 a(\032)2650 3731 y Fr(\024)2790 3664 y Fq(1)p 2790 3708 49 4 v 2790 3799 a(2)2870 3731 y Fs(;)258 b Fv(\(III.102\))257 4001 y(since)27 b Fs(\016)34 b Fr(\024)e Fs(")721 4016 y Fn(0)790 4001 y Fq(:=)f(\(8)17 b Fs(C)1098 4016 y Fn(\002)1156 4001 y Fq(\))1194 3965 y Fl(\000)p Fn(1)1289 4001 y Fs(\032)p Fv(.)36 b(Thus)25 b(the)i(geometric)f(series)g(in)g(the)h(last)e(line)h (of)h(\(III.100\))257 4122 y(is)e(con)l(v)o(er)n(gent.)30 b(W)-8 b(e)25 b(obtain)372 4280 y Fl(1)335 4310 y Fo(X)338 4521 y Fm(L)p Fn(=1)479 4404 y Fq(\()p Fs(L)e Fq(+)f(1\))17 b Fs(B)887 4363 y Fm(L)991 4404 y Fq(=)1169 4337 y Fs(d)p 1129 4381 130 4 v 1129 4473 a(dB)1269 4264 y Fo(\024)1375 4280 y Fl(1)1338 4310 y Fo(X)1341 4521 y Fm(L)p Fn(=0)1499 4404 y Fs(B)1578 4363 y Fm(L)1630 4264 y Fo(\025)1733 4404 y Fr(\000)50 b Fq(1)i(=)2258 4337 y(1)p 2100 4381 365 4 v 2100 4473 a(\(1)22 b Fr(\000)g Fs(B)5 b Fq(\))2425 4444 y Fn(2)2524 4404 y Fr(\000)51 b Fq(1)h Fr(\024)h Fq(8)17 b Fs(B)27 b(:)78 b Fv(\(III.103\))257 4697 y(Inserting)24 b(\(III.101\)\226\(III.103\))i(into)e(\(III.100\),)h(we)g(arri)n(v)o(e) f(at)280 4836 y Fo(\015)280 4896 y(\015)335 4921 y Fq(\()c(^)-69 b Fs(w)443 4936 y Fm(M)s(;N)601 4921 y Fq([)p Fs(\020)8 b Fq(]\))744 4936 y Fm(M)f Fn(+)p Fm(N)g Fl(\025)p Fn(1)1030 4921 y Fr(k)1080 4874 y Fn(#)1080 4949 y Fm(\026;\030)1263 4921 y Fr(\024)83 b Fq(32)17 b Fs(C)1608 4936 y Fn(\002)1683 4921 y Fs(\032)1733 4880 y Fn(\(1+)p Fm(\026)p Fn(\))1942 4921 y Fs(B)57 b Fq(=)c(128)17 b Fs(C)2443 4880 y Fn(2)2436 4946 y(\002)2510 4921 y Fs(\032)2560 4880 y Fm(\026)2629 4836 y Fo(\015)2629 4896 y(\015)2684 4921 y Fq(\()p Fs(w)2792 4936 y Fm(m;n)2921 4921 y Fq([)p Fs(z)t Fq(]\))3062 4936 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)3317 4836 y Fo(\015)3317 4896 y(\015)3373 4863 y Fn(#)3373 4960 y Fm(\026;\030)1263 5143 y Fr(\024)1434 5075 y Fs(\016)p 1433 5120 49 4 v 1433 5211 a Fq(2)1514 5143 y Fs(;)1614 b Fv(\(III.104\))257 5388 y(as)25 b Fq(128)p Fs(C)589 5352 y Fn(2)582 5413 y(\002)640 5388 y Fs(\032)690 5352 y Fm(\026)765 5388 y Fq(=)j(128)p Fs(C)1093 5352 y Fn(2)1086 5413 y(\002)1144 5388 y Fq(\(16)17 b Fs(C)1367 5403 y Fn(\002)1425 5388 y Fq(\))1463 5352 y Fl(\000)p Fn(2)1585 5388 y Fq(=)27 b(1)p Fs(=)p Fq(2)p Fv(.)p eop %%Page: 41 42 41 41 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(41)404 573 y(Ne)o(xt,)23 b(we)h(estimate)42 b Fq(^)-69 b Fs(w)1210 588 y Fn(0)p Fm(;)p Fn(0)1304 573 y Fq([)p Fs(\020)8 b Fq(])p Fv(,)24 b(which,)f(according)g(to)h(\(III.81\))g (and)f(\(III.83\)\226\(III.85\),)i(is)257 693 y(gi)n(v)o(en)f(by)327 938 y Fq(^)-69 b Fs(w)377 953 y Fn(0)p Fm(;)p Fn(0)471 938 y Fq([)p Fs(\020)8 b Fq(;)33 b Fs(r)s Fq(])49 b(=)g Fs(\032)907 897 y Fl(\000)p Fn(1)1018 938 y Fs(w)1088 953 y Fn(0)p Fm(;)p Fn(0)1182 938 y Fq([)p Fs(z)t Fq(;)34 b Fs(\032)17 b(r)s Fq(])h(+)h Fs(\032)1623 897 y Fl(\000)p Fn(1)1771 814 y Fl(1)1734 844 y Fo(X)1737 1055 y Fm(L)p Fn(=2)1878 938 y Fq(\()p Fr(\000)p Fq(1\))2080 897 y Fm(L)p Fl(\000)p Fn(1)2393 844 y Fo(X)2249 1031 y Fk(p)2281 1046 y Fh(1)2316 1031 y Fk(;q)2363 1046 y Fh(1)2398 1031 y Fk(;:::)o(;p)2524 1048 y(L)2570 1031 y(;q)2617 1048 y(L)2664 1031 y Fh(:)2343 1092 y Fk(p)2375 1109 y(`)2405 1092 y Fh(+)p Fk(q)2480 1109 y(`)2510 1092 y Fi(\025)p Fh(1)2708 938 y Fs(V)2765 953 y Fn(0)p Fm(;p;)p Fn(0)p Fm(;q)p 2765 978 203 3 v 2968 938 a Fq([)p Fs(r)s Fq(])9 b Fs(:)50 b Fv(\(III.105\))257 1254 y(The)36 b(function)f Fs(V)873 1269 y Fn(0)p Fm(;p;)p Fn(0)p Fm(;q)p 873 1294 V 1076 1254 a Fq([)p Fs(r)s Fq(])g Fv(and)h(its)f(partial)h(deri)n(v)n (ati)n(v)o(e)d Fs(@)2289 1269 y Fm(r)2327 1254 y Fs(V)2384 1269 y Fn(0)p Fm(;p;)p Fn(0)p Fm(;q)p 2384 1294 V 2587 1254 a Fq([)p Fs(r)s Fq(])j Fv(are)g(estimated)f(as)h(in)257 1375 y(Eqs.)25 b(\(III.95\)\226\(III.98\),)g(with)f(the)h(result)f (that)533 1652 y Fs(\032)583 1611 y Fl(\000)p Fn(1)700 1567 y Fo(\015)700 1627 y(\015)755 1652 y Fs(V)812 1667 y Fn(0)p Fm(;p;)p Fn(0)p Fm(;q)p 812 1692 V 1015 1652 a Fr(k)1065 1611 y Fn(#)1065 1677 y Fm(\026)1180 1652 y Fr(\024)53 b Fq(2\()p Fs(L)23 b Fq(+)f(1\))17 b Fs(C)1765 1601 y Fn(\()p Fm(L)p Fn(+1\))1758 1679 y(\002)1978 1652 y Fs(\032)2028 1611 y Fl(\000)p Fm(L)2208 1527 y(L)2168 1557 y Fo(Y)2172 1769 y Fm(`)p Fn(=1)2322 1476 y Fo(\015)2322 1536 y(\015)2377 1561 y Fs(w)2447 1576 y Fm(p)2483 1588 y Fk(`)2513 1576 y Fm(;q)2565 1588 y Fk(`)2598 1561 y Fq([)p Fs(z)t Fq(])2701 1476 y Fo(\015)2701 1536 y(\015)2757 1503 y Fn(#)2757 1600 y Fm(\026)p 2322 1629 499 4 v 2397 1649 a Fo(p)p 2497 1649 248 4 v 84 x Fs(p)2546 1686 y Fm(p)2582 1698 y Fk(`)2546 1761 y Fm(`)2632 1733 y Fs(q)2679 1686 y Fm(q)2711 1698 y Fk(`)2675 1761 y Fm(`)2852 1652 y Fs(:)276 b Fv(\(III.106\))257 1937 y(Writing)608 1912 y Fo(b)592 1937 y Fs(T)14 b Fq([)p Fs(\020)8 b Fq(;)17 b Fs(r)s Fq(])26 b(:=)48 b(^)-69 b Fs(w)1086 1952 y Fn(0)p Fm(;)p Fn(0)1180 1937 y Fq([)p Fs(\020)8 b Fq(;)17 b Fs(r)s Fq(])k Fr(\000)42 b Fq(^)-69 b Fs(w)1566 1952 y Fn(0)p Fm(;)p Fn(0)1661 1937 y Fq([)p Fs(\020)8 b Fq(;)17 b(0])p Fv(,)23 b(we)i(therefore)h(\002nd)f(that)313 2037 y Fo(\015)313 2097 y(\015)16 b(b)368 2122 y Fs(T)e Fq([)p Fs(\020)8 b Fq(;)17 b Fs(r)s Fq(])k Fr(\000)h Fs(r)802 2037 y Fo(\015)802 2097 y(\015)857 2161 y Fl(T)998 2122 y Fq(=)80 b(sup)1169 2203 y Fm(r)r Fl(2)p Fm(I)1318 2037 y Fo(\014)1318 2097 y(\014)1351 2122 y Fs(@)1402 2137 y Fm(r)1460 2122 y Fq(^)-69 b Fs(w)1510 2137 y Fn(0)p Fm(;)p Fn(0)1604 2122 y Fq([)p Fs(\020)8 b Fq(;)33 b Fs(r)s Fq(])38 b Fr(\000)i Fq(1)2020 2037 y Fo(\014)2020 2097 y(\014)3155 2122 y Fv(\(III.107\))396 2414 y Fr(\024)83 b Fq(sup)572 2495 y Fm(r)r Fl(2)p Fm(I)720 2329 y Fo(\014)720 2389 y(\014)753 2414 y Fs(@)804 2429 y Fm(r)859 2414 y Fs(w)929 2429 y Fn(0)p Fm(;)p Fn(0)1023 2414 y Fq([)p Fs(z)t Fq(;)34 b Fs(\032)17 b(r)s Fq(])39 b Fr(\000)g Fq(1)1505 2329 y Fo(\014)1505 2389 y(\014)1582 2414 y Fq(+)44 b Fs(\032)1752 2373 y Fl(\000)p Fn(1)1900 2289 y Fl(1)1864 2319 y Fo(X)1867 2531 y Fm(L)p Fn(=2)2179 2319 y Fo(X)2034 2507 y Fk(p)2066 2522 y Fh(1)2101 2507 y Fk(;q)2148 2522 y Fh(1)2183 2507 y Fk(;:::)o(;p)2309 2524 y(L)2356 2507 y(;q)2403 2524 y(L)2449 2507 y Fh(:)2128 2568 y Fk(p)2160 2585 y(`)2190 2568 y Fh(+)p Fk(q)2265 2585 y(`)2295 2568 y Fi(\025)p Fh(1)2493 2414 y Fq(sup)2508 2495 y Fm(r)r Fl(2)p Fm(I)2657 2329 y Fo(\014)2657 2389 y(\014)2690 2414 y Fs(@)2741 2429 y Fm(r)2779 2414 y Fs(V)2836 2429 y Fn(0)p Fm(;p;)p Fn(0)p Fm(;q)p 2836 2454 203 3 v 3039 2414 a Fq([)p Fs(r)s Fq(])3140 2329 y Fo(\014)3140 2389 y(\014)3195 2414 y Fs(:)396 2792 y Fr(\024)556 2707 y Fo(\015)556 2767 y(\015)612 2792 y Fs(T)14 b Fq([)p Fs(z)t Fq(;)j Fs(r)s Fq(])22 b Fr(\000)h Fs(r)1046 2707 y Fo(\015)1046 2767 y(\015)1101 2831 y Fl(T)1205 2792 y Fq(+)44 b(2)17 b Fs(C)1461 2807 y Fn(\002)1589 2667 y Fl(1)1553 2697 y Fo(X)1556 2909 y Fm(L)p Fn(=2)1697 2792 y Fq(\()p Fs(L)22 b Fq(+)g(1\))2025 2681 y Fo(\020)2094 2724 y Fs(C)2164 2739 y Fn(\002)p 2094 2769 129 4 v 2133 2860 a Fs(\032)2233 2681 y Fo(\021)2292 2704 y Fm(L)2361 2651 y Fo(\032)2488 2697 y(X)2452 2908 y Fm(p)p Fn(+)p Fm(q)r Fl(\025)p Fn(1)2683 2707 y Fo(\015)2683 2767 y(\015)2739 2792 y Fs(w)2809 2807 y Fm(p;q)2902 2792 y Fq([)p Fs(z)t