Content-Type: multipart/mixed; boundary="-------------0212170750530" This is a multi-part message in MIME format. ---------------0212170750530 Content-Type: text/plain; name="02-520.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-520.keywords" semi-classique matrix resonances ---------------0212170750530 Content-Type: application/postscript; name="tra.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="tra.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: tra.dvi %%Pages: 26 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o tra.ps tra %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.09.12:1900 %%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 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Fo(h)3055 4174 y Fj(\000)p Fk(n)3157 4210 y Fq(\))g(res-)386 4326 y(onances)i(near)f(an)m(y)g(p)s(oin)m(t)f(of)g(the)h(analytic)e (singular)h(supp)s(ort)h(of)f(the)h(distribu-)386 4442 y(tion)f Fo(\027)34 b Fq(de\014ned)c(as)e Fo(\027)6 b Fq(\()p Fo(E)g Fq(\))28 b(=)1450 4362 y Fi(R)1497 4477 y Fk(x;V)1598 4486 y Fh(0)1632 4477 y Fn(\()p Fk(x)p Fn(\))p Fj(\025)p Fk(E)1858 4442 y Fo(dx)g Fq(\(if)f Fo(V)2172 4457 y Fn(0)2211 4442 y Fq(\()p Fo(x)p Fq(\))h Fm(!)f Fq(0)h(when)h Fm(j)p Fo(x)p Fm(j)f(!)f(1)h Fq(and)386 4577 y Fo(E)41 b(>)34 b Fq(0\).)56 b(\(There)37 b(could)g(also)e(exist) i(man)m(y)g(resonances)h(near)f(regular)f(p)s(oin)m(ts)386 4693 y(of)g(the)i(distribution)d Fo(\027)6 b Fq(.\))57 b(This)38 b(result)f(is)f(an)h(application)e(of)h(the)i(lo)s(cal)c (trace)386 4809 y(form)m(ula)d(of)h(Sj\177)-49 b(ostrand,)33 b(see)g([15])g(and)f([16].)486 4925 y(In)k(the)h(pap)s(er)g([11],)g(w)m (e)h(pro)m(v)m(ed)g(a)e(similar)e(trace)j(form)m(ula)d(for)i(matrix)f (op-)386 5042 y(erators)i(and)g(that)g(there)g(exist)g Fm(O)s Fq(\()p Fo(h)1790 5005 y Fj(\000)p Fk(n)1893 5042 y Fq(\))f(resonances)j(near)e(an)m(y)h(p)s(oin)m(t)e(of)g(the)386 5158 y(analytic)e(singular)h(supp)s(ort)h(of)f(a)h(distribution)e Fo(\027)42 b Fq(when)37 b(the)g(dimension)d Fo(n)i Fq(is)386 5274 y(ev)m(en.)486 5390 y(W)-8 b(e)46 b(are)g(in)m(terested)h(in)e (the)i(n)m(um)m(b)s(er)f(of)g(resonances)i(near)e(a)g(p)s(oin)m(t)f (that)386 5506 y(is)d(not)g(in)f(the)h(singular)f(supp)s(ort)i(of)e (the)i(distribution)d Fo(\027)6 b Fq(.)73 b(The)43 b(considered)386 5623 y(case)29 b(is)f(the)g(generic)g(crossing)h(of)e(t)m(w)m(o)i (eigen)m(v)-5 b(alues)28 b(of)g(a)g(Hermitian)e(matrix)g Fo(V)3335 5638 y Fn(0)1861 5722 y Fg(1)p eop %%Page: 2 2 2 1 bop 386 459 a Fq(when)31 b Fo(n)d Fq(=)f(3.)43 b(With)29 b(this)h(pap)s(er,)h(w)m(e)g(con)m(tin)m(ue)f(the)h(classi\014cation)d (whic)m(h)j(w)m(as)386 575 y(started)i(in)f([11].)486 691 y(The)41 b(pro)s(of)e(of)h(the)g(lo)m(w)m(er)g(b)s(ound)h(on)f(the) g(n)m(um)m(b)s(er)h(of)e(resonances)j(uses)g(a)386 807 y(computation)d(of)h(the)h(trace)g(included)g(in)f(Theorem)h(2.1,)h (using)f(mainly)d(the)386 924 y(residue)31 b(theorem.)43 b(The)32 b(estimate)e(on)h(the)h(n)m(um)m(b)s(er)f(of)g(resonances)h (is)f(in)f(The-)386 1040 y(orem)d(2.2.)42 b(The)29 b(n)m(um)m(b)s(er)g (of)e(resonances,)k(near)d(the)h(p)s(oin)m(t)e(studied,)j(is)d(at)h (least)386 1164 y Fo(C)7 b(h)519 1127 y Fj(\000)p Fn(1)613 1164 y Fm(j)p Fq(ln)o Fo(h)p Fm(j)806 1127 y Fj(\000)871 1100 y Fh(3)p 871 1112 31 3 v 871 1153 a(2)916 1164 y Fq(.)65 b(F)-8 b(or)40 b(related)f(results)i(on)f(the)g(same)g(mo)s (del,)g(see)i([4],)g([5)o(])f(and)386 1280 y([9].)1280 1517 y(2.)49 b Fp(not)-7 b(a)g(tion)39 b(and)f(resul)-7 b(ts)486 1691 y Fq(Let)28 b Fo(n)g(>)g Fq(0)g(b)s(e)g(an)g(o)s(dd)h(in) m(teger.)42 b(Denote)28 b(b)m(y)h Fm(M)2311 1706 y Fk(r)2349 1691 y Fq(\()p Fl(C)20 b Fq(\))34 b(the)29 b(set)g(of)f(Hermitian)386 1808 y(matrices)43 b(of)g(size)h(\()p Fo(r)m(;)17 b(r)s Fq(\),)46 b(and)d(let)g Fm(k)p Fo(M)10 b Fm(k)47 b Fq(=)f(sup)2259 1831 y Fj(k)p Fk(u)p Fj(k)p Fn(=1)2482 1808 y Fm(k)p Fo(M)10 b(u)p Fm(k)p Fq(.)76 b(There)45 b(exists)386 1931 y Fo(C)60 b(>)52 b Fq(0)47 b(suc)m(h)i(that)e Fm(j)p Fq(T)-8 b(r)p Fo(M)10 b Fm(j)53 b(\024)g Fo(C)7 b Fm(k)p Fo(M)j Fm(k)p Fq(.)88 b(Let)48 b Fo(x)53 b Fm(7!)g Fo(V)2548 1946 y Fn(0)2587 1931 y Fq(\()p Fo(x)p Fq(\))47 b(b)s(e)h(a)f(b)s (ounded)386 2047 y(smo)s(oth)28 b(function)h(of)f Fl(R)1279 2011 y Fk(n)1361 2047 y Fq(with)h(v)-5 b(alues)28 b(in)h Fm(M)2097 2062 y Fk(r)2134 2047 y Fq(\()p Fl(C)20 b Fq(\).)48 b(Let)30 b Fo(x)e Fq(=)f Fo(r)9 b Fq(^)-55 b Fo(x)29 b Fq(with)g Fo(r)h Fq(=)e Fm(j)p Fo(x)p Fm(j)p Fq(,)392 2163 y(^)-55 b Fo(x)28 b Fm(2)g Fl(S)624 2127 y Fk(n)p Fj(\000)p Fn(1)756 2163 y Fq(.)486 2279 y(W)-8 b(e)33 b(mak)m(e)f(the)h(follo)m(wing)d(assumptions)j(on)f Fo(V)2235 2294 y Fn(0)2307 2279 y Fq(:)601 2426 y Fy(\(H1\):)40 b Fq(There)32 b(exist)g Fo(\022)1459 2441 y Fn(0)1526 2426 y Fm(2)c Fq([0)p Fo(;)1750 2386 y Fk(\031)p 1750 2403 43 4 v 1754 2460 a Fn(2)1803 2426 y Fq([,)j Fo(")c(>)h Fq(0,)j Fo(R)e(>)e Fq(0,)2507 2401 y(~)2485 2426 y Fo(R)i(>)f(R)g(>)g Fq(0)j(suc)m(h)h(that)700 2547 y(for)f(all)k(^)-55 b Fo(x)28 b Fm(2)g Fl(S)1220 2511 y Fk(n)p Fj(\000)p Fn(1)1351 2547 y Fq(,)k Fo(r)e Fm(7!)d Fo(V)1668 2562 y Fn(0)1708 2547 y Fq(\()p Fo(r)9 b Fq(^)-55 b Fo(x)p Fq(\))30 b(extends)j(to)e(an) g(analytic)e(function)i(on)700 2664 y(S)d Fm(\032)h Fl(C)19 b Fq(,)39 b(where)396 2837 y(S)28 b(=)581 2756 y Fi(\010)639 2837 y Fo(r)j Fm(2)d Fl(C)20 b Fq(;)d Fo(r)36 b Fm(2)28 b Fl(C)20 b Fo(;)d Fm(j)p Fo(r)s Fm(j)32 b Fo(>)c(R)q(;)17 b Fq(arg)g Fo(r)30 b Fm(2)e Fq([)p Fm(\000)p Fo(";)17 b(\022)2120 2852 y Fn(0)2182 2837 y Fq(+)22 b Fo(")p Fq([)2370 2756 y Fi(\011)2450 2837 y Fm([)2539 2756 y Fi(\010)2597 2837 y Fo(r)s Fq(;)17 b Fo(r)29 b Fm(2)g Fl(C)20 b Fo(;)d Fm(j)p Fo(r)s Fm(j)32 b Fo(<)3226 2812 y Fq(~)3204 2837 y Fo(R)3280 2756 y Fi(\011)3338 2837 y Fo(:)601 3014 y Fy(\(H2\):)40 b Fq(There)f(exist)f Fo(V)1484 3029 y Fj(1)1595 3014 y Fm(2)f(M)1818 3029 y Fk(r)1856 3014 y Fq(\()p Fl(C)20 b Fq(\),)50 b(~)-54 b Fo(n)37 b Fm(2)f Fl(R)54 b Fq(~)-54 b Fo(n)37 b(>)f(n)i Fq(and)g Fo(C)43 b(>)37 b Fq(0)g(suc)m(h)700 3130 y(that)c(:)1180 3299 y Fm(8)p Fo(r)d Fm(2)e Fq(S)p Fo(;)45 b Fm(k)p Fo(V)1636 3314 y Fn(0)1675 3299 y Fq(\()p Fo(x)p Fq(\))22 b Fm(\000)h Fo(V)1985 3314 y Fj(1)2060 3299 y Fm(k)k(\024)h Fo(C)7 b Fm(h)p Fo(x)p Fm(i)2452 3258 y Fj(\000)t Fn(~)-39 b Fk(n)2554 3299 y Fo(:)386 3477 y Fq(Let)45 b Fo(I)616 3492 y Fk(r)698 3477 y Fq(b)s(e)f(the)h(iden)m(tit)m(y)g(matrix)d(of)i (size)h(\()p Fo(r)m(;)17 b(r)s Fq(\).)79 b(Let)44 b Fo(P)2606 3492 y Fn(0)2690 3477 y Fq(b)s(e)g(the)h(op)s(erator)386 3594 y Fm(\000)p Fo(h)519 3557 y Fn(2)559 3594 y Fq(\001)s Fm(\012)s Fo(I)766 3609 y Fk(r)808 3594 y Fq(+)s Fo(V)944 3609 y Fn(0)983 3594 y Fq(\()p Fo(x)p Fq(\))23 b(with)g(domain)e Fm(D)31 b Fq(=)c Fo(H)1987 3557 y Fn(2)2026 3594 y Fq(\()p Fl(R)2130 3557 y Fk(n)2183 3594 y Fq(\).)40 b(Set)24 b Fo(P)2510 3609 y Fj(1)2612 3594 y Fq(=)k Fm(\000)p Fo(h)2849 3557 y Fn(2)2889 3594 y Fq(\001)s Fm(\012)s Fo(I)3096 3609 y Fk(r)3137 3594 y Fq(+)s Fo(V)3273 3609 y Fj(1)3348 3594 y Fq(.)386 3710 y(Let)g Fo(\025)613 3725 y Fn(1)p Fk(;)p Fj(1)770 3710 y Fm(\024)g Fo(\025)932 3725 y Fn(2)p Fk(;)p Fj(1)1089 3710 y Fm(\024)h(\001)17 b(\001)g(\001)26 b(\024)i Fo(\025)1501 3725 y Fk(r)n(;)p Fj(1)1653 3710 y Fq(b)s(e)g(the)g(eigen)m(v)-5 b(alues)28 b(of)f Fo(V)2607 3725 y Fj(1)2682 3710 y Fq(.)42 b(F)-8 b(or)27 b(all)e Fo(x)k Fm(2)f Fl(R)3295 3674 y Fk(n)3348 3710 y Fq(,)386 3826 y(denote)46 b(b)m(y)h Fo(\025)919 3841 y Fn(1)p Fk(;)p Fn(0)1013 3826 y Fq(\()p Fo(x)p Fq(\))p Fo(;)67 b(\025)1295 3841 y Fn(2)p Fk(;)p Fn(0)1389 3826 y Fq(\()p Fo(x)p Fq(\))p Fo(;)17 b Fm(\001)g(\001)g(\001)31 b Fo(;)17 b(\025)1814 3841 y Fk(r)n(;)p Fn(0)1903 3826 y Fq(\()p Fo(x)p Fq(\))46 b(the)g(eigen)m(v)-5 b(alues)45 b(of)g Fo(V)2959 3841 y Fn(0)2999 3826 y Fq(\()p Fo(x)p Fq(\).)82 b(Set)386 3958 y Fo(\030)32 b Fq(=)c Fo(\032)625 3931 y Fq(^)615 3958 y Fo(\030)37 b Fq(with)927 3931 y(^)917 3958 y Fo(\030)32 b Fm(2)c Fl(S)1148 3921 y Fk(n)o Fj(\000)p Fn(1)1279 3958 y Fq(.)386 4140 y Fy(Remark)48 b(1.)f Ff(Under)d(these)g(assumptions,)h(the)g(op)-5 b(er)g(ators)43 b Fo(P)2780 4155 y Fn(0)2864 4140 y Ff(and)g Fo(P)3125 4155 y Fj(1)3244 4140 y Ff(ar)-5 b(e)386 4256 y(self-adjoint.)42 b(F)-7 b(or)30 b(al)5 b(l)37 b Fq(^)-55 b Fo(x)28 b Fm(2)g Fl(S)1483 4220 y Fk(n)p Fj(\000)p Fn(1)1614 4256 y Ff(,)k(the)f(functions)g Fo(r)f Fm(7!)e Fo(\025)2513 4271 y Fk(j;)p Fn(0)2600 4256 y Fq(\()p Fo(r)9 b Fq(^)-55 b Fo(x)p Fq(\))31 b Ff(c)-5 b(an)31 b(b)-5 b(e)31 b(lab)-5 b(ele)g(d)386 4372 y(so)34 b(that)i(they)f(ar)-5 b(e)35 b(holomorphic)e(functions)h(in)h(a)f(neighb)-5 b(ourho)g(o)g(d)34 b(of)g Fq(0)h Ff(in)g Fl(C)19 b Ff(.)601 4555 y Fy(\(H3\):)40 b Fq(Let)33 b Fo(E)1151 4570 y Fn(0)1218 4555 y Fm(2)c Fl(R)k Fm(n)22 b(f)p Fq(0)p Fm(g)31 b Fq(and)i Fo(E)1921 4570 y Fn(0)2000 4555 y Fo(=)-61 b Fm(2)29 b(f)p Fo(\025)2190 4570 y Fk(j;)p Fj(1)2312 4555 y Fm(g)2362 4519 y Fk(n)2362 4579 y(j)t Fn(=1)2489 4555 y Fq(.)43 b(Assume)33 b(that)808 4675 y Fm(\017)41 b(8)p Fo(j)6 b Fq(,)33 b Fo(\025)1117 4690 y Fk(j;)p Fn(0)1204 4675 y Fq(\(0\))28 b(=)f Fo(E)1532 4690 y Fn(0)1572 4675 y Fq(.)808 4791 y Fm(\017)41 b Fq(There)36 b(exists)f(a)f(neigh)m(b)s (orho)s(o)s(d)f Fo(S)2211 4806 y Fn(0)2281 4791 y Fm(\032)f Fo(S)40 b Fq(of)34 b(0)g(suc)m(h)i(that)e(for)g(all)899 4908 y Fo(r)c Fm(2)f Fo(S)1128 4923 y Fn(0)1189 4908 y Fm(n)22 b(f)p Fq(0)p Fm(g)p Fq(,)32 b Fo(\025)1526 4923 y Fk(j;)p Fn(0)1614 4908 y Fq(\()p Fo(r)9 b Fq(^)-55 b Fo(x)p Fq(\))32 b(is)g(a)g(simple)f(eigen)m(v)-5 b(alue)32 b(of)h Fo(V)2942 4923 y Fn(0)2981 4908 y Fq(\()p Fo(r)9 b Fq(^)-55 b Fo(x)p Fq(\).)808 5024 y Fm(\017)41 b Fq(F)-8 b(or)32 b(all)e Fo(r)36 b Fq(suc)m(h)e(that)e Fo(\025)1777 5039 y Fk(j;)p Fn(0)1864 5024 y Fq(\()p Fo(r)9 b Fq(^)-55 b Fo(x)p Fq(\))28 b(=)g(0)k(w)m(e)h(ha)m(v)m(e)h Fm(r)2706 5039 y Fk(r)2744 5024 y Fo(\025)2801 5039 y Fk(j;)p Fn(0)2889 5024 y Fq(\()p Fo(r)9 b Fq(^)-55 b Fo(x)p Fq(\))27 b Fm(6)p Fq(=)h(0.)386 5167 y(In)33 b(what)f(follo)m(ws,)g(the)g(brac)m (k)m(et)i([)p Fo(L)1703 5182 y Fj(\017)1743 5167 y Fq(])1770 5131 y Fj(1)1770 5192 y Fn(0)1877 5167 y Fq(will)d(denote)i(the)g (di\013erence)g Fo(L)3046 5182 y Fj(1)3143 5167 y Fm(\000)22 b Fo(L)3308 5182 y Fn(0)3348 5167 y Fq(.)386 5297 y(Denote)34 b Fm(h)p Fo(w)s Fm(i)c Fq(=)h(\(1)23 b(+)g Fo(w)1292 5261 y Fn(2)1331 5297 y Fq(\))1379 5233 y Fh(1)p 1379 5245 31 3 v 1379 5286 a(2)1423 5297 y Fq(.)49 b(Let)35 b Fo(\025)f Fq(b)s(e)g(a)g(p)s(ositiv)m(e)g(constan)m(t)h(\(that)g (will)d(later)386 5413 y(dep)s(end)i(on)e Fo(h)p Fq(\).)44 b(Let)32 b Fo(a)c(>)g Fq(0,)k Fo(b)c(>)g Fq(0,)k Fo(b)1812 5428 y Fn(0)1880 5413 y Fo(>)27 b Fq(0,)1282 5584 y(\012)i(=)e([)p Fm(\000)p Fo(a;)17 b(a)p Fq(])23 b(+)f([)p Fm(\000)p Fo(ib;)17 b(ib)2178 5599 y Fn(0)2219 5584 y Fq(])28 b Fm(\032)g Fl(C)20 b Fo(:)1861 5722 y Fg(2)p eop %%Page: 3 3 3 2 bop 386 459 a Fq(suc)m(h)24 b(that)f Fo(\025)855 474 y Fk(j;)p Fj(1)1017 459 y Fo(=)-61 b Fm(2)28 b Fq(\012)r(+)r Fo(E)1321 474 y Fn(0)1361 459 y Fq(.)40 b(Let)23 b(\()p Fo(z)t(;)17 b(\025)p Fq(\))28 b Fm(7!)g Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\))22 b(b)s(e)h(a)g(holomorphic)d(function)386 575 y(in)32 b Fo(z)37 b Fq(de\014ned)d(on)e(\012)23 b(+)f Fo(E)1316 590 y Fn(0)1356 575 y Fq(,)32 b(satisfying)g(the)h(follo)m (wing)d(four)i(conditions)g(:)601 734 y Fy(\(H4\):)40 b Fq(F)-8 b(or)32 b(all)f(0)c Fo(<)1391 708 y Fq(\024)1395 734 y Fo(b)h(<)f(b)p Fq(,)33 b(there)h(exist)f Fo(C)2216 741 y Fn(\024)2219 759 y Fk(b)2280 734 y Fo(>)28 b Fq(0)k(and)2677 709 y(~)2655 734 y Fo(C)2722 741 y Fn(\024)2725 759 y Fk(b)2787 734 y Fo(>)27 b Fq(0)32 b(suc)m(h)j(that)1349 940 y Fm(j)p Fo(f)11 b Fq(\()p Fo(z)26 b Fq(+)c Fo(E)1715 955 y Fn(0)1755 940 y Fo(;)17 b(\025)p Fq(\))p Fm(j)26 b(\024)2076 915 y Fq(~)2054 940 y Fo(C)2121 947 y Fn(\024)2124 965 y Fk(b)2158 940 y Fo(e)2203 899 y Fj(\000)p Fk(C)2306 907 y Fh(\024)2308 920 y Fe(b)2340 899 y Fk(\025)2385 940 y Fo(:)700 1157 y Fq(uniformly)31 b(in)h Fo(\025)27 b Fm(\025)h Fq(0)33 b(and)f(for)g(all)f Fo(z)h Fm(2)c Fq([)p Fm(\000)p Fo(a;)17 b(a)p Fq(])23 b(+)f([)p Fm(\000)p Fo(i)2706 1131 y Fq(\024)2710 1157 y Fo(b)r(;)17 b Fm(\000)p Fo(ib)p Fq(].)601 1281 y Fy(\(H5\):)40 b Fq(There)f(exist)e(0)f Fo(<)h Fq(~)-51 b Fo(a)36 b(<)g(a)p Fq(,)i Fo(b)1978 1296 y Fn(0)2054 1281 y Fo(>)e Fq(0,)i Fo(C)2350 1296 y Fn(0)2425 1281 y Fo(>)d Fq(0,)j(and)2867 1256 y(~)2845 1281 y Fo(C)2915 1296 y Fn(0)2990 1281 y Fo(>)d Fq(0)i(suc)m(h)700 1398 y(that)1358 1595 y Fm(j)p Fo(f)11 b Fq(\()p Fo(z)26 b Fq(+)c Fo(E)1724 1610 y Fn(0)1764 1595 y Fo(;)17 b(\025)p Fq(\))p Fm(j)27 b(\024)2085 1570 y Fq(~)2063 1595 y Fo(C)2133 1610 y Fn(0)2172 1595 y Fo(e)2217 1554 y Fj(\000)p Fk(C)2322 1563 y Fh(0)2357 1554 y Fk(\025)700 1792 y Fq(uniformly)k(in)h Fo(\025)27 b Fm(\025)h Fq(0)33 b(and)f(for)g(all)f Fo(z)h Fm(2)c Fq(\012)2245 1807 y Fn(0)2285 1792 y Fq(,)853 1989 y(\012)923 2004 y Fn(0)990 1989 y Fq(=)g(\([)p Fm(\000)p Fo(a;)17 b Fm(\000)q Fq(~)-50 b Fo(a)q Fq(])22 b(+)g([)p Fm(\000)p Fo(ib;)17 b(ib)1903 2004 y Fn(0)1944 1989 y Fq(]\))23 b Fm([)f Fq(\([)q(~)-50 b Fo(a;)17 b(a)p Fq(])23 b(+)f([)p Fm(\000)p Fo(ib;)17 b(ib)2775 2004 y Fn(0)2816 1989 y Fq(]\))p Fo(:)601 2209 y Fy(\(H6\):)40 b Fq(There)34 b(exists)1478 2184 y(~)1456 2209 y Fo(C)g(>)28 b Fq(0)k(and)h Fo(C)2005 2224 y Fn(1)2072 2209 y Fo(>)27 b Fq(0)32 b(suc)m(h)i(that) 1388 2414 y Fm(j)p Fo(f)11 b Fq(\()p Fo(z)26 b Fq(+)c Fo(E)1754 2429 y Fn(0)1794 2414 y Fo(;)17 b(\025)p Fq(\))p Fm(j)27 b(\024)2115 2389 y Fq(~)2093 2414 y Fo(C)7 b(e)2215 2373 y Fk(C)2265 2382 y Fh(1)2300 2373 y Fk(\025)2345 2414 y Fo(:)700 2611 y Fq(uniformly)31 b(in)h Fo(\025)27 b Fm(\025)h Fq(0)33 b(for)f(all)e Fo(z)j Fm(2)28 b Fq(\012.)601 2728 y Fy(\(H7\):)40 b Fo(f)11 b Fq(\()p Fo(E)1073 2743 y Fn(0)1113 2728 y Fq(\))27 b Fm(6)p Fq(=)h(0.)386 2893 y(An)39 b(example)f(of)g(suc)m(h)i(a)e(function)g(is)g Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\))37 b(=)h Fo(e)2343 2857 y Fj(\000)p Fk(i\025)p Fn(\()p Fk(z)s Fj(\000)p Fk(E)2633 2866 y Fh(0)2668 2857 y Fn(\))2699 2893 y Fo(e)2744 2857 y Fj(\000)p Fk(\025)p Fn(\()p Fk(z)s Fj(\000)p Fk(E)3010 2866 y Fh(0)3045 2857 y Fn(\))3072 2833 y Fh(2)3149 2893 y Fq(for)g(\012)386 3009 y(a)e(small)e(enough)j (neigh)m(b)s(orho)s(o)s(d)e(of)h(0.)55 b(W)-8 b(e)37 b(will)d(c)m(ho)s(ose)k(0)c Fo(<)g(b)g(<)h(a)p Fq(,)i(and)g Fo(b)3335 3024 y Fn(0)386 3126 y Fq(arbitrary)32 b(small.)44 b(Let)35 b(~)-51 b Fo(a)30 b(<)g Fq(^)-50 b Fo(a)29 b(<)g(a)p Fq(.)46 b(Let)33 b Fo(\037)d Fm(2)f(C)2186 3089 y Fj(1)2180 3150 y Fn(0)2261 3126 y Fq(\()17 b(])22 b Fm(\000)j Fq(^)-50 b Fo(a)22 b Fq(+)h Fo(E)2710 3141 y Fn(0)2750 3126 y Fo(;)18 b Fq(^)-50 b Fo(a)22 b Fq(+)h Fo(E)3038 3141 y Fn(0)3077 3126 y Fq([)17 b(\))33 b(b)s(e)h(a)386 3254 y(function)f(suc)m(h)i(that)e Fo(\037)p Fq(\()p Fo(x)23 b Fq(+)g Fo(E)1550 3269 y Fn(0)1589 3254 y Fq(\))30 b(=)f(1)k(for)g (all)e Fo(x)f Fm(2)f Fq([)p Fm(\000)q Fq(~)-50 b Fo(a)q(;)18 b Fq(~)-50 b Fo(a)p Fq(].)46 b(Let)2833 3228 y(~)2837 3254 y Fo(b)2878 3269 y Fn(0)2947 3254 y Fo(<)3048 3228 y Fq(^)3052 3254 y Fo(b)3093 3269 y Fn(0)3162 3254 y Fo(<)29 b(b)3308 3269 y Fn(0)3348 3254 y Fq(,)386 3383 y(and)574 3357 y(~)578 3383 y Fo(b)j(<)754 3357 y Fq(^)758 3383 y Fo(b)g(<)f(b)p Fq(.)51 b(Let)46 b(~)-60 b Fo(\037)34 b Fq(b)s(e)h(an)g(almost-analytic)c(extension)36 b(of)e Fo(\037)h Fq(of)f(supp)s(ort)386 3499 y(strictly)e(included)g(in)g (\012,)h(satisfying)p 1414 3616 57 4 v 1414 3697 a Fo(@)18 b Fq(~)-61 b Fo(\037)p Fq(\()p Fo(z)t Fq(\))29 b(=)e Fm(O)s Fq(\()p Fm(j)p Fq(Im)o(\()p Fo(z)t Fq(\))p Fm(j)2207 3648 y Fj(1)2281 3697 y Fq(\))p Fo(;)1482 3998 y Fq(~)-61 b Fo(\037)p Fq(\()p Fo(z)27 b Fq(+)22 b Fo(E)1811 4013 y Fn(0)1851 3998 y Fq(\))27 b(=)h(1)97 b(for)g Fo(z)33 b Fm(2)28 b Fq([)p Fm(\000)q Fq(~)-50 b Fo(a)q(;)18 b Fq(~)-50 b Fo(a)p Fq(])22 b(+)g([)p Fm(\000)p Fo(i)3083 3972 y Fq(~)3087 3998 y Fo(b)q(;)17 b(i)3202 3972 y Fq(~)3206 3998 y Fo(b)3247 4013 y Fn(0)3287 3998 y Fq(])p Fo(;)431 4170 y Fq(~)-61 b Fo(\037)p Fq(\()p Fo(z)27 b Fq(+)22 b Fo(E)760 4185 y Fn(0)800 4170 y Fq(\))28 b(=)f(0)97 b(for)g(Im\()p Fo(z)t Fq(\))28 b Fm(\025)1701 4144 y Fq(^)1704 4170 y Fo(b)1745 4185 y Fn(0)1883 4170 y Fq(or)97 b(Im\()p Fo(z)t Fq(\))28 b Fm(\024)g(\000)2515 4144 y Fq(^)2519 4170 y Fo(b)99 b Fq(or)e Fm(j)p Fq(Re\(z\))p Fm(j)27 b(\025)h Fq(^)-49 b(a)p Fo(:)386 4367 y Fq(Let)35 b Fo(m)d Fm(2)h Fl(N)50 b Fq(b)s(e)35 b(a)g(constan)m(t)h(suc)m(h)g (that)f Fo(m)24 b Fm(\000)g Fq(1)32 b Fo(>)2346 4328 y Fk(n)p 2346 4345 43 4 v 2350 4402 a Fn(2)2399 4367 y Fq(.)51 b(Let)35 b Fo(z)2699 4382 y Fn(0)2774 4367 y Fq(b)s(e)g(an)g(elemen)m(t)386 4485 y(of)d Fl(C)48 b Fm(n)22 b Fq(\012)33 b(suc)m(h)h(that)e(Im\()p Fo(z)1397 4500 y Fn(0)1437 4485 y Fq(\))27 b Fo(>)h Fq(0.)43 b(Let)1338 4665 y(~)1317 4691 y Fo(f)1376 4692 y Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\))27 b(=)h Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)\()p Fo(z)27 b Fm(\000)22 b Fo(z)2272 4707 y Fn(0)2312 4692 y Fq(\))2350 4650 y Fk(m)2417 4692 y Fo(:)386 4690 y Fq(\(2.1\))386 4889 y(Denote)33 b(b)m(y)1292 5086 y Fm(k)p Fo(f)11 b Fm(k)1451 5101 y Fj(1)1553 5086 y Fq(=)27 b(sup)1661 5167 y Fk(x)p Fj(2)p Fn(\012)1820 5086 y Fm(k)p Fo(f)11 b Fq(\()p Fo(x)22 b Fq(+)g Fo(E)2214 5101 y Fn(0)2253 5086 y Fo(;)17 b(\025)p Fq(\))p Fm(k)p Fo(:)386 5356 y Fq(Denote)48 b Fo(r)780 5371 y Fn(0)p Fk(;)p Fn(0)874 5356 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))53 b(=)h(\()p Fo(\030)1459 5320 y Fn(2)1530 5356 y Fq(+)32 b Fo(V)1695 5371 y Fn(0)1735 5356 y Fq(\()p Fo(x)p Fq(\))h Fm(\000)f Fo(z)t Fq(\))2095 5320 y Fj(\000)p Fn(1)2238 5356 y Fq(and)48 b Fo(D)56 b Fq(=)2720 5317 y Fk(@)p 2720 5333 42 4 v 2729 5390 a(i)2771 5356 y Fq(.)89 b(Recall)46 b(that)386 5494 y Fo(x)41 b Fq(=)f Fo(r)9 b Fq(^)-55 b Fo(x;)17 b(r)42 b Fq(=)e Fm(j)p Fo(x)p Fm(j)p Fo(;)23 b Fq(^)-55 b Fo(x)40 b Fq(=)g Fo(x=)p Fm(j)p Fo(x)p Fm(j)g(2)h Fq(S)1728 5458 y Fk(n)p Fj(\000)p Fn(1)1865 5494 y Fo(;)17 b(\030)45 b Fq(=)40 b Fo(\032)2173 5468 y Fq(^)2163 5494 y Fo(\030)t(;)17 b(\032)41 b Fq(=)f Fm(j)p Fo(\030)5 b Fm(j)p Fo(;)2617 5468 y Fq(^)2609 5494 y Fo(\030)43 b Fq(=)d Fo(\030)5 b(=)p Fm(j)p Fo(\030)g Fm(j)38 b(2)i Fq(S)3210 5458 y Fk(n)p Fj(\000)p Fn(1)3348 5494 y Fo(:)386 5623 y Fq(Denote)24 b Fo(L)p Fq(\()p Fo(d)6 b Fq(^)-55 b Fo(x)p Fq(\))24 b(the)h(measure)f(giv)m(en)g(b)m(y) h Fo(dx)i Fq(=)h Fo(r)2172 5587 y Fk(n)p Fj(\000)p Fn(1)2309 5623 y Fo(dr)s(L)p Fq(\()p Fo(d)6 b Fq(^)-55 b Fo(x)p Fq(\),)25 b(and)f Fo(L)p Fq(\()p Fo(d)3053 5596 y Fq(^)3043 5623 y Fo(\030)5 b Fq(\))24 b(giv)m(en)1861 5722 y Fg(3)p eop %%Page: 4 4 4 3 bop 386 471 a Fq(b)m(y)33 b Fo(d\030)f Fq(=)c Fo(\032)801 435 y Fk(n)p Fj(\000)p Fn(1)938 471 y Fo(d\032L)p Fq(\()p Fo(d)1204 445 y Fq(^)1194 471 y Fo(\030)5 b Fq(\),)32 b(and)h Fm(k)p Fo(A)p Fm(k)1702 486 y Fn(T)-6 b(r)1811 471 y Fq(the)33 b(trace)g(norm)e(of)i Fo(A)p Fq(.)43 b(Let)464 670 y Fo(\026)p Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))28 b(=)970 576 y Fi(X)1030 786 y Fk(i)1131 560 y Fi(n)1197 670 y Fq(T)-8 b(r)1297 560 y Fi(\020)1379 670 y Fm(\000)23 b Fo(r)1526 629 y Fn(3)1523 695 y(0)p Fk(;)p Fn(0)1617 670 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(D)2052 685 y Fk(x)2092 695 y Fe(i)2122 670 y Fo(V)2179 685 y Fn(0)2218 670 y Fq(\()p Fo(x)p Fq(\)\))p Fo(r)2431 685 y Fn(0)p Fk(;)p Fn(0)2526 670 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(D)2961 685 y Fk(x)3001 695 y Fe(i)3031 670 y Fo(V)3088 685 y Fn(0)3127 670 y Fq(\()p Fo(x)p Fq(\)\))545 965 y(+)653 897 y(1)p 653 942 49 4 v 653 1033 a(2)712 965 y Fo(r)759 923 y Fn(2)756 989 y(0)p Fk(;)p Fn(0)850 965 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(D)1285 980 y Fk(x)1325 990 y Fe(i)1355 965 y Fo(V)1412 980 y Fn(0)1451 965 y Fq(\()p Fo(x)p Fq(\)\))p Fo(r)1667 923 y Fn(2)1664 989 y(0)p Fk(;)p Fn(0)1759 965 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(D)2194 980 y Fk(x)2234 990 y Fe(i)2264 965 y Fo(V)2321 980 y Fn(0)2360 965 y Fq(\()p Fo(x)p Fq(\)\))2529 854 y Fi(\021)q(o)545 1233 y Fq(+)694 1109 y Fk(n)643 1139 y Fi(X)658 1349 y Fk(i)p Fn(=1)804 1123 y Fi(n)870 1233 y Fq(2)p Fo(\030)967 1192 y Fn(2)962 1258 y Fk(i)1006 1233 y Fq(T)-8 b(r)1106 1123 y Fi(\020)1188 1233 y Fm(\000)22 b Fo(r)1334 1192 y Fn(2)1331 1258 y(0)p Fk(;)p Fn(0)1426 1233 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(D)1861 1248 y Fk(x)1901 1258 y Fe(i)1931 1233 y Fo(V)1988 1248 y Fn(0)2027 1233 y Fq(\()p Fo(x)p Fq(\)\))p Fo(r)2243 1192 y Fn(3)2240 1258 y(0)p Fk(;)p Fn(0)2334 1233 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(D)2769 1248 y Fk(x)2809 1258 y Fe(i)2839 1233 y Fo(V)2896 1248 y Fn(0)2936 1233 y Fq(\()p Fo(x)p Fq(\)\))545 1511 y(+)22 b Fo(r)690 1469 y Fn(4)687 1535 y(0)p Fk(;)p Fn(0)781 1511 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(D)1216 1526 y Fk(x)1256 1536 y Fe(i)1286 1511 y Fo(V)1343 1526 y Fn(0)1383 1511 y Fq(\()p Fo(x)p Fq(\)\))p Fo(r)1596 1526 y Fn(0)p Fk(;)p Fn(0)1690 1511 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(D)2125 1526 y Fk(x)2165 1536 y Fe(i)2195 1511 y Fo(V)2252 1526 y Fn(0)2292 1511 y Fq(\()p Fo(x)p Fq(\)\))2461 1400 y Fi(\021o)2587 1511 y Fo(:)386 1710 y Fq(Let)32 b Fo(@)5 b(D)s Fq(\()p Fo(z)t(;)17 b(")p Fq(\))29 b(=)e Fm(f)p Fo(z)1146 1674 y Fj(0)1198 1710 y Fm(2)h Fl(C)20 b Fo(;)d Fm(j)p Fo(z)31 b Fm(\000)22 b Fo(z)1654 1674 y Fj(0)1678 1710 y Fm(j)27 b Fq(=)h Fo(")p Fm(g)p Fq(,)k(and)g Fo(D)s Fq(\()p Fo(z)t(;)17 b(")p Fq(\))27 b(=)h Fm(f)p Fo(z)2710 1674 y Fj(0)2761 1710 y Fm(2)h Fl(C)20 b Fo(;)d Fm(j)p Fo(z)31 b Fm(\000)22 b Fo(z)3218 1674 y Fj(0)3242 1710 y Fm(j)27 b(\024)386 1826 y Fo(")p Fm(g)p Fq(.)423 2007 y Fy(Theorem)73 b(2.1.)42 b Ff(The)34 b(op)-5 b(er)g(ator)34 b Fo(\037f)11 b Fq(\()p Fo(P)1953 2022 y Fj(1)2028 2007 y Fo(;)17 b(\025)p Fq(\))j Fm(\000)i Fo(\037f)11 b Fq(\()p Fo(P)2507 2022 y Fn(0)2546 2007 y Fo(;)17 b(\025)p Fq(\))34 b Ff(is)h(of)f(tr)-5 b(ac)g(e)34 b(class)386 2123 y(and)g(ther)-5 b(e)35 b(exists)g Fo(C)1155 2138 y Fn(2)1221 2123 y Fo(>)28 b Fq(0)35 b Ff(and)f Fo(<)27 b(\027)q(;)17 b(f)39 b(>)p Fm(2)28 b Fl(C)60 b Ff(such)35 b(that)g(:)488 2411 y Fq(T)-8 b(r[\()p Fo(\037f)11 b Fq(\)\()p Fo(P)912 2426 y Fj(\017)951 2411 y Fo(;)17 b(\025)p Fq(\)])1117 2370 y Fj(1)1117 2436 y Fn(0)1219 2411 y Fq(=)p Fo(<)28 b(\027)q(;)17 b(f)38 b(>)27 b(h)1737 2370 y Fn(2)p Fj(\000)p Fk(n)1897 2411 y Fq(+)22 b Fm(k)p Fo(f)11 b Fm(k)2154 2426 y Fj(1)2228 2411 y Fm(O)s Fq(\()p Fo(h)2404 2370 y Fn(4)p Fj(\000)p Fk(n)2541 2411 y Fq(\))22 b(+)g Fm(O)s Fq(\()p Fo(e)2864 2370 y Fj(\000)p Fk(C)2969 2379 y Fh(2)3004 2370 y Fk(\025)3050 2411 y Fo(h)3106 2370 y Fj(\000)p Fk(n)3208 2411 y Fq(\))p Fo(:)386 2277 y Fq(\(2.2\))386 2580 y Ff(F)-7 b(urthermor)i(e,)48 b(ther)-5 b(e)46 b(exist)f Fo(\032)1534 2595 y Fn(0)1574 2580 y Ff(,)53 b Fq(~)-53 b Fo(r)1697 2595 y Fn(0)1782 2580 y Ff(such)46 b(that)g(for)g(al)5 b(l)45 b Fo(r)2586 2595 y Fn(1)2674 2580 y Fm(\024)53 b Fq(~)-53 b Fo(r)2844 2595 y Fn(0)2929 2580 y Ff(ther)-5 b(e)46 b(exist)386 2696 y Fo(\026)445 2711 y Fn(1)484 2696 y Fq(\()p Fo(r)566 2711 y Fn(1)605 2696 y Fq(\))35 b Ff(such)g(that)556 2944 y Fo(<)27 b(\027)q(;)17 b(f)38 b(>)p Fq(=)28 b Fm(\000)1105 2877 y Fq(\(2)p Fo(\031)t Fq(\))1289 2840 y Fj(\000)p Fk(n)p 1105 2921 286 4 v 1153 3012 a Fq(16)p Fo(i\031)1468 2819 y Fk(n)1417 2849 y Fi(X)1428 3059 y Fk(j)t Fn(=1)1578 2944 y Fq(sgn\()p Fo(@)1808 2959 y Fk(r)1846 2944 y Fo(\025)1903 2959 y Fk(j;)p Fn(0)1991 2944 y Fq(\(0\)\))2171 2808 y Fi(Z)10 b(Z)2296 3034 y Fn(^)-39 b Fk(x;)2359 3016 y Fn(^)2352 3034 y Fk(\030)r Fj(2)p Fk(S)2479 3015 y Fe(n)p Fd(\000)p Fh(1)2620 2808 y Fi(Z)2675 3034 y Fk(\032)p Fj(2)p Fk(e)2791 3003 y Fd(\000)p Fe(i)2871 2984 y(\031)p 2871 2992 38 3 v 2875 3027 a Fh(4)2924 3034 y Fc(R)p Fk(;)p Fj(j)p Fk(\032)p Fj(j\024)p Fk(\032)3159 3043 y Fh(0)544 3119 y Fi(Z)12 b(Z)824 3303 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)1120 3312 y Fh(1)1154 3303 y Fn(\))666 3384 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)918 3394 y Fe(j;)p Fh(0)997 3384 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1180 3361 y Fh(2)1215 3384 y Fk(;\026)1277 3393 y Fh(1)1311 3384 y Fn(\))1360 3255 y Fo(f)f Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\))p Fo(\026)p Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\))27 b Fo(r)2094 3214 y Fk(n)p Fj(\000)p Fn(1)2231 3255 y Fo(dr)s(L)p Fq(\()p Fo(d)6 b Fq(^)-55 b Fo(x)o Fq(\))p Fo(L)p Fq(\()p Fo(d)2741 3229 y Fq(^)2731 3255 y Fo(\030)5 b Fq(\))p Fo(\032)2867 3214 y Fk(n)p Fj(\000)p Fn(1)3004 3255 y Fo(d\032dz)t(:)386 3538 y Ff(for)35 b(al)5 b(l)34 b Fo(\026)740 3553 y Fn(1)807 3538 y Fm(\024)28 b Fo(\026)971 3553 y Fn(1)1010 3538 y Fq(\()p Fo(r)1092 3553 y Fn(1)1131 3538 y Fq(\))386 3719 y(A)34 b(similar)c(form)m(ula)i (to)i(2.2)f(w)m(as)i(obtained)f(in)f([11)o(])h(with)g(the)g(main)e (term)i Fo(h)3273 3683 y Fj(\000)p Fk(n)386 3835 y Fq(instead)e(of)h Fo(h)890 3799 y Fn(2)p Fj(\000)p Fk(n)1059 3835 y Fq(for)f(the)h(case)h Fo(n)f Fq(ev)m(en.)486 3951 y(F)-8 b(or)23 b(the)h(sp)s(ecial)f(case)i Fo(r)30 b Fq(=)e(2,)d Fo(V)1654 3966 y Fn(0)1718 3951 y Fq(is)e(Hermitian)f(so)i(there)h(exist)f(\()p Fo(a)2925 3966 y Fk(j)2962 3951 y Fq(\()p Fo(x)p Fq(\)\))3131 3966 y Fn(1)p Fj(\024)p Fk(j)t Fj(\024)p Fn(3)3348 3951 y Fq(,)386 4067 y Fo(b)p Fq(\()p Fo(x)p Fq(\))33 b(suc)m(h)h(that)737 4297 y Fo(V)794 4312 y Fn(0)833 4297 y Fq(\()p Fo(x)p Fq(\))28 b(=)g Fo(b)p Fq(\()p Fo(x)p Fq(\))p Fo(I)1311 4312 y Fn(2)1373 4297 y Fq(+)1471 4156 y Fi(\022)1586 4237 y Fo(a)1637 4252 y Fn(1)1677 4237 y Fq(\()p Fo(x)p Fq(\))458 b Fo(a)2317 4252 y Fn(2)2357 4237 y Fq(\()p Fo(x)p Fq(\))22 b Fm(\000)h Fo(ia)2694 4252 y Fn(3)2734 4237 y Fq(\()p Fo(x)p Fq(\))1586 4353 y Fo(a)1637 4368 y Fn(2)1677 4353 y Fq(\()p Fo(x)p Fq(\))f(+)g Fo(ia)2012 4368 y Fn(3)2052 4353 y Fq(\()p Fo(x)p Fq(\))83 b Fm(\000)p Fo(a)2394 4368 y Fn(1)2434 4353 y Fq(\()p Fo(x)p Fq(\))2907 4156 y Fi(\023)2997 4297 y Fo(:)386 4530 y Fq(W)-8 b(e)28 b(denote)h(V)-8 b(ol)o(\(S)1092 4494 y Fn(2)1132 4530 y Fq(\))28 b(the)g(v)m(olume)g(of)f(S)1853 4494 y Fn(2)1921 4530 y Fq(with)h(resp)s(ect)h(to)f(Leb)s(esgue)h(measure.)432 4710 y Fy(Theorem)93 b(2.2.)46 b Ff(L)-5 b(et)43 b Fo(n)e Fq(=)h(3)g Ff(and)f Fo(r)j Fq(=)d(2)p Ff(.)67 b(Supp)-5 b(ose)41 b Fo(V)2649 4725 y Fn(0)2731 4710 y Ff(satisfy)h Fo(a)3093 4725 y Fn(1)3133 4710 y Fq(\(0\))f(=)386 4827 y Fo(a)437 4842 y Fn(2)477 4827 y Fq(\(0\))d(=)h Fo(a)806 4842 y Fn(3)846 4827 y Fq(\(0\))g(=)g(0)p Ff(,)j Fo(b)p Fq(\(0\))e(=)f Fo(E)1639 4842 y Fn(0)1678 4827 y Ff(,)k Fm(fr)p Fo(a)1935 4842 y Fk(j)1971 4827 y Fq(\(0\))p Fm(g)2146 4790 y Fn(3)2146 4851 y Fk(j)t Fn(=1)2314 4827 y Ff(ar)-5 b(e)41 b(line)-5 b(arly)40 b(indep)-5 b(endent.)386 4947 y(L)g(et)31 b Fq(\()p Fo(a)639 4962 y Fn(1)678 4947 y Fo(;)17 b(a)773 4962 y Fn(2)812 4947 y Fo(;)g(a)907 4962 y Fn(3)947 4947 y Fq(\))30 b Ff(b)-5 b(e)30 b(lo)-5 b(c)g(al)30 b(c)-5 b(o)g(or)g(dinates)29 b(and)g(denote)h(by)h Fm(r)2544 4962 y Fk(a)2581 4971 y Fh(1)2616 4962 y Fk(;a)2673 4971 y Fh(2)2707 4962 y Fk(;a)2764 4971 y Fh(3)2803 4947 y Fo(b)g Ff(the)f(gr)-5 b(adient)386 5063 y(of)33 b Fo(b)h Ff(for)g(these)f(c)-5 b(o)g(or)g(dinates.)43 b(Supp)-5 b(ose)33 b Fm(jr)2001 5078 y Fk(a)2038 5087 y Fh(1)2073 5078 y Fk(;a)2130 5087 y Fh(2)2164 5078 y Fk(;a)2221 5087 y Fh(3)2260 5063 y Fo(b)p Fq(\(0\))p Fm(j)28 b Fo(<)f Fq(1)33 b Ff(and)g(supp)-5 b(ose)33 b(that)386 5180 y Fo(r)d Fm(7!)e Fq(\()p Fo(\025)683 5195 y Fn(1)p Fk(;)p Fn(0)799 5180 y Fm(\000)22 b Fo(\025)955 5195 y Fn(2)p Fk(;)p Fn(0)1050 5180 y Fq(\)\()p Fo(r)9 b Fq(^)-55 b Fo(x)p Fq(\))35 b Ff(has)f(a)h(zer)-5 b(o)34 b(of)h(or)-5 b(der)34 b(one)g(at)h Fo(r)c Fq(=)c(0)p Ff(.)486 5296 y(Then)34 b(ther)-5 b(e)34 b(exist)h Fo(C)7 b Fq(\()p Fo(V)1381 5311 y Fn(0)1420 5296 y Fq(\))28 b Fo(>)f Fq(0)p Ff(,)35 b(and)f Fo(\032)1942 5311 y Fn(0)2010 5296 y Fo(>)27 b Fq(0)35 b Ff(such)g(that)g(:)718 5516 y Fo(<)28 b(\027)q(;)17 b(f)38 b(>)p Fq(=)1267 5449 y Fo(i)p 1190 5493 187 4 v 1190 5584 a Fq(2)1239 5556 y Fn(5)1279 5584 y Fo(\031)1338 5556 y Fn(2)1387 5516 y Fq(V)-8 b(ol)o(\(S)1620 5475 y Fn(2)1659 5516 y Fq(\))p Fo(C)7 b Fq(\()p Fo(V)1869 5531 y Fn(0)1908 5516 y Fq(\))1963 5381 y Fi(Z)2018 5606 y Fj(j)p Fk(\032)p Fj(j\024)p Fk(\032)2185 5615 y Fh(0)2240 5516 y Fo(f)k Fq(\()p Fo(E)2409 5531 y Fn(0)2471 5516 y Fq(+)22 b Fo(\032)2619 5475 y Fn(2)2658 5516 y Fo(;)17 b(\025)p Fq(\))27 b Fo(\032)2874 5475 y Fn(2)2914 5516 y Fo(d\032;)1861 5722 y Fg(4)p eop %%Page: 5 5 5 4 bop 386 459 a Ff(with)710 702 y Fo(C)7 b Fq(\()p Fo(V)882 717 y Fn(0)921 702 y Fq(\))27 b(=)1100 634 y Fo(\031)p 1100 679 59 4 v 1105 770 a Fq(3)1169 702 y(det\()p Fm(r)p Fo(a)1476 717 y Fn(1)1516 702 y Fq(\(0\))p Fo(;)17 b Fm(r)p Fo(a)1819 717 y Fn(2)1858 702 y Fq(\(0\))p Fo(;)g Fm(r)p Fo(a)2161 717 y Fn(3)2200 702 y Fq(\(0\)\))2363 660 y Fj(\000)p Fn(1)2528 577 y(3)2473 607 y Fi(X)2484 817 y Fk(j)t Fn(=1)2634 702 y Fm(jr)p Fo(a)2796 717 y Fk(j)2832 702 y Fq(\(0\))p Fm(j)2985 660 y Fn(2)3024 702 y Fo(:)486 990 y Fq(Denote)37 b(Res)q(\()p Fo(P)1080 1005 y Fn(0)1119 990 y Fq(\))h(the)g(set)g(of)f(resonances)i(of)f Fo(P)2309 1005 y Fn(0)2348 990 y Fq(.)58 b(W)-8 b(e)38 b(then)h(ha)m(v)m(e)f(the)h(fol-)386 1107 y(lo)m(wing)418 1291 y Fy(Theorem)63 b(2.3.)38 b Ff(With)30 b(the)h(same)e(hyp)-5 b(otheses)30 b(as)g(in)g(The)-5 b(or)g(em)29 b(2.2,)i(for)f(any)386 1407 y(neighb)-5 b(orho)g(o)g(d)33 b Fo(W)49 b Ff(of)34 b Fo(E)1292 1422 y Fn(0)1332 1407 y Ff(,)g(ther)-5 b(e)35 b(exist)g Fo(h)1922 1422 y Fn(0)1989 1407 y Fo(>)28 b Fq(0)34 b Ff(and)h Fo(C)f(>)28 b Fq(0)35 b Ff(such)f(that)802 1592 y Fm(8)p Fo(h)28 b Fm(2)17 b Fq(]0)p Fo(;)g(h)1200 1607 y Fn(0)1239 1592 y Fq([)p Fo(;)144 b Fq(card)q Fm(f)p Fq(Res\()p Fo(P)1926 1607 y Fn(0)1965 1592 y Fq(\))22 b Fm(\\)h Fo(W)14 b Fm(g)27 b(\025)h Fo(C)7 b(h)2535 1551 y Fj(\000)p Fn(1)2630 1592 y Fm(j)p Fq(ln)o Fo(h)p Fm(j)2823 1551 y Fj(\000)2888 1524 y Fh(3)p 2887 1536 31 3 v 2887 1577 a(2)2932 1592 y Fo(:)386 1776 y Fq(Throughout)33 b(the)g(pro)s(ofs,)f(w)m(e)i(will)c(denote)j(b)m(y)h Fo(C)39 b Fq(an)m(y)34 b(constan)m(ts.)870 2021 y(3.)49 b Fp(reduction)38 b(using)f(functional)i(calculus)486 2196 y Fq(In)27 b(this)h(section)f(w)m(e)i(use)g(the)f(functional)e (calculus)h(to)g(get)h(an)f(expression)i(for)386 2312 y([\()p Fo(\037f)11 b Fq(\)\()p Fo(P)710 2327 y Fj(1)784 2312 y Fo(;)17 b(\025)p Fq(\))26 b Fm(\000)h Fq(\()p Fo(\037f)11 b Fq(\)\()p Fo(P)1350 2327 y Fn(0)1389 2312 y Fo(;)17 b(\025)p Fq(\)],)40 b(using)f(only)f(the)i(op)s(erator)e(\()p Fo(P)2779 2327 y Fj(\017)2845 2312 y Fm(\000)27 b Fo(z)t Fq(\))3036 2276 y Fj(\000)p Fn(1)3170 2312 y Fq(for)38 b Fo(z)386 2428 y Fq(suc)m(h)27 b(that)f(Im\()p Fo(z)t Fq(\))i(=)g Fo(\016)t Fq(,)f(up)f(to)g(a)g(remainder.)40 b(The)27 b(main)d(result)i(of)g(this)g(section)386 2544 y(is)33 b(Lemma)g(3.1.)47 b(A)34 b(similar)d(reduction)j(can)g(b)s(e)g (found)g(in)f(Dimassi-Sj\177)-49 b(ostrand)386 2661 y([3].)53 b(F)-8 b(or)35 b(the)h(functional)e(calculus,)i(w)m(e)g(use)h(the)f (Hel\013er-Sj\177)-49 b(ostrand)35 b(form)m(ula,)386 2777 y(see)f([7)o(].)386 2906 y(Let)40 b Fo(b)g(>)762 2879 y Fq(\024)765 2906 y Fo(b)h(>)959 2879 y Fq(^)963 2906 y Fo(b)p Fq(.)65 b(Let)42 b(^)-51 b Fo(a)41 b Fm(2)p Fq(])q(~)-50 b Fo(a;)18 b Fq(\024)-50 b Fo(a)p Fq([.)65 b(Denote)40 b(b)m(y)h Fo(L)p Fq(\()p Fo(dz)t Fq(\))g(=)f Fo(d)p Fq(\(Re)o(\()p Fo(z)t Fq(\)\))p Fo(d)p Fq(\(Im\()p Fo(z)t Fq(\)\).)386 3031 y(Let)33 b Fo(\016)j Fq(b)s(e)d(a)g(constan)m (t)g(suc)m(h)i(that)d(0)c Fo(<)g(\016)j(<)d Fq(Min)p Fm(f)2256 3005 y Fq(~)2260 3031 y Fo(b)2301 3046 y Fn(0)2341 3031 y Fo(;)2381 3005 y Fq(~)2385 3031 y Fo(b;)17 b Fq(1)p Fm(g)p Fo(:)32 b Fq(By)h(the)g(functional)386 3157 y(calculus)f(w)m(e)i (get)e(\(for)1269 3130 y(~)1248 3157 y Fo(f)43 b Fq(see)34 b(\(2.1\)\))391 3505 y([\()p Fo(\037f)11 b Fq(\)\()p Fo(P)715 3520 y Fj(\017)754 3505 y Fo(;)17 b(\025)p Fq(\)])920 3464 y Fj(1)920 3530 y Fn(0)1022 3505 y Fq(=)1140 3438 y(1)p 1135 3482 59 4 v 1135 3573 a Fo(\031)1221 3369 y Fi(Z)1276 3595 y Fk(z)s Fj(2)p Fn(\012)1452 3479 y Fq(~)1430 3505 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\))p 1715 3424 57 4 v Fo(@)h Fq(~)-61 b Fo(\037)p Fq(\()p