Fq(])3005 2707 y Fo(\015)3005 2767 y(\015)3061 2733 y Fn(#)3061 2831 y Fm(\026)3124 2651 y Fo(\033)3199 2674 y Fm(L)396 3127 y Fr(\024)556 3042 y Fo(\015)556 3102 y(\015)612 3127 y Fs(T)14 b Fq([)p Fs(z)t Fq(;)j Fs(r)s Fq(])22 b Fr(\000)h Fs(r)1046 3042 y Fo(\015)1046 3102 y(\015)1101 3166 y Fl(T)1205 3127 y Fq(+)44 b(2)17 b Fs(C)1461 3142 y Fn(\002)1589 3002 y Fl(1)1553 3032 y Fo(X)1556 3244 y Fm(L)p Fn(=2)1697 3127 y Fq(\()p Fs(L)22 b Fq(+)g(1\))2025 3016 y Fo(\020)2094 3059 y Fs(C)2164 3074 y Fn(\002)2239 3059 y Fs(\030)p 2094 3104 193 4 v 2165 3195 a(\032)2297 3016 y Fo(\021)2356 3038 y Fm(L)2425 3016 y Fo(\020)2485 3042 y(\015)2485 3102 y(\015)2540 3127 y Fq(\()p Fs(w)2648 3142 y Fm(m;n)2777 3127 y Fq([)p Fs(z)t Fq(]\))2918 3142 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)3173 3042 y Fo(\015)3173 3102 y(\015)3228 3068 y Fn(#)3228 3166 y Fm(\026;\030)3328 3016 y Fo(\021)3388 3038 y Fm(L)396 3423 y Fr(\024)556 3339 y Fo(\015)556 3398 y(\015)612 3423 y Fs(T)14 b Fq([)p Fs(z)t Fq(;)j Fs(r)s Fq(])22 b Fr(\000)h Fs(r)1046 3339 y Fo(\015)1046 3398 y(\015)1101 3462 y Fl(T)1205 3423 y Fq(+)44 b(24)17 b Fs(C)1510 3438 y Fn(\002)1585 3313 y Fo(\020)1654 3356 y Fs(C)1724 3371 y Fn(\002)1800 3356 y Fs(\030)p 1654 3400 V 1726 3492 a(\032)1874 3339 y Fo(\015)1874 3398 y(\015)1929 3423 y Fq(\()p Fs(w)2037 3438 y Fm(m;n)2166 3423 y Fq([)p Fs(z)t Fq(]\))2307 3438 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)2562 3339 y Fo(\015)2562 3398 y(\015)2618 3365 y Fn(#)2618 3462 y Fm(\026;\030)2717 3313 y Fo(\021)2777 3335 y Fn(2)396 3681 y Fr(\024)83 b Fs(")44 b Fq(+)777 3613 y Fs(\016)p 777 3658 49 4 v 777 3749 a Fq(2)857 3681 y Fs(;)257 3891 y Fv(where)d(we)f(are)h(using)e(that)1304 3816 y Fo(P)1409 3843 y Fl(1)1409 3920 y Fm(L)p Fn(=2)1552 3891 y Fq(\()p Fs(L)33 b Fq(+)h(1\))p Fs(D)1970 3855 y Fm(L)2077 3891 y Fr(\024)56 b Fq(12)p Fs(D)2392 3855 y Fn(2)2431 3891 y Fv(,)44 b(for)c Fs(D)j Fv(de\002ned)d(by)g Fs(D)58 b Fq(:=)257 4022 y Fs(C)327 4037 y Fn(\002)386 4022 y Fs(\030)5 b(\032)484 3986 y Fl(\000)p Fn(1)578 4022 y Fr(k)p Fq(\()p Fs(w)736 4037 y Fm(m;n)865 4022 y Fq([)p Fs(z)t Fq(]\))1006 4037 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)1261 4022 y Fr(k)1311 3974 y Fn(#)1311 4048 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)1593 4022 y Fr(\024)28 b Fs(C)1768 4037 y Fn(\002)1827 4022 y Fs(\030)5 b(\032)1925 3986 y Fl(\000)p Fn(1)2019 4022 y Fs(")2065 4037 y Fn(0)2132 4022 y Fq(=)27 b Fs(\030)5 b(=)p Fq(8)27 b Fr(\024)h Fq(1)p Fs(=)p Fq(2)p Fv(,)20 b(and)e Fq(24)p Fs(C)3042 3986 y Fn(3)3035 4047 y(\002)3094 4022 y Fs(\030)3142 3986 y Fn(2)3181 4022 y Fs(")3227 4037 y Fn(0)3266 4022 y Fs(\032)3316 3986 y Fl(\000)p Fn(2)3438 4022 y Fq(=)257 4142 y(3)p Fs(C)383 4106 y Fn(2)376 4167 y(\002)435 4142 y Fs(\030)483 4106 y Fn(2)522 4142 y Fs(\032)572 4106 y Fl(\000)p Fn(1)694 4142 y Fq(=)28 b(3)p Fs(=)p Fq(16)e Fr(\024)j Fq(1)p Fs(=)p Fq(2)p Fv(,)24 b(similar)f(to)i(\(III.103\).) 404 4273 y(In)31 b(the)g(third)g(step)g(we)g(set)1392 4248 y Fo(b)1372 4273 y Fs(E)6 b Fq([)p Fs(\020)i Fq(])39 b(:=)60 b(^)-69 b Fs(w)1807 4288 y Fn(0)p Fm(;)p Fn(0)1901 4273 y Fq([)p Fs(\020)8 b Fq(;)17 b(0])p Fv(.)49 b(Since)31 b Fs(E)6 b Fq([)p Fs(z)t Fq(])41 b(=)f Fs(w)2833 4288 y Fn(0)p Fm(;)p Fn(0)2927 4273 y Fq([)p Fs(z)t Fq(;)17 b(0])31 b Fv(and)h Fs(\020)46 b Fq(=)257 4393 y Fs(E)329 4408 y Fm(\032)370 4393 y Fq([)p Fs(z)t Fq(])28 b(=)g Fs(\032)655 4357 y Fl(\000)p Fn(1)750 4393 y Fs(E)6 b Fq([)p Fs(z)t Fq(])p Fv(,)25 b(Eq.)g(\(III.105\))g(yields)663 4619 y Fo(b)644 4644 y Fs(E)6 b Fq([)p Fs(\020)i Fq(])52 b(=)g Fs(\020)f Fq(+)45 b Fs(\032)1272 4603 y Fl(\000)p Fn(1)1419 4520 y Fl(1)1383 4550 y Fo(X)1386 4761 y Fm(L)p Fn(=2)1527 4644 y Fq(\()p Fr(\000)p Fq(1\))1729 4603 y Fm(L)p Fl(\000)p Fn(1)2042 4550 y Fo(X)1898 4737 y Fk(p)1930 4752 y Fh(1)1964 4737 y Fk(;q)2011 4752 y Fh(1)2046 4737 y Fk(;:::;p)2173 4754 y(L)2219 4737 y(;q)2266 4754 y(L)2313 4737 y Fh(:)1991 4798 y Fk(p)2023 4815 y(`)2053 4798 y Fh(+)p Fk(q)2128 4815 y(`)2158 4798 y Fi(\025)p Fh(1)2357 4644 y Fs(V)2414 4659 y Fn(0)p Fm(;p;)p Fn(0)p Fm(;q)p 2414 4684 203 3 v 2617 4644 a Fq([0])22 b Fs(:)386 b Fv(\(III.108\))257 4954 y(Inserting)24 b(the)h(estimates)f (\(III.106\)\226\(III.107\),)h(we)g(\002nally)f(obtain)415 5087 y Fo(\014)415 5147 y(\014)c(b)448 5172 y Fs(E)6 b Fq([)p Fs(\020)i Fq(])38 b Fr(\000)h Fs(\020)836 5087 y Fo(\014)836 5147 y(\014)921 5172 y Fr(\024)53 b Fq(24)17 b Fs(C)1236 5187 y Fn(\002)1311 5062 y Fo(\020)1380 5105 y Fs(C)1450 5120 y Fn(\002)1526 5105 y Fs(\030)p 1380 5149 193 4 v 1452 5240 a(\032)1600 5087 y Fo(\015)1600 5147 y(\015)1655 5172 y Fq(\()p Fs(w)1763 5187 y Fm(m;n)1892 5172 y Fq([)p Fs(z)t Fq(]\))2033 5187 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)2288 5087 y Fo(\015)2288 5147 y(\015)2344 5114 y Fn(#)2344 5211 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)2598 5062 y Fo(\021)2658 5084 y Fn(2)2750 5172 y Fr(\024)2890 5105 y Fs(\016)p 2890 5149 49 4 v 2890 5240 a Fq(2)2970 5172 y Fs(;)158 b Fv(\(III.109\))257 5388 y(which)25 b(completes)f(the)g(proof.)2076 b Fd(\003)p eop %%Page: 42 43 42 42 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(42)257 573 y Fe(III.4)118 b(Construction)31 b(of)e(Eigen)-5 b(v)o(alues)30 b(and)h(Eigen)-5 b(v)o(ectors)257 760 y Fv(In)36 b(this)e(section)g(we)h(apply)g(the)g(renormalization)f (transformation)g Fr(R)2809 775 y Fm(\032)2884 760 y Fv(to)h(Hamiltonian)257 881 y(operators)25 b Fs(H)35 b Fr(2)28 b(B)s Fq([)p Fr(H)1042 896 y Fn(red)1145 881 y Fq(])d Fv(of)g(the)g(form)1259 1101 y Fs(H)60 b Fq(=)53 b Fs(T)14 b Fq([)p Fs(H)1708 1116 y Fm(f)1753 1101 y Fq(])50 b(+)f Fs(W)64 b Fr(\000)50 b Fs(E)28 b Fr(\001)22 b Fp(1)g Fs(;)662 b Fv(\(III.110\))257 1321 y(assumed)33 b(to)g(be)h(small)e(perturbations)g(of)i Fs(H)8 b Fq(\()p Fs(w)p 1925 1337 73 4 v 1997 1345 a Fn(0)2036 1321 y Fq(\))44 b(=)f Fs(H)2318 1336 y Fm(f)2392 1321 y Fr(\000)29 b Fs(z)t Fv(,)36 b(in)d(the)g(sense)h(that)f Fs(H)41 b Fv(is)257 1441 y(represented)25 b(as)g Fs(H)36 b Fq(=)27 b Fs(H)8 b Fq(\()p Fs(w)p 1196 1457 V 2 w Fq(\))p Fv(,)25 b(with)1481 1689 y Fs(w)p 1481 1705 V 55 w Fr(2)53 b(D)1805 1578 y Fo(\020)1874 1621 y Fs(")1920 1636 y Fn(0)p 1874 1666 86 4 v 1893 1757 a Fq(2)1986 1689 y Fs(;)2067 1621 y(")2113 1636 y Fn(0)p 2067 1666 V 2086 1757 a Fq(2)2162 1578 y Fo(\021)2244 1689 y Fs(;)884 b Fv(\(III.111\))257 1951 y(where)30 b Fs(")576 1966 y Fn(0)650 1951 y Fq(:=)36 b(\(8)17 b Fs(C)963 1966 y Fn(\002)1021 1951 y Fq(\))1059 1915 y Fl(\000)p Fn(1)1153 1951 y Fq(\(16)g Fs(C)1376 1966 y Fn(\002)1434 1951 y Fq(\))1472 1915 y Fl(\000)p Fn(2)p Fm(=\026)1644 1951 y Fv(,)30 b(as)f(in)g(Theorem)g(III.8.)43 b(In)29 b(Eqs.)g(\(III.59\)\226\(III.60\))257 2071 y(it)c(is)f(sho)n (wn)g(that)g(\(III.111\))h(implies)e(that)1340 2248 y Fo(\015)1340 2307 y(\015)1395 2332 y Fs(w)p 1395 2348 73 4 v 24 w Fr(\000)g Fs(w)p 1589 2348 V 1662 2356 a Fn(0)1701 2248 y Fo(\015)1701 2307 y(\015)1756 2371 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(0)2063 2332 y Fr(\024)2203 2265 y Fq(3)17 b Fs(")2315 2280 y Fn(0)p 2203 2309 151 4 v 2254 2401 a Fq(2)2386 2332 y Fs(:)742 b Fv(\(III.112\))257 2577 y(The)25 b(main)f(result)h(of)g(this)e(section)i(is)f(the)h(follo) n(wing)d(theorem.)257 2780 y Ff(Theor)n(em)36 b(III.12)e (\(Construction)h(of)g(eigen)l(v)o(alues)f(and)h(eigen)l(v)o(ectors\).) 