Fo(z)t Fq(\))1975 3424 y Fi(\002)2017 3505 y Fq(\()p Fo(P)2118 3520 y Fj(\017)2180 3505 y Fm(\000)22 b Fo(z)t Fq(\))2366 3464 y Fj(\000)p Fn(1)2461 3505 y Fq(\()p Fo(P)2562 3520 y Fj(\017)2624 3505 y Fm(\000)g Fo(z)2768 3520 y Fn(0)2808 3505 y Fq(\))2846 3464 y Fj(\000)p Fk(m)2967 3424 y Fi(\003)3009 3447 y Fj(1)3009 3544 y Fn(0)3100 3505 y Fo(L)p Fq(\()p Fo(dz)t Fq(\))p Fo(:)386 3322 y Fq(\(3.1\))386 3734 y(W)-8 b(e)33 b(ha)m(v)m(e)h(the)f(follo)m (wing)d(lemma)423 3918 y Fy(Lemma)75 b(3.1.)42 b Ff(With)35 b Fm(Z)h Fq(=)27 b Fm(f)p Fo(z)32 b Fm(2)d Fq(\012;)17 b(Im)o(\()p Fo(z)t Fq(\))28 b(=)g Fo(\016)t Fm(g)471 4144 y Fq([\()p Fo(\037f)11 b Fq(\)\()p Fo(P)795 4159 y Fj(\017)834 4144 y Fo(;)17 b(\025)p Fq(\)])1000 4103 y Fj(1)1000 4168 y Fn(0)1102 4144 y Fq(=)22 b Fm(\000)1356 4076 y Fq(1)p 1310 4121 141 4 v 1310 4212 a(2)p Fo(i\031)1477 4008 y Fi(Z)1533 4234 y Fj(Z)1631 4118 y Fq(~)1609 4144 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\))12 b(~)-61 b Fo(\037)q Fq(\()p Fo(z)t Fq(\))2081 4063 y Fi(\002)2123 4144 y Fq(\()p Fo(P)2224 4159 y Fj(\017)2285 4144 y Fm(\000)23 b Fo(z)t Fq(\))2472 4103 y Fj(\000)p Fn(1)2567 4144 y Fq(\()p Fo(P)2668 4159 y Fj(\017)2729 4144 y Fm(\000)g Fo(z)2874 4159 y Fn(0)2913 4144 y Fq(\))2951 4103 y Fj(\000)p Fk(m)3073 4063 y Fi(\003)3114 4085 y Fj(1)3114 4183 y Fn(0)3189 4144 y Fo(dz)1300 4362 y Fq(+)f Fm(O)s Fq(\()p Fo(h)1574 4321 y Fj(\000)p Fk(n)1676 4362 y Fq(\))p Fo(e)1759 4321 y Fj(\000)12 b Fn(min)n Fj(f)p Fk(C)2028 4330 y Fh(0)2063 4321 y Fk(;C)2131 4332 y Fh(~)2133 4345 y Fe(b)2165 4321 y Fj(g)p Fk(\025)2245 4362 y Fo(:)386 4536 y Ff(The)34 b(r)-5 b(emainder)34 b(is)h(estimate)-5 b(d)34 b(with)h(the)g(tr)-5 b(ac)g(e)34 b(norm.)386 4720 y Fq(Pro)s(of)p 386 4736 235 4 v 36 w(:)51 b(Denote)36 b Fo($)s Fq(\()p Fo(z)t Fq(\))f(=)1448 4693 y(~)1427 4720 y Fo(f)10 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\))p 1711 4639 57 4 v Fo(@)h Fq(~)-60 b Fo(\037)p Fq(\()p Fo(z)t Fq(\))17 b([\()p Fo(P)2100 4735 y Fj(\017)2161 4720 y Fm(\000)23 b Fo(z)t Fq(\))2348 4684 y Fj(\000)p Fn(1)2443 4720 y Fq(\()p Fo(P)2544 4735 y Fj(\017)2605 4720 y Fm(\000)g Fo(z)2750 4735 y Fn(0)2790 4720 y Fq(\))2828 4684 y Fj(\000)p Fk(m)2949 4720 y Fq(])2976 4667 y Fj(1)2976 4749 y Fn(0)3051 4720 y Fq(.)55 b(Using)386 4836 y(Stok)m(es)34 b(form)m(ula)d(and)h (\(3.1\),)g(w)m(e)i(get)423 5062 y([\()p Fo(\037f)11 b Fq(\)\()p Fo(P)747 5077 y Fj(\017)786 5062 y Fo(;)17 b(\025)p Fq(\))o(])951 5021 y Fj(1)951 5087 y Fn(0)1054 5062 y Fq(=)1172 4995 y(1)p 1167 5039 59 4 v 1167 5130 a Fo(\031)1253 4926 y Fi(Z)1373 5118 y Fk(z)s Fj(2)p Fn(\012)1308 5181 y(Im)o(\()p Fk(z)s Fn(\))p Fj(\025)p Fk(\016)1592 5062 y Fo($)s Fq(\()p Fo(z)t Fq(\))p Fo(L)p Fq(\()p Fo(dz)t Fq(\))946 5371 y(+)1059 5303 y(1)p 1054 5348 V 1054 5439 a Fo(\031)1140 5235 y Fi(Z)1195 5461 y Fk(z)s Fj(2)p Fn(\012;)p Fj(j)p Fn(Im)n(\()p Fk(z)s Fn(\))p Fj(j\024)p Fk(\016)1672 5371 y Fo($)s Fq(\()p Fo(z)t Fq(\))p Fo(L)p Fq(\()p Fo(dz)t Fq(\))22 b(+)2258 5303 y(1)p 2253 5348 V 2253 5439 a Fo(\031)2339 5235 y Fi(Z)2394 5461 y Fk(z)s Fj(2)p Fn(\012;Im)o(\()p Fk(z)s Fn(\))p Fj(\024\000)p Fk(\016)2886 5371 y Fo($)s Fq(\()p Fo(z)t Fq(\))p Fo(L)p Fq(\()p Fo(dz)t Fq(\))821 5584 y(=:)p Fo(A)997 5599 y Fn(0)1059 5584 y Fq(+)g Fo(A)1230 5599 y Fn(1)1292 5584 y Fq(+)g Fo(A)1463 5599 y Fn(2)1502 5584 y Fo(:)1861 5722 y Fg(5)p eop %%Page: 6 6 6 5 bop 386 459 a Fq(Stok)m(es)34 b(theorem)e(again)f(giv)m(es)726 685 y Fo(A)799 700 y Fn(0)866 685 y Fq(=)c Fm(\000)1103 618 y Fq(1)p 1056 662 141 4 v 1056 753 a(2)p Fo(i\031)1224 549 y Fi(Z)1280 775 y Fj(Z)1378 659 y Fq(~)1357 685 y Fo(f)10 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\))12 b(~)-61 b Fo(\037)q Fq(\()p Fo(z)t Fq(\)[\()p Fo(P)1956 700 y Fj(\017)2018 685 y Fm(\000)22 b Fo(z)t Fq(\))2204 644 y Fj(\000)p Fn(1)2299 685 y Fq(\()p Fo(P)2400 700 y Fj(\017)2462 685 y Fm(\000)g Fo(z)2606 700 y Fn(0)2646 685 y Fq(\))2684 644 y Fj(\000)p Fk(m)2806 685 y Fq(])2833 644 y Fj(1)2833 710 y Fn(0)2907 685 y Fo(dz)t(:)386 915 y Fq(Prop)s(osition)31 b(A.3)h(implies)684 1142 y Fm(k)p Fo(A)807 1157 y Fn(1)846 1142 y Fm(k)896 1157 y Fn(T)-6 b(r)1001 1142 y Fq(=)1104 1027 y Fi(\015)1104 1087 y(\015)1104 1146 y(\015)1174 1074 y Fq(1)p 1169 1119 59 4 v 1169 1210 a Fo(\031)1255 1006 y Fi(Z)1310 1231 y Fk(z)s Fj(2)p Fn(\012;)p Fj(j)p Fn(Im)n(\()p Fk(z)s Fn(\))p Fj(j\024)p Fk(\016)1787 1142 y Fo($)s Fq(\()p Fo(z)t Fq(\))p Fo(L)p Fq(\()p Fo(dz)t Fq(\))2238 1027 y Fi(\015)2238 1087 y(\015)2238 1146 y(\015)2294 1210 y Fn(T)g(r)2398 1142 y Fm(\024)28 b(O)s Fq(\()p Fo(h)2679 1100 y Fj(\000)p Fk(n)2781 1142 y Fq(\))p Fo(e)2864 1100 y Fj(\000)p Fk(C)2969 1109 y Fh(0)3004 1100 y Fk(\025)3050 1142 y Fo(:)386 1388 y Fq(and)677 1596 y Fm(k)p Fo(A)800 1611 y Fn(2)839 1596 y Fm(k)889 1611 y Fn(T)-6 b(r)993 1596 y Fm(\024)1099 1481 y Fi(\015)1099 1540 y(\015)1099 1600 y(\015)1169 1528 y Fq(1)p 1164 1572 V 1164 1664 a Fo(\031)1249 1460 y Fi(Z)1305 1685 y Fk(z)s Fj(2)p Fn(\012;Im)n(\()p Fk(z)s Fn(\))p Fj(\024\000)p Fk(\016)1797 1596 y Fo($)s Fq(\()p Fo(z)t Fq(\))p Fo(L)p Fq(\()p Fo(dz)t Fq(\))2248 1481 y Fi(\015)2248 1540 y(\015)2248 1600 y(\015)2304 1664 y Fn(T)g(r)2408 1596 y Fm(\024)29 b(O)s Fq(\()p Fo(h)2690 1554 y Fj(\000)p Fk(n)2792 1596 y Fq(\))p Fo(e)2875 1554 y Fj(\000)p Fk(C)2978 1565 y Fh(~)2980 1578 y Fe(b)3011 1554 y Fk(\025)3057 1596 y Fo(:)386 1842 y Fq(This)k(\014nishes)g(the)g(pro)s(ofs)f(of)g(Lemma)g (3.1)p 3316 1842 60 60 v 1354 2103 a(4.)48 b Fp(symbolic)38 b(calculus)486 2278 y Fq(In)29 b(this)g(section)g(w)m(e)h(use)g(the)f (pseudo-di\013eren)m(tial)f(calculus)h(to)g(simplify)d(the)386 2394 y(righ)m(t)33 b(hand)h(side)g(in)g(Lemma)e(3.1.)48 b(Some)33 b(notation)g(and)h(results)h(are)f(recalled)386 2510 y(in)41 b(the)i(app)s(endix.)72 b(T)-8 b(o)42 b(b)s(e)g(able)f(to) h(compute)g(the)g(trace)h(of)e([\()p Fo(\037f)11 b Fq(\)\()p Fo(P)3095 2525 y Fj(\017)3134 2510 y Fo(;)17 b(\025)p Fq(\)])3300 2474 y Fj(1)3300 2535 y Fn(0)386 2626 y Fq(w)m(e)35 b(reduce)g(the)f(op)s(erator)f(\()p Fo(P)1505 2641 y Fj(\017)1567 2626 y Fm(\000)24 b Fo(z)t Fq(\))1755 2590 y Fj(\000)p Fn(1)1850 2626 y Fq(\()p Fo(P)1951 2641 y Fj(\017)2013 2626 y Fm(\000)f Fo(z)2158 2641 y Fn(0)2198 2626 y Fq(\))2236 2590 y Fj(\000)p Fk(m)2391 2626 y Fq(to)33 b(a)h(pseudo-di\013eren)m(tial)386 2743 y(op)s(erator)24 b(with)h(a)f(sym)m(b)s(ol)g(of)h(\014nite)f(expansion)h(in)g Fo(h)p Fq(.)41 b(The)25 b(follo)m(wing)e(lemma)f(is)386 2859 y(the)34 b(main)d(result)i(of)g(this)g(section.)46 b(Denote)33 b(b)m(y)h(Op)2341 2817 y Fk(w)2398 2859 y Fq(\()p Fo(r)2483 2823 y Fk(N)2480 2883 y Fj(\017)2550 2859 y Fq(\))f(an)g(appro)m(ximation)386 2975 y(of)f(\()p Fo(P)598 2990 y Fj(\017)659 2975 y Fm(\000)23 b Fo(z)t Fq(\))846 2939 y Fj(\000)p Fn(1)974 2975 y Fq(giv)m(en)32 b(b)m(y)i Fo(r)1411 2939 y Fk(N)1408 3000 y Fj(\017)1478 2975 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))27 b(=)1925 2900 y Fi(P)2030 3004 y Fk(j)t()h Fq(0)35 b Ff(such)f(that)462 1889 y Fq(T)-8 b(r[\()p Fo(\037f)11 b Fq(\)\()p Fo(P)886 1904 y Fj(\017)925 1889 y Fo(;)17 b(\025)p Fq(\)])1091 1848 y Fj(1)1091 1914 y Fn(0)1193 1889 y Fq(=)22 b Fm(\000)1401 1822 y Fq(\(2)p Fo(\031)t(h)p Fq(\))1640 1773 y Fj(\000)p Fk(n)p 1401 1866 342 4 v 1477 1957 a Fq(16)p Fo(i\031)1769 1753 y Fi(Z)1824 1979 y Fj(D)1901 1753 y Fi(Z)1957 1979 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)2253 1988 y Fh(1)2287 1979 y Fn(\))2386 1765 y Fk(n)2336 1794 y Fi(X)2346 2004 y Fk(j)t Fn(=1)2496 1889 y Fq(sgn)q(\()p Fo(@)2727 1904 y Fk(r)2765 1889 y Fo(\025)2822 1904 y Fk(j;)p Fn(0)2909 1889 y Fq(\(0\)\))1291 2200 y Fm(\002)1391 2065 y Fi(Z)1446 2290 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1698 2300 y Fe(j;)p Fh(0)1777 2290 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1960 2271 y Fh(2)1995 2290 y Fk(;\026)2057 2299 y Fh(1)2091 2290 y Fn(\))2140 2200 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r)2541 2120 y Fi(\000)2587 2200 y Fo(r)2634 2159 y Fk(N)2631 2225 y Fn(0)2701 2200 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))3017 2120 y Fi(\001)3079 2200 y Fo(dz)t(d)p Fm(L)1391 2435 y Fq(+)22 b Fm(O)s Fq(\()p Fo(h)1665 2393 y Fj(\000)p Fk(n)1767 2435 y Fq(\))p Fo(e)1850 2393 y Fj(\000)p Fk(C)1955 2402 y Fh(2)1990 2393 y Fk(\025)2057 2435 y Fq(+)g Fm(k)p Fo(f)11 b Fm(k)2314 2450 y Fj(1)2388 2435 y Fm(O)s Fq(\()p Fo(h)2564 2393 y Fk(N)c Fj(\000)p Fk(n)2729 2435 y Fq(\))p Fo(:)386 2601 y Ff(for)35 b(al)5 b(l)34 b Fo(\026)740 2616 y Fn(1)807 2601 y Fm(\024)28 b Fo(\026)971 2616 y Fn(1)1010 2601 y Fq(\()p Fo(r)1092 2616 y Fn(1)1131 2601 y Fq(\))p Ff(,)35 b(with)g Fo(C)1516 2616 y Fn(2)1583 2601 y Fq(=)27 b(inf)6 b Fm(f)p Fo(C)1924 2616 y Fn(0)1963 2601 y Fo(;)17 b(C)2074 2611 y Fn(~)2077 2629 y Fk(b)2111 2601 y Fo(;)g(C)2222 2611 y Fn(~)2225 2629 y Fk(b)p Fj(\000)p Fk(\026)2352 2638 y Fh(0)2391 2601 y Fm(g)28 b Fo(>)f Fq(0)p Fo(:)430 2780 y Fy(Lemma)89 b(5.2.)45 b Ff(Ther)-5 b(e)40 b(exist)h Fo(\032)1667 2795 y Fn(0)1747 2780 y Ff(and)j Fq(~)-53 b Fo(r)1986 2795 y Fn(0)2066 2780 y Ff(such)41 b(that)g(for)f(al)5 b(l)40 b Fo(r)2849 2795 y Fn(1)2927 2780 y Fm(\024)j Fq(~)-53 b Fo(r)3087 2795 y Fn(0)3167 2780 y Ff(ther)-5 b(e)386 2896 y(exists)34 b Fo(\026)711 2911 y Fn(1)751 2896 y Fq(\()p Fo(r)833 2911 y Fn(1)872 2896 y Fq(\))27 b Fo(>)h Fq(0)35 b Ff(such)f(that)541 3137 y Fq(T)-8 b(r)q([\()p Fo(\037f)11 b Fq(\)\()p Fo(P)966 3152 y Fj(\017)1005 3137 y Fo(;)17 b(\025)p Fq(\)])1171 3096 y Fj(1)1171 3161 y Fn(0)1273 3137 y Fq(=)22 b Fm(\000)1480 3069 y Fq(\(2)p Fo(\031)t(h)p Fq(\))1720 3021 y Fj(\000)p Fk(n)p 1480 3114 V 1556 3205 a Fq(16)p Fo(i\031)1848 3001 y Fi(Z)1904 3227 y Fj(D)1981 3001 y Fi(Z)2036 3227 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)2332 3236 y Fh(1)2367 3227 y Fn(\))2466 3012 y Fk(n)2415 3042 y Fi(X)2426 3252 y Fk(j)t Fn(=1)2576 3137 y Fq(sgn\()p Fo(@)2806 3152 y Fk(r)2844 3137 y Fo(\025)2901 3152 y Fk(j;)p Fn(0)2989 3137 y Fq(\(0\)\))1365 3312 y Fi(Z)1420 3538 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1672 3548 y Fe(j;)p Fh(0)1751 3538 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1934 3519 y Fh(2)1969 3538 y Fk(;\026)2031 3547 y Fh(1)2066 3538 y Fn(\))2135 3422 y Fq(~)2114 3448 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r\()p Fo(b)2578 3407 y Fk(N)2578 3473 y Fn(0)2646 3448 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))p Fo(dz)t(d)p Fm(L)1371 3682 y Fq(+)22 b Fm(O)s Fq(\()p Fo(h)1645 3641 y Fj(\000)p Fk(n)1747 3682 y Fq(\))p Fo(e)1830 3641 y Fj(\000)p Fk(C)1935 3650 y Fh(2)1970 3641 y Fk(\025)2037 3682 y Fq(+)g Fm(k)p Fo(f)11 b Fm(k)2294 3697 y Fj(1)2368 3682 y Fm(O)s Fq(\()p Fo(h)2544 3641 y Fk(N)c Fj(\000)p Fk(n)2709 3682 y Fq(\))p Fo(;)386 3849 y Ff(for)35 b(al)5 b(l)34 b Fo(\026)740 3864 y Fn(1)807 3849 y Fm(\024)28 b Fo(\026)971 3864 y Fn(1)1010 3849 y Fq(\()p Fo(r)1092 3864 y Fn(1)1131 3849 y Fq(\))p Ff(,)35 b(with)g Fo(C)1516 3864 y Fn(2)1583 3849 y Fq(=)27 b(inf)6 b Fm(f)p Fo(C)1924 3864 y Fn(0)1963 3849 y Fo(;)17 b(C)2074 3859 y Fn(~)2077 3877 y Fk(b)2111 3849 y Fo(;)g(C)2222 3859 y Fn(~)2225 3877 y Fk(b)p Fj(\000)p Fk(\026)2352 3886 y Fh(0)2391 3849 y Fm(g)28 b Fo(>)f Fq(0)p Fo(:)386 4027 y Fq(Pro)s(of)p 386 4043 235 4 v 32 w(:)43 b(Denote)921 4236 y Fo(B)995 4251 y Fj(\017)1062 4236 y Fq(=)28 b Fm(\000)1300 4168 y Fq(1)p 1253 4213 141 4 v 1253 4304 a(2)p Fo(i\031)1421 4100 y Fi(Z)1476 4326 y Fj(Z)1565 4236 y Fq(~)-61 b Fo(\037)p Fq(\()p Fo(z)t Fq(\))1761 4210 y(~)1739 4236 y Fo(f)12 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)Op)2155 4194 y Fk(w)2211 4236 y Fq(\()p Fo(b)2290 4195 y Fk(N)2290 4260 y Fj(\017)2358 4236 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\)\))p Fo(dz)t(:)386 4453 y Fq(F)-8 b(rom)32 b(Lemma)g(4.1)h(and)h(the)g (standard)g(expression)h(\([12)o(]\))f(for)f(the)h(trace)g(of)f(a)386 4570 y(pseudo-di\013eren)m(tial)e(op)s(erator,)h(w)m(e)i(get)400 4777 y(T)-8 b(r[)p Fo(B)601 4792 y Fj(\017)641 4777 y Fq(])668 4736 y Fj(1)668 4802 y Fn(0)771 4777 y Fq(=)1035 4710 y Fm(\000)p Fq(1)p 884 4755 428 4 v 884 4846 a(2)p Fo(i\031)t Fq(\(2)p Fo(\031)t(h)p Fq(\))1265 4797 y Fk(n)1338 4642 y Fi(Z)12 b(Z)f(Z)1527 4867 y Fk(x;\030)s Fj(2)p Fc(R)1716 4849 y Fe(n)1757 4867 y Fk(;)p Fj(Z)1865 4777 y Fq(~)-61 b Fo(\037)p Fq(\()p Fo(z)t Fq(\))2061 4751 y(~)2039 4778 y Fo(f)2099 4777 y Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r[)p Fo(b)2493 4736 y Fk(N)2493 4802 y Fj(\017)2561 4777 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)])2904 4736 y Fj(1)2904 4802 y Fn(0)2995 4777 y Fo(dz)k(dx)c(d\030)5 b(:)386 5029 y Fq(Let)33 b Fo(x)28 b Fq(=)f Fo(r)9 b Fq(^)-55 b Fo(x)p Fq(,)33 b Fo(\030)f Fq(=)1098 5002 y(^)1088 5029 y Fo(\030)t(\032)p Fq(.)44 b(F)-8 b(or)32 b(the)h(\014rst)g(time)e(w)m(e)j(use)f(the)g (fact)f(that)h Fo(n)g Fq(is)f(o)s(dd:)647 5257 y(T)-8 b(r[)p Fo(B)848 5272 y Fj(\017)888 5257 y Fq(])915 5216 y Fj(1)915 5282 y Fn(0)1017 5257 y Fq(=)28 b Fm(\000)1208 5190 y Fq(\(2)p Fo(\031)t(h)p Fq(\))1448 5141 y Fj(\000)p Fk(n)p 1208 5234 342 4 v 1308 5325 a Fq(8)p Fo(i\031)1576 5121 y Fi(Z)1632 5347 y Fk(r)r Fj(2)p Fc(R)1781 5121 y Fi(Z)12 b(Z)1907 5347 y Fn(^)-39 b Fk(x;)1970 5329 y Fn(^)1963 5347 y Fk(\030)r Fj(2)p Fn(S)2082 5328 y Fe(n)p Fd(\000)p Fh(1)2224 5121 y Fi(Z)2279 5347 y Fk(\032)p Fj(2)p Fc(R)1576 5397 y Fi(Z)1632 5623 y Fj(Z)1720 5533 y Fq(~)-61 b Fo(\037)p Fq(\()p Fo(z)t Fq(\))1916 5506 y(~)1894 5533 y Fo(f)12 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r[)p Fo(b)2348 5492 y Fk(N)2348 5557 y Fj(\017)2416 5533 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)])2759 5492 y Fj(1)2759 5557 y Fn(0)2850 5533 y Fo(d)p Fm(L)g Fo(dz)t(:)1842 5722 y Fg(10)p eop %%Page: 11 11 11 10 bop 386 459 a Fq(The)33 b(function)776 698 y(\()p Fo(r)m(;)17 b(\032)p Fq(\))28 b Fm(7!)1142 562 y Fi(Z)1197 788 y Fj(Z)1286 698 y Fq(~)-61 b Fo(\037)p Fq(\()p Fo(z)t Fq(\))1482 671 y(~)1460 698 y Fo(f)12 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r[)p Fo(b)1914 657 y Fk(N)1914 722 y Fj(\017)1982 698 y Fq(\()p Fo(r)9 b Fq(^)-55 b Fo(x;)17 b(\032)2226 671 y Fq(^)2216 698 y Fo(\030)t(;)g(z)t Fq(\)])2421 657 y Fj(1)2421 722 y Fn(0)2497 698 y Fo(r)2544 657 y Fk(n)p Fj(\000)p Fn(1)2680 698 y Fo(\032)2730 657 y Fk(n)p Fj(\000)p Fn(1)2884 698 y Fo(dz)386 970 y Fq(is)35 b(analytic)f(in)g (ev)m(ery)j(con)m(v)m(ex)g(domain)d(in)g(\()p Fo(r)m(;)17 b(\032)p Fq(\))33 b Fm(2)f Fl(C)2433 934 y Fn(2)2514 970 y Fq(where)k(Im\()p Fo(\025)3010 985 y Fk(j;)p Fn(0)3097 970 y Fq(\()p Fo(r)9 b Fq(^)-55 b Fo(x)p Fq(\))24 b(+)386 1086 y Fo(\032)436 1050 y Fn(2)476 1086 y Fq(\))37 b Fm(6)p Fq(=)g Fo(\016)42 b Fq(for)c(all)e Fo(j)44 b Fq(and)38 b Fo(r)i Fm(2)e Fo(S)1572 1101 y Fn(0)1650 1086 y Fq(\(w)m(e)h(use)g Fy(\(H3\))p Fq(\).)59 b(Studying)38 b Fo(b)2828 1101 y Fk(j;)p Fj(\017)2954 1086 y Fq(w)m(e)i(obtain)386 1202 y(uniformly)21 b(in)h Fo(\023)i Fq(in)e(the)i(domain)d(in)i Fo(x;)17 b(\030)5 b(;)17 b(z)27 b Fq(suc)m(h)d(that)f Fm(8)p Fo(k)s(;)34 b(d)p Fq(\()p Fo(\025)2679 1217 y Fk(k)r(;)p Fj(\017)2776 1202 y Fq(\()p Fo(x)p Fq(\))s(+)s Fo(\030)3037 1166 y Fn(2)3076 1202 y Fo(;)17 b(z)t Fq(\))28 b Fm(\025)g Fo(\023)386 1319 y Fq(and)33 b Fm(8)p Fo(k)s(;)g(d)p Fq(\()p Fo(\025)891 1334 y Fk(k)r(;)p Fj(\017)988 1319 y Fq(\()p Fo(x)p Fq(\))23 b(+)f Fo(\030)1288 1283 y Fn(2)1327 1319 y Fo(;)17 b(z)1416 1334 y Fn(0)1455 1319 y Fq(\))28 b Fm(\025)g Fo(C)7 b Fm(h)p Fo(\030)e Fm(i)1829 1283 y Fn(2)1867 1319 y Fq(,)33 b Fo(x)28 b Fm(2)g Fq(S,)640 1450 y Fi(\015)640 1510 y(\015)640 1570 y(\015)695 1565 y Fo(@)751 1523 y Fk(\013)746 1589 y(x)802 1565 y Fo(@)858 1517 y Fk(\014)853 1592 y(\030)906 1565 y Fq([)p Fo(b)974 1580 y Fk(j;)p Fj(\017)1062 1565 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)])1405 1523 y Fj(1)1405 1589 y Fn(0)1480 1450 y Fi(\015)1480 1510 y(\015)1480 1570 y(\015)1563 1565 y Fm(\024)28 b Fo(C)23 b Fq(Min)p Fm(f)p Fo(\023;)17 b(\023)2094 1523 y Fk(r)r Fn(\(2)p Fk(j)t Fn(+)p Fk(\013)p Fn(+)p Fk(\014)s Fn(+1\))2543 1565 y Fm(g)2593 1500 y Fj(\000)p Fn(1)2687 1565 y Fm(h)p Fo(x)p Fm(i)2820 1523 y Fj(\000)t Fn(~)-39 b Fk(n)2922 