48 b Ft(F)l(ix)33 b Fs(\026)45 b(>)h Fq(0)p Ft(,)257 2900 y(and)26 b(c)o(hoose)f Fs(\032;)17 b(\030)5 b(;)17 b(")966 2915 y Fn(0)1034 2900 y Fs(>)29 b Fq(0)d Ft(suf)n(\002ciently)f (small)g(suc)o(h)g(that)g(\(III.82\))h(holds,)f(for)h(all)f Fq(0)k Fr(\024)h Fs(")g Fr(\024)257 3021 y Fs(")303 3036 y Fn(0)369 3021 y Ft(and)c Fq(0)k Fr(\024)h Fs(\016)j Fr(\024)d Fs(")963 3036 y Fn(0)1002 3021 y Ft(.)36 b(Suppose)25 b(that)h Fs(w)p 1603 3037 73 4 v 33 w Fr(2)31 b(D)s Fq(\()p Fs(")1967 3036 y Fn(0)2005 3021 y Fs(=)p Fq(2)17 b Fs(;)33 b(")2226 3036 y Fn(0)2265 3021 y Fs(=)p Fq(2\))p Ft(.)i(Then)26 b(the)g(comple)n(x)h(number)257 3141 y Fs(e)302 3157 y Fn(\(0)p Fm(;)p Fl(1)p Fn(\))515 3141 y Fr(2)h Fs(D)690 3157 y Fn(1)p Fm(=)p Fn(2)825 3141 y Ft(de\002ned)c(in)h(\(III.126\))g (is)f(an)h(eig)o(en)l(value)f(of)h Fs(H)8 b Fq(\()p Fs(w)p 2502 3157 V 2 w Fq(\))p Ft(,)24 b(in)h(the)f(sense)h(that)1258 3361 y Fq(dim)15 b(Ker)p Fr(f)p Fs(H)8 b Fq(\()p Fs(w)p 1771 3377 V 2 w Fq([)p Fs(e)1915 3377 y Fn(\(0)p Fm(;)p Fl(1)p Fn(\))2099 3361 y Fq(]\))p Fr(g)53 b(\025)g Fq(1)22 b Fs(:)660 b Fv(\(III.113\))257 3581 y Ft(Mor)l(eo)o(ver)-11 b(,)30 b(the)f(vector)g Fq(\011)1194 3597 y Fn(\()p Fl(1)p Fn(\))1358 3581 y Fr(2)36 b(H)1544 3596 y Fn(red)1675 3581 y Ft(de\002ned)28 b(in)h(\(III.146\))g(is)f(a)h(corr)l(esponding)e (eig)o(en-)257 3702 y(vector)-11 b(,)25 b(i.e)o(.,)f(it)g(is)h(a)g (nonzer)l(o)e(element)i(of)f Fq(Ker)p Fr(f)p Fs(H)8 b Fq(\()p Fs(w)p 2074 3717 V 2 w Fq([)p Fs(e)2218 3717 y Fn(\(0)p Fm(;)p Fl(1)p Fn(\))2403 3702 y Fq(]\))p Fr(g)p Ft(.)257 3905 y(Pr)l(oof)o(.)29 b Fv(First,)c(we)g(observ)o(e)f(that)g (thanks)g(to)h(Eq.)f(\(III.82\))i(in)e(Theorem)h(III.8,)418 4149 y Fr(R)502 4164 y Fm(\032)598 4149 y Fq(:)53 b Fr(D)758 4038 y Fo(\020)817 4149 y Fq(2)866 4108 y Fl(\000)p Fn(1)977 4149 y Fs(")1023 4164 y Fn(0)1078 4149 y Fs(;)45 b Fq(2)1199 4108 y Fl(\000)p Fn(1)1309 4149 y Fs(")1355 4164 y Fn(0)1395 4038 y Fo(\021)1537 4149 y Fr(!)83 b(D)1800 4038 y Fo(\020)1859 4068 y(\002)1900 4149 y Fq(2)1949 4108 y Fl(\000)p Fn(1)2066 4149 y Fq(+)22 b(2)2213 4108 y Fl(\000)p Fn(2)2307 4068 y Fo(\003)2348 4149 y Fs(")2394 4164 y Fn(0)2450 4149 y Fs(;)44 b Fq(2)2570 4108 y Fl(\000)p Fn(2)2681 4149 y Fs(")2727 4164 y Fn(0)2766 4038 y Fo(\021)2848 4149 y Fs(;)280 b Fv(\(III.114\))418 4362 y Fr(R)502 4320 y Fn(2)502 4386 y Fm(\032)598 4362 y Fq(:)53 b Fr(D)758 4251 y Fo(\020)817 4362 y Fq(2)866 4320 y Fl(\000)p Fn(1)977 4362 y Fs(")1023 4377 y Fn(0)1078 4362 y Fs(;)45 b Fq(2)1199 4320 y Fl(\000)p Fn(1)1309 4362 y Fs(")1355 4377 y Fn(0)1395 4251 y Fo(\021)1537 4362 y Fr(!)83 b(D)1800 4251 y Fo(\020)1859 4281 y(\002)1900 4362 y Fq(2)1949 4320 y Fl(\000)p Fn(1)2066 4362 y Fq(+)22 b(2)2213 4320 y Fl(\000)p Fn(2)2329 4362 y Fq(+)g(2)2476 4320 y Fl(\000)p Fn(3)2570 4281 y Fo(\003)2611 4362 y Fs(")2657 4377 y Fn(0)2713 4362 y Fs(;)45 b Fq(2)2834 4320 y Fl(\000)p Fn(3)2944 4362 y Fs(")2990 4377 y Fn(0)3029 4251 y Fo(\021)3111 4362 y Fs(;)17 b Fv(\(III.115\))1574 4519 y(.)1574 4553 y(.)1574 4586 y(.)409 4731 y Fr(R)493 4690 y Fm(\013)493 4756 y(\032)598 4731 y Fq(:)53 b Fr(D)758 4621 y Fo(\020)817 4731 y Fq(2)866 4690 y Fl(\000)p Fn(1)977 4731 y Fs(")1023 4746 y Fn(0)1078 4731 y Fs(;)45 b Fq(2)1199 4690 y Fl(\000)p Fn(1)1309 4731 y Fs(")1355 4746 y Fn(0)1395 4621 y Fo(\021)1537 4731 y Fr(!)83 b(D)1800 4621 y Fo(\020)1859 4650 y(\002)1900 4731 y Fq(2)1949 4690 y Fl(\000)p Fn(1)2066 4731 y Fq(+)22 b Fs(:)17 b(:)g(:)k Fq(+)h(2)2447 4690 y Fl(\000)p Fm(\013)2551 4650 y Fo(\003)2593 4731 y Fs(")2639 4746 y Fn(0)2695 4731 y Fs(;)44 b Fq(2)2815 4690 y Fl(\000)p Fm(\013)2936 4731 y Fs(")2982 4746 y Fn(0)3021 4621 y Fo(\021)3102 4731 y Fs(:)26 b Fv(\(III.116\))257 4972 y(It)f(follo)n(ws)e(that)i(the)f(iteration)g(of)h Fr(R)1534 4987 y Fm(\032)1600 4972 y Fv(on)f Fs(w)p 1724 4988 V 3 w Fv(,)1119 5222 y Fs(w)p 1119 5238 V 1191 5181 a Fn(\()p Fm(\013)p Fn(\))1348 5222 y Fq(:=)53 b Fr(R)1588 5181 y Fm(\013)1588 5247 y(\032)1638 5222 y Fq(\()p Fs(w)p 1676 5238 V 2 w Fq(\))g Fr(2)g(D)2038 5112 y Fo(\020)2097 5222 y Fs(")2143 5237 y Fn(0)2199 5222 y Fs(;)44 b Fq(2)2319 5181 y Fl(\000)p Fm(\013)2440 5222 y Fs(")2486 5237 y Fn(0)2525 5112 y Fo(\021)2606 5222 y Fs(;)522 b Fv(\(III.117\))p eop %%Page: 43 44 43 43 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(43)257 574 y(is)25 b(de\002ned,)g(for)g(all)f Fs(\013)29 b Fr(2)f Fg(N)1208 589 y Fn(0)1253 574 y Fv(.)j(W)-8 b(e)25 b(set)g Fs(E)1672 589 y Fn(\()p Fm(\013)p Fn(\))1776 574 y Fq([)p Fs(z)t Fq(])k(:=)f Fs(w)2112 523 y Fn(\()p Fm(\013)p Fn(\))2109 598 y(0)p Fm(;)p Fn(0)2215 574 y Fq([)p Fs(z)t Fq(;)17 b(0])p Fv(,)682 834 y Fr(U)744 849 y Fn(\()p Fm(\013)p Fn(\))902 834 y Fq(:=)52 b Fr(U)10 b Fq([)p Fs(w)p 1156 850 73 4 v 1229 793 a Fn(\()p Fm(\013)p Fn(\))1333 834 y Fq(])53 b(=)1541 723 y Fo(n)1608 834 y Fs(z)32 b Fr(2)c Fs(D)1860 849 y Fn(1)p Fm(=)p Fn(2)1995 719 y Fo(\014)1995 779 y(\014)1995 839 y(\014)2053 834 y Fr(j)p Fs(E)2153 849 y Fn(\()p Fm(\013)p Fn(\))2258 834 y Fq([)p Fs(z)t Fq(])p Fr(j)g(\024)2532 766 y Fs(\032)p 2532 811 51 4 v 2533 902 a Fq(2)2614 723 y Fo(o)2703 834 y Fs(;)425 b Fv(\(III.118\))257 1078 y(and)647 1198 y Fs(J)701 1214 y Fn(\()p Fm(\013)p Fn(\))833 1198 y Fq(:=)27 b Fs(E)1035 1214 y Fm(\032;)p Fn(\()p Fm(\013)p Fn(\))1248 1198 y Fq(:)53 b Fr(U)1390 1214 y Fn(\()p Fm(\013)p Fn(\))1545 1198 y Fr(!)c Fs(D)1775 1214 y Fn(1)p Fm(=)p Fn(2)1907 1198 y Fs(;)92 b(z)54 b Fr(7!)c Fs(\032)2325 1157 y Fl(\000)p Fn(1)2436 1198 y Fs(E)2508 1214 y Fn(\()p Fm(\013)p Fn(\))2612 1198 y Fq([)p Fs(z)t Fq(])23 b Fs(:)390 b Fv(\(III.119\))257 1373 y(W)-8 b(e)29 b(recall)g(from)g(Lemma)f (III.4)g(that)g Fs(J)1656 1388 y Fn(\()p Fm(\013)p Fn(\))1795 1373 y Fq(:)35 b Fr(U)1919 1388 y Fn(\()p Fm(\013)p Fn(\))2058 1373 y Fr(!)g Fs(D)2274 1388 y Fn(1)p Fm(=)p Fn(2)2412 1373 y Fv(is)28 b(an)h(analytic)f(bijection)f(and)257 1493 y(that)1359 1529 y Fo(\014)1359 1589 y(\014)1392 1614 y Fs(\032)17 b(@)1510 1629 y Fm(z)1550 1614 y Fs(J)1604 1629 y Fn(\()p Fm(\013)p Fn(\))1708 1614 y Fq([)p Fs(z)t Fq(])23 b Fr(\000)g Fq(1)1983 1529 y Fo(\014)1983 1589 y(\014)2068 1614 y Fr(\024)54 b Fq(1)p Fs(=)p Fq(5)21 b Fs(;)761 b Fv(\(III.120\))257 1788 y(for)25 b(all)g Fs(z)32 b Fr(2)c(U)756 1803 y Fn(\()p Fm(\013)p Fn(\))861 1788 y Fv(,)d(which)f(implies)g(that)1311 1923 y Fo(\014)1311 1983 y(\014)1344 2008 y Fs(\032)1394 1967 y Fl(\000)p Fn(1)1506 2008 y Fs(@)1557 2023 y Fm(\020)1597 2008 y Fs(J)1660 1967 y Fl(\000)p Fn(1)1651 2039 y(\()p Fm(\013)p Fn(\))1755 2008 y Fq([)p Fs(\020)8 b Fq(])22 b Fr(\000)g Fq(1)2030 1923 y Fo(\014)2030 1983 y(\014)2116 2008 y Fr(\024)53 b Fq(1)p Fs(=)p Fq(4)21 b Fs(;)714 b Fv(\(III.121\))257 2248 y(for)25 b(all)g Fs(\020)35 b Fr(2)28 b Fs(D)776 2263 y Fn(1)p Fm(=)p Fn(2)886 2248 y Fv(,)d(where)g Fs(J)1267 2207 y Fl(\000)p Fn(1)1258 2279 y(\()p Fm(\013)p Fn(\))1387 2248 y Fv(denotes)f(the)h(in)l(v)o(erse)f(of)h Fs(J)2332 2263 y Fn(\()p Fm(\013)p Fn(\))2436 2248 y Fv(,)1019 2488 y Fs(J)1082 2447 y Fl(\000)p Fn(1)1073 2519 y(\()p Fm(\013)p Fn(\))1221 2488 y Fq(:)53 b Fs(D)1382 2503 y Fn(1)p Fm(=)p Fn(2)1547 2488 y Fr(!)i(U)1764 2503 y Fn(\()p Fm(\013)p Fn(\))1891 2488 y Fs(;)158 b(\020)63 b Fr(7!)55 b Fs(E)2415 2447 y Fl(\000)p Fn(1)2409 2519 y(\()p Fm(\013)p Fn(\))2513 2488 y Fq([)p Fs(\032)17 b(\020)8 b Fq(])22 b Fs(:)421 b Fv(\(III.122\))257 2717 y(Thus,)24 b(for)i(an)o(y)e Fs(\020)8 b(;)17 b(\020)961 2681 y Fl(0)1010 2717 y Fr(2)28 b Fs(D)1185 2732 y Fn(1)p Fm(=)p Fn(2)1295 2717 y Fv(,)688 2921 y Fq(3)17 b Fs(\032)p 688 2966 116 4 v 721 3057 a Fq(4)830 2989 y Fr(j)p Fs(\020)29 b Fr(\000)22 b Fs(\020)1080 2947 y Fl(0)1103 2989 y Fr(j)52 b(\024)1313 2904 y Fo(\014)1313 2964 y(\014)1346 2989 y Fs(J)1409 2947 y Fl(\000)p Fn(1)1400 3020 y(\()p Fm(\013)p Fn(\))1505 2989 y Fq([)p Fs(\020)8 b Fq(])21 b Fr(\000)i Fs(J)1794 2947 y Fl(\000)p Fn(1)1785 3020 y(\()p Fm(\013)p Fn(\))1889 2989 y Fq([)p Fs(\020)1967 2947 y Fl(0)1990 2989 y Fq(])2017 2904 y Fo(\014)2017 2964 y(\014)2102 2989 y Fr(\024)2243 2921 y Fq(5)17 b Fs(\032)p 2243 2966 V 2276 3057 a Fq(4)2385 2989 y Fr(j)p Fs(\020)29 b Fr(\000)22 b Fs(\020)2635 2947 y Fl(0)2658 2989 y Fr(j)g Fs(:)420 b Fv(\(III.123\))257 3234 y(Here)30 b(we)g(use)f(that)g(\(III.121\))h (applies)e(to)h(all)g(points)g(of)g(the)g(line)g(se)o(gment)f(joining)g Fs(\020)36 b Fv(and)257 3354 y Fs(\020)308 3318 y Fl(0)331 3354 y Fv(,)25 b(since)f Fs(D)691 3370 y Fn(1)p Fm(=)p Fn(2)826 3354 y Fv(is)h(con)l(v)o(e)o(x.)404 3475 y(For)g Fq(0)i Fr(\024)i Fs(\013)f Fr(\024)g Fs(\014)6 b Fv(,)25 b(we)g(de\002ne)1280 3695 y Fs(e)1325 3710 y Fn(\()p Fm(\013;\014)s Fn(\))1545 3695 y Fq(:=)52 b Fs(J)1763 3654 y Fl(\000)p Fn(1)1754 3726 y(\()p Fm(\013)p Fn(\))1881 3695 y Fr(\016)22 b(\001)17 b(\001)g(\001)j(\016)i Fs(J)2226 3654 y Fl(\000)p Fn(1)2217 3726 y(\()p Fm(\014)s Fn(\))2320 3695 y Fq([0])h Fs(:)682 b Fv(\(III.124\))257 3925 y(Our)39 b(ne)o(xt)f(goal)h(is)f(to)g(pro)o(v)o(e)g(that)g Fq(\()p Fs(e)1626 3941 y Fn(\()p Fm(\013;\014)s Fn(\))1793 3925 y Fq(\))1831 3940 y Fm(\014)s Fl(2)p Fa(N)2012 3925 y Fv(con)l(v)o(er)n(ges,)k(as)d Fs(\014)59 b Fr(!)53 b(1)p Fv(.)73 b(Indeed,)42 b(an)257 4046 y(iterated)25 b(application)f(of)h (\(III.123\))g(yields)506 4266 y Fr(j)p Fs(e)579 4281 y Fn(\()p Fm(\013;\014)s Fn(\))768 4266 y Fr(\000)d Fs(e)912 4281 y Fn(\()p Fm(\013;\014)s Fn(+1\))1170 4266 y Fr(j)83 b Fq(=)1441 4181 y Fo(\014)1441 4241 y(\014)1474 4266 y Fs(J)1537 4225 y Fl(\000)p Fn(1)1528 4297 y(\()p Fm(\013)p Fn(\))1632 4185 y Fo(\002)1674 4266 y Fs(e)1719 4281 y Fn(\()p Fm(\013)p Fn(+1)p Fm(;\014)s Fn(\))1998 4266 y Fr(\000)23 b Fs(e)2143 4281 y Fn(\()p Fm(\013)p Fn(+1)p Fm(;\014)s Fn(+1\))2491 4185 y Fo(\003)2532 4181 y(\014)2532 4241 y(\014)1280 4479 y Fr(\024)1451 4411 y Fq(5)17 b Fs(\032)p 1451 4456 V 1484 4547 a Fq(4)1593 4394 y Fo(\014)1593 4454 y(\014)1626 4479 y Fs(e)1671 4494 y Fn(\()p Fm(\013)p Fn(+1)p Fm(;\014)s Fn(\))1951 4479 y Fr(\000)22 b Fs(e)2095 4494 y Fn(\()p Fm(\013)p Fn(+1)p Fm(;\014)s Fn(+1\))2443 4394 y Fo(\014)2443 4454 y(\014)2556 4479 y Fr(\024)81 b(\001)17 b(\001)g(\001)323 b Fv(\(III.125\))1280 4715 y Fr(\024)1441 4604 y Fo(\020)1510 4647 y Fq(5)17 b Fs(\032)p 1510 4692 V 1544 4783 a Fq(4)1636 4604 y Fo(\021)1696 4626 y Fm(\014)s Fl(\000)p Fm(\013)1860 4630 y Fo(\014)1860 4690 y(\014)1893 4715 y Fs(e)1938 4730 y Fn(\()p Fm(\014)s(;\014)s Fn(\))2125 4715 y Fr(\000)22 b Fs(e)2269 4730 y Fn(\()p Fm(\014)s(;\014)s Fn(+1\))2524 4630 y Fo(\014)2524 4690 y(\014)2638 4715 y Fr(\024)2796 4604 y Fo(\020)2865 4647 y Fq(5)17 b Fs(\032)p 2865 4692 V 2899 4783 a Fq(4)2991 4604 y Fo(\021)3050 4626 y Fm(\014)s Fl(\000)p Fm(\013)3220 4715 y Fs(;)257 4966 y Fv(using)29 b(that)g Fs(e)727 4981 y Fn(\()p Fm(\014)s(;\014)s Fn(\))892 4966 y Fs(;)17 b(e)981 4981 y Fn(\()p Fm(\014)s(;\014)s Fn(+1\))1272 4966 y Fr(2)37 b Fs(D)1456 4981 y Fn(1)p Fm(=)p Fn(2)1566 4966 y Fv(.)45 b(Since)30 b Fq(5)p Fs(\032=)p Fq(4)36 b Fs(<)g Fq(1)p Fv(,)30 b(Eq.)g(\(III.125\))f(implies)f(that)h(the)257 5086 y(sequences)686 5005 y Fo(\000)732 5086 y Fs(e)777 5102 y Fn(\()p Fm(\013;\014)s Fn(\))944 5005 y Fo(\001)990 5028 y Fl(1)990 5125 y Fm(\014)s Fn(=)p Fm(\013)1162 5086 y Fv(are)c(con)l(v)o(er)n(gent,)g(for)g(all)f Fs(\013)29 b Fr(2)f Fg(N)2310 5101 y Fn(0)2355 5086 y Fv(.)j(Hence,)25 b(the)g(limits)1277 5315 y Fs(e)1322 5331 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))1569 5315 y Fq(:=)77 b(lim)1725 5378 y Fm(\014)s Fl(!1)1925 5315 y Fs(e)1970 5331 y Fn(\()p Fm(\013;\014)s Fn(\))2190 5315 y Fr(2)53 b(U)2371 5331 y Fn(\()p Fm(\013)p Fn(\))3155 5315 y Fv(\(III.126\))p eop %%Page: 44 45 44 44 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(44)257 573 y(e)o(xist)24 b(and)h(obe)o(y)964 780 y Fs(\032)1014 739 y Fl(\000)p Fn(1)1125 780 y Fs(E)1197 795 y Fn(\()p Fm(\013)p Fn(\))1302 780 y Fq([)p Fs(e)1374 795 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))1568 780 y Fq(])53 b(=)f Fs(J)1830 795 y Fn(\()p Fm(\013)p Fn(\))1935 780 y Fq([)p Fs(e)2007 795 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))2202 780 y Fq(])g(=)h Fs(e)2455 795 y Fn(\()p Fm(\013)p Fn(+1)p Fm(;)p Fl(1)p Fn(\))2762 780 y Fs(:)366 b Fv(\(III.127\))257 987 y(W)-8 b(e)25 b(observ)o(e)f(that)g(this)g(identity)f(and)h(the)h (bound)e Fr(j)p Fs(E)2105 1003 y Fn(\()p Fm(\013)p Fn(\))2209 987 y Fq([)p Fs(z)t Fq(])f Fr(\000)g Fs(z)t Fr(j)28 b(\024)g Fq(2)2692 951 y Fl(\000)p Fm(\013)2796 987 y Fs(")2842 1002 y Fn(0)2881 987 y Fv(,)d(which)f(follo)n(ws)257 1108 y(from)h(\(III.117\),)g(imply)e(that)511 1230 y Fo(\014)511 1290 y(\014)544 1315 y Fs(e)589 1330 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))806 1315 y Fr(\000)f Fs(\032)17 b(e)1017 1330 y Fn(\()p Fm(\013)p Fn(+1)p Fm(;)p Fl(1)p Fn(\))1302 1230 y Fo(\014)1302 1290 y(\014)1388 1315 y Fq(=)1517 1230 y Fo(\014)1517 1290 y(\014)1550 1315 y Fs(e)1595 1330 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))1812 1315 y Fr(\000)22 b Fs(E)1983 1330 y Fn(\()p Fm(\013)p Fn(\))2088 1315 y Fq([)p Fs(e)2160 1330 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))2355 1315 y Fq(])2382 1230 y Fo(\014)2382 1290 y(\014)2468 1315 y Fr(\024)53 b Fq(2)2647 1274 y Fl(\000)p Fm(\013)2767 1315 y Fs(")2813 1330 y Fn(0)2875 1315 y Fs(:)253 b Fv(\(III.128\))404 1522 y(Our)26 b(ne)o(xt)g(task)g (is)g(to)h(sho)n(w)e(that)h Fs(e)1622 1537 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))1848 1522 y Fr(!)31 b Fq(0)p Fv(,)26 b(as)h Fs(\013)32 b Fr(!)e(1)p Fv(.)36 b(T)-8 b(o)27 b(this)e(end,)i(we)g(remark)257 1642 y(that,)e(clearly)-6 b(,)24 b Fs(\032)821 1606 y Fm(j)858 1642 y Fs(e)903 1658 y Fn(\()p Fm(\013)p Fn(+)p Fm(j;)p Fl(1)p Fn(\))1209 1642 y Fr(!)j Fq(0)p Fv(,)e(as)f Fs(j)34 b Fr(!)27 b(1)p Fv(,)e(since)f Fs(\032)k(<)g Fq(1)p Fv(.)j(Therefore,)932 1932 y Fr(j)p Fs(e)1005 1948 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))1199 1932 y Fr(j)84 b Fq(=)1470 1818 y Fo(\014)1470 1878 y(\014)1470 1937 y(\014)1557 1808 y Fl(1)1520 1838 y Fo(X)1531 2048 y Fm(j)t Fn(=0)1681 1932 y Fs(\032)1731 1891 y Fm(j)1767 1852 y Fo(\000)1813 1932 y Fs(e)1858 1948 y Fn(\()p Fm(\013)p