1565 y Fm(h)p Fo(\030)5 b Fm(i)3048 1523 y Fj(\000)p Fn(2)p Fk(m)p Fj(\000)p Fn(2)3294 1565 y Fo(:)-2935 b Fq(\(5.1\))386 1807 y(There)34 b(exists)f Fo(r)981 1822 y Fn(0)1048 1807 y Fo(>)28 b Fq(0)k(suc)m(h)i(that)e(for)g(all)f Fo(s)d Fm(2)g Fq([0)p Fo(;)17 b(r)2281 1822 y Fn(0)2320 1807 y Fq(])32 b(w)m(e)i(get)997 2072 y(\()p Fy(C1)p Fq(\))28 b Fm(8)p Fo(j;)17 b(r)31 b Fm(2)d Fl(R)5 b Fo(;)50 b Fm(j)p Fq(Im)o(\()p Fo(\025)1928 2087 y Fk(j;)p Fj(\017)2016 2072 y Fq(\(\()p Fo(r)24 b Fq(+)e Fo(is)p Fq(\))6 b(^)-55 b Fo(x)q Fq(\)\))p Fm(j)27 b(\024)2678 2005 y Fo(\016)p 2677 2049 49 4 v 2677 2141 a Fq(2)2736 2072 y Fo(:)386 2313 y Fq(Using)39 b(the)g(analyticit)m(y)e (in)i Fo(r)i Fq(and)e Fo(\032)h Fq(to)e(c)m(hange)i(the)g(in)m (tegration)d(path,)k(and)386 2429 y(\(5.1\),)32 b(w)m(e)i(get)651 2714 y(T)-8 b(r[)p Fo(B)852 2729 y Fj(\017)891 2714 y Fq(])918 2673 y Fj(1)918 2738 y Fn(0)1021 2714 y Fq(=)27 b Fm(\000)1211 2646 y Fq(\(2)p Fo(\031)t(h)p Fq(\))1452 2598 y Fj(\000)p Fk(n)p 1211 2691 342 4 v 1288 2782 a Fq(16)p Fo(i\031)1580 2578 y Fi(Z)1635 2804 y Fk(r)r Fj(2f)p Fc(R)p Fn(+)p Fk(ir)1910 2813 y Fh(0)1945 2804 y Fj(g[f)p Fc(R)p Fj(\000)p Fk(ir)2221 2813 y Fh(0)2256 2804 y Fj(g)2312 2578 y Fi(Z)11 b(Z)2438 2804 y Fn(^)-39 b Fk(x)o(;)2500 2786 y Fn(^)2493 2804 y Fk(\030)s Fj(2)p Fn(S)2613 2785 y Fe(n)p Fd(\000)p Fh(1)2754 2578 y Fi(Z)2810 2804 y Fk(\032)p Fj(2)p Fk(e)2926 2773 y Fd(\000)p Fe(i)3006 2754 y(\031)p 3006 2762 38 3 v 3010 2797 a Fh(4)3058 2804 y Fc(R)1633 2858 y Fi(Z)1688 3083 y Fj(Z)1776 2993 y Fq(~)-60 b Fo(\037)p Fq(\()p Fo(z)t Fq(\))1973 2967 y(~)1951 2993 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r)q([)p Fo(b)2405 2952 y Fk(N)2405 3018 y Fj(\017)2473 2993 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)])2816 2952 y Fj(1)2816 3018 y Fn(0)2890 2993 y Fo(d)p Fm(L)p Fo(dz)386 3261 y Fq(Denote)33 b(T)-8 b(r[)p Fo(B)922 3276 y Fj(\017)961 3261 y Fq(])988 3225 y Fj(1)988 3286 y Fn(0)1091 3261 y Fq(=:)28 b Fo(A)1295 3276 y Fn(1)1356 3261 y Fq(+)22 b Fo(A)1527 3276 y Fn(2)1599 3261 y Fq(with)674 3546 y Fo(A)747 3561 y Fn(1)814 3546 y Fq(=)g Fm(\000)1022 3478 y Fq(\(2)p Fo(\031)t(h)p Fq(\))1261 3430 y Fj(\000)p Fk(n)p 1022 3523 342 4 v 1098 3614 a Fq(16)p Fo(i\031)1390 3410 y Fi(Z)1445 3635 y Fk(r)r Fj(2f)p Fc(R)p Fn(+)p Fk(ir)1720 3644 y Fh(0)1755 3635 y Fj(g[f)p Fc(R)p Fj(\000)p Fk(ir)2031 3644 y Fh(0)2066 3635 y Fj(g)2122 3410 y Fi(Z)11 b(Z)2248 3635 y Fn(^)-40 b Fk(x;)2310 3618 y Fn(^)2303 3635 y Fk(\030)r Fj(2)p Fn(S)2422 3617 y Fe(n)p Fd(\000)p Fh(1)2564 3410 y Fi(Z)2620 3635 y Fk(\032)p Fj(2)p Fk(e)2736 3605 y Fd(\000)p Fe(i)2816 3586 y(\031)p 2816 3594 38 3 v 2820 3628 a Fh(4)2868 3635 y Fc(R)907 3689 y Fi(Z)962 3915 y Fk(z)s Fj(2)p Fn(\012;Im)n(\()p Fk(z)s Fn(\)=)p Fk(\016)r Fn(+)1442 3888 y Fh(1)p 1443 3900 31 3 v 1443 3941 a(2)1484 3915 y Fn(Im)o(\()p Fk(\032)1631 3896 y Fh(2)1666 3915 y Fn(\))1726 3825 y Fq(~)-61 b Fo(\037)p Fq(\()p Fo(z)t Fq(\))1922 3799 y(~)1900 3825 y Fo(f)12 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r[)p Fo(b)2354 3784 y Fk(N)2354 3850 y Fj(\017)2422 3825 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)])2765 3784 y Fj(1)2765 3850 y Fn(0)2840 3825 y Fo(d)p Fm(L)p Fo(dz)t(;)520 4296 y(A)593 4311 y Fn(2)633 4296 y Fq(\()p Fo(r)s Fq(\))27 b(=)22 b Fm(\000)990 4228 y Fq(\(2)p Fo(\031)t(h)p Fq(\))1230 4180 y Fj(\000)p Fk(n)p 990 4273 342 4 v 1066 4364 a Fq(16)p Fo(i\031)1358 4160 y Fi(Z)1414 4385 y Fk(r)r Fj(2f)p Fc(R)p Fn(+)p Fk(ir)1689 4394 y Fh(0)1723 4385 y Fj(g[f)p Fc(R)p Fj(\000)p Fk(ir)1999 4394 y Fh(0)2034 4385 y Fj(g)2090 4160 y Fi(Z)12 b(Z)2216 4385 y Fn(^)-39 b Fk(x;)2279 4368 y Fn(^)2272 4385 y Fk(\030)r Fj(2)p Fn(S)2391 4367 y Fe(n)p Fd(\000)p Fh(1)2533 4160 y Fi(Z)2588 4385 y Fk(\032)p Fj(2)p Fk(e)2704 4355 y Fd(\000)p Fe(i)2784 4335 y(\031)p 2784 4343 38 3 v 2788 4378 a Fh(4)2836 4385 y Fc(R)875 4439 y Fi(Z)931 4665 y Fk(z)s Fj(2)p Fn(\012;)p Fk(\016)r Fn(+)1183 4638 y Fh(1)p 1182 4650 31 3 v 1182 4691 a(2)1223 4665 y Fn(Im)o(\()p Fk(\032)1370 4646 y Fh(2)1405 4665 y Fn(\))p Fj(\024)p Fn(Im\()p Fk(z)s Fn(\))p Fj(\024)p Fk(\016)p 1772 4494 57 4 v 1772 4575 a Fo(@)1829 4590 y Fk(z)1880 4575 y Fq(~)-61 b Fo(\037)p Fq(\()p Fo(z)t Fq(\))2076 4549 y(~)2054 4575 y Fo(f)12 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r[)p Fo(b)2508 4534 y Fk(N)2508 4600 y Fj(\017)2576 4575 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)])2919 4534 y Fj(1)2919 4600 y Fn(0)2994 4575 y Fo(d)p Fm(L)p Fo(dz)t(:)386 4869 y Fq(W)-8 b(e)33 b(obtain,)e(using)i(\(5.1\),)1324 5088 y Fo(A)1397 5103 y Fn(2)1464 5088 y Fq(=)28 b Fm(O)s Fq(\()p Fo(h)1744 5047 y Fj(\000)p Fk(n)1846 5088 y Fq(\))p Fo(e)1929 5047 y Fj(\000)p Fn(Min)o Fj(f)p Fk(C)2192 5056 y Fh(0)2228 5047 y Fk(;C)2296 5058 y Fh(~)2298 5071 y Fe(b)2329 5047 y Fj(g)p Fk(\025)2409 5088 y Fo(:)386 5300 y Fq(Let)33 b Fo(\032)611 5315 y Fn(0)683 5300 y Fq(b)s(e)g(a)f(p)s(ositiv)m(e)g(constan)m(t)h(suc)m(h)h(that)1464 5554 y(\()p Fy(C2)p Fq(\))1770 5528 y(~)1774 5554 y Fo(b)28 b Fm(\024)1958 5487 y Fq(1)p 1958 5531 49 4 v 1958 5623 a(2)2017 5554 y Fo(\032)2067 5513 y Fn(2)2067 5579 y(0)2129 5554 y Fm(\000)22 b Fo(\016)n(:)1842 5722 y Fg(11)p eop %%Page: 12 12 12 11 bop 386 459 a Fq(W)-8 b(e)33 b(cut)g Fo(A)795 474 y Fn(1)867 459 y Fq(in)f(t)m(w)m(o)747 959 y Fo(A)820 974 y Fn(1)888 959 y Fq(=)22 b Fm(\000)1095 891 y Fq(\(2)p Fo(\031)t(h)p Fq(\))1335 843 y Fj(\000)p Fk(n)p 1095 936 342 4 v 1171 1027 a Fq(16)p Fo(i\031)1463 823 y Fi(Z)1519 1049 y Fk(r)r Fj(2f)p Fc(R)p Fn(+)p Fk(ir)1794 1058 y Fh(0)1828 1049 y Fj(g[f)p Fc(R)p Fj(\000)p Fk(ir)2104 1058 y Fh(0)2139 1049 y Fj(g)2195 823 y Fi(Z)2250 1049 y Fj(D)980 1102 y Fi(Z)1035 1328 y Fk(z)s Fj(2)p Fn(\012;Im)o(\()p Fk(z)s Fn(\)=)p Fk(\016)r Fn(+)1516 1301 y Fh(1)p 1517 1313 31 3 v 1517 1354 a(2)1557 1328 y Fn(Im)o(\()p Fk(\032)1704 1309 y Fh(2)1740 1328 y Fn(\))1799 1238 y Fq(~)-60 b Fo(\037)p Fq(\()p Fo(z)t Fq(\))1996 1212 y(~)1974 1238 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r)q([)p Fo(b)2428 1197 y Fk(N)2428 1263 y Fj(\017)2496 1238 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)])2839 1197 y Fj(1)2839 1263 y Fn(0)2913 1238 y Fo(d)p Fm(L)p Fo(dz)986 1547 y Fm(\000)1095 1480 y Fq(\(2)p Fo(\031)t(h)p Fq(\))1335 1432 y Fj(\000)p Fk(n)p 1095 1524 342 4 v 1171 1616 a Fq(16)p Fo(i\031)1463 1412 y Fi(Z)1519 1637 y Fk(r)r Fj(2f)p Fc(R)p Fn(+)p Fk(ir)1794 1646 y Fh(0)1828 1637 y Fj(g[f)p Fc(R)p Fj(\000)p Fk(ir)2104 1646 y Fh(0)2139 1637 y Fj(g)2195 1412 y Fi(Z)2250 1637 y Fk(\032)p Fj(2)p Fk(e)2366 1607 y Fd(\000)p Fe(i)2446 1587 y(\031)p 2446 1595 38 3 v 2450 1630 a Fh(4)2499 1637 y Fc(R)p Fk(;)p Fj(j)p Fk(\032)p Fj(j\025)p Fk(\032)2734 1646 y Fh(0)2788 1412 y Fi(Z)11 b(Z)2914 1637 y Fn(^)-39 b Fk(x)o(;)2976 1620 y Fn(^)2969 1637 y Fk(\030)s Fj(2)p Fn(S)3089 1618 y Fe(n)p Fd(\000)p Fh(1)980 1691 y Fi(Z)1035 1917 y Fk(z)s Fj(2)p Fn(\012;Im)o(\()p Fk(z)s Fn(\)=)p Fk(\016)r Fn(+)1516 1890 y Fh(1)p 1517 1902 31 3 v 1517 1943 a(2)1557 1917 y Fn(Im)o(\()p Fk(\032)1704 1898 y Fh(2)1740 1917 y Fn(\))1799 1827 y Fq(~)-60 b Fo(\037)p Fq(\()p Fo(z)t Fq(\))1996 1801 y(~)1974 1827 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r)q([)p Fo(b)2428 1786 y Fk(N)2428 1852 y Fj(\017)2496 1827 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)])2839 1786 y Fj(1)2839 1852 y Fn(0)2913 1827 y Fo(d)p Fm(L)p Fo(dz)860 2049 y Fq(=:)p Fo(A)1036 2064 y Fn(11)1133 2049 y Fq(+)22 b Fo(A)1304 2064 y Fn(12)1379 2049 y Fo(:)386 1470 y Fq(\(5.2\))386 2492 y(W)-8 b(e)33 b(observ)m(e,)h(using)f Fy(\(C2\))p Fq(,)e(that)1456 2933 y Fo(A)1529 2948 y Fn(12)1631 2933 y Fq(=)d Fm(O)s Fq(\()p Fo(h)1911 2892 y Fj(\000)p Fk(n)2013 2933 y Fq(\))p Fo(e)2096 2892 y Fj(\000)p Fk(C)2199 2903 y Fh(~)2201 2916 y Fe(b)2233 2892 y Fk(\025)2278 2933 y Fo(:)386 3376 y Fq(Let)53 b(~)-53 b Fo(r)621 3391 y Fn(0)716 3376 y Fm(\024)55 b Fo(r)892 3391 y Fn(0)932 3376 y Fq(.)92 b(Let)49 b Fo(\015)54 b Fq(b)s(e)49 b(a)f(path)h(in)f Fo(z)54 b Fq(included)48 b(in)60 b(~)-61 b Fo(\037)56 b Fq(=)f(0)48 b(and)h Fm(f)p Fo(z)60 b Fm(2)386 3492 y Fl(C)20 b Fo(;)50 b Fq(Im\()p Fo(z)t Fq(\))28 b Fm(\024)g Fo(\016)t Fm(g)p Fq(,)c(starting)d(in)g(the)h(region)f Fm(f)p Fo(z)32 b Fm(2)c Fl(C)20 b Fo(;)50 b Fq(Im\()p Fo(z)t Fq(\))28 b(=)g Fo(\016)t Fq(+)2818 3453 y Fn(1)p 2818 3469 36 4 v 2818 3526 a(2)2863 3492 y Fq(Im)o(\()p Fo(\032)3067 3456 y Fn(2)3107 3492 y Fq(\))p Fo(;)17 b Fq(Re\()p Fo(z)t Fq(\))28 b(=)386 3617 y Fm(\000)p Fo(a)p Fm(g)p Fq(,)36 b(and)f(ending)g(in)f(the)h(region)f Fm(f)p Fo(z)i Fm(2)c Fl(C)20 b Fo(;)55 b Fq(Im)o(\()p Fo(z)t Fq(\))33 b(=)e Fo(\016)d Fq(+)2655 3577 y Fn(1)p 2655 3594 V 2655 3651 a(2)2701 3617 y Fq(Im)o(\()p Fo(\032)2905 3580 y Fn(2)2945 3617 y Fq(\))p Fo(;)17 b Fq(Re\()p Fo(z)t Fq(\))32 b(=)386 3734 y Fo(a)p Fm(g)p Fq(.)72 b(Let)42 b Fo(\026)829 3749 y Fn(0)912 3734 y Fo(>)h Fq(0)f(b)s(e)g(a)g(constan) m(t)h(small)c(enough)k(suc)m(h)g(that)f(the)g(follo)m(wing)386 3850 y(h)m(yp)s(otheses)35 b(are)d(satis\014ed)h(:)609 4127 y Fm(\017)41 b Fy(\(C3\))130 b Fo(d)p Fq(\()p Fo(\015)5 b(;)17 b Fq(Supp)11 b(~)-60 b Fo(\037)p Fq(\))28 b Fo(>)f Fq(2)p Fo(\026)1798 4142 y Fn(0)1837 4127 y Fq(,)609 4245 y Fm(\017)41 b Fy(\(C4\))97 b Fo(\026)1081 4260 y Fn(0)1148 4245 y Fm(\024)1264 4206 y Fk(\016)p 1263 4222 V 1263 4280 a Fn(2)1308 4245 y Fq(,)609 4363 y Fm(\017)41 b Fy(\(C5\))200 b Fq(inf)1022 4425 y Fc(R)p Fj(\006)p Fk(ir)1181 4434 y Fh(0)1216 4425 y Fk(;i)p Fj(6)p Fn(=)p Fk(j)1363 4363 y Fo(d)p Fq(\()p Fo(\025)1509 4378 y Fk(j;)p Fn(0)1596 4363 y Fq(\()p Fo(x)p Fq(\))p Fo(;)17 b(\025)1828 4378 y Fk(i;)p Fn(0)1911 4363 y Fq(\()p Fo(x)p Fq(\)\))28 b Fm(\025)g Fq(2)p Fo(\026)2321 4378 y Fn(0)2360 4363 y Fq(,)609 4531 y Fm(\017)41 b Fy(\(C6\))97 b Fq(Supp)q(\(1)1326 4546 y Fn(Supp)9 b(~)-44 b Fk(\037)1553 4531 y Fm(\000)34 b Fq(~)-61 b Fo(\037)p Fq(\))22 b(+)g Fo(D)s Fq(\(0)p Fo(;)17 b(\026)2145 4546 y Fn(0)2184 4531 y Fq(\))27 b Fm(2)h Fq(\012)23 b(+)f Fo(E)2606 4546 y Fn(0)2646 4531 y Fq(,)609 4648 y Fm(\017)41 b Fy(\(C7\))97 b Fq(Supp)13 b(~)-61 b Fo(\037)22 b Fq(+)g Fo(D)s Fq(\(0)p Fo(;)17 b Fq(2)p Fo(\026)1743 4663 y Fn(0)1781 4648 y Fq(\))28 b Fm(2)g Fq(\012)23 b(+)f Fo(E)2204 4663 y Fn(0)2243 4648 y Fq(,)609 4764 y Fm(\017)41 b Fy(\(C8\))97 b Fo(D)s Fq(\()p Fo(\025)1201 4779 y Fk(j)t Fj(1)1308 4764 y Fo(;)17 b(\026)1411 4779 y Fn(0)1450 4764 y Fq(\))22 b(+)g Fo(i)p Fl(R)50 b Fo(=)-60 b Fm(2)28 b Fq(Supp)12 b(~)-60 b Fo(\037)p Fq(,)800 4880 y(and)609 4996 y Fm(\017)41 b Fy(\(C9\))97 b Fq(inf)1140 5012 y Fk(r)10 b(=)-43 b Fj(2)p Fk(D)r Fn(\(0)p Fk(;)s Fn(~)-38 b Fk(r)1395 5021 y Fh(0)1430 5012 y Fn(\))1478 4996 y Fo(d)p Fq(\()p Fo(\025)1624 5011 y Fk(j;)p Fn(0)1711 4996 y Fq(\()p Fo(r)9 b Fq(^)-55 b Fo(x)p Fq(\))p Fo(;)17 b(\025)1990 5011 y Fk(i;)p Fn(0)2073 4996 y Fq(\()p Fo(r)9 b Fq(^)-55 b Fo(x)p Fq(\)\))27 b Fm(\025)h Fq(2)p Fo(\026)2529 5011 y Fn(0)2568 4996 y Fo(:)386 5273 y Fq(Mostly)36 b(these)h(conditions)e(are)h(necessary)i (to)e(de\014ne)h(the)f(future)g(in)m(tegration)386 5389 y(that)23 b(w)m(e)h(will)d(use.)41 b(The)24 b(main)d(p)s(oin)m(t)h(w)m (e)i(use)g(is)f(that)g(the)g(discs)h Fo(@)5 b(D)s Fq(\()p Fo(\025)2972 5404 y Fn(0)p Fk(;j)3064 5389 y Fq(\()p Fo(x)p Fq(\))p Fo(;)17 b(\026)3298 5404 y Fn(0)3337 5389 y Fq(\))386 5505 y(for)41 b Fm(j)p Fq(Im)o(\()p Fo(x)p Fq(\))p Fm(j)j Fq(=)f Fo(r)1054 5520 y Fn(0)1136 5505 y Fq(are)f(disjoin)m(t)e(\()p Fy(\(C5\))p Fq(\))h(and)h(included)g(in)f Fm(f)p Fo(z)t(;)17 b Fq(Im\()p Fo(z)t Fq(\))44 b Fm(\024)g Fo(\016)t Fm(g)386 5622 y Fq(\()p Fy(\(C1\))p Fq(,)g Fy(\(C4\))p Fq(\).)71 b(Changing)42 b(the)h(in)m(tegration)d(path)j(Im) o(\()p Fo(z)t Fq(\))i(=)f Fo(\016)33 b Fq(+)3047 5582 y Fn(1)p 3047 5599 V 3047 5656 a(2)3093 5622 y Fq(Im)o(\()p Fo(\032)3297 5585 y Fn(2)3337 5622 y Fq(\))1842 5722 y Fg(12)p eop %%Page: 13 13 13 12 bop 386 459 a Fq(in)m(to)32 b(the)h(path)f Fo(\015)38 b Fq(in)32 b(the)h(form)m(ula)d(\(5.2\),)j(using)f Fy(\(C7\))f Fq(and)i Fy(\(C3\))p Fq(,)f(giv)m(es)h(:)470 710 y(T)-8 b(r[)p Fo(B)671 725 y Fj(\017)711 710 y Fq(])738 668 y Fj(1)738 734 y Fn(0)840 710 y Fq(=)22 b Fm(\000)1048 642 y Fq(\(2)p Fo(\031)t(h)p Fq(\))1287 594 y Fj(\000)p Fk(n)p 1048 687 342 4 v 1124 778 a Fq(16)p Fo(i\031)1416 574 y Fi(Z)1471 800 y Fk(r)r Fj(2f)p Fc(R)p Fj(\006)p Fk(ir)1746 809 y Fh(0)1781 800 y Fj(g)1837 574 y Fi(Z)1892 800 y Fj(D)1969 710 y Fo(d)p Fm(L)916 908 y Fi(\002)974 854 y(Z)1097 1037 y Fn(Im)o(\()p Fk(z)s Fn(\))p Fj(\024)p Fk(\016)r Fn(+)1424 1010 y Fh(1)p 1425 1022 31 3 v 1425 1063 a(2)1465 1037 y Fn(Im)o(\()p Fk(\032)1612 1013 y Fh(2)1648 1037 y Fn(\))1030 1127 y Fk(z)11 b(=)-43 b Fj(2)o([)1159 1137 y Fe(j)1192 1127 y Fk(D)r Fn(\()p Fk(\025)1320 1137 y Fe(j;)p Fd(\017)1400 1127 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1583 1104 y Fh(2)1618 1127 y Fk(;\026)1680 1136 y Fh(0)1715 1127 y Fn(\))p 1763 908 57 4 v 1763 989 a Fo(@)1820 1004 y Fk(z)1871 989 y Fq(~)-60 b Fo(\037)p Fq(\()p Fo(z)t Fq(\))2067 963 y(~)2046 989 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r\()p Fo(b)2510 948 y Fk(N)2510 1014 y Fj(\017)2578 989 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))p Fo(L)p Fq(\()p Fo(dz)t Fq(\))3174 908 y Fi(\003)3216 931 y Fj(1)3216 1028 y Fn(0)938 1341 y Fm(\000)1048 1273 y Fq(\(2)p Fo(\031)t(h)p Fq(\))1287 1225 y Fj(\000)p Fk(n)p 1048 1318 342 4 v 1124 1409 a Fq(16)p Fo(i\031)1416 1205 y Fi(Z)1471 1431 y Fk(r)r Fj(2f)p Fc(R)p Fj(\006)p Fk(ir)1746 1440 y Fh(0)1781 1431 y Fj(g)1837 1205 y Fi(Z)1892 1431 y Fj(D)1969 1341 y Fo(d)p Fm(L)916 1539 y Fi(\002)974 1485 y(Z)1030 1710 y Fk(z)s Fj(2)p Fn(\()p Fj([)1187 1720 y Fe(j)1219 1710 y Fk(@)t(D)r Fn(\()p Fk(\025)1388 1720 y Fe(j;)p Fd(\017)1469 1710 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1652 1691 y Fh(2)1686 1710 y Fk(;\026)1748 1719 y Fh(0)1783 1710 y Fn(\)\))1870 1620 y Fq(~)-60 b Fo(\037)p Fq(\()p Fo(z)t Fq(\))2067 1594 y(~)2045 1620 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r)q(\()p Fo(b)2510 1579 y Fk(N)2510 1645 y Fj(\017)2577 1620 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))p Fo(dz)3031 1539 y Fi(\003)3073 1562 y Fj(1)3073 1659 y Fn(0)938 1854 y Fq(+)22 b Fm(O)s Fq(\()p Fo(h)1212 1813 y Fj(\000)p Fk(n)1314 1854 y Fq(\))p Fo(e)1397 1813 y Fj(\000)p Fk(C)1502 1822 y Fh(2)1537 1813 y Fk(\025)1582 1854 y Fo(:)386 2038 y Fq(W)-8 b(e)33 b(get)f(from)g(\(5.1\))g(that)473 2289 y Fm(\000)560 2222 y Fq(\(2)p Fo(\031)t(h)p Fq(\))801 2173 y Fj(\000)p Fk(n)p 560 2266 V 637 2358 a Fq(16)p Fo(i\031)929 2154 y Fi(Z)984 2379 y Fk(r)r Fj(2f)p Fc(R)p Fj(\006)p Fk(ir)1259 2388 y Fh(0)1294 2379 y Fj(g)1350 2154 y Fi(Z)1405 2379 y Fj(D)1482 2289 y Fo(d)p Fm(L)912 2488 y Fi(\002)971 2433 y(Z)1093 2616 y Fn(Im)o(\()p Fk(z)s Fn(\))p Fj(\024)p Fk(\016)r Fn(+)1420 2589 y Fh(1)p 1421 2601 31 3 v 1421 2643 a(2)1462 2616 y Fn(Im)o(\()p Fk(\032)1609 2593 y Fh(2)1644 2616 y Fn(\))1026 2707 y Fk(z)11 b(=)-43 b Fj(2[)1156 2717 y Fe(j)1188 2707 y Fk(D)r Fn(\()p Fk(\025)1316 2717 y Fe(j;)p Fd(\017)1397 2707 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1580 2684 y Fh(2)1614 2707 y Fk(;\026)1676 2716 y Fh(0)1711 2707 y Fn(\))p 1759 2488 57 4 v 1759 2569 a Fo(@)1816 2584 y Fk(z)1867 2569 y Fq(~)-60 b Fo(\037)p Fq(\()p Fo(z)t Fq(\))2064 2542 y(~)2042 2569 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r)q(\()p Fo(b)2507 2528 y Fk(N)2507 2593 y Fj(\017)2574 2569 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))2928 2488 y Fi(\003)2970 2510 y Fj(1)2970 2608 y Fn(0)3045 2569 y Fo(L)p Fq(\()p Fo(dz)t Fq(\))940 2851 y(=)28 b Fm(O)s Fq(\()p Fo(h)1220 2810 y Fj(\000)p Fk(n)1322 2851 y Fq(\))p Fo(e)1405 2810 y Fj(\000)p Fk(C)1510 2819 y Fh(2)1545 2810 y Fk(\025)1590 2851 y Fo(:)386 3035 y Fq(The)33 b(condition)f Fy(\(C9\))f Fq(implies)f(that)724 3286 y(T)-8 b(r[)p Fo(B)925 3301 y Fj(\017)965 3286 y Fq(])992 3245 y Fj(1)992 3311 y Fn(0)1094 3286 y Fq(=)1180 3219 y(\(2)p Fo(\031)t(h)p Fq(\))1420 3170 y Fj(\000)p Fk(n)p 1180 3263 342 4 v 1256 3354 a Fq(16)p Fo(i\031)1548 3151 y Fi(Z)12 b(Z)1901 3334 y Fk(r)r Fj(2f)p Fc(R)p Fj(\006)p Fk(ir)2176 3343 y Fh(0)2211 