Fn(+)p Fm(j;)p Fl(1)p Fn(\))2158 1932 y Fr(\000)23 b Fs(\032)17 b(e)2370 1948 y Fn(\()p Fm(\013)p Fn(+)p Fm(j)t Fn(+1)p Fm(;)p Fl(1)p Fn(\))2742 1852 y Fo(\001)2788 1818 y(\014)2788 1878 y(\014)2788 1937 y(\014)3155 1932 y Fv(\(III.129\))1310 2267 y Fr(\024)1507 2142 y Fl(1)1470 2172 y Fo(X)1481 2382 y Fm(j)t Fn(=0)1631 2267 y Fs(\032)1681 2226 y Fm(j)1734 2267 y Fq(2)1783 2226 y Fl(\000)p Fm(\013)p Fl(\000)p Fm(j)1991 2267 y Fs(")2037 2282 y Fn(0)2157 2267 y Fr(\024)80 b Fq(2)2363 2226 y Fl(\000)p Fm(\013)p Fn(+1)2574 2267 y Fs(")2620 2282 y Fn(0)2681 2267 y Fs(;)257 2575 y Fv(using)24 b(\(III.128\),)h (which)g(implies)e(that)h Fq(\()p Fs(e)1732 2590 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))1927 2575 y Fq(\))1965 2590 y Fm(\013)p Fl(2)p Fa(N)2101 2599 y Fh(0)2169 2575 y Fv(con)l(v)o(er)n (ges)g(to)h Fq(0)p Fv(.)404 2709 y(Ne)o(xt,)f(we)h(de\002ne)g(a)g (sequence)g(of)g(operators)g Fq(\()p Fs(H)2137 2724 y Fn(\()p Fm(\013)p Fn(\))2241 2709 y Fq(\))2279 2672 y Fl(1)2279 2733 y Fm(\013)p Fn(=0)2443 2709 y Fv(in)g Fs(H)8 b Fq(\()p Fr(W)2779 2661 y Fn(#)2771 2735 y Fl(\025)p Fn(0)2865 2709 y Fq(\))28 b Fr(\022)g(B)s Fq(\()p Fr(H)3226 2724 y Fn(red)3329 2709 y Fq(\))c Fv(by)488 2932 y Fs(H)569 2947 y Fn(\()p Fm(\013)p Fn(\))726 2932 y Fq(:=)53 b Fs(T)939 2947 y Fn(\()p Fm(\013)p Fn(\))1043 2932 y Fq([)p Fs(H)1151 2947 y Fm(f)1197 2932 y Fq(])38 b(+)h Fs(W)1469 2947 y Fn(\()p Fm(\013)p Fn(\))1612 2932 y Fq(+)g Fs(E)1799 2947 y Fn(\()p Fm(\013)p Fn(\))1926 2932 y Fr(\001)21 b Fp(1)53 b Fq(:=)g Fs(H)8 b Fq(\()p Fs(w)p 2367 2948 73 4 v 2438 2891 a Fn(\()p Fm(\013)p Fn(\))2543 2932 y Fq(\)[)p Fs(e)2653 2947 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))2848 2932 y Fq(])22 b Fs(;)231 b Fv(\(III.130\))257 3139 y(with)668 3346 y Fs(T)725 3362 y Fn(\()p Fm(\013)p Fn(\))829 3346 y Fq([)p Fs(r)s Fq(])83 b(:=)g Fs(w)1272 3295 y Fn(\()p Fm(\013)p Fn(\))1269 3371 y(0)p Fm(;)p Fn(0)1376 3346 y Fq([)p Fs(e)1448 3362 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))1643 3346 y Fq(;)17 b Fs(r)s Fq(])k Fr(\000)i Fs(w)1955 3295 y Fn(\()p Fm(\013)p Fn(\))1952 3371 y(0)p Fm(;)p Fn(0)2059 3346 y Fq([)p Fs(e)2131 3362 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))2326 3346 y Fq(;)17 b(0])k Fs(;)661 b Fv(\(III.131\))734 3521 y Fs(W)826 3537 y Fn(\()p Fm(\013)p Fn(\))1013 3521 y Fq(:=)1252 3427 y Fo(X)1199 3637 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)1466 3521 y Fs(W)1558 3536 y Fm(m;n)1687 3441 y Fo(\002)1729 3521 y Fs(w)p 1729 3537 V 1801 3480 a Fn(\()p Fm(\013)p Fn(\))1905 3521 y Fq([)p Fs(e)1977 3537 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))2172 3521 y Fq(])2199 3441 y Fo(\003)2263 3521 y Fs(;)865 b Fv(\(III.132\))754 3792 y Fs(E)826 3808 y Fn(\()p Fm(\013)p Fn(\))1013 3792 y Fq(:=)83 b Fs(w)1272 3741 y Fn(\()p Fm(\013)p Fn(\))1269 3817 y(0)p Fm(;)p Fn(0)1376 3792 y Fq([)p Fs(e)1448 3808 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))1643 3792 y Fq(;)17 b(0])80 b(=)g Fs(E)2071 3808 y Fn(\()p Fm(\013)p Fn(\))2175 3792 y Fq([)p Fs(e)2247 3808 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))2442 3792 y Fq(])h(=)f Fs(e)2751 3808 y Fn(\()p Fm(\013)p Fn(+1)p Fm(;)p Fl(1)p Fn(\))3058 3792 y Fs(;)70 b Fv(\(III.133\))257 3999 y(for)28 b(all)f Fs(\013)33 b Fr(2)g Fg(N)788 4014 y Fn(0)833 3999 y Fv(,)28 b(noting)e(that)h Fs(H)1426 4015 y Fn(\(0\))1553 3999 y Fq(=)32 b Fs(H)8 b Fq(\()p Fs(w)p 1788 4015 V 2 w Fq(\)[)p Fs(e)1970 4015 y Fn(\(0)p Fm(;)p Fl(1)p Fn(\))2155 3999 y Fq(])p Fv(.)38 b(Morero)o(v)o(er)l(,)27 b(we)h(observ)o(e)f(that)f(the)257 4120 y(de\002nition)e(\(III.65\))i (of)e Fr(R)1171 4084 y Fm(H)1171 4144 y(\032)1264 4120 y Fv(then)g(implies)g(that,)g(for)h(all)g Fs(\013)j Fr(2)g Fg(N)9 b Fv(,)461 4343 y Fs(H)542 4359 y Fn(\()p Fm(\013)p Fn(\))729 4343 y Fq(=)83 b Fs(H)8 b Fq(\()p Fs(w)p 1015 4359 V 1087 4302 a Fn(\()p Fm(\013)p Fn(\))1191 4343 y Fq(\)[)p Fs(e)1301 4359 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))1496 4343 y Fq(])53 b(=)f Fr(R)1788 4302 y Fm(H)1788 4368 y(\032)1856 4263 y Fo(\000)1902 4343 y Fs(w)p 1902 4359 V 1974 4302 a Fn(\()p Fm(\013)p Fl(\000)p Fn(1\))2168 4263 y Fo(\001)2214 4343 y Fq([)p Fs(e)2286 4359 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))2481 4343 y Fq(])729 4529 y(=)83 b Fs(S)948 4544 y Fm(\032)988 4418 y Fo(\020)1048 4529 y Fs(F)1111 4544 y Fm(\037)1155 4552 y Fk(\032)1191 4544 y Fn([)p Fm(H)1269 4556 y Fk(f)1308 4544 y Fn(])1332 4448 y Fo(\000)1394 4529 y Fs(H)8 b Fq(\()p Fs(w)p 1521 4544 V 1593 4487 a Fn(\()p Fm(\013)p Fl(\000)p Fn(1\))1788 4529 y Fq([)p Fs(e)1860 4544 y Fn(\()p Fm(\013)p Fl(\000)p Fn(1)p Fm(;)p Fl(1)p Fn(\))2145 4529 y Fq(]\))p Fs(;)41 b(W)2370 4544 y Fn(0)p Fm(;)p Fn(0)2465 4529 y Fq([)p Fs(w)2565 4478 y Fn(\()p Fm(\013)p Fl(\000)p Fn(1\))2562 4553 y(0)p Fm(;)p Fn(0)2759 4529 y Fq([)p Fs(e)2831 4544 y Fn(\()p Fm(\013)p Fl(\000)p Fn(1)p Fm(;)p Fl(1)p Fn(\))3116 4529 y Fq(]])3187 4448 y Fo(\001)3232 4418 y(\021)729 4758 y Fq(=)899 4691 y(1)p 898 4735 51 4 v 898 4826 a Fs(\032)975 4758 y Fq(\000)1036 4773 y Fm(\032)1098 4758 y Fs(F)1161 4774 y Fn(\()p Fm(\013)p Fl(\000)p Fn(1\))1378 4758 y Fq(\000)1439 4717 y Fl(\003)1439 4783 y Fm(\032)1501 4758 y Fs(;)1627 b Fv(\(III.134\))257 5009 y(where)26 b Fs(F)589 5024 y Fn(\()p Fm(\013)p Fl(\000)p Fn(1\))808 5009 y Fv(denotes)389 5227 y Fs(F)452 5243 y Fn(\()p Fm(\013)p Fl(\000)p Fn(1\))729 5227 y Fq(:=)83 b Fs(F)978 5243 y Fm(\037)1022 5251 y Fk(\032)1058 5243 y Fn([)p Fm(H)1136 5255 y Fk(f)1176 5243 y Fn(])1199 5146 y Fo(\000)1262 5227 y Fs(H)8 b Fq(\()p Fs(w)p 1389 5243 73 4 v 1461 5186 a Fn(\()p Fm(\013)p Fl(\000)p Fn(1\))1655 5227 y Fq([)p Fs(e)1727 5243 y Fn(\()p Fm(\013)p Fl(\000)p Fn(1)p Fm(;)p Fl(1)p Fn(\))2012 5227 y Fq(]\))p Fs(;)42 b(W)2238 5242 y Fn(0)p Fm(;)p Fn(0)2332 5227 y Fq([)p Fs(w)p 2359 5243 V 2432 5186 a Fn(\()p Fm(\013)p Fl(\000)p Fn(1\))2626 5227 y Fq([)p Fs(e)2698 5243 y Fn(\()p Fm(\013)p Fl(\000)p Fn(1)p Fm(;)p Fl(1)p Fn(\))2983 5227 y Fq(]])3054 5146 y Fo(\001)3122 5227 y Fs(;)6 b Fv(\(III.135\))375 5388 y Fs(Q)452 5404 y Fn(\()p Fm(\013)p Fl(\000)p Fn(1\))729 5388 y Fq(:=)83 b Fs(Q)992 5404 y Fm(\037)1036 5412 y Fk(\032)1072 5404 y Fn([)p Fm(H)1150 5416 y Fk(f)1190 5404 y Fn(])1213 5307 y Fo(\000)1276 5388 y Fs(H)8 b Fq(\()p Fs(w)p 1403 5404 V 1475 5347 a Fn(\()p Fm(\013)p Fl(\000)p Fn(1\))1669 5388 y Fq([)p Fs(e)1741 5404 y Fn(\()p Fm(\013)p Fl(\000)p Fn(1)p Fm(;)p Fl(1)p Fn(\))2026 5388 y Fq(]\))p Fs(;)42 b(W)2252 5403 y Fn(0)p Fm(;)p Fn(0)2346 5388 y Fq([)p Fs(w)p 2373 5404 V 2446 5347 a Fn(\()p Fm(\013)p Fl(\000)p Fn(1\))2640 5388 y Fq([)p Fs(e)2712 5404 y Fn(\()p Fm(\013)p Fl(\000)p Fn(1)p Fm(;)p Fl(1)p Fn(\))2997 5388 y Fq(]])3068 5307 y Fo(\001)3136 5388 y Fs(;)-8 b Fv(\(III.136\))p eop %%Page: 45 46 45 45 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(45)257 573 y(and)25 b Fq(\000)487 588 y Fm(\032)552 573 y Fv(is)g(the)f(unitary)g(dilatation)g(introduced)g(in)g (\(III.49\).)31 b(Then)25 b(Eq.)f(\(II.16\))i(reads)1190 759 y Fs(H)1271 774 y Fn(\()p Fm(\013)p Fl(\000)p Fn(1\))1493 759 y Fs(Q)1570 774 y Fn(\()p Fm(\013)p Fl(\000)p Fn(1\))1818 759 y Fq(=)52 b