3334 y Fj(g)1670 3415 y Fk(z)s Fj(2)p Fn(\()p Fj([)1827 3425 y Fe(j)1860 3415 y Fk(@)t(D)r Fn(\()p Fk(\025)2029 3425 y Fe(j;)p Fh(0)2108 3415 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)2291 3392 y Fh(2)2326 3415 y Fk(;\026)2388 3424 y Fh(0)2422 3415 y Fn(\)\))2498 3151 y Fi(Z)2553 3376 y Fj(D)2631 3286 y Fo(d)p Fm(L)1182 3565 y Fq(~)-61 b Fo(\037)p Fq(\()p Fo(z)t Fq(\))1378 3538 y(~)1356 3565 y Fo(f)12 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r\()p Fo(b)1821 3523 y Fk(N)1821 3589 y Fn(0)1889 3565 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))p Fo(dz)26 b Fq(+)c Fm(O)s Fq(\()p Fo(h)2639 3523 y Fj(\000)p Fk(n)2741 3565 y Fq(\))p Fo(e)2824 3523 y Fj(\000)p Fk(C)2929 3532 y Fh(2)2964 3523 y Fk(\025)3010 3565 y Fo(:)386 3748 y Fq(Since)43 b(Supp\(1)954 3763 y Fn(Supp)9 b(~)-43 b Fk(\037)1188 3748 y Fm(\000)41 b Fq(~)-61 b Fo(\037)p Fq(\))29 b(+)g Fo(D)s Fq(\(0)p Fo(;)17 b(\026)1801 3763 y Fn(0)1839 3748 y Fq(\))42 b(is)g(included)h(in)e(a)i(region)e(where)j(f)e(is)386 3867 y(de\014ned)37 b Fy(\(C6\))p Fq(,)f(and)g(included)f(in)g(a)h (region)e(where)j Fo(f)47 b Fq(is)35 b(b)s(ounded)i(b)m(y)f Fo(e)3162 3831 y Fj(\000)p Fk(C)3267 3840 y Fh(2)3302 3831 y Fk(\025)3348 3867 y Fq(,)386 3983 y(w)m(e)e(get)e(from)f(\(5\))i (and)f Fy(\(C5\))g Fq(that)508 4241 y(T)-8 b(r[)p Fo(B)709 4256 y Fj(\017)749 4241 y Fq(])776 4200 y Fj(1)776 4266 y Fn(0)878 4241 y Fq(=)964 4174 y(\(2)p Fo(\031)t(h)p Fq(\))1204 4125 y Fj(\000)p Fk(n)p 964 4218 V 1040 4309 a Fq(16)p Fo(i\031)1332 4106 y Fi(Z)1388 4331 y Fj(D)1515 4117 y Fk(n)1465 4146 y Fi(X)1476 4356 y Fk(j)t Fn(=1)1625 4106 y Fi(Z)12 b(Z)1911 4289 y Fk(r)r Fj(2f)p Fc(R)p Fj(\006)p Fk(ir)2186 4298 y Fh(0)2221 4289 y Fj(g)1747 4370 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1999 4380 y Fe(j;)p Fh(0)2078 4370 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)2261 4347 y Fh(2)2296 4370 y Fk(;\026)2358 4379 y Fh(0)2392 4370 y Fn(\))2441 4241 y Fq(1)2490 4256 y Fn(Supp)c(~)-43 b Fk(\037)2694 4241 y Fq(\()p Fo(\025)2789 4256 y Fk(j;)p Fn(0)2876 4241 y Fq(\()p Fo(x)p Fq(\))23 b(+)f Fo(\030)3176 4200 y Fn(2)3215 4241 y Fq(\))1063 4520 y(~)-60 b Fo(\037)p Fq(\()p Fo(z)t Fq(\))1260 4493 y(~)1238 4520 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r)q(\()p Fo(b)1703 4478 y Fk(N)1703 4544 y Fn(0)1770 4520 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))p Fo(dz)t(d)p Fm(L)22 b Fq(+)g Fm(O)s Fq(\()p Fo(h)2640 4478 y Fj(\000)p Fk(n)2742 4520 y Fq(\))p Fo(e)2825 4478 y Fj(\000)p Fk(C)2930 4487 y Fh(2)2965 4478 y Fk(\025)878 4759 y Fq(=)964 4692 y(\(2)p Fo(\031)t(h)p Fq(\))1204 4643 y Fj(\000)p Fk(n)p 964 4736 V 1040 4827 a Fq(16)p Fo(i\031)1383 4635 y Fk(n)1332 4664 y Fi(X)1343 4874 y Fk(j)t Fn(=1)1493 4623 y Fi(Z)11 b(Z)1778 4807 y Fk(r)r Fj(2f)p Fc(R)p Fj(\006)p Fk(ir)2053 4816 y Fh(0)2088 4807 y Fj(g)1615 4888 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1867 4898 y Fe(j;)p Fh(0)1945 4888 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)2128 4865 y Fh(2)2163 4888 y Fk(;\026)2225 4897 y Fh(0)2260 4888 y Fn(\))2308 4623 y Fi(Z)2363 4849 y Fj(D)2452 4759 y Fq(~)-60 b Fo(\037)p Fq(\()p Fo(\025)2597 4774 y Fk(j;)p Fn(0)2684 4759 y Fq(\()p Fo(r)9 b Fq(^)-55 b Fo(x)p Fq(\))22 b(+)g Fo(\030)3030 4718 y Fn(2)3069 4759 y Fq(\))1073 5011 y(~)1052 5038 y Fo(f)10 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r)q(\()p Fo(b)1516 4996 y Fk(N)1516 5062 y Fn(0)1584 5038 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))p Fo(dz)t(d)p Fm(L)k Fq(+)h Fm(O)s Fq(\()p Fo(h)2453 4996 y Fj(\000)p Fk(n)2555 5038 y Fq(\))p Fo(e)2638 4996 y Fj(\000)p Fk(C)2743 5005 y Fh(2)2778 4996 y Fk(\025)2824 5038 y Fo(:)386 5232 y Fq(Let,)33 b(for)f Fo(\032)c Fm(2)g Fo(e)954 5195 y Fj(\000)p Fk(i)1043 5168 y Fe(\031)p 1043 5180 38 3 v 1047 5221 a Fh(4)1095 5232 y Fl(R)5 b Fq(,)426 5485 y Fo(g)t Fq(\()p Fo(r)s Fq(\))26 b(=)780 5360 y Fk(n)730 5390 y Fi(X)740 5600 y Fk(j)t Fn(=1)902 5485 y Fq(~)-61 b Fo(\037)p Fq(\()p Fo(\025)1046 5500 y Fk(j;)p Fn(0)1134 5485 y Fq(\()p Fo(r)9 b Fq(^)-55 b Fo(x)p Fq(\))22 b(+)g Fo(\030)1480 5443 y Fn(2)1519 5485 y Fq(\))1574 5349 y Fi(Z)1628 5574 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1880 5584 y Fe(j;)p Fh(0)1959 5574 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)2142 5556 y Fh(2)2177 5574 y Fk(;\026)2239 5583 y Fh(0)2274 5574 y Fn(\))2343 5458 y Fq(~)2322 5485 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r\()p Fo(b)2786 5443 y Fk(N)2786 5509 y Fn(0)2854 5485 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))p Fo(dz)t(:)1842 5722 y Fg(13)p eop %%Page: 14 14 14 13 bop 386 459 a Fq(W)-8 b(e)38 b(get)f(lim)861 474 y Fj(j)p Fk(r)r Fj(j!1)p Fk(;r)r Fj(2)p Fk(S)1243 459 y Fo(\025)1300 474 y Fk(j;)p Fn(0)1388 459 y Fq(\()p Fo(r)9 b Fq(^)-55 b Fo(x)p Fq(\))35 b(=)h Fo(\025)1770 474 y Fk(j)t Fj(1)1914 459 y Fq(and)i Fy(\(C8\))e Fq(so)h Fo(g)k Fq(is)c(of)f(compact)h(sup-)386 575 y(p)s(ort.)42 b Fy(\(H3\))28 b Fq(implies)e(that)j(there)h(exist)j(~)-53 b Fo(r)1949 590 y Fn(1)2017 575 y Fq(and)29 b Fo(C)2273 590 y Fk(V)2314 599 y Fh(0)2382 575 y Fq(\()g(dep)s(enden)m(t)i(of)d Fo(V)3081 590 y Fn(0)3120 575 y Fq(\))h(suc)m(h)386 691 y(that)39 b(for)f(all)f Fo(r)945 706 y Fn(1)1024 691 y Fo(<)43 b Fq(~)-54 b Fo(r)1182 706 y Fn(1)1222 691 y Fq(,)41 b Fo(r)g Fm(2)e Fo(S)1540 706 y Fn(0)1619 691 y Fq(and)g(all)e Fo(j)45 b Fq(satisfying)38 b Fo(@)2534 706 y Fk(r)2573 691 y Fo(\025)2630 706 y Fk(j)2666 691 y Fq(\(0\))h Fo(>)f Fq(0)h(w)m(e)h(ha)m(v)m(e)386 807 y(Im)o(\()p Fo(\025)597 822 y Fk(j;)p Fn(0)685 807 y Fq(\(\()p Fo(r)27 b Fm(\000)f Fo(ir)1012 822 y Fn(1)1051 807 y Fq(\))6 b(^)-55 b Fo(x)q Fq(\)\))34 b Fm(\024)h(\000)p Fo(C)1514 822 y Fk(V)1555 831 y Fh(0)1594 807 y Fo(r)1638 822 y Fn(1)1714 807 y Fq(\(resp.)57 b(for)36 b(all)e Fo(j)43 b Fq(satisfying)35 b Fo(@)2873 822 y Fk(r)2912 807 y Fo(\025)2969 822 y Fk(j;)p Fn(0)3056 807 y Fq(\(0\))g Fo(<)f Fq(0)386 924 y(w)m(e)g(ha)m(v)m(e)g(Im)o(\()p Fo(\025)966 939 y Fk(j;)p Fn(0)1054 924 y Fq(\(\()p Fo(r)24 b Fq(+)f Fo(ir)1374 939 y Fn(1)1413 924 y Fq(\))6 b(^)-55 b Fo(x)q Fq(\)\))28 b Fm(\024)g(\000)p Fo(C)1863 939 y Fk(V)1904 948 y Fh(0)1943 924 y Fo(r)1987 939 y Fn(1)2027 924 y Fq(.\))44 b(Let)33 b(us)g(add)g(the)g(last)f(condition)386 1055 y(on)g Fo(\026)580 1070 y Fn(0)652 1055 y Fq(and)838 1029 y(^)842 1055 y Fo(b)h Fq(:)1144 1235 y(\()p Fy(C10)p Fq(\))98 b Fo(C)1581 1250 y Fk(V)1622 1259 y Fh(0)1665 1235 y Fq(~)-54 b Fo(r)1704 1250 y Fn(1)1772 1235 y Fm(\025)1873 1208 y Fq(~)1877 1235 y Fo(b;)17 b(C)2032 1250 y Fk(V)2073 1259 y Fh(0)2116 1235 y Fq(~)-53 b Fo(r)2156 1250 y Fn(1)2217 1235 y Fm(\000)23 b Fo(\026)2376 1250 y Fn(0)2442 1235 y Fm(\025)2544 1208 y Fq(~)2548 1235 y Fo(b:)386 1416 y Fq(There)39 b(exist)g(paths)g Fo(r)h Fm(7!)d Fo(\015)1454 1431 y Fk(j)1490 1416 y Fq(\()p Fo(r)s Fq(\))h(suc)m(h)i(that)e Fo(\015)2145 1431 y Fk(j)2181 1416 y Fq(\()p Fo(r)s Fq(\))f(=)g Fo(r)k Fq(for)d(large)f Fo(r)s Fq(,)j Fo(\015)3103 1431 y Fk(j)3139 1416 y Fq(\()p Fo(r)s Fq(\))d(=)386 1532 y Fo(r)25 b Fm(\006)d Fo(i)t Fq(~)-53 b Fo(r)631 1547 y Fn(1)704 1532 y Fq(for)32 b Fo(r)j Fq(near)e(0)f(and)1325 1708 y(Im)o(\()p Fo(\025)1536 1723 y Fk(j;)p Fn(0)1623 1708 y Fq(\()p Fo(\015)1712 1723 y Fk(j)1748 1708 y Fq(\()p Fo(r)s Fq(\))6 b(^)-55 b Fo(x)p Fq(\))28 b Fm(\024)g(\000)2170 1682 y Fq(~)2174 1708 y Fo(b)23 b Fm(\000)g Fo(\026)2397 1723 y Fn(0)386 1894 y Fq(for)h Fo(r)j Fq(suc)m(h)f(that)e Fo(\025)1070 1909 y Fk(j;)p Fn(0)1157 1894 y Fq(\()p Fo(r)9 b Fq(^)-55 b Fo(x)p Fq(\))5 b(+)g Fo(\030)1469 1857 y Fn(2)1536 1894 y Fm(2)28 b Fq(Supp)13 b(~)-61 b Fo(\037)p Fq(.)41 b(Changing)24 b(one)g(of)g(the)h(paths)g Fl(R)10 b Fm(\006)c Fo(ir)3329 1909 y Fn(0)386 2010 y Fq(in)m(to)32 b Fo(\015)635 2025 y Fk(j)671 2010 y Fq(,)g(using)h Fy(\(C9\))p Fq(,)e(giv)m(es)i(:)425 2268 y(T)-8 b(r[)p Fo(B)626 2283 y Fj(\017)666 2268 y Fq(])693 2227 y Fj(1)693 2293 y Fn(0)795 2268 y Fq(=)909 2201 y(\(2)p Fo(\031)t(h)p Fq(\))1149 2153 y Fj(\000)p Fk(n)p 909 2246 342 4 v 1009 2337 a Fq(8)p Fo(i\031)1328 2144 y Fk(n)1277 2174 y Fi(X)1288 2384 y Fk(j)t Fn(=1)1438 2133 y Fi(Z)1493 2358 y Fk(r)r Fj(2)p Fk(\015)1610 2368 y Fe(j)1663 2133 y Fi(Z)1719 2358 y Fj(D)1808 2268 y Fq(~)-61 b Fo(\037)p Fq(\()p Fo(\025)1952 2283 y Fk(j;)p Fn(0)2040 2268 y Fq(\()p Fo(r)9 b Fq(^)-55 b Fo(x)p Fq(\))22 b(+)g Fo(\032)2388 2227 y Fn(2)2427 2268 y Fq(\))780 2444 y Fi(Z)835 2670 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1087 2680 y Fe(j;)p Fh(0)1166 2670 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1349 2651 y Fh(2)1384 2670 y Fk(;\026)1446 2679 y Fh(0)1481 2670 y Fn(\))1550 2553 y Fq(~)1529 2580 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r\()p Fo(b)1993 2539 y Fk(N)1993 2604 y Fn(0)2061 2580 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))p Fo(dz)t(d)p Fm(L)688 2890 y(\000)798 2822 y Fq(\(2)p Fo(\031)t(h)p Fq(\))1037 2774 y Fj(\000)p Fk(n)p 798 2867 V 874 2958 a Fq(16)p Fo(i\031)1216 2765 y Fk(n)1166 2795 y Fi(X)1177 3005 y Fk(j)t Fn(=1)1326 2890 y Fq(sgn)q(\()p Fo(@)1557 2905 y Fk(r)1595 2890 y Fo(\025)1652 2905 y Fk(j;)p Fn(0)1740 2890 y Fq(\(0\)\))786 3201 y Fm(\002)885 3065 y Fi(Z)941 3291 y Fj(D)1018 3065 y Fi(Z)11 b(Z)1297 3249 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;)s Fn(~)-38 b Fk(r)1593 3258 y Fh(0)1628 3249 y Fn(\))1140 3330 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1392 3340 y Fe(j;)p Fh(0)1471 3330 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1654 3306 y Fh(2)1688 3330 y Fk(;\026)1750 3339 y Fh(0)1785 3330 y Fn(\))1845 3201 y Fq(~)-61 b Fo(\037)p Fq(\()p Fo(\025)1989 3216 y Fk(j;)p Fn(0)2077 3201 y Fq(\()p Fo(r)9 b Fq(^)-55 b Fo(x)p Fq(\))22 b(+)g Fo(\032)2425 3160 y Fn(2)2464 3201 y Fq(\))688 3523 y Fm(\002)788 3387 y Fi(Z)843 3613 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1095 3623 y Fe(j;)p Fh(0)1174 3613 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1357 3594 y Fh(2)1392 3613 y Fk(;\026)1454 3622 y Fh(0)1488 3613 y Fn(\))1558 3496 y Fq(~)1537 3523 y Fo(f)10 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r)q(\()p Fo(b)2001 3482 y Fk(N)2001 3547 y Fn(0)2068 3523 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))p Fo(dz)t(d)p Fm(L)22 b Fq(+)g Fm(O)s Fq(\()p Fo(h)2938 3482 y Fj(\000)p Fk(n)3040 3523 y Fq(\))p Fo(e)3123 3482 y Fj(\000)p Fk(C)3228 3491 y Fh(2)3263 3482 y Fk(\025)3308 3523 y Fo(:)386 3775 y Fq(Assumption)32 b Fy(\(H4\))g Fq(giv)m(es)438 4030 y(T)-8 b(r)q([)p Fo(B)640 4045 y Fj(\017)679 4030 y Fq(])706 3989 y Fj(1)706 4055 y Fn(0)809 4030 y Fq(=)22 b Fm(\000)1016 3963 y Fq(\(2)p Fo(\031)t(h)p Fq(\))1256 3915 y Fj(\000)p Fk(n)p 1016 4008 V 1092 4099 a Fq(16)p Fo(i\031)1435 3906 y Fk(n)1384 3936 y Fi(X)1395 4146 y Fk(j)t Fn(=1)1545 4030 y Fq(sgn\()p Fo(@)1775 4045 y Fk(r)1814 4030 y Fo(\025)1871 4045 y Fk(j;)p Fn(0)1958 4030 y Fq(\(0\)\))901 4206 y Fi(Z)957 4432 y Fj(D)1034 4206 y Fi(Z)1089 4432 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;)s Fn(~)-38 b Fk(r)1385 4441 y Fh(0)1420 4432 y Fn(\))1468 4206 y Fi(Z)1523 4432 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1775 4442 y Fe(j;)p Fh(0)1854 4432 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)2037 4413 y Fh(2)2072 4432 y Fk(;\026)2134 4441 y Fh(0)2169 4432 y Fn(\))2238 4315 y Fq(~)2217 4342 y Fo(f)10 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r)q(\()p Fo(b)2681 4300 y Fk(N)2681 4366 y Fn(0)2749 4342 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))p Fo(dz)t(d)p Fm(L)907 4576 y Fq(+)22 b Fm(O)s Fq(\()p Fo(h)1181 4535 y Fj(\000)p Fk(n)1283 4576 y Fq(\))p Fo(e)1366 4535 y Fj(\000)p Fk(C)1471 4544 y Fh(2)1506 4535 y Fk(\025)1551 4576 y Fo(:)386 4775 y Fq(The)31 b(function)e Fo(z)k Fm(7!)1189 4749 y Fq(~)1168 4775 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r\()p Fo(b)1632 4739 y Fk(N)1632 4800 y Fn(0)1700 4775 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))29 b(is)h(holomorphic)d(in)i Fo(z)35 b Fq(in)29 b(a)h(neigh-)386 4891 y(b)s(orho)s(o)s(d)g(of)h Fo(\025)939 4906 y Fk(j;)p Fn(0)1026 4891 y Fq(\()p Fo(x)p Fq(\))20 b(+)f Fo(\030)1320 4855 y Fn(2)1390 4891 y Fq(except)33 b(at)d(the)i(p)s(oin)m(ts)f Fo(\025)2328 4906 y Fk(j;)p Fn(0)2415 4891 y Fq(\()p Fo(x)p Fq(\))20 b(+)f Fo(\030)2709 4855 y Fn(2)2748 4891 y Fq(.)43 b(Therefore)32 b(for)386 5008 y(all)e Fo(\026)580 5023 y Fn(1)647 5008 y Fm(\024)e Fo(\026)811 5023 y Fn(0)417 5095 y Fi(Z)472 5320 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)724 5330 y Fe(j;)p Fh(0)803 5320 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)986 5301 y Fh(2)1021 5320 y Fk(;\026)1083 5329 y Fh(0)1117 5320 y Fn(\))1187 5204 y Fq(~)1166 5230 y Fo(f)10 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\))q(T)-8 b(r\()p Fo(b)1630 5189 y Fk(N)1630 5255 y Fn(0)1697 5230 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))p Fo(dz)1478 5513 y Fq(=)1582 5377 y Fi(Z)1637 5603 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1889 5613 y Fe(j;)p Fh(0)1968 5603 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)2151 5584 y Fh(2)2186 5603 y Fk(;\026)2248 5612 y Fh(1)2282 5603 y Fn(\))2352 5487 y Fq(~)2331 5513 y Fo(f)10 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r)q(\()p Fo(b)2795 5472 y Fk(N)2795 5538 y Fn(0)2862 5513 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))p Fo(dz)t(:)1842 5722 y Fg(14)p eop %%Page: 15 15 15 14 bop 386 459 a Fq(Due)32 b(to)h(analyticit)m(y)e(in)h Fo(r)s Fq(,)g(w)m(e)h(get)g(for)f(all)f Fo(r)2050 474 y Fn(1)2117 459 y Fm(\024)h Fq(~)-53 b Fo(r)2266 474 y Fn(0)2338 459 y Fq(and)1111 715 y Fo(\026)1170 730 y Fn(1)1237 715 y Fm(\024)1352 648 y Fq(1)p 1352 693 49 4 v 1352 784 a(2)1613 715 y(inf)1427 782 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)1723 791 y Fh(1)1758 782 y Fn(\))p Fk(;i)p Fj(6)p Fn(=)p Fk(j)1933 715 y Fo(d)p Fq(\()p Fo(\025)2079 730 y Fk(j;)p Fn(0)2166 715 y Fq(\()p Fo(x)p Fq(\))p Fo(;)17 b(\025)2398 730 y Fk(i;)p Fn(0)2481 715 y Fq(\()p Fo(x)p Fq(\)\))622 1176 y(T)-8 b(r[)p Fo(B)823 1191 y Fj(\017)863 1176 y Fq(])890 1135 y Fj(1)890 1201 y Fn(0)993 1176 y Fq(=)22 b Fm(\000)1200 1108 y Fq(\(2)p Fo(\031)t(h)p Fq(\))1440 1060 y Fj(\000)p Fk(n)p 1200 1153 342 4 v 1276 1244 a Fq(16)p Fo(i\031)1619 1051 y Fk(n)1568 1081 y Fi(X)1579 1291 y Fk(j)t Fn(=1)1729 1176 y Fq(sgn\()p Fo(@)1959 1191 y Fk(r)1998 1176 y Fo(\025)2055 1191 y Fk(j;)p Fn(0)2142 1176 y Fq(\(0\)\))1085 1351 y Fi(Z)1140 1577 y Fj(D)1218 1351 y Fi(Z)11 b(Z)1497 1535 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)1793 1544 y Fh(1)1827 1535 y Fn(\))1339 1616 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1591 1626 y Fe(j;)p Fh(0)1670 1616 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1853 1592 y Fh(2)1888 1616 y Fk(;\026)1950 1625 y Fh(1)1985 1616 y Fn(\))2054 1461 y Fq(~)2033 1487 y Fo(f)g Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r\()p Fo(b)2497 1446 y Fk(N)2497 1512 y Fn(0)2565 1487 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))p Fo(dz)t(d)p Fm(L)1091 1760 y Fq(+)22 b Fm(O)s Fq(\()p Fo(h)1365 1719 y Fj(\000)p Fk(n)1467 1760 y Fq(\))p Fo(e)1550 1719 y Fj(\000)p Fk(C)1655 1728 y Fh(2)1690 1719 y Fk(\025)1735 1760 y Fo(:)386 1975 y Fq(This)33 b(giv)m(es)g(Lemma)e(5.2.)p 3316 1975 60 60 v 386 2091 a(Pro)s(of)h(of)g(Lemma)f(5.1)p 386 2107 851 4 v 32 w(:)44 b(An)33 b(iteration)d(will)g(pro)m(v)m(e)409 2339 y Fi(Z)464 2565 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)716 2575 y Fe(j;)p Fh(0)795 2565 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)978 2546 y Fh(2)1012 2565 y Fk(;\026)1074 2574 y Fh(1)1109 2565 y Fn(\))1157 2475 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r)q(\()p Fo(r)1628 2433 y Fk(N)1625 2499 y Fn(0)1694 2475 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))g Fo(dz)1470 2757 y Fq(=)1573 2622 y Fi(Z)1629 2847 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1881 2857 y Fe(j;)p Fh(0)1960 2847 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)2143 2828 y Fh(2)2177 2847 y Fk(;\026)2239 2856 y Fh(1)2274 2847 y Fn(\))2344 2731 y Fq(~)2322 2757 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r\()p