Fs(\037)2007 774 y Fm(\032)2047 759 y Fq([)p Fs(H)2155 774 y Fm(f)2201 759 y Fq(])28 b Fs(F)2319 774 y Fn(\()p Fm(\013)p Fl(\000)p Fn(1\))2535 759 y Fs(;)593 b Fv(\(III.137\))257 944 y(which,)25 b(together)f(with)g(\(III.134\),)h (implies)e(the)i(intertwining)e(property)1059 1130 y Fs(H)1140 1146 y Fn(\()p Fm(\013)p Fl(\000)p Fn(1\))1362 1130 y Fs(Q)1439 1146 y Fn(\()p Fm(\013)p Fl(\000)p Fn(1\))1661 1130 y Fq(\000)1722 1089 y Fl(\003)1722 1155 y Fm(\032)1815 1130 y Fq(=)53 b Fs(\032)28 b Fq(\000)2083 1089 y Fl(\003)2083 1155 y Fm(\032)2151 1130 y Fs(\037)2212 1145 y Fn(1)2251 1130 y Fq([)p Fs(H)2359 1145 y Fm(f)2404 1130 y Fq(])g Fs(H)2540 1146 y Fn(\()p Fm(\013)p Fn(\))2667 1130 y Fs(:)461 b Fv(\(III.138\))257 1316 y(Eq.)28 b(\(III.138\))f(is)g(the)h Ft(k)o(e)m(y)g(identity)e Fv(for)i(the)f(proof)h(of)f(the)h Ft(e)n(xistence)f(of)g(an)h(eig)o(en)l(vector)h Fv(at)257 1436 y Fs(e)302 1452 y Fn(\(0)p Fm(;)p Fl(1)p Fn(\))487 1436 y Fv(.)404 1557 y(F)o(or)24 b(the)g(construction)f(of)i(this)e (eigen)l(v)o(ector)l(,)h(we)g(\002x)h Fs(\013)q(;)17 b(\014)33 b Fr(2)28 b Fg(N)2634 1572 y Fn(0)2704 1557 y Fv(with)23 b Fs(\013)29 b Fr(\024)f Fs(\014)6 b Fv(,)24 b(and)g(we)257 1677 y(de\002ne)i(v)o(ectors)e Fq(\011)913 1692 y Fn(\()p Fm(\013;\014)s Fn(\))1105 1677 y Fv(in)g Fr(H)1291 1692 y Fn(red)1418 1677 y Fv(by)h(setting)e Fq(\011)1909 1692 y Fn(\()p Fm(\013;\013)p Fn(\))2106 1677 y Fq(:=)28 b(\012)d Fv(and)998 1863 y Fq(\011)1074 1878 y Fn(\()p Fm(\013;\014)s Fn(\))1294 1863 y Fq(:=)52 b Fs(Q)1526 1878 y Fn(\()p Fm(\013)p Fn(\))1658 1863 y Fq(\000)1719 1822 y Fl(\003)1719 1887 y Fm(\032)1787 1863 y Fs(Q)1864 1878 y Fn(\()p Fm(\013)p Fn(+1\))2087 1863 y Fq(\000)2148 1822 y Fl(\003)2148 1887 y Fm(\032)2205 1863 y Fr(\001)17 b(\001)g(\001)e Fs(Q)2415 1878 y Fn(\()p Fm(\014)s Fl(\000)p Fn(1\))2635 1863 y Fq(\012)22 b Fs(:)401 b Fv(\(III.139\))257 2048 y(W)-8 b(e)21 b(\002rst)g(sho)n(w)e(that)h (this)f(sequence)h(is)g(con)l(v)o(er)n(gent,)h(as)f Fs(\014)34 b Fr(!)27 b(1)p Fv(.)i(T)-8 b(o)20 b(this)f(end,)i(we)g(observ)o(e)257 2169 y(that)k Fq(\012)j(=)f(\000)694 2133 y Fl(\003)694 2193 y Fm(\032)751 2169 y Fs(\037)812 2184 y Fm(\032)853 2169 y Fq([)p Fs(H)961 2184 y Fm(f)1006 2169 y Fq(])17 b(\012)25 b Fv(and)g(hence)380 2365 y Fq(\011)456 2380 y Fn(\()p Fm(\013;\014)s Fn(+1\))735 2365 y Fr(\000)e Fq(\011)911 2380 y Fn(\()p Fm(\013;\014)s Fn(\))1130 2365 y Fq(=)53 b Fs(Q)1336 2380 y Fn(\()p Fm(\013)p Fn(\))1468 2365 y Fq(\000)1529 2324 y Fl(\003)1529 2389 y Fm(\032)1597 2365 y Fs(Q)1674 2380 y Fn(\()p Fm(\013)p Fn(+1\))1896 2365 y Fq(\000)1957 2324 y Fl(\003)1957 2389 y Fm(\032)2014 2365 y Fr(\001)17 b(\001)g(\001)e Fs(Q)2224 2380 y Fn(\()p Fm(\014)s Fl(\000)p Fn(1\))2444 2365 y Fq(\000)2505 2324 y Fl(\003)2505 2389 y Fm(\032)2562 2284 y Fo(\000)2608 2365 y Fs(Q)2685 2380 y Fn(\()p Fm(\014)s Fn(\))2809 2365 y Fr(\000)23 b Fs(\037)2970 2380 y Fm(\032)3010 2365 y Fq([)p Fs(H)3118 2380 y Fm(f)3164 2365 y Fq(])3191 2284 y Fo(\001)3253 2365 y Fq(\012)f Fs(:)3155 2485 y Fv(\(III.140\))257 2605 y(Since)k Fr(k)p Fs(\037)615 2620 y Fm(\032)655 2605 y Fq([)p Fs(H)763 2620 y Fm(f)808 2605 y Fq(])p Fr(k)i(\024)g Fq(1)p Fv(,)c(this)g(implies)g(that)328 2807 y Fo(\015)328 2867 y(\015)383 2892 y Fq(\011)459 2907 y Fn(\()p Fm(\013;\014)s Fn(+1\))738 2892 y Fr(\000)f Fq(\011)914 2907 y Fn(\()p Fm(\013;\014)s Fn(\))1081 2807 y Fo(\015)1081 2867 y(\015)1188 2892 y Fr(\024)1319 2807 y Fo(\015)1319 2867 y(\015)1374 2892 y Fs(Q)1451 2907 y Fn(\()p Fm(\014)s Fn(\))1575 2892 y Fr(\000)g Fs(\037)1736 2907 y Fm(\032)1776 2892 y Fq([)p Fs(H)1884 2907 y Fm(f)1930 2892 y Fq(])1957 2807 y Fo(\015)1957 2867 y(\015)2012 2931 y Fn(op)2129 2763 y Fm(\014)s Fl(\000)p Fn(1)2132 2797 y Fo(Y)2130 3007 y Fm(j)t Fn(=)p Fm(\013)2279 2781 y Fo(n)2346 2892 y Fq(1)38 b(+)2548 2807 y Fo(\015)2548 2867 y(\015)2603 2892 y Fs(Q)2680 2907 y Fn(\()p Fm(j)t Fn(\))2794 2892 y Fr(\000)23 b Fs(\037)2955 2907 y Fm(\032)2995 2892 y Fq([)p Fs(H)3103 2907 y Fm(f)3148 2892 y Fq(])3175 2807 y Fo(\015)3175 2867 y(\015)3231 2931 y Fn(op)3309 2781 y Fo(o)3398 2892 y Fs(:)3155 3106 y Fv(\(III.141\))257 3226 y(Recalling)i(the)g(de\002nition)f(\(II.6\))h(of)g Fs(Q)1635 3241 y Fn(\010)1690 3226 y Fv(,)g(we)g(ha)n(v)o(e)378 3412 y Fs(\037)439 3427 y Fm(\032)479 3412 y Fq([)p Fs(H)587 3427 y Fm(f)633 3412 y Fq(])d Fr(\000)g Fs(Q)858 3428 y Fn(\()p Fm(j)t Fn(\))1003 3412 y Fq(=)2076 b Fv(\(III.142\))p 426 3524 62 4 v 426 3579 a Fs(\037)487 3602 y Fm(\032)527 3579 y Fq([)p Fs(H)635 3594 y Fm(f)680 3579 y Fq(])707 3498 y Fo(\000)753 3579 y Fs(T)810 3594 y Fn(\()p Fm(j)t Fn(\))902 3579 y Fq([)p Fs(H)1010 3594 y Fm(f)1055 3579 y Fq(])22 b Fr(\000)h Fs(E)1276 3594 y Fn(\()p Fm(j)t Fn(\))1368 3579 y Fq([)p Fs(z)t Fq(])39 b(+)p 1625 3524 V 39 w Fs(\037)1686 3602 y Fm(\032)1726 3579 y Fq([)p Fs(H)1834 3594 y Fm(f)1880 3579 y Fq(])17 b Fs(W)2016 3594 y Fn(\()p Fm(j)t Fn(\))p 2123 3524 V 2123 3579 a Fs(\037)2185 3602 y Fm(\032)2225 3579 y Fq([)p Fs(H)2333 3594 y Fm(f)2378 3579 y Fq(])2405 3498 y Fo(\001)2451 3520 y Fl(\000)p Fn(1)p 2545 3524 V 2545 3579 a Fs(\037)2606 3602 y Fm(\032)2647 3579 y Fq([)p Fs(H)2755 3594 y Fm(f)2800 3579 y Fq(])g Fs(W)2936 3594 y Fn(\()p Fm(j)t Fn(\))3044 3579 y Fs(\037)3105 3594 y Fm(\032)3145 3579 y Fq([)p Fs(H)3253 3594 y Fm(f)3298 3579 y Fq(])23 b Fs(:)257 3764 y Fv(Furthermore,)i(for)g Fs(j)34 b Fr(2)28 b Fg(N)40 b Fv(suf)n(\002ciently)24 b(lar)n(ge,)1308 3970 y Fr(k)p Fs(W)1450 3985 y Fn(\()p Fm(j)t Fn(\))1542 3970 y Fr(k)1592 3985 y Fn(op)1723 3970 y Fr(\024)53 b Fq(2)1902 3929 y Fl(\000)p Fm(j)2010 3970 y Fs(")2056 3985 y Fn(0)2147 3970 y Fr(\024)2311 3902 y Fs(\032)p 2287 3947 98 4 v 2287 4038 a Fq(16)2417 3970 y Fs(:)711 b Fv(\(III.143\))257 4191 y(Indeed,)26 b(since)g Fs(")856 4206 y Fn(0)925 4191 y Fq(=)j(\(8)p Fs(C)1187 4206 y Fn(\002)1246 4191 y Fq(\))1284 4155 y Fl(\000)p Fn(1)1378 4191 y Fs(\032)h Fr(\024)g Fs(\032=)p Fq(8)p Fv(,)25 b(Eq.)h(\(III.143\))g(holds)f(true) h(for)g(all)f Fs(j)36 b Fr(\025)30 b Fq(1)p Fv(.)j(Thus,)257 4311 y(for)25 b(all)g Fs(j)34 b Fr(2)28 b Fg(N)9 b Fv(,)30 b(we)c(may)e(estimate)346 4538 y Fr(k)p Fs(\037)457 4553 y Fm(\032)498 4538 y Fq([)p Fs(H)606 4553 y Fm(f)651 4538 y Fq(])e Fr(\000)h Fs(Q)877 4554 y Fn(\()p Fm(j)t Fn(\))968 4538 y Fr(k)1018 4553 y Fn(op)1149 4538 y Fr(\024)1280 4427 y Fo(\020)1349 4471 y Fs(\032)p 1349 4515 51 4 v 1350 4606 a Fq(8)1448 4538 y Fr(\000)39 b(k)p Fs(W)1706 4554 y Fn(\()p Fm(j)t Fn(\))1798 4538 y Fr(k)1848 4553 y Fn(op)1926 4427 y Fo(\021)1985 4450 y Fl(\000)p Fn(1)2096 4538 y Fr(k)p Fs(W)2238 4554 y Fn(\()p Fm(j)t Fn(\))2330 4538 y Fr(k)2380 4553 y Fn(op)2511 4538 y Fr(\024)2651 4471 y Fq(16)17 b Fs(")2812 4486 y Fn(0)p 2651 4515 200 4 v 2725 4606 a Fs(\032)2877 4538 y Fq(2)2926 4497 y Fl(\000)p Fm(j)3039 4538 y Fs(:)89 b Fv(\(III.144\))257 4767 y(Inserting)24 b(this)g(estimate)g(into)g(\(III.141\),)h(we)g (obtain)707 4963 y Fo(\015)707 5022 y(\015)763 5047 y Fq(\011)839 5063 y Fn(\()p Fm(\013;\014)s Fn(+1\))1118 5047 y Fr(\000)d Fq(\011)1293 5063 y Fn(\()p Fm(\013;\014)s Fn(\))1460 4963 y Fo(\015)1460 5022 y(\015)1599 5047 y Fr(\024)1769 4980 y Fq(16)17 b Fs(")1930 4995 y Fn(0)p 1769 5024 V 1844 5116 a Fs(\032)1995 5047 y Fq(2)2044 5006 y Fl(\000)p Fm(\014)2185 4919 y(\014)s Fl(\000)p Fn(1)2187 4953 y Fo(Y)2185 5163 y Fm(j)t Fn(=)p Fm(\013)2334 4937 y Fo(n)2401 5047 y Fq(1)38 b(+)2613 4980 y(16)17 b Fs(")2774 4995 y Fn(0)p 2613 5024 V 2687 5116 a Fs(\032)2839 5047 y Fq(2)2888 5006 y Fl(\000)p Fm(j)2979 4937 y Fo(o)1599 5350 y Fr(\024)83 b Fq(2)1808 5309 y Fl(\000)p Fm(\014)1937 5283 y Fq(16)17 b Fs(")2098 5298 y Fn(0)p 1937 5327 V 2011 5419 a Fs(\032)2179 5350 y Fq(exp)2345 5270 y Fo(\002)2386 5350 y Fq(2)2435 5309 y Fl(\000)p Fm(\013)2556 5350 y Fq(32)g Fs(")2717 5365 y Fn(0)2772 5350 y Fs(\032)2822 5309 y Fl(\000)p Fn(1)2916 5270 y Fo(\003)2980 5350 y Fs(;)148 b Fv(\(III.145\))p eop %%Page: 46 47 46 46 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(46)257 588 y(using)24 b(that)673 513 y Fo(Q)767 539 y Fl(1)767 617 y Fm(j)t Fn(=0)893 588 y Fq(\(1)e(+)g Fs(\025)1157 603 y Fm(j)1194 588 y Fq(\))28 b Fr(\024)g Fq(exp)1531 477 y Fo(h)1594 513 y(P)1699 539 y Fl(1)1699 617 y Fm(j)t Fn(=0)1843 588 y Fs(\025)1900 603 y Fm(j)1936 477 y Fo(i)1983 588 y Fv(,)d(for)g Fs(\025)2231 603 y Fm(j)2296 588 y Fr(\025)j Fq(0)p Fv(.)j(Hence,)25 b(for)g(\002x)o(ed)g Fs(\013)k Fr(2)f Fg(N)3425 603 y Fn(0)3470 588 y Fv(,)257 736 y(the)d(sequence)g Fq(\(\011)908 751 y Fn(\()p Fm(\013;\014)s Fn(\))1075 736 y Fq(\))1113 751 y Fm(\014)s Fl(2)p Fa(N)1247 760 y Fh(0)1314 736 y Fv(of)g(v)o(ectors)f(in)h Fr(H)1920 751 y Fn(red)2047 736 y Fv(is)g(con)l(v)o(er)n(gent,)f(and)g (de\002ning)1391 956 y Fq(\011)1467 971 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))1714 956 y Fq(:=)76 b(lim)1869 1018 y Fm(\014)s Fl(!1)2070 956 y Fq(\011)2146 971 y Fn(\()p Fm(\013;\014)s Fn(\))2335 956 y Fs(;)793 b Fv(\(III.146\))257 1217 y(we)25 b(ha)n(v)o(e)g(the)g(norm)f(estimate)364 1373 y Fo(\015)364 1433 y(\015)419 1458 y Fq(\011)495 1473 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))711 1458 y Fr(\000)f Fq(\012)881 1373 y Fo(\015)881 1433 y(\015)989 1458 y Fq(=)1118 1373 y Fo(\015)1118 1433 y(\015)1173 1458 y Fq(\011)1249 1473 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))1466 1458 y Fr(\000)f Fq(\011)1641 1473 y Fn(\()p Fm(\013;\013)p Fl(\000)p Fn(1\))1901 1373 y Fo(\015)1901 1433 y(\015)2009 1458 y Fr(\024)53 b Fq(2)2188 1417 y Fl(\000)p Fm(\013)2318 1391 y Fq(32)17 b Fs(")2479 1406 y Fn(0)p 2318 1435 200 4 v 2393 1526 a Fs(\032)2561 1458 y Fq(exp)2727 1377 y Fo(\002)2768 1458 y Fq(2)2817 1417 y Fl(\000)p Fm(\013)2938 1458 y Fq(32)g Fs(")3099 1473 y Fn(0)3154 1458 y Fs(\032)3204 1417 y Fl(\000)p Fn(1)3298 1377 y Fo(\003)3362 1458 y Fs(;)3155 1622 y Fv(\(III.147\))257 1742 y(which)25 b(guarantees)g(that)f Fq(\011)1227 1758 y Fn(\()p Fm(\013)1299 1767 y Fh(0)1334 1758 y Fm(;)p Fl(1)p Fn(\))1484 1742 y Fr(6)p Fq(=)j(0)p Fv(,)e(pro)o(vided)e Fs(\013)2125 1757 y Fn(0)2193 1742 y Fr(2)28 b Fg(N)40 b Fv(is)24 b(chosen)g(suf)n(\002ciently)g(lar)n (ge.)404 1863 y(The)29 b(v)o(ector)g Fq(\011)941 1878 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))1165 1863 y Fv(constructed)g(abo)o(v) o(e)g(is)g(an)h(element)f(of)g(the)h(k)o(ernel)f(of)h Fs(H)3219 1878 y Fn(\()p Fm(\013)p Fn(\))3323 1863 y Fv(,)h(for)257 1983 y(e)n(v)o(ery)f Fs(\013)39 b Fr(2)f Fg(N)776 1998 y Fn(0)822 1983 y Fv(,)31 b(as)g(we)f(will)g(no)n(w)f (demonstrate.)47 b(F)o(or)30 b Fs(\014)44 b(>)38 b(\013)q Fv(,)31 b(observ)o(e)f(that,)h(thanks)f(to)257 2103 y(\(III.138\),)390 2323 y Fs(H)471 2339 y Fn(\()p Fm(\013)p Fn(\))592 2323 y Fq(\011)668 2339 y Fn(\()p Fm(\013;\014)s Fn(\))917 2323 y Fq(=)1076 2243 y Fo(\000)1122 2323 y Fs(H)1203 2339 y Fn(\()p Fm(\013)p Fn(\))1335 2323 y Fs(Q)1412 2339 y Fn(\()p Fm(\013)p Fn(\))1544 2323 y Fq(\000)1605 2282 y Fl(\003)1605 2348 y Fm(\032)1645 2243 y Fo(\001)17 b(\000)1753 2323 y Fs(Q)1830 2339 y Fn(\()p Fm(\013)p Fn(+1\))2053 2323 y Fq(\000)2114 2282 y Fl(\003)2114 2348 y Fm(\032)2171 2323 y Fr(\001)g(\001)g(\001)d Fs(Q)2380 2339 y Fn(\()p Fm(\014)s Fl(\000)p Fn(1\))2601 2323 y Fq(\012)2671 2243 y Fo(\001)917 2479 y Fq(=)83 b Fs(\032)17 b Fq(\000)1204 2438 y Fl(\003)1204 2504 y Fm(\032)1261 2479 y Fs(\037)1322 2494 y Fn(1)1362 2479 y Fq([)p Fs(H)1470 2494 y Fm(f)1515 2479 y Fq(])1559 2398 y Fo(\000)1604 2479 y Fs(H)1685 2494 y Fn(\()p Fm(\013)p Fn(+1\))1907 2479 y Fs(Q)1984 2494 y Fn(\()p Fm(\013)p Fn(+1\))2207 2479 y Fq(\000)2268 2438 y Fl(\003)2268 2504 y Fm(\032)2308 2398 y Fo(\001\000)2399 2479 y Fs(Q)2476 2494 y Fn(\()p Fm(\013)p Fn(+2\))2699 2479 y Fq(\000)2760 2438 y Fl(\003)2760 2504 y Fm(\032)2817 2479 y Fr(\001)g(\001)g(\001)e Fs(Q)3027 2494 y Fn(\()p Fm(\014)s Fl(\000)p Fn(1\))3247 2479 y Fq(\012)3317 2398 y Fo(\001)943 2610 y Fv(.)943 2643 y(.)943 2676 y(.)917 2821 y Fq(=)83 b Fs(\032)1126 2780 y Fm(\014)s Fl(\000)p Fm(\013)1291 2741 y Fo(\000)1336 2821 y Fq(\000)1397 2780 y Fl(\003)1397 2846 y Fm(\032)1465 2821 y Fs(\037)1526 2836 y Fn(1)1566 2821 y Fq([)p Fs(H)1674 2836 y Fm(f)1719 2821 y Fq(])1746 2741 y Fo(\001)1792 2763 y Fm(\014)s Fl(\000)p Fm(\013)1967 2821 y Fs(H)2048 2837 y Fn(\()p Fm(\014)s Fn(\))2172 2821 y Fq(\012)22 b Fs(:)864 b Fv(\(III.148\))257 3041 y(Since)26 b Fs(T)561 3057 y Fn(\()p Fm(\014)s Fn(\))663 3041 y Fq([)p Fs(H)771 3056 y Fm(f)816 3041 y Fq(]\012)i(=)g(0)p Fv(,)d(we)g(furthermore)f (observ)o(e)h(that)686 3177 y Fo(\015)686 3236 y(\015)741 3261 y Fs(\037)802 3276 y Fn(1)842 3261 y Fq([)p Fs(H)950 3276 y Fm(f)995 3261 y Fq(])d Fs(H)1125 3277 y Fn(\()p Fm(\014)s Fn(\))1249 3261 y Fq(\012)1319 3177 y Fo(\015)1319 3236 y(\015)1459 3261 y Fq(=)1619 3177 y Fo(\015)1619 3236 y(\015)1674 3261 y Fs(\037)1735 3276 y Fn(1)1775 3261 y Fq([)p Fs(H)1883 3276 y Fm(f)1928 3261 y Fq(])g Fs(W)2069 3277 y Fn(\()p Fm(\014)s Fn(\))2188 3261 y Fq(\012)50 b(+)g Fs(E)2506 3277 y Fn(\()p Fm(\014)s Fn(\))2625 3261 y Fq(\012)2695 3177 y Fo(\015)2695 3236 y(\015)3155 3261 y Fv(\(III.149\))1458 3434 y Fr(\024)1619 3349 y Fo(\015)1619 3409 y(\015)1674 3434 y Fq(\()p Fs(w)1785 3393 y Fn(\()p Fm(\014)s Fn(\))1782 3458 y Fm(m;n)1911 3434 y Fq([)p Fs(e)1983 3449 y Fn(\()p Fm(\014)s(;)p Fl(1)p Fn(\))2175 3434 y Fq(]\))2240 3449 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)2495 3349 y Fo(\015)2495 3409 y(\015)2550 3375 y Fn(#)2550 3473 y Fm(\014)s(;\030)2700 3434 y Fq(+)2826 3349 y Fo(\014)2826 3409 y(\014)2859 3434 y Fs(E)2931 3449 y Fn(\()p Fm(\014)s Fn(\))3033 3349 y Fo(\014)3033 3409 y(\014)1458 3606 y Fr(\024)84 b Fq(2)1668 3565 y Fl(\000)p Fm(\014)1786 3606 y Fs(")1832 3621 y Fn(0)1916 3606 y Fq(+)44 b Fs(\032)17 b Fr(j)p Fs(e)2176 3622 y Fn(\()p Fm(\014)s Fn(+1)p Fm(;)p Fl(1)p Fn(\))2458 3606 y Fr(j)52 b(\024)h Fq(2)2717 3565 y Fl(\000)p Fm(\014)s Fn(+1)2926 3606 y Fs(")2972 3621 y Fn(0)3033 3606 y Fs(:)257 3826 y Fv(Here)25 b(we)g(also)f(use)g(Theorem)h(III.3,)f(the)h(f)o(act) f(that)g Fs(w)p 2059 3842 73 4 v 2132 3790 a Fn(\()p Fm(\014)s Fn(\))2261 3826 y Fr(2)29 b(D)s Fq(\()p Fs(")2520 3841 y Fn(0)2558 3826 y Fs(;)k Fq(2)2667 3790 y Fl(\000)p Fm(\014)2769 3826 y Fs(")2815 3841 y Fn(0)2854 3826 y Fq(\))p Fv(,)25 b(which,)f(in)g(par)n(-)257 3960 y(ticular)l(,)31 b(implies)d(that)h Fr(k)p Fq(\()p Fs(w)1227 3909 y Fn(\()p