Fo(b)2786 2716 y Fk(N)2786 2782 y Fn(0)2854 2757 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))g Fo(dz)t(:)386 2292 y Fq(\(5.3\))386 3055 y(Let)28 b Fo(c)598 3019 y Fk(N)s(;m)743 3055 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))27 b(b)s(e)h(the)g(sum)g(of)f(the)h Fo(N)22 b Fm(\000)12 b Fq(1)28 b(\014rst)g(terms)f(of)g(the)h (expansion)g(in)386 3171 y Fo(h)i Fq(of)g(the)h(sym)m(b)s(ol)e(of)h(\() p Fo(P)1288 3186 y Fn(0)1344 3171 y Fm(\000)18 b Fo(z)t Fq(\))1526 3135 y Fj(\000)p Fn(1)1621 3171 y Fq(\()p Fo(P)1722 3186 y Fn(0)1779 3171 y Fm(\000)g Fo(z)1919 3186 y Fn(0)1958 3171 y Fq(\))1996 3135 y Fj(\000)p Fk(m)2118 3171 y Fo(:)30 b Fq(Let)g Fo(d)2398 3135 y Fk(N)s(;m)2543 3171 y Fq(\()p Fo(x;)17 b(\030)5 b Fq(\))30 b(b)s(e)g(the)h(sum)f(of) 386 3287 y(the)d Fo(N)19 b Fm(\000)9 b Fq(1)27 b(\014rst)f(terms)g(of)g (the)h(expansion)f(in)g Fo(h)g Fq(of)g(the)g(sym)m(b)s(ol)g(of)g(\()p Fo(P)2969 3302 y Fn(0)3017 3287 y Fm(\000)9 b Fo(z)3148 3302 y Fn(0)3188 3287 y Fq(\))3226 3251 y Fj(\000)p Fk(m)3348 3287 y Fq(.)386 3403 y(W)-8 b(e)30 b(get)g(b)m(y)h(the)f(W)-8 b(eyl)30 b(calculus)f(:)42 b Fo(c)1730 3367 y Fk(N)s(;m)1875 3403 y Fq(#\()p Fo(p)2043 3418 y Fn(0)2100 3403 y Fm(\000)17 b Fo(z)2239 3418 y Fn(0)2278 3403 y Fq(\))28 b(=)g Fo(c)2490 3367 y Fk(N)s(;m)p Fj(\000)p Fn(1)2725 3403 y Fq(,)i Fo(c)2824 3367 y Fk(N)s(;)p Fn(0)2970 3403 y Fq(=)e Fo(r)3121 3367 y Fk(N)3118 3428 y Fn(0)3218 3403 y Fq(and)386 3530 y Fo(c)428 3494 y Fk(N)s(;m)601 3530 y Fq(=)f Fo(b)745 3494 y Fk(N)745 3555 y Fn(0)813 3530 y Fq(.)44 b(Let)32 b Fo(c)1100 3494 y Fk(N)s(;m)1273 3530 y Fq(=)1377 3456 y Fi(P)1482 3482 y Fk(j)t Fn(=)p Fk(N)7 b Fj(\000)p Fn(1)1482 3560 y Fk(j)t Fn(=0)1743 3530 y Fo(h)1799 3494 y Fk(j)1836 3530 y Fo(c)1878 3483 y Fk(N)s(;m)1878 3556 y(j)2023 3530 y Fq(.)44 b(The)33 b(functions)g Fo(c)2757 3483 y Fk(N)s(;m)2757 3556 y(j)2935 3530 y Fq(satisfy)830 3828 y Fo(c)872 3781 y Fk(N)s(;m)p Fj(\000)p Fn(1)872 3854 y Fk(j)1108 3828 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))27 b(=)1591 3734 y Fi(X)1555 3946 y Fk(k)r Fn(+)p Fk(p)p Fn(=)p Fk(j)1846 3761 y Fo(i)1879 3725 y Fk(k)p 1797 3805 173 4 v 1797 3897 a Fo(k)s Fq(!2)1927 3868 y Fk(k)1980 3828 y Fq(\()p Fo(\033)t Fq(\()p Fo(D)2196 3843 y Fk(x)2240 3828 y Fo(;)17 b(D)2365 3843 y Fk(\030)2402 3828 y Fq(;)g Fo(D)2527 3843 y Fk(y)2568 3828 y Fo(;)g(D)2693 3843 y Fk(\021)2735 3828 y Fq(\)\))2811 3787 y Fk(k)2853 3828 y Fo(c)2895 3787 y Fk(N)s(;m)2895 3853 y(p)3041 3828 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\))2376 4081 y Fm(\002)p Fq(\()p Fo(p)2540 4096 y Fn(0)2602 4081 y Fm(\000)23 b Fo(z)2747 4096 y Fn(0)2787 4081 y Fq(\)\()p Fo(y)t(;)17 b(\021)t Fq(\))p Fm(j)3077 4096 y Fk(y)r Fn(=)p Fk(x;\021)r Fn(=)p Fk(\030)404 4307 y Fq(=)616 4212 y Fi(X)507 4424 y Fk(k)r Fn(+)p Fk(p)p Fn(=)p Fk(j;k)r(>)p Fn(0)943 4239 y Fo(i)976 4203 y Fk(k)p 895 4284 V 895 4375 a Fo(k)s Fq(!2)1025 4346 y Fk(k)1077 4307 y Fq(\()p Fo(\033)t Fq(\()p Fo(D)1293 4322 y Fk(x)1337 4307 y Fo(;)g(D)1462 4322 y Fk(\030)1499 4307 y Fq(;)g Fo(D)1624 4322 y Fk(y)1666 4307 y Fo(;)g(D)1791 4322 y Fk(\021)1832 4307 y Fq(\)\))1908 4266 y Fk(k)1951 4307 y Fo(c)1993 4266 y Fk(N)s(;m)1993 4331 y(p)2138 4307 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(p)2541 4322 y Fn(0)2602 4307 y Fm(\000)23 b Fo(z)2747 4322 y Fn(0)2787 4307 y Fq(\)\()p Fo(y)t(;)17 b(\021)t Fq(\))p Fm(j)3077 4322 y Fk(y)r Fn(=)p Fk(x;\021)r Fn(=)p Fk(\030)2157 4577 y Fq(+)p Fo(c)2275 4530 y Fk(N)s(;m)2275 4603 y(j)2420 4577 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(p)2823 4592 y Fn(0)2885 4577 y Fm(\000)22 b Fo(z)3029 4592 y Fn(0)3069 4577 y Fq(\)\()p Fo(x;)17 b(\030)5 b Fq(\))p Fo(:)386 4792 y Fq(F)-8 b(or)32 b Fo(k)e(>)e Fq(0)k(w)m(e)i(ha)m(v)m(e)727 4988 y Fo(\033)t Fq(\()p Fo(D)905 5003 y Fk(x)949 4988 y Fo(;)17 b(D)1074 5003 y Fk(\030)1111 4988 y Fq(;)g Fo(D)1236 5003 y Fk(y)1277 4988 y Fo(;)g(D)1402 5003 y Fk(\021)1444 4988 y Fq(\))1482 4946 y Fk(k)1524 4988 y Fo(c)1566 4946 y Fk(N)s(;m)1566 5012 y(p)1712 4988 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(p)2115 5003 y Fn(0)2176 4988 y Fm(\000)23 b Fo(z)2321 5003 y Fn(0)2360 4988 y Fq(\)\()p Fo(y)t(;)17 b(\021)t Fq(\))p Fm(j)2650 5003 y Fk(y)r Fn(=)p Fk(x;\021)r Fn(=)p Fk(\030)2958 4988 y Fq(=)838 5151 y Fo(\033)t Fq(\()p Fo(D)1016 5166 y Fk(x)1060 5151 y Fo(;)g(D)1185 5166 y Fk(\030)1222 5151 y Fq(;)g Fo(D)1347 5166 y Fk(y)1389 5151 y Fo(;)g(D)1514 5166 y Fk(\021)1555 5151 y Fq(\))1593 5110 y Fk(k)1636 5151 y Fo(c)1678 5110 y Fk(N)s(;m)1678 5176 y(p)1823 5151 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(p)2226 5166 y Fn(0)2287 5151 y Fm(\000)23 b Fo(z)t Fq(\)\()p Fo(y)t(;)17 b(\021)t Fq(\))p Fm(j)2726 5166 y Fk(y)r Fn(=)p Fk(x;\021)r Fn(=)p Fk(\030)3007 5151 y Fo(:)386 5366 y Fq(Therefore,)34 b(the)f(di\013erence)1105 5584 y Fo(c)1147 5536 y Fk(N)s(;m)p Fj(\000)p Fn(1)1147 5609 y Fk(j)1383 5584 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))22 b Fm(\000)g Fo(c)1862 5536 y Fk(N)s(;m)1862 5609 y(j)2008 5584 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(z)26 b Fm(\000)d Fo(z)2578 5599 y Fn(0)2618 5584 y Fq(\))1842 5722 y Fg(15)p eop %%Page: 16 16 16 15 bop 386 459 a Fq(is)32 b(a)g(function)g(indep)s(enden)m(t)i(of)e Fo(z)t Fq(.)44 b(This)33 b(pro)m(v)m(es)i(that)397 548 y Fi(Z)452 774 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)704 784 y Fe(j;)p Fh(0)783 774 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)966 755 y Fh(2)1001 774 y Fk(;\026)1063 783 y Fh(1)1098 774 y Fn(\))1129 684 y Fq(\()p Fo(z)27 b Fm(\000)c Fo(z)1384 699 y Fn(0)1424 684 y Fq(\))1462 643 y Fk(m)p Fj(\000)p Fn(1)1618 684 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r\()p Fo(c)2083 637 y Fk(N)s(;m)p Fj(\000)p Fn(1)2083 710 y Fk(j)2319 684 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))27 b Fo(dz)t(r)2847 643 y Fk(n)p Fj(\000)p Fn(1)3001 684 y Fo(dr)19 b(L)p Fq(\()p Fo(d)6 b Fq(^)-55 b Fo(x)q Fq(\))447 967 y(=)550 831 y Fi(Z)606 1057 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)858 1067 y Fe(j;)p Fh(0)937 1057 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1120 1038 y Fh(2)1154 1057 y Fk(;\026)1216 1066 y Fh(1)1251 1057 y Fn(\))1283 967 y Fq(\()p Fo(z)26 b Fm(\000)d Fo(z)1537 982 y Fn(0)1577 967 y Fq(\))1615 926 y Fk(m)1681 967 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r\()p Fo(c)2146 920 y Fk(N)s(;m)2146 992 y(j)2292 967 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))27 b Fo(dz)t(r)2820 926 y Fk(n)p Fj(\000)p Fn(1)2974 967 y Fo(dr)19 b(L)p Fq(\()p Fo(d)6 b Fq(^)-55 b Fo(x)p Fq(\))p Fo(:)386 1222 y Fq(The)33 b(iteration)e(pro)m(v)m(es)j(the)f (relation)e(\(5.3\).)p 3316 1222 60 60 v 1462 1480 a(6.)49 b Fp(cancella)-7 b(tions)486 1655 y Fq(W)f(e)52 b(compute)h(the)g (\014rst)f(terms)h(of)f(the)g(expansion)h(in)f(p)s(o)m(w)m(ers)i(of)e Fo(h)g Fq(of)386 1771 y(T)-8 b(r[)p Fo(\037f)11 b Fq(\()p Fo(P)734 1786 y Fj(\017)773 1771 y Fo(;)17 b(\025)p Fq(\)])939 1735 y Fj(1)939 1795 y Fn(0)1046 1771 y Fq(up)33 b(to)f(errors)h(of)f (order)h Fm(O)s Fq(\()p Fo(h)2125 1735 y Fj(\000)p Fk(n)2227 1771 y Fq(\))p Fo(e)2310 1735 y Fj(\000)p Fk(C)2415 1744 y Fh(2)2450 1735 y Fk(\025)2495 1771 y Fq(.)486 1887 y(De\014ne)g(the)g(n)m(um)m(b)s(ers)g Fo(T)1406 1902 y Fk(s)1476 1887 y Fq(b)m(y)g(the)g(equation)457 2081 y(T)-8 b(r[\()p Fo(\037f)11 b Fq(\)\()p Fo(P)881 2096 y Fj(\017)920 2081 y Fo(;)17 b(\025)p Fq(\)])1086 2040 y Fj(1)1086 2106 y Fn(0)1188 2081 y Fq(=)1295 1987 y Fi(X)1292 2198 y Fk(s)p Fj(\024)p Fk(N)1459 2081 y Fo(h)1515 2040 y Fj(\000)p Fk(n)p Fn(+)p Fk(s)1704 2081 y Fo(T)1761 2096 y Fk(s)1820 2081 y Fq(+)22 b Fm(O)s Fq(\()p Fo(h)2094 2040 y Fj(\000)p Fk(n)2196 2081 y Fq(\))p Fo(e)2279 2040 y Fj(\000)p Fk(C)2384 2049 y Fh(2)2419 2040 y Fk(\025)2487 2081 y Fq(+)g Fm(O)s Fq(\()p Fo(h)2761 2040 y Fk(N)7 b Fn(+1)p Fj(\000)p Fk(s)3006 2081 y Fq(\))p Fm(k)p Fo(f)k Fm(k)3203 2096 y Fj(1)3277 2081 y Fo(:)386 2359 y Fq(W)-8 b(e)33 b(pro)m(v)m(e)h(that)e Fo(T)1085 2374 y Fn(0)1152 2359 y Fq(=)c(0)k(:)800 2612 y Fo(T)857 2627 y Fn(0)924 2612 y Fq(=)22 b Fm(\000)1132 2544 y Fq(\(2)p Fo(\031)t Fq(\))1315 2496 y Fj(\000)p Fk(n)p 1132 2589 286 4 v 1180 2680 a Fq(16)p Fo(i\031)1494 2487 y Fk(n)1444 2517 y Fi(X)1454 2727 y Fk(j)t Fn(=1)1604 2612 y Fq(sgn)q(\()p Fo(@)1835 2627 y Fk(r)1873 2612 y Fo(\025)1930 2627 y Fk(j;)p Fn(0)2017 2612 y Fq(\(0\)\))2197 2476 y Fi(Z)2252 2702 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)2548 2711 y Fh(1)2583 2702 y Fn(\))2631 2476 y Fi(Z)2686 2702 y Fj(D)1017 2787 y Fi(Z)1072 3013 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1324 3023 y Fe(j;)p Fh(0)1403 3013 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1586 2994 y Fh(2)1621 3013 y Fk(;\026)1683 3022 y Fh(1)1717 3013 y Fn(\))1765 2923 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r)q(\()p Fo(r)2233 2938 y Fn(0)p Fk(;)p Fn(0)2327 2923 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))g Fo(dz)j(d)p Fm(L)p Fo(:)386 3182 y Fq(Since)33 b Fo(r)685 3197 y Fn(0)p Fk(;)p Fj(\017)779 3182 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))27 b(=)h(\()p Fo(\030)1312 3146 y Fn(2)1373 3182 y Fq(+)22 b Fo(V)1528 3197 y Fj(\017)1567 3182 y Fq(\()p Fo(x)p Fq(\))h Fm(\000)f Fo(z)t Fq(\))1907 3146 y Fj(\000)p Fn(1)2035 3182 y Fq(w)m(e)33 b(ha)m(v)m(e)h(that)689 3348 y(1)p 643 3393 141 4 v 643 3484 a(2)p Fo(i\031)810 3280 y Fi(Z)866 3506 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1118 3516 y Fe(j;)p Fh(0)1196 3506 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1379 3487 y Fh(2)1414 3506 y Fk(;\026)1476 3515 y Fh(1)1511 3506 y Fn(\))1559 3416 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r\()p Fo(r)2026 3431 y Fn(0)p Fk(;)p Fn(0)2120 3416 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))28 b Fo(dz)718 3726 y Fq(=)850 3658 y(1)p 804 3703 V 804 3794 a(2)p Fo(i\031)971 3590 y Fi(Z)1026 3816 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1278 3826 y Fe(j;)p Fh(0)1357 3816 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1540 3797 y Fh(2)1575 3816 y Fk(;\026)1637 3825 y Fh(1)1672 3816 y Fn(\))1720 3726 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\))2076 3601 y Fk(r)2021 3631 y Fi(X)2029 3843 y Fk(k)r Fn(=1)2165 3726 y Fq(\()p Fo(\030)2251 3685 y Fn(2)2312 3726 y Fq(+)22 b Fo(\025)2467 3741 y Fk(k)r(;)p Fn(0)2565 3726 y Fq(\()p Fo(x)p Fq(\))g Fm(\000)h Fo(z)t Fq(\))2905 3685 y Fj(\000)p Fn(1)3027 3726 y Fo(dz)718 3975 y Fq(=)f Fm(\000)g Fo(f)11 b Fq(\()p Fo(\030)1060 3933 y Fn(2)1121 3975 y Fq(+)22 b Fo(\025)1276 3990 y Fk(j;)p Fn(0)1363 3975 y Fq(\()p Fo(x)p Fq(\))p Fo(;)17 b(\025)p Fq(\))p Fo(:)386 4153 y Fq(Then)586 4359 y Fo(T)643 4374 y Fn(0)711 4359 y Fq(=)939 4292 y(1)p 796 4336 335 4 v 796 4437 a(8\(2)p Fo(\031)t Fq(\))1029 4388 y Fj(\000)p Fk(n)1295 4264 y Fi(X)1157 4480 y Fk(j;@)1243 4488 y Fe(r)1277 4480 y Fk(\025)1318 4490 y Fe(j;)p Fh(0)1397 4480 y Fn(\(0\))p Fk(>)p Fn(0)1594 4223 y Fi(Z)1649 4449 y Fj(D)1726 4223 y Fi(Z)1782 4449 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)2078 4458 y Fh(1)2112 4449 y Fn(\))2161 4359 y Fo(f)11 b Fq(\()p Fo(\030)2306 4318 y Fn(2)2366 4359 y Fq(+)22 b Fo(\025)2521 4374 y Fk(j;)p Fn(0)2609 4359 y Fq(\()p Fo(x)p Fq(\))p Fo(;)17 b(\025)p Fq(\))27 b Fo(d)p Fm(L)809 4683 y(\000)918 4616 y Fq(1)p 918 4660 49 4 v 918 4751 a(8)977 4683 y(\(2)p Fo(\031)t Fq(\))1160 4635 y Fj(\000)p Fk(n)1417 4588 y Fi(X)1279 4804 y Fk(j;@)1365 4812 y Fe(r)1399 4804 y Fk(\025)1440 4814 y Fe(j;)p Fh(0)1519 4804 y Fn(\(0\))p Fk(<)p Fn(0)1716 4547 y Fi(Z)1771 4773 y Fj(D)1848 4547 y Fi(Z)1904 4773 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)2200 4782 y Fh(1)2234 4773 y Fn(\))2282 4683 y Fo(f)11 b Fq(\()p Fo(\030)2427 4642 y Fn(2)2488 4683 y Fq(+)22 b Fo(\025)2643 4698 y Fk(j;)p Fn(0)2730 4683 y Fq(\()p Fo(x)p Fq(\))p Fo(;)17 b(\025)p Fq(\))28 b Fo(d)p Fm(L)p Fo(:)386 4983 y Fq(The)33 b(function)g Fo(r)d Fm(7!)d Fo(f)11 b Fq(\()p Fo(\030)1315 4947 y Fn(2)1376 4983 y Fq(+)22 b Fo(\025)1531 4998 y Fk(j;)p Fn(0)1618 4983 y Fq(\()p Fo(r)9 b Fq(^)-55 b Fo(x)p Fq(\))p Fo(;)17 b(\025)p Fq(\))32 b(is)g(analytic)g(so,)g Fo(T)2640 4998 y Fn(0)2708 4983 y Fq(=)27 b(0.)p 3316 4983 60 60 v 386 5100 a(Next,)33 b(w)m(e)h(pro)m(v)m(e)g Fo(T)1115 5115 y Fn(1)1182 5100 y Fq(=)28 b(0.)43 b(A)33 b(computation)e(giv)m (es)i(that)935 5352 y Fo(r)979 5367 y Fn(1)p Fk(;)p Fn(0)1101 5352 y Fq(=)1255 5228 y Fk(n)1205 5258 y Fi(X)1220 5468 y Fk(i)p Fn(=1)1365 5352 y Fo(\030)1408 5367 y Fk(i)1436 5352 y Fq(\()p Fo(r)1521 5311 y Fn(2)1518 5377 y(0)p Fk(;)p Fn(0)1612 5352 y Fq(\()p Fo(D)1731 5367 y Fk(x)1771 5377 y Fe(i)1802 5352 y Fo(V)21 b Fq(\))p Fo(r)1962 5367 y Fn(0)p Fk(;)p Fn(0)2078 5352 y Fm(\000)i Fo(r)2222 5367 y Fn(0)p Fk(;)p Fn(0)2316 5352 y Fq(\()p Fo(D)2435 5367 y Fk(x)2475 5377 y Fe(i)2505 5352 y Fo(V)f Fq(\))p Fo(r)2669 5311 y Fn(2)2666 5377 y(0)p Fk(;)p Fn(0)2760 5352 y Fq(\))p Fo(;)-2439 b Fq(\(6.1\))386 5623 y(so)33 b(T)-8 b(r)17 b(\()p Fo(r)705 5638 y Fn(1)p Fk(;)p Fn(0)799 5623 y Fq(\))27 b(=)h(0.)p 3316 5623 V 1842 5722 a Fg(16)p eop %%Page: 17 17 17 16 bop 386 459 a Fq(Let)33 b(us)g(compute)f(the)h(term)g Fo(T)1540 474 y Fn(2)1612 459 y Fq(:)800 709 y Fo(T)857 724 y Fn(2)924 709 y Fq(=)22 b Fm(\000)1132 642 y Fq(\(2)p Fo(\031)t Fq(\))1315 594 y Fj(\000)p Fk(n)p 1132 687 286 4 v 1180 778 a Fq(16)p Fo(i\031)1494 585 y Fk(n)1444 615 y Fi(X)1454 825 y Fk(j)t Fn(=1)1604 709 y Fq(sgn)q(\()p Fo(@)1835 724 y Fk(r)1873 709 y Fo(\025)1930 724 y Fk(j;)p Fn(0)2017 709 y Fq(\(0\)\))2197 574 y Fi(Z)2252 799 y Fj(D)2329 574 y Fi(Z)2385 799 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)2681 808 y Fh(1)2715 799 y Fn(\))1017 885 y Fi(Z)1072 1111 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1324 1121 y Fe(j;)p Fh(0)1403 1111 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1586 1092 y Fh(2)1621 1111 y Fk(;\026)1683 1120 y Fh(1)1717 1111 y Fn(\))1765 1021 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r)q(\()p Fo(r)2233 1036 y Fn(2)p Fk(;)p Fn(0)2327 1021 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))g Fo(dz)j(d)p Fm(L)p Fo(:)386 1267 y Fq(W)-8 b(e)33 b(get)f(:)835 1486 y Fo(r)879 1501 y Fn(2)p Fk(;)p Fn(0)973 1486 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))28 b(=)22 b Fm(\000)h Fo(r)1559 1501 y Fn(0)p Fk(;)p Fn(0)1653 1486 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))22 b Fm(\002)2101 1419 y Fq(1)p 2101 1463 49 4 v 2101 1554 a(8)2159 1375 y Fi(\020)2235 1391 y(X)2295 1601 y Fk(i)2396 1486 y Fq(2\()p Fo(D)2567 1445 y Fn(2)2564 1511 y Fk(x)2604 1521 y Fe(i)2634 1486 y Fo(r)2678 1501 y Fn(0)p Fk(;)p Fn(0)2772 1486 y Fq(\)\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))1415 1763 y(+)1513 1669 y Fi(X)1547 1878 y Fk(i;j)1657 1763 y Fq(\()p Fo(D)1779 1722 y Fn(2)1776 1788 y Fk(x)1816 1798 y Fe(i)1842 1788 y Fk(;x)1902 1798 y Fe(j)1938 1763 y Fo(V)22 b Fq(\)\()p Fo(x)p Fq(\))p Fo(@)2237 1778 y Fk(\030)2268 1788 y Fe(i)2299 1763 y Fo(@)2350 1778 y Fk(\030)2381 1788 y Fe(j)2418 1763 y Fo(r)2462 1778 y Fn(0)p Fk(;)p Fn(0)2556 1763 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))2872 1653 y Fi(\021)1415 2070 y Fm(\000)23 b Fo(r)1559 2085 y Fn(0)p Fk(;)p Fn(0)1653 2070 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))1979 2003 y(1)p 1979 2047 V 1979 2139 a(2)2038 1960 y Fi(\020)2114 1976 y(X)2174 2186 y Fk(i)2274 2070 y Fq(2)p Fo(\030)2366 2085 y Fk(i)2394 2070 y Fq(\()p Fo(D)2513 2085 y Fk(x)2553 2095 y Fe(i)2583 2070 y Fo(r)2627 2085 y Fn(1)p Fk(;)p Fn(0)2721 2070 y Fq(\)\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\))1415 2348 y Fm(\000)23 b Fq(\()p Fo(D)1634 2363 y Fk(x)1674 2373 y Fe(i)1704 2348 y Fo(V)1761 2363 y Fn(0)1800 2348 y Fq(\))p Fo(@)1889 2363 y Fk(\030)1920 2373 y Fe(i)1951 2348 y Fo(r)1995 2363 y Fn(1)p Fk(;)p Fn(0)2089 2348 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))2405 2237 y Fi(\021)2465 2348 y Fo(:)386 1908 y Fq(\(6.2\))386 2549 y(Let)555 2791 y Fm(A)635 2806 y Fn(1)701 2791 y Fq(=)28 b Fm(\000)p Fo(r)926 2806 y Fn(0)p Fk(;)p Fn(0)1030 2724 y Fq(1)p 1030 2768 