Fm(\014)s Fn(\))1224 3970 y Fm(m;n)1353 3960 y Fq([)p Fs(e)1425 3976 y Fn(\()p Fm(\014)s(;)p Fl(1)p Fn(\))1617 3960 y Fq(]\))1682 3975 y Fm(m)p Fn(+)p Fm(n)p Fl(\025)p Fn(1)1937 3960 y Fr(k)1987 3913 y Fn(#)1987 3988 y Fm(\014)s(;\030)2124 3960 y Fr(\024)37 b Fq(2)2287 3924 y Fl(\000)p Fm(\014)2389 3960 y Fs(")2435 3975 y Fn(0)2474 3960 y Fv(,)31 b(and)e(Eq.)h (\(III.133\).)45 b(Sum-)257 4081 y(marizing)24 b (\(III.148\)\226\(III.149\),)i(we)f(ha)n(v)o(e)f(that)1185 4216 y Fo(\015)1185 4276 y(\015)1240 4301 y Fs(H)1321 4316 y Fn(\()p Fm(\013)p Fn(\))1442 4301 y Fq(\011)1518 4316 y Fn(\()p Fm(\013;\014)s Fn(\))1685 4216 y Fo(\015)1685 4276 y(\015)1792 4301 y Fr(\024)54 b Fq(2)1972 4259 y Fl(\000)p Fm(\014)s Fn(+1)2180 4301 y Fs(")2226 4316 y Fn(0)2318 4301 y Fr(!)e Fq(0)22 b Fs(;)587 b Fv(\(III.150\))257 4521 y(as)27 b Fs(\014)36 b Fr(!)30 b(1)p Fv(,)c(and)h(we)f(ha)n(v)o(e) g(used)h(that)e(the)i(operator)f(norm)g(of)g Fq(\000)2562 4484 y Fl(\003)2562 4545 y Fm(\032)2626 4521 y Fs(\037)2687 4536 y Fn(1)2726 4521 y Fq([)p Fs(H)2834 4536 y Fm(f)2880 4521 y Fq(])g Fv(is)g(bounded)g(by)257 4641 y Fq(1)p Fv(.)31 b(Since)25 b Fs(H)689 4656 y Fn(\()p Fm(\013)p Fn(\))821 4641 y Fr(2)j(B)s Fq(\()p Fr(H)1105 4656 y Fn(red)1208 4641 y Fq(\))d Fv(is)f(continuous,)f(\(III.150\))i(implies) f(that)1029 4861 y Fs(H)1110 4876 y Fn(\()p Fm(\013)p Fn(\))1231 4861 y Fq(\011)1307 4876 y Fn(\()p Fm(\013;)p Fl(1)p Fn(\))1554 4861 y Fq(=)77 b(lim)1683 4924 y Fm(\014)s Fl(!1)1883 4780 y Fo(\010)1941 4861 y Fs(H)2022 4876 y Fn(\()p Fm(\013)p Fn(\))2143 4861 y Fq(\011)2219 4876 y Fn(\()p Fm(\013;\014)s Fn(\))2386 4780 y Fo(\011)2497 4861 y Fq(=)52 b(0)22 b Fs(;)432 b Fv(\(III.151\))257 5123 y(for)29 b(all)f Fs(\013)35 b Fr(2)g Fg(N)794 5138 y Fn(0)839 5123 y Fv(.)42 b(Thus)27 b Fs(H)1214 5139 y Fn(\(0\))1308 5123 y Fq(\011)1384 5139 y Fn(\(0)p Fm(;)p Fl(1)p Fn(\))1603 5123 y Fq(=)34 b(0)28 b Fv(and)h Fs(H)2044 5139 y Fn(\()p Fm(\013)2116 5148 y Fh(0)2151 5139 y Fn(\))2183 5123 y Fq(\011)2259 5139 y Fn(\()p Fm(\013)2331 5148 y Fh(0)2366 5139 y Fm(;)p Fl(1)p Fn(\))2522 5123 y Fq(=)34 b(0)p Fv(,)29 b(as)f(well.)41 b(Moreo)o(v)o(er)l(,)257 5244 y Fs(H)338 5259 y Fn(\()p Fm(\013)410 5268 y Fh(0)446 5259 y Fn(\))500 5244 y Fv(deri)n(v)o(es)21 b(from)i Fs(H)1104 5259 y Fn(\(0\))1221 5244 y Fv(by)f Fs(\013)1405 5259 y Fn(0)1472 5244 y Fs(<)28 b Fr(1)22 b Fv(applications)g(of)g(the) h(Feshbach)g(map)f(and)h(scaling)257 5364 y(transformations.)28 b(Since)20 b Fq(\011)1254 5379 y Fn(\()p Fm(\013)1326 5388 y Fh(0)1361 5379 y Fm(;)p Fl(1)p Fn(\))1510 5364 y Fr(6)p Fq(=)28 b(0)p Fv(,)20 b(Theorem)f(II.1\(iii\))g(implies)f (that)h(there)h(is)f(a)h(v)o(ector)p eop %%Page: 47 48 47 47 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(47)257 573 y Fq(\011)333 537 y Fl(0)402 573 y Fr(6)p Fq(=)46 b(0)34 b Fv(such)h(that)f Fs(H)1090 588 y Fn(\(0\))1184 573 y Fq(\011)1260 537 y Fl(0)1329 573 y Fq(=)46 b(0)p Fv(,)37 b(and)d(a)h(closer)g(look)e(at)i(the)g(proof)f(of)h(this)e (statement)257 693 y(re)n(v)o(eals)24 b(that)h Fq(\011)811 709 y Fn(\(0)p Fm(;)p Fl(1)p Fn(\))1023 693 y Fq(=)i(\011)1202 657 y Fl(0)1253 693 y Fr(6)p Fq(=)h(0)p Fv(.)1987 b Fd(\003)257 884 y Ff(Ackno)o(wledgements:)30 b Fv(W)-8 b(e)21 b(thank)e(M.)g(M)8 b(\250)-41 b(uck)19 b(for)h(careful)h(proofreading)e(the)h(manuscript.) 257 1005 y(W)-8 b(e)23 b(also)g(thank)f(M.)g(Salmhofer)g(for)h(useful)f (discussions.)28 b(V)-13 b(.B.)23 b(gratefully)f(ackno)n(wledges)257 1125 y(support)i(by)h(DFG)g(Gerhard)g(Hess-Grant)g(Ba)g(1744/03.)257 1531 y Fu(Refer)m(ences)307 1755 y Fv([1])49 b(V)-13 b(.)26 b(Bach,)g(Th.)f(Chen,)h(J.)g(Fr)8 b(\250)-41 b(ohlich,)25 b(and)h(I.)f(M.)g(Sigal.)38 b(Determination)24 b(of)h(the)h(renor)n(-) 472 1875 y(malized)e(mass)h(for)g(the)f(Nelson)g(model.)35 b Ft(in)25 b(pr)l(epar)o(ation)p Fv(,)d(2002.)307 2070 y([2])49 b(V)-13 b(.)25 b(Bach,)h(J.)f(Fr)8 b(\250)-41 b(ohlich,)25 b(and)g(I.)g(M.)g(Sigal.)36 b(Quantum)24 b(electrodynamics)g(of)h(con\002ned)472 2190 y(non-relati)n(vistic)e (particles.)35 b Ft(Adv)-7 b(.)24 b(in)h(Math.)f Fv(,)h (137:299\226395,)d(1998.)307 2385 y([3])49 b(V)-13 b(.)36 b(Bach,)k(J.)35 b(Fr)8 b(\250)-41 b(ohlich,)39 b(and)d(I.)g(M.)g (Sigal.)70 b(Renormalization)36 b(group)f(analysis)g(of)472 2505 y(spectral)i(problems)e(in)h(quantum)f(\002eld)i(theory)-6 b(.)72 b Ft(Adv)-7 b(.)36 b(in)g(Math.)g Fv(,)j(137:205\226298,)472 2626 y(1998.)307 2821 y([4])49 b(V)-13 b(.)52 b(Bach,)60 b(J.)52 b(Fr)8 b(\250)-41 b(ohlich,)59 b(and)52 b(I.)g(M.)g(Sigal.)122 b(Spectral)53 b(analysis)e(for)i(systems)472 2941 y(of)43 b(atoms)e(and)h(molecules)g(coupled)g(to)g(the)g(quantized)g(radiation) g(\002eld.)92 b Ft(Com-)472 3061 y(mun.)24 b(Math.)h(Phys.)p Fv(,)f(207\(2\):249\226290,)f(1999.)307 3256 y([5])49 b(V)-13 b(.)43 b(Bach,)48 b(J.)43 b(Fr)8 b(\250)-41 b(ohlich,)47 b(I.)d(M.)e(Sigal,)47 b(and)c(A.)g(Sof)n(fer)-5 b(.)94 b(Positi)n(v)o(e)41 b(commutators)472 3377 y(and)c(spectrum)g(of)g (Pauli-Fierz)i(Hamiltonian)c(of)i(atoms)g(and)g(molecules.)75 b Ft(Com-)472 3497 y(mun.)24 b(Math.)h(Phys.)p Fv(,)f (207\(3\):557\226587,)f(1999.)307 3692 y([6])49 b(T)-7 b(.)38 b(Chen.)79 b(Operator)n(-theoretic)39 b(infrared)g (renormalization)e(and)h(construction)f(of)472 3812 y(dressed)25 b(1-particle)f(states.)36 b Ft(pr)l(eprint)23 b(mp-ar)l(c)i(01-310)p Fv(,)f(2001.)307 4007 y([7])49 b(C.)28 b(Cohen-T)-8 b(annoudji,)26 b(J.)h(Dupont-Roc,)g(and)g(G.)f(Grynber)n(g.)43 b Ft(Photons)26 b(and)g(Atoms)472 4127 y(\226)f(Intr)l(oduction)d(to)j(Quantum)f (Electr)l(odynamics)p Fv(.)33 b(John)24 b(W)l(ile)o(y)-6 b(,)24 b(Ne)n(w)g(Y)-11 b(ork,)25 b(1991.)307 4322 y([8])49 b(C.)27 b(Cohen-T)-8 b(annoudji,)26 b(J.)g(Dupont-Roc,)h(and)f(G.)h (Grynber)n(g.)41 b Ft(Atom-Photon)25 b(Inter)n(-)472 4443 y(actions)f(\226)h(Basic)f(Pr)l(ocesses)g(and)g(Applications)p Fv(.)33 b(John)25 b(W)l(ile)o(y)-6 b(,)23 b(Ne)n(w)h(Y)-11 b(ork,)25 b(1992.)307 4637 y([9])49 b(J.)i(Derezinski)g(and)g(V)-13 b(.)51 b(Jaksic.)119 b(Spectral)52 b(theory)f(of)g(Pauli-Fierz)h (operators.)472 4758 y Ft(J)n(.)25 b(Funct.)f(Anal.)p Fv(,)g(180\(2\):243\226327,)f(2001.)257 4953 y([10])49 b(H.)31 b(Feshbach.)56 b(Uni\002ed)31 b(theory)g(of)g(nuclear)g (reactions.)55 b Ft(Ann.)31 b(Phys.)p Fv(,)h(5:357\226390,)472 5073 y(1958.)257 5268 y([11])49 b(J.S.)21 b(Ho)n(wland.)k(The)c(Li)n (vsic)e(matrix)h(in)g(perturbation)g(theory)-6 b(.)25 b Ft(J)n(.)c(Math.)e(Anal.)h(Appl.)p Fv(,)472 5388 y(50:415\226437,)j (1975.)p eop %%Page: 48 49 48 48 bop 257 266 a Fw(BCFS-I,)27 b(May)e(6,)f(2002)2272 b Fv(48)257 573 y([12])49 b(T)-7 b(.)59 b(Kato.)147 b Ft(P)-8 b(erturbation)57 b(Theory)j(of)f(Linear)g(Oper)o(ator)o(s)p Fv(,)67 b(v)n(olume)58 b(132)h(of)472 693 y Ft(Grundlehr)l(en)33 b(der)h(mathematisc)o(hen)e(W)-5 b(issensc)o(haften)p Fv(.)62 b(Springer)n(-V)-11 b(erlag,)37 b(2)c(edi-)472 814 y(tion,)24 b(1976.)257 1017 y([13])49 b(M.)20 b(Merkli.)26 b(Positi)n(v)o(e)19 b(commutators)g(in)h(non-equilibrium)e(statistical) h(quantum)g(me-)472 1137 y(chanics.)36 b Ft(Commun.)24 b(Math.)g(Phys.)p Fv(,)h(223\(2\):327\226362,)e(2001.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0205071010298--