V 1030 2860 a(8)1106 2621 y Fi(")1164 2697 y(X)1224 2907 y Fk(i)1324 2791 y Fq(2\()p Fo(D)1495 2750 y Fn(2)1492 2816 y Fk(x)1532 2826 y Fe(i)1562 2791 y Fo(r)1606 2806 y Fn(0)p Fk(;)p Fn(0)1701 2791 y Fq(\))22 b(+)1859 2697 y Fi(X)1893 2907 y Fk(i;j)2003 2791 y Fq(\()p Fo(D)2125 2750 y Fn(2)2122 2816 y Fk(x)2162 2826 y Fe(i)2188 2816 y Fk(;x)2248 2826 y Fe(j)2284 2791 y Fo(V)2341 2806 y Fn(0)2380 2791 y Fq(\))p Fo(@)2469 2806 y Fk(\030)2500 2816 y Fe(i)2531 2791 y Fo(@)2582 2806 y Fk(\030)2613 2816 y Fe(j)2650 2791 y Fo(r)2694 2806 y Fn(0)p Fk(;)p Fn(0)2788 2621 y Fi(#)2863 2791 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))p Fo(:)386 3072 y Fq(Let)33 b Fm(A)641 3087 y Fn(2)707 3072 y Fq(=)28 b Fo(r)855 3087 y Fn(2)p Fk(;)p Fn(0)949 3072 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))22 b Fm(\000)h Fo(A)1460 3087 y Fn(1)1499 3072 y Fq(.)44 b(Then)581 3300 y Fm(A)661 3315 y Fn(1)727 3300 y Fq(=)813 3233 y(1)p 813 3277 V 813 3369 a(4)888 3206 y Fi(X)948 3416 y Fk(i)1049 3300 y Fo(r)1096 3259 y Fn(2)1093 3325 y(0)p Fk(;)p Fn(0)1187 3300 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(D)1625 3259 y Fn(2)1622 3325 y Fk(x)1662 3335 y Fe(i)1692 3300 y Fo(V)1749 3315 y Fn(0)1789 3300 y Fq(\))p Fo(r)1871 3315 y Fn(0)p Fk(;)p Fn(0)1965 3300 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\))825 3594 y Fm(\000)935 3527 y Fq(1)p 935 3572 V 935 3663 a(2)1010 3500 y Fi(X)1070 3710 y Fk(i)1171 3594 y Fo(r)1218 3553 y Fn(2)1215 3619 y(0)p Fk(;)p Fn(0)1309 3594 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(D)1744 3609 y Fk(x)1784 3619 y Fe(i)1814 3594 y Fo(V)1871 3609 y Fn(0)1910 3594 y Fq(\))p Fo(r)1992 3609 y Fn(0)p Fk(;)p Fn(0)2087 3594 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(D)2522 3609 y Fk(x)2562 3619 y Fe(i)2592 3594 y Fo(V)2649 3609 y Fn(0)2688 3594 y Fq(\))p Fo(r)2770 3609 y Fn(0)p Fk(;)p Fn(0)2864 3594 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\))825 3862 y Fm(\000)925 3767 y Fi(X)959 3977 y Fk(i;j)1085 3862 y Fo(\030)1128 3877 y Fk(i)1156 3862 y Fo(\030)1199 3877 y Fk(j)1235 3862 y Fo(r)1279 3877 y Fn(0)p Fk(;)p Fn(0)1374 3862 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(D)1812 3821 y Fn(2)1809 3886 y Fk(x)1849 3896 y Fe(i)1874 3886 y Fk(;x)1934 3896 y Fe(j)1970 3862 y Fo(V)2027 3877 y Fn(0)2067 3862 y Fq(\))p Fo(r)2152 3821 y Fn(3)2149 3886 y(0)p Fk(;)p Fn(0)2243 3862 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\))825 4169 y(+)933 4101 y(1)p 933 4146 V 933 4237 a(4)1009 4074 y Fi(X)1069 4284 y Fk(i)1169 4169 y Fo(r)1213 4184 y Fn(0)p Fk(;)p Fn(0)1307 4169 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(D)1745 4128 y Fn(2)1742 4193 y Fk(x)1782 4203 y Fe(i)1812 4169 y Fo(V)1869 4184 y Fn(0)1909 4169 y Fq(\))p Fo(r)1994 4128 y Fn(2)1991 4193 y(0)p Fk(;)p Fn(0)2085 4169 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\))p Fo(:)386 4430 y Fq(So)641 4673 y(T)-8 b(r)p Fm(A)821 4688 y Fj(1)923 4673 y Fq(=T)g(r)1115 4503 y Fi( )1204 4606 y Fq(1)p 1204 4650 V 1204 4741 a(2)1280 4578 y Fi(X)1340 4788 y Fk(i)1440 4673 y Fo(r)1487 4632 y Fn(3)1484 4698 y(0)p Fk(;)p Fn(0)1578 4673 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(D)2016 4632 y Fn(2)2013 4698 y Fk(x)2053 4708 y Fe(i)2084 4673 y Fo(V)2141 4688 y Fn(0)2180 4673 y Fq(\))2218 4503 y Fi(!)1021 5010 y Fm(\000)22 b Fq(T)-8 b(r)1237 4840 y Fi( )1326 4943 y Fq(1)p 1326 4987 V 1326 5078 a(2)1401 4915 y Fi(X)1461 5125 y Fk(i)1562 5010 y Fo(r)1609 4969 y Fn(3)1606 5035 y(0)p Fk(;)p Fn(0)1700 5010 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(D)2135 5025 y Fk(x)2175 5035 y Fe(i)2205 5010 y Fo(V)2262 5025 y Fn(0)2302 5010 y Fq(\))p Fo(r)2384 5025 y Fn(0)p Fk(;)p Fn(0)2478 5010 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\))p Fo(D)2875 5025 y Fk(x)2915 5035 y Fe(i)2945 5010 y Fo(V)3002 5025 y Fn(0)3041 4840 y Fi(!)1021 5347 y Fm(\000)22 b Fq(T)-8 b(r)1237 5177 y Fi( )1316 5252 y(X)1350 5462 y Fk(i;j)1477 5347 y Fo(\030)1520 5362 y Fk(i)1547 5347 y Fo(\030)1590 5362 y Fk(j)1627 5347 y Fq(\()p Fo(D)1749 5306 y Fn(2)1746 5372 y Fk(x)1786 5382 y Fe(i)1812 5372 y Fk(;x)1872 5382 y Fe(j)1908 5347 y Fo(V)1965 5362 y Fn(0)2004 5347 y Fq(\))p Fo(r)2089 5306 y Fn(4)2086 5372 y(0)p Fk(;)p Fn(0)2180 5347 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))2496 5177 y Fi(!)2591 5347 y Fo(:)386 5623 y Fq(No)m(w)33 b(w)m(e)h(need)f(the)g(follo)m(wing)d(remark:)1842 5722 y Fg(17)p eop %%Page: 18 18 18 17 bop 386 459 a Fy(Remark)42 b(3.)j Ff(If)38 b Fo(D)s Fq(\()p Fo(x;)17 b(\030)5 b Fq(\))39 b Ff(is)g(a)g(matrix)g (holomorphic)f(in)h Fo(r)s Ff(,)h(then)f(ther)-5 b(e)39 b(exist)386 575 y Fo(d)437 590 y Fk(j)473 575 y Fq(\()p Fo(x;)17 b(\030)5 b Fq(\))p Ff(,)34 b(holomorphic)f(in)i Fo(r)30 b Fm(2)e Fo(S)1662 590 y Fn(0)1724 575 y Fm(n)22 b(f)p Fq(0)p Fm(g)p Ff(,)34 b(such)h(that)g(for)g(al)5 b(l)34 b Fo(p)h Ff(:)670 813 y Fq(T)-8 b(r)786 733 y Fi(\000)832 813 y Fo(D)s Fq(\()p Fo(x;)17 b(\030)5 b Fq(\))p Fo(r)1186 766 y Fk(p)1183 838 y Fn(0)p Fk(;)p Fn(0)1276 813 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))1592 733 y Fi(\001)1666 813 y Fq(=)1769 719 y Fi(X)1825 929 y Fk(j)1930 813 y Fo(d)1981 828 y Fk(j)2017 813 y Fq(\()p Fo(x;)g(\030)5 b Fq(\)\()p Fo(\030)2326 772 y Fn(2)2386 813 y Fq(+)22 b Fo(\025)2541 828 y Fk(j;)p Fn(0)2629 813 y Fq(\()p Fo(x)p Fq(\))g Fm(\000)h Fo(z)t Fq(\))2969 772 y Fj(\000)p Fk(p)3064 813 y Fo(;)386 1126 y Ff(and)34 b(the)h(function)648 1387 y Fo(r)30 b Fm(7!)905 1320 y Fq(1)p 859 1364 141 4 v 859 1456 a(2)p Fo(i\031)1027 1252 y Fi(Z)1082 1477 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1334 1487 y Fe(j;)p Fh(0)1413 1477 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1596 1458 y Fh(2)1631 1477 y Fk(;\026)1693 1486 y Fh(1)1727 1477 y Fn(\))1776 1387 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r)2161 1307 y Fi(\000)2207 1387 y Fo(D)s Fq(\()p Fo(x;)17 b(\030)5 b Fq(\))p Fo(r)2561 1340 y Fk(p)2558 1412 y Fn(0)p Fk(;)p Fn(0)2651 1387 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))2967 1307 y Fi(\001)3012 1387 y Fo(dz)386 1672 y Ff(is)35 b(holomorphic,)e(for)h(al)5 b(l)35 b Fo(j)6 b Ff(.)386 1879 y Fq(This)33 b(remark)f(pro)m(v)m(es)i(that)711 2004 y Fi(Z)766 2230 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)1062 2239 y Fh(1)1097 2230 y Fn(\))1145 2004 y Fi(Z)1201 2230 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1453 2240 y Fe(j;)p Fh(0)1531 2230 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1714 2211 y Fh(2)1749 2230 y Fk(;\026)1811 2239 y Fh(1)1846 2230 y Fn(\))1894 2140 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r)2279 2059 y Fi(\000)2325 2140 y Fo(A)2398 2155 y Fn(1)2437 2140 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))2753 2059 y Fi(\001)2799 2140 y Fo(dz)t(r)2946 2099 y Fk(n)p Fj(\000)p Fn(1)3083 2140 y Fo(dr)1977 2422 y Fq(=)2080 2287 y Fi(Z)12 b(Z)2360 2470 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)2656 2479 y Fh(1)2690 2470 y Fn(\))2202 2551 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)2454 2561 y Fe(j;)p Fh(0)2533 2551 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)2716 2528 y Fh(2)2751 2551 y Fk(;\026)2813 2560 y Fh(1)2847 2551 y Fn(\))2896 2422 y Fo(f)f Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\))580 2740 y Fm(\002)p Fq(T)-8 b(r)758 2659 y Fi(\000)826 2740 y Fm(\000)935 2673 y Fq(1)p 935 2717 49 4 v 935 2808 a(2)1011 2645 y Fi(X)1070 2855 y Fk(i)1171 2740 y Fo(r)1218 2699 y Fn(3)1215 2765 y(0)p Fk(;)p Fn(0)1309 2740 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(D)1744 2755 y Fk(x)1784 2765 y Fe(i)1814 2740 y Fo(V)1871 2755 y Fn(0)1911 2740 y Fq(\))p Fo(r)1993 2755 y Fn(0)p Fk(;)p Fn(0)2087 2740 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(D)2522 2755 y Fk(x)2562 2765 y Fe(i)2592 2740 y Fo(V)2649 2755 y Fn(0)2688 2740 y Fq(\))2726 2659 y Fi(\001)2772 2740 y Fo(r)2819 2699 y Fk(n)p Fj(\000)p Fn(1)2956 2740 y Fo(dr)s(dz)t(:)386 3046 y Fq(Next,)33 b(let)f(us)i(compute)e(the)h(next)h(term)e(in)f(\(6.2\):)521 3430 y Fm(A)601 3445 y Fn(2)667 3430 y Fq(=)d Fm(\000)858 3363 y Fq(1)p 858 3407 V 858 3499 a(4)917 3430 y Fo(r)961 3445 y Fn(0)p Fk(;)p Fn(0)1055 3430 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))1388 3336 y Fi(X)1448 3546 y Fk(i)1548 3350 y Fi(\000)1594 3430 y Fq(2)p Fo(\030)1686 3445 y Fk(i)1714 3430 y Fq(\()p Fo(D)1833 3445 y Fk(x)1873 3455 y Fe(i)1903 3430 y Fo(r)1947 3445 y Fn(1)p Fk(;)p Fn(0)2041 3430 y Fq(\))22 b Fm(\000)h Fq(\()p Fo(D)2320 3445 y Fk(x)2360 3455 y Fe(i)2390 3430 y Fo(V)2447 3445 y Fn(0)2486 3430 y Fq(\)\()p Fo(@)2613 3445 y Fk(\030)2644 3455 y Fe(i)2675 3430 y Fo(r)2719 3445 y Fn(1)p Fk(;)p Fn(0)2813 3430 y Fq(\))2851 3350 y Fi(\001)2897 3430 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))p Fo(:)386 3251 y Fq(\(6.3\))386 3736 y(Recall)31 b(that)h Fo(r)935 3751 y Fn(1)p Fk(;)p Fn(0)1062 3736 y Fq(is)g(giv)m(en)h(in)f(\(6.1\).) 43 b(Therefore)33 b(:)546 4033 y Fo(D)627 4048 y Fk(x)667 4058 y Fe(i)697 4033 y Fo(r)741 4048 y Fn(1)p Fk(;)p Fn(0)835 4033 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))28 b(=)1333 3908 y Fk(n)1283 3938 y Fi(X)1293 4148 y Fk(j)t Fn(=1)1443 4033 y Fo(\030)1486 4048 y Fk(j)1522 3922 y Fi(\020)1604 4033 y Fm(\000)22 b Fo(r)1747 4048 y Fn(0)p Fk(;)p Fn(0)1842 4033 y Fq(\()p Fo(D)1961 4048 y Fk(x)2001 4058 y Fe(i)2031 4033 y Fo(V)2088 4048 y Fn(0)2127 4033 y Fq(\))p Fo(r)2212 3992 y Fn(2)2209 4057 y(0)p Fk(;)p Fn(0)2303 4033 y Fq(\()p Fo(D)2422 4048 y Fk(x)2462 4058 y Fe(j)2499 4033 y Fo(V)2556 4048 y Fn(0)2595 4033 y Fq(\))p Fo(r)2677 4048 y Fn(0)p Fk(;)p Fn(0)858 4292 y Fm(\000)g Fo(r)1004 4251 y Fn(2)1001 4317 y(0)p Fk(;)p Fn(0)1096 4292 y Fq(\()p Fo(D)1215 4307 y Fk(x)1255 4317 y Fe(i)1285 4292 y Fo(V)1342 4307 y Fn(0)1381 4292 y Fq(\))p Fo(r)1463 4307 y Fn(0)p Fk(;)p Fn(0)1557 4292 y Fq(\()p Fo(D)1676 4307 y Fk(x)1716 4317 y Fe(j)1753 4292 y Fo(V)1810 4307 y Fn(0)1849 4292 y Fq(\))p Fo(r)1931 4307 y Fn(0)p Fk(;)p Fn(0)2047 4292 y Fm(\000)h Fo(r)2194 4251 y Fn(2)2191 4317 y(0)p Fk(;)p Fn(0)2285 4292 y Fq(\()p Fo(D)2404 4307 y Fk(x)2444 4317 y Fe(j)2480 4292 y Fo(V)2537 4307 y Fn(0)2577 4292 y Fq(\))p Fo(r)2659 4307 y Fn(0)p Fk(;)p Fn(0)2753 4292 y Fq(\()p Fo(D)2872 4307 y Fk(x)2912 4317 y Fe(i)2942 4292 y Fo(V)2999 4307 y Fn(0)3038 4292 y Fq(\))p Fo(r)3120 4307 y Fn(0)p Fk(;)p Fn(0)858 4453 y Fq(+)f Fo(r)1003 4412 y Fn(2)1000 4477 y(0)p Fk(;)p Fn(0)1094 4453 y Fq(\()p Fo(D)1216 4412 y Fn(2)1213 4477 y Fk(x)1253 4487 y Fe(j)1285 4477 y Fk(;x)1345 4487 y Fe(i)1375 4453 y Fo(V)1432 4468 y Fn(0)1471 4453 y Fq(\))p Fo(r)1553 4468 y Fn(0)p Fk(;)p Fn(0)1670 4453 y Fq(+)g Fo(r)1812 4468 y Fn(0)p Fk(;)p Fn(0)1906 4453 y Fq(\()p Fo(D)2025 4468 y Fk(x)2065 4478 y Fe(i)2095 4453 y Fo(V)2152 4468 y Fn(0)2191 4453 y Fq(\))p Fo(r)2273 4468 y Fn(0)p Fk(;)p Fn(0)2368 4453 y Fq(\()p Fo(D)2487 4468 y Fk(x)2527 4478 y Fe(j)2563 4453 y Fo(V)2620 4468 y Fn(0)2659 4453 y Fq(\))p Fo(r)2744 4412 y Fn(2)2741 4477 y(0)p Fk(;)p Fn(0)858 4618 y Fm(\000)g Fo(r)1001 4633 y Fn(0)p Fk(;)p Fn(0)1096 4618 y Fq(\()p Fo(D)1218 4577 y Fn(2)1215 4643 y Fk(x)1255 4653 y Fe(j)1287 4643 y Fk(;x)1347 4653 y Fe(i)1377 4618 y Fo(V)1434 4633 y Fn(0)1473 4618 y Fq(\))p Fo(r)1558 4577 y Fn(2)1555 4643 y(0)p Fk(;)p Fn(0)1671 4618 y Fq(+)g Fo(r)1813 4633 y Fn(0)p Fk(;)p Fn(0)1907 4618 y Fq(\()p Fo(D)2026 4633 y Fk(x)2066 4643 y Fe(j)2103 4618 y Fo(V)2160 4633 y Fn(0)2199 4618 y Fq(\))p Fo(r)2281 4633 y Fn(0)p Fk(;)p Fn(0)2375 4618 y Fq(\()p Fo(D)2494 4633 y Fk(x)2534 4643 y Fe(i)2565 4618 y Fo(V)2622 4633 y Fn(0)2661 4618 y Fq(\))p Fo(r)2746 4577 y Fn(2)2743 4643 y(0)p Fk(;)p Fn(0)858 4815 y Fq(+)g Fo(r)1000 4830 y Fn(0)p Fk(;)p Fn(0)1094 4815 y Fq(\()p Fo(D)1213 4830 y Fk(x)1253 4840 y Fe(j)1289 4815 y Fo(V)1346 4830 y Fn(0)1386 4815 y Fq(\))p Fo(r)1471 4774 y Fn(2)1468 4839 y(0)p Fk(;)p Fn(0)1562 4815 y Fq(\()p Fo(D)1681 4830 y Fk(x)1721 4840 y Fe(i)1751 4815 y Fo(V)1808 4830 y Fn(0)1847 4815 y Fq(\))p Fo(r)1929 4830 y Fn(0)p Fk(;)p Fn(0)2023 4704 y Fi(\021)2083 4815 y Fo(;)386 5055 y Fq(and)476 5245 y Fo(@)527 5260 y Fk(\030)558 5270 y Fe(i)589 5245 y Fo(r)633 5260 y Fn(1)p Fk(;)p Fn(0)727 5245 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))28 b(=)1163 5164 y Fi(\000)1209 5245 y Fo(r)1256 5204 y Fn(2)1253 5270 y(0)p Fk(;)p Fn(0)1347 5245 y Fq(\()p Fo(D)1466 5260 y Fk(x)1506 5270 y Fe(i)1536 5245 y Fo(V)1593 5260 y Fn(0)1632 5245 y Fq(\))p Fo(r)1714 5260 y Fn(0)p Fk(;)p Fn(0)1831 5245 y Fm(\000)22 b Fo(r)1974 5260 y Fn(0)p Fk(;)p Fn(0)2069 5245 y Fq(\()p Fo(D)2188 5260 y Fk(x)2228 5270 y Fe(i)2258 5245 y Fo(V)2315 5260 y Fn(0)2354 5245 y Fq(\))p Fo(r)2439 5204 y Fn(2)2436 5270 y(0)p Fk(;)p Fn(0)2530 5164 y Fi(\001)1169 5485 y Fq(+)1317 5360 y Fk(n)1267 5390 y Fi(X)1277 5600 y Fk(j)t Fn(=1)1427 5485 y Fq(2)p Fo(\030)1519 5500 y Fk(j)1555 5485 y Fo(\030)1598 5500 y Fk(i)1642 5404 y Fi(\000)1688 5485 y Fm(\000)p Fq(2)p Fo(r)1861 5443 y Fn(3)1858 5509 y(0)p Fk(;)p Fn(0)1953 5485 y Fq(\()p Fo(D)2072 5500 y Fk(x)2112 5510 y Fe(j)2148 5485 y Fo(V)2205 5500 y Fn(0)2244 5485 y Fq(\))p Fo(r)2326 5500 y Fn(0)p Fk(;)p Fn(0)2443 5485 y Fq(+)g(2)p Fo(r)2634 5500 y Fn(0)p Fk(;)p Fn(0)2728 5485 y Fq(\()p Fo(D)2847 5500 y Fk(x)2887 5510 y Fe(j)2923 5485 y Fo(V)2980 5500 y Fn(0)3019 5485 y Fq(\))p Fo(r)3104 5443 y Fn(3)3101 5509 y(0)p Fk(;)p Fn(0)3196 5404 y Fi(\001)3258 5485 y Fo(:)1842 5722 y Fg(18)p eop %%Page: 19 19 19 18 bop 386 459 a Fq(Plugging)31 b(this)h(in)m(to)g(\(6.3\),)g(w)m(e) h(get)387 701 y(T)-8 b(r)p Fm(A)567 716 y Fn(2)634 701 y Fq(=)747 634 y(1)p 747 678 49 4 v 747 770 a(2)873 577 y Fk(n)822 607 y Fi(X)837 816 y Fk(i)p Fn(=1)983 701 y Fq(T)g(r)1083 591 y Fi(\020)1143 701 y Fo(r)1190 660 y Fn(2)1187 726 y(0)p Fk(;)p Fn(0)1281 701 y Fq(\()p Fo(D)1400 716 y Fk(x)1440 726 y Fe(i)1470 701 y Fo(V)1527 716 y Fn(0)1567 701 y Fq(\))p Fo(r)1652 660 y Fn(2)1649 726 y(0)p Fk(;)p Fn(0)1743 701 y Fq(\()p Fo(D)1862 716 y Fk(x)1902 726 y Fe(i)1932 701 y Fo(V)1989 716 y Fn(0)2028 701 y Fq(\))22 b Fm(\000)h Fo(r)2235 660 y Fn(3)2232 726 y(0)p Fk(;)p Fn(0)2326 701 y Fq(\()p Fo(D)2445 716 y Fk(x)2485 726 y Fe(i)2515 701 y Fo(V)2572 716 y Fn(0)2612 701 y Fq(\))p Fo(r)2694 716 y Fn(0)p Fk(;)p Fn(0)2788 701 y Fq(\()p Fo(D)2907 716 y Fk(x)2947 726 y Fe(i)2977 701 y Fo(V)3034 716 y Fn(0)3073 701 y Fq(\))3111 591 y Fi(\021)409 1027 y Fq(+)557 902 y Fk(n)507 932 y Fi(X)522 1142 y Fk(i)p Fn(=1)718 902 y Fk(n)667 932 y Fi(X)678 1142 y Fk(j)t Fn(=1)828 1027 y Fq(2)p Fo(\030)920 1042 y Fk(j)956 1027 y Fo(\030)999 1042 y Fk(i)1027 1027 y Fq(T)-8 b(r)1127 916 y Fi(\020)1208 1027 y Fm(\000)23 b Fo(r)1355 986 y Fn(2)1352 1052 y(0)p Fk(;)p Fn(0)1446 1027 y Fq(\()p Fo(D)1565 1042 y Fk(x)1605 1052 y Fe(i)1636 1027 y Fo(V)1693 1042 y Fn(0)1732 1027 y Fq(\))p Fo(r)1817 986 y Fn(3)1814 1052 y(0)p Fk(;)p Fn(0)1908 1027 y Fq(\()p Fo(D)2027 1042 y Fk(x)2067 1052 y Fe(j)2103 1027 y Fo(V)2160 1042 y Fn(0)2200 1027 y Fq(\))f(+)g Fo(r)2405 986 y Fn(4)2402 1052 y(0)p Fk(;)p Fn(0)2496 1027 y Fq(\()p Fo(D)2615 1042 y Fk(x)2655 1052 y Fe(i)2685 1027 y Fo(V)2742 1042 y Fn(0)2782 1027 y Fq(\))p Fo(r)2864 1042 y Fn(0)p Fk(;)p Fn(0)2958 1027 y Fq(\()p Fo(D)3077 1042 y Fk(x)3117 1052 y Fe(j)3153 1027 y Fo(V)3210 1042 y Fn(0)3249 1027 y Fq(\))3287 916 y Fi(\021)3347 1027 y Fo(:)386 1289 y Fq(So)32 b(w)m(e)i(obtain)547 1343 y Fi(Z)11 b(Z)826 1527 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)1122 1536 y Fh(1)1157 1527 y Fn(\))669 1608 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)921 1618 y Fe(j;)p Fh(0)1000 1608 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1183 1584 y Fh(2)1217 1608 y Fk(;\026)1279 1617 y Fh(1)1314 1608 y Fn(\))1362 1479 y Fo(f)g Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r)1747 1398 y Fi(\000)1793 1479 y Fo(r)1837 1494 y Fn(2)p Fk(;)p Fn(0)1931 1479 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))2247 1398 y Fi(\001)2293 1479 y Fo(r)2340 1438 y Fk(n)p Fj(\000)p Fn(1)2477 1479 y Fo(dr)s(dz)454 1800 y Fq(=)547 1665 y Fi(Z)11 b(Z)826 1848 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)1122 1857 y Fh(1)1157 1848 y Fn(\))669 1929 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)921 1939 y Fe(j;)p Fh(0)1000 1929 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1183 1906 y Fh(2)1217 1929 y Fk(;\026)1279 1938 y Fh(1)1314 1929 y Fn(\))1362 1706 y Fi(X)1422 1916 y Fk(i)1523 1800 y Fq(T)-8 b(r)1623 1690 y Fi(\020)1692 1733 y Fq(1)p 1692 1777 V 1692 1869 a(2)1751 1800 y Fo(r)1798 1759 y Fn(2)1795 1825 y(0)p Fk(;)p Fn(0)1889 1800 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(D)2324 1815 y Fk(x)2364 1825 y Fe(i)2394 1800 y Fo(V)2451 1815 y Fn(0)2491 1800 y Fq(\))p Fo(r)2576 1759 y Fn(2)2573 1825 y(0)p Fk(;)p Fn(0)2667 1800 y Fq(\()p Fo(x;)g(\030)5 b(;)17 b(z)t Fq(\)\()p Fo(D)3102 1815 y Fk(x)3142 1825 y Fe(i)3172 1800 y Fo(V)3229 1815 y Fn(0)3268 1800 y Fq(\))650 2101 y Fm(\000)23 b Fo(r)797 2060 y Fn(3)794 2126 y(0)p Fk(;)p Fn(0)888 2101 y Fq(\()p Fo(D)1007 2116 y Fk(x)1047 2126 y Fe(i)1077 2101 y Fo(V)1134 2116 y Fn(0)1173 2101 y Fq(\))p Fo(r)1255 2116 y Fn(0)p Fk(;)p Fn(0)1350 2101 y Fq(\()p Fo(D)1469 2116 y Fk(x)1509 2126 y Fe(i)1539 2101 y Fo(V)1596 2116 y Fn(0)1635 2101 y Fq(\))1673 1990 y Fi(\021)1732 2101 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\))p Fo(r)2064 2060 y Fk(n)p Fj(\000)p Fn(1)2201 2101 y Fo(dr)s(dz)552 2366 y Fq(+)650 2231 y Fi(Z)12 b(Z)930 2414 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)1226 2423 y Fh(1)1260 2414 y Fn(\))772 2495 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1024 2505 y Fe(j;)p Fh(0)1103 2495 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1286 2472 y Fh(2)1321 2495 y Fk(;\026)1383 2504 y Fh(1)1417 2495 y Fn(\))1516 2242 y Fk(n)1466 2272 y Fi(X)1481 2482 y Fk(i)p Fn(=1)1677 2242 y Fk(n)1626 2272 y Fi(X)1637 2482 y Fk(j)t Fn(=1)1787 2366 y Fq(2)p Fo(\030)1879 2381 y Fk(j)1915 2366 y Fo(\030)1958 2381 y Fk(i)1985 2366 y Fq(T)-8 b(r)2086 2256 y Fi(\020)2167 2366 y Fm(\000)23 b Fo(r)2314 2325 y Fn(2)2311 2391 y(0)p Fk(;)p Fn(0)2405 2366 y Fq(\()p Fo(D)2524 2381 y Fk(x)2564 2391 y Fe(i)2594 2366 y Fo(V)2651 2381 y Fn(0)2691 2366 y Fq(\))p Fo(r)2776 2325 y Fn(3)2773 2391 y(0)p Fk(;)p Fn(0)2867 2366 y Fq(\()p Fo(D)2986 2381 y Fk(x)3026 2391 y Fe(j)3062 2366 y Fo(V)3119 2381 y Fn(0)3158 2366 y Fq(\))650 2667 y(+)f Fo(r)795 2626 y Fn(4)792 2692 y(0)p Fk(;)p Fn(0)886 2667 y Fq(\()p Fo(D)1005 2682 y Fk(x)1045 2692 y Fe(i)1075 2667 y Fo(V)1132 2682 y Fn(0)1172 2667 y Fq(\))p Fo(r)1254 2682 y Fn(0)p Fk(;)p Fn(0)1348 2667 y Fq(\()p Fo(D)1467 2682 y Fk(x)1507 2692 y Fe(j)1543 2667 y Fo(V)1600 2682 y Fn(0)1640 2667 y Fq(\))1678 2556 y Fi(\021)1737 2667 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\))p Fo(r)2069 2626 y Fk(n)p Fj(\000)p Fn(1)2206 2667 y Fo(dr)s(dz)t(:)386 2855 y Fq(W)-8 b(e)33 b(get)f(b)m(y)i(symmetry)-8 b(,)33 b(for)f Fo(i)c Fm(6)p Fq(=)f Fo(j)6 b Fq(,)570 2929 y Fi(Z)632 3137 y Fn(^)625 3155 y Fk(\030)680 2929 y Fi(Z)735 3155 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)987 3165 y Fe(j;)p Fh(0)1066 3155 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1249 3136 y Fh(2)1284 3155 y Fk(;\026)1346 3164 y Fh(1)1380 3155 y Fn(\))1429 3065 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\))p Fo(L)p Fq(\()p Fo(d)1879 3038 y Fq(^)1869 3065 y Fo(\030)t Fq(\))p Fo(dz)t Fq(2)p Fo(\030)2146 3080 y Fk(j)2182 3065 y Fo(\030)2225 3080 y Fk(i)663 3326 y Fq(T)-8 b(r)763 3216 y Fi(\020)845 3326 y Fm(\000)23 b Fo(r)992 3285 y Fn(2)989 3351 y(0)p Fk(;)p Fn(0)1083 3326 y Fq(\()p Fo(D)1202 3341 y Fk(x)1242 3351 y Fe(i)1272 3326 y Fo(V)1329 3341 y Fn(0)1368 3326 y Fq(\))p Fo(r)1453 3285 y Fn(3)1450 3351 y(0)p Fk(;)p Fn(0)1545 3326 y Fq(\()p Fo(D)1664 3341 y Fk(x)1704 3351 y Fe(j)1740 3326 y Fo(V)1797 3341 y Fn(0)1836 3326 y Fq(\))f(+)g Fo(r)2041 3285 y Fn(4)2038 3351 y(0)p Fk(;)p Fn(0)2133 3326 y Fq(\()p Fo(D)2252 3341 y Fk(x)2292 3351 y Fe(i)2322 3326 y Fo(V)2379 3341 y Fn(0)2418 3326 y Fq(\))p Fo(r)2500 3341 y Fn(0)p Fk(;)p Fn(0)2594 3326 y Fq(\()p Fo(D)2713 3341 y Fk(x)2753 3351 y Fe(j)2790 3326 y Fo(V)2847 3341 y Fn(0)2886 3326 y Fq(\))2924 3216 y Fi(\021)3011 3326 y Fq(=)28 b(0)p Fo(:)386 3515 y Fq(W)-8 b(e)36 b(then)g(get)g(the)g(expression)h(of)e Fo(T)1763 3530 y Fn(2)1803 3515 y Fq(.)53 b(Next,)37 b(w)m(e)g(pro)m(v)m(e,)h(b)m(y)e(induction,)g(that)386 3631 y(for)c(all)e Fo(j)39 b Fq(:)1057 3791 y Fo(r)1101 3806 y Fk(j)t Fn(+1)p Fk(;)p Fj(\017)1282 3791 y Fq(\()p Fo(x;)17 b Fm(\000)p Fo(\030)5 b(;)17 b(z)t Fq(\))28 b(=)g(\()p Fm(\000)p Fq(1\))2009 3750 y Fk(j)t Fn(+1)2136 3791 y Fo(r)2180 3806 y Fk(j)t Fn(+1)p Fk(;)p Fj(\017)2361 3791 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))p Fo(:)386 3966 y Fq(The)37 b(relation)d(is)h(true)h(for)f Fo(j)40 b Fq(=)33 b(0)i(:)50 b Fo(r)1807 3981 y Fn(0)p Fk(;)p Fj(\017)1901 3966 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\))33 b(=)h(\()p Fo(p)2447 3981 y Fj(\017)2486 3966 y Fq(\()p Fo(x;)17 b(\030)5 b Fq(\))24 b Fm(\000)g Fo(z)t Fq(I)2918 3981 y Fk(r)2957 3966 y Fq(\))2995 3930 y Fj(\000)p Fn(1)3090 3966 y Fo(:)36 b Fq(Then)386 4082 y(the)d(iteration)e(is)h(obtained)g(b)m(y)h(the)g(relation)e(\(A.3\).) 486 4198 y(F)-8 b(or)31 b Fo(s)i Fq(o)s(dd,)777 4435 y Fo(T)834 4450 y Fk(s)899 4435 y Fq(=)22 b Fm(\000)1107 4368 y Fq(\(2)p Fo(\031)t Fq(\))1290 4319 y Fj(\000)p Fk(n)p 1107 4412 286 4 v 1154 4503 a Fq(16)p Fo(i\031)1419 4300 y Fi(Z)1474 4525 y Fj(D)1551 4300 y Fi(Z)1607 4525 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)1903 4534 y Fh(1)1937 4525 y Fn(\))2036 4311 y Fk(n)1985 4341 y Fi(X)1996 4550 y Fk(j)t Fn(=1)2146 4435 y Fq(sgn\()p Fo(@)2376 4450 y Fk(r)2415 4435 y Fo(\025)2472 4450 y Fk(j;)p Fn(0)2559 4435 y Fq(\(0\)\))991 4611 y Fi(Z)1047 4836 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1299 4846 y Fe(j;)p Fh(0)1378 4836 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1561 4817 y Fh(2)1595 4836 y Fk(;\026)1657 4845 y Fh(1)1692 4836 y Fn(\))1740 4746 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r\()p Fo(r)2207 4761 y Fk(s;)p Fn(0)2299 4746 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))g Fo(dz)j(d)p Fm(L)1002 5057 y Fq(=)28 b Fm(\000)1193 4989 y Fq(\(2)p Fo(\031)t Fq(\))1377 4941 y Fj(\000)p Fk(n)p 1193 5034 V 1241 5125 a Fq(16)p Fo(i\031)1505 4921 y Fi(Z)1561 5146 y Fj(D)1638 4921 y Fi(Z)1693 5146 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)1989 5155 y Fh(1)2024 5146 y Fn(\))2123 4932 y Fk(n)2072 4962 y Fi(X)2083 5172 y Fk(j)t Fn(=1)2233 5057 y Fq(sgn\()p Fo(@)2463 5072 y Fk(r)2502 5057 y Fo(\025)2559 5072 y Fk(j;)p Fn(0)2646 5057 y Fq(\(0\)\))991 5232 y Fi(Z)1047 5458 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1299 5468 y Fe(j;)p Fh(0)1378 5458 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1561 5439 y Fh(2)1595 5458 y Fk(;\026)1657 5467 y Fh(1)1692 5458 y Fn(\))1740 5368 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r\()p Fo(r)2207 5383 y Fk(s;)p Fn(0)2299 5368 y Fq(\()p Fo(x;)17 b Fm(\000)p Fo(\030)5 b(;)17 b(z)t Fq(\)\))g Fo(dz)k(d)p Fm(L)899 5584 y Fq(=)h Fm(\000)h Fo(T)1154 5599 y Fk(s)1842 5722 y Fg(19)p eop %%Page: 20 20 20 19 bop 386 459 a Fq(So)32 b Fo(T)578 474 y Fk(s)643 459 y Fq(=)c(0)k(for)g Fo(s)g Fq(o)s(dd.)44 b(This)32 b(\014nishes)i(the)f(pro)s(of)f(of)g(Theorem)h(2.1.)774 850 y(7.)49 b Fp(Resonances)36 b(due)i(to)g(the)f(ma)-7 b(trix)38 b(potential)486 1025 y Fq(In)g(this)g(section)g(w)m(e)i(pro)m (v)m(e)f(that)f(the)h(p)s(oten)m(tial)d Fo(V)2423 1040 y Fn(0)2501 1025 y Fq(generate)j(resonances,)386 1141 y(using)32 b(tec)m(hniques)j(similar)29 b(to)j(the)h(ones)g(used)h(in)e ([14].)386 1257 y(Pro)s(of)g(of)g(Theorem)h(2.2)p 386 1273 915 4 v 38 w(:)56 b(W)-8 b(e)39 b(impro)m(v)m(e)g(the)g (computation)f(obtained)g(ab)s(o)m(v)m(e)386 1373 y(in)f(Theorem)h(2.1) g(in)f(the)h(case)h Fo(r)g Fq(=)e(2)g(and)h Fo(n)f Fq(=)g(3.)59 b(This)38 b(is)f(done)i(using)e(the)386 1490 y(residue)c(theorem)f(in)g Fo(r)s Fq(.)386 1703 y Fy(Remark)41 b(4.)j Ff(L)-5 b(et)39 b Fo(B)1192 1718 y Fk(i)1220 1703 y Fq(\()p Fo(x;)17 b(\030)5 b Fq(\))38 b Ff(and)g Fo(B)1748 1718 y Fk(j)1785 1703 y Fq(\()p Fo(x;)17 b(\030)5 b Fq(\))38 b Ff(b)-5 b(e)38 b(two)h(matric)-5 b(es)38 b(indep)-5 b(endent)37 b(of)386 1819 y Fo(z)t Ff(.)53 b(L)-5 b(et)37 b Fo(\013)c(>)f Fq(0)37 b Ff(and)g Fo(\014)h(>)32 b Fq(0)p Ff(.)52 b(F)-7 b(or)36 b(\014xe)-5 b(d)37 b Fo(x)p Ff(,)h(and)f(for)g Fo(P)50 b Ff(such)38 b(that)f Fo(P)3001 1783 y Fj(\000)p Fn(1)3095 1819 y Fo(V)3152 1834 y Fn(0)3191 1819 y Fo(P)51 b Ff(is)386 1935 y(diagonal,)33 b(denote)1337 2216 y Fo(P)1414 2175 y Fj(\000)p Fn(1)1508 2216 y Fo(B)1582 2231 y Fk(i)1610 2216 y Fo(P)41 b Fq(=)1818 2076 y Fi(\022)1933 2156 y Fo(b)1974 2171 y Fn(1)p Fk(;i)2140 2156 y Fo(b)2181 2171 y Fn(2)p Fk(;i)1933 2273 y Fo(b)1974 2288 y Fn(3)p Fk(;i)2140 2273 y Fo(b)2181 2288 y Fn(4)p Fk(;i)2306 2076 y Fi(\023)2396 2216 y Fo(;)622 2640 y Fq(T)-8 b(r)28 b Fo(r)797 2599 y Fk(\013)794 2665 y Fn(0)p Fk(;)p Fn(0)888 2640 y Fo(B)962 2655 y Fk(i)991 2640 y Fo(r)1038 2593 y Fk(\014)1035 2664 y Fn(0)p Fk(;)p Fn(0)1129 2640 y Fo(B)1203 2655 y Fk(j)1267 2640 y Fq(=)p Fo(b)1384 2655 y Fn(1)p Fk(;i)1468 2640 y Fo(b)1509 2655 y Fn(1)p Fk(;j)1601 2640 y Fq(\()p Fo(\030)1687 2599 y Fn(2)1748 2640 y Fq(+)22 b Fo(\025)1903 2655 y Fn(1)p Fk(;)p Fn(0)2019 2640 y Fm(\000)h Fo(z)t Fq(\))2206 2582 y Fj(\000)p Fk(\013)p Fj(\000)p Fk(\014)1365 2820 y Fq(+)f Fo(b)1504 2835 y Fn(2)p Fk(;i)1588 2820 y Fo(b)1629 2835 y Fn(3)p Fk(;j)1721 2820 y Fq(\()p Fo(\030)1807 2779 y Fn(2)1868 2820 y Fq(+)g Fo(\025)2023 2835 y Fn(1)p Fk(;)p Fn(0)2139 2820 y Fm(\000)h Fo(z)t Fq(\))2326 2763 y Fj(\000)p Fk(\013)2431 2820 y Fq(\()p Fo(\030)2517 2779 y Fn(2)2578 2820 y Fq(+)f Fo(\025)2733 2835 y Fn(2)p Fk(;)p Fn(0)2849 2820 y Fm(\000)h Fo(z)t Fq(\))3036 2763 y Fj(\000)p Fk(\014)1365 3000 y Fq(+)f Fo(b)1504 3015 y Fn(3)p Fk(;i)1588 3000 y Fo(b)1629 3015 y Fn(2)p Fk(;j)1721 3000 y Fq(\()p Fo(\030)1807 2959 y Fn(2)1868 3000 y Fq(+)g Fo(\025)2023 3015 y Fn(2)p Fk(;)p Fn(0)2139 3000 y Fm(\000)h Fo(z)t Fq(\))2326 2943 y Fj(\000)p Fk(\013)2431 3000 y Fq(\()p Fo(\030)2517 2959 y Fn(2)2578 3000 y Fq(+)f Fo(\025)2733 3015 y Fn(1)p Fk(;)p Fn(0)2849 3000 y Fm(\000)h Fo(z)t Fq(\))3036 2943 y Fj(\000)p Fk(\014)1365 3180 y Fq(+)f Fo(b)1504 3195 y Fn(4)p Fk(;i)1588 3180 y Fo(b)1629 3195 y Fn(4)p Fk(;j)1721 3180 y Fq(\()p Fo(\030)1807 3139 y Fn(2)1868 3180 y Fq(+)g Fo(\025)2023 3195 y Fn(2)p Fk(;)p Fn(0)2139 3180 y Fm(\000)h Fo(z)t Fq(\))2326 3123 y Fj(\000)p Fk(\013)p Fj(\000)p Fk(\014)2529 3180 y Fo(:)386 3406 y Ff(Then,)34 b(for)h Fo(\013)28 b Fm(6)p Fq(=)f(1)35 b Ff(and)f Fo(\014)g Fm(6)p Fq(=)27 b(1)p Ff(,)877 3542 y Fi(Z)932 3767 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)1184 3777 y Fe(j;)p Fh(0)1263 3767 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1446 3749 y Fh(2)1481 3767 y Fk(;\026)1543 3776 y Fh(1)1577 3767 y Fn(\))1626 3677 y Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r\()p Fo(r)2096 3636 y Fk(\013)2093 3702 y Fn(0)p Fk(;)p Fn(0)2187 3677 y Fo(B)2261 3692 y Fk(i)2289 3677 y Fo(r)2336 3630 y Fk(\014)2333 3702 y Fn(0)p Fk(;)p Fn(0)2428 3677 y Fo(B)2502 3692 y Fk(j)2538 3677 y Fq(\))p Fo(dz)32 b Fq(=)c(0)p Fo(:)486 3962 y Fq(This)k(remark)h(pro)m(v)m(es)h(:)405 4098 y Fi(Z)12 b(Z)684 4281 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)980 4290 y Fh(1)1015 4281 y Fn(\))527 4362 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)779 4372 y Fe(j;)p Fh(0)858 4362 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1041 4339 y Fh(2)1075 4362 y Fk(;\026)1137 4371 y Fh(1)1172 4362 y Fn(\))1220 4234 y Fo(f)f Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\)T)-8 b(r)q(\()p Fo(r)1688 4249 y Fn(2)p Fk(;)p Fn(0)1782 4234 y Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(z)t Fq(\)\))27 b Fo(r)2210 4192 y Fn(2)2249 4234 y Fo(dr)s(dz)32 b Fq(=)411 4583 y Fm(\000)510 4447 y Fi(Z)12 b(Z)789 4630 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)1085 4639 y Fh(1)1120 4630 y Fn(\))632 4712 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)884 4722 y Fe(j;)p Fh(0)963 4712 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1146 4688 y Fh(2)1181 4712 y Fk(;\026)1243 4721 y Fh(1)1277 4712 y Fn(\))1325 4583 y Fo(f)f Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\))1678 4458 y Fk(n)1627 4488 y Fi(X)1638 4698 y Fk(j)t Fn(=1)1788 4583 y Fq(T)-8 b(r\()p Fo(r)1973 4542 y Fn(3)1970 4607 y(0)p Fk(;)p Fn(0)2064 4583 y Fq(\()p Fo(D)2183 4598 y Fk(x)2223 4608 y Fe(j)2259 4583 y Fo(V)2316 4598 y Fn(0)2356 4583 y Fq(\))p Fo(r)2438 4598 y Fn(0)p Fk(;)p Fn(0)2532 4583 y Fq(\()p Fo(D)2651 4598 y Fk(x)2691 4608 y Fe(j)2727 4583 y Fo(V)2784 4598 y Fn(0)2824 4583 y Fq(\)\))27 b Fo(r)2974 4542 y Fn(2)3013 4583 y Fo(dr)s(dz)411 4932 y Fq(+)509 4796 y Fi(Z)11 b(Z)788 4980 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)1084 4989 y Fh(1)1118 4980 y Fn(\))630 5061 y Fk(z)s Fj(2)p Fk(@)t(D)r Fn(\()p Fk(\025)882 5071 y Fe(j;)p Fh(0)961 5061 y Fn(\()p Fk(x)p Fn(\)+)p Fk(\030)1144 5037 y Fh(2)1179 5061 y Fk(;\026)1241 5070 y Fh(1)1276 5061 y Fn(\))1324 4932 y Fo(f)g Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\))1676 4807 y Fk(n)1625 4837 y Fi(X)1636 5047 y Fk(j)t Fn(=1)1786 4932 y Fq(2)p Fo(\030)1883 4891 y Fn(2)1878 4956 y Fk(j)1922 4932 y Fq(T)-8 b(r\()p Fo(r)2107 4891 y Fn(4)2104 4956 y(0)p Fk(;)p Fn(0)2198 4932 y Fq(\()p Fo(D)2317 4947 y Fk(x)2357 4957 y Fe(j)2393 4932 y Fo(V)2450 4947 y Fn(0)2490 4932 y Fq(\))p Fo(r)2572 4947 y Fn(0)p Fk(;)p Fn(0)2666 4932 y Fq(\()p Fo(D)2785 4947 y Fk(x)2825 4957 y Fe(j)2861 4932 y Fo(V)2918 4947 y Fn(0)2958 4932 y Fq(\)\))27 b Fo(r)3108 4891 y Fn(2)3147 4932 y Fo(dr)s(dz)t(:)386 5267 y Fq(De\014ne)1261 5528 y Fo(P)1338 5487 y Fj(\000)p Fn(1)1432 5528 y Fq(\()p Fo(D)1551 5543 y Fk(x)1591 5553 y Fe(i)1621 5528 y Fo(V)1678 5543 y Fn(0)1717 5528 y Fq(\))p Fo(P)41 b Fq(=)1963 5388 y Fi(\022)2077 5468 y Fo(a)2128 5432 y Fn(0)2128 5493 y Fk(i)2256 5468 y Fo(b)2297 5432 y Fn(0)2297 5493 y Fk(i)2082 5585 y Fo(c)2124 5548 y Fn(0)2124 5609 y Fk(i)2251 5585 y Fo(d)2302 5548 y Fn(0)2302 5609 y Fk(i)2383 5388 y Fi(\023)2473 5528 y Fo(:)1842 5722 y Fg(20)p eop %%Page: 21 21 21 20 bop 386 471 a Fq(Denote)35 b Fo(d)p Fm(L)843 435 y Fn(3)915 471 y Fq(=)d Fo(r)1070 435 y Fn(2)1110 471 y Fo(dr)s(L)p Fq(\()p Fo(d)6 b Fq(^)-55 b Fo(x)o Fq(\))p Fo(\032)1505 435 y Fn(2)1545 471 y Fo(d\032L)p Fq(\()p Fo(d)1811 445 y Fq(^)1801 471 y Fo(\030)t Fq(\),)37 b Fm(D)2030 435 y Fn(3)2101 471 y Fq(=)c Fm(f)p Fo(x;)17 b(\030)37 b Fm(2)c Fq(S)2592 435 y Fn(2)2631 471 y Fo(;)17 b(\032)33 b Fm(2)g Fo(e)2902 434 y Fj(\000)p Fk(i)2991 407 y Fe(\031)p 2991 419 38 3 v 2995 460 a Fh(4)3043 471 y Fl(R)5 b Fo(;)17 b Fm(j)p Fo(\032)p Fm(j)38 b(\024)386 587 y Fo(\032)436 602 y Fn(0)476 587 y Fm(g)p Fo(:)32 b Fq(Supp)s(ose)i Fo(@)1019 602 y Fk(r)1057 587 y Fo(\025)1114 602 y Fn(1)p Fk(;)p Fn(0)1208 587 y Fq(\(0\))28 b Fo(>)f Fq(0,)33 b(then)g Fo(@)1846 602 y Fk(r)1884 587 y Fo(\025)1941 602 y Fn(2)p Fk(;)p Fn(0)2036 587 y Fq(\(0\))27 b Fo(<)g Fq(0)33 b(and)f(so)427 812 y Fm(h)p Fo(\027)q(;)17 b(f)11 b Fm(i)26 b Fq(=)838 744 y(1)p 769 789 187 4 v 769 880 a(4)818 851 y Fn(3)857 880 y Fo(\031)916 851 y Fn(3)982 676 y Fi(Z)1037 902 y Fj(D)1094 883 y Fh(3)1149 676 y Fi(Z)1205 902 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)1501 911 y Fh(1)1535 902 y Fn(\))1583 731 y Fi(\010)1642 812 y Fq(\()1680 717 y Fi(X)1739 927 y Fk(i)1840 812 y Fo(d)1891 770 y Fn(0)1891 836 y Fk(i)1930 812 y Fo(b)1971 770 y Fn(0)1971 836 y Fk(i)2011 812 y Fq(\)\()p Fo(x)p Fq(\))759 1065 y(\()p Fo(f)11 b Fq(\()p Fo(\025)951 1080 y Fn(2)p Fk(;)p Fn(0)1045 1065 y Fq(\()p Fo(x)p Fq(\))22 b(+)g Fo(\030)1344 1024 y Fn(2)1383 1065 y Fo(;)17 b(\025)p Fq(\))22 b(+)g Fo(f)11 b Fq(\()p Fo(\025)1796 1080 y Fn(1)p Fk(;)p Fn(0)1890 1065 y Fq(\()p Fo(x)p Fq(\))22 b(+)g Fo(\030)2189 1024 y Fn(2)2228 1065 y Fo(;)17 b(\025)p Fq(\)\)\()p Fo(\025)2500 1080 y Fn(1)p Fk(;)p Fn(0)2616 1065 y Fm(\000)22 b Fo(\025)2772 1080 y Fn(2)p Fk(;)p Fn(0)2867 1065 y Fq(\))2905 1017 y Fj(\000)p Fn(3)2999 985 y Fi(\011)3057 1065 y Fo(d)p Fm(L)3177 1024 y Fn(3)781 1277 y Fq(+)958 1210 y(1)p 889 1254 V 889 1346 a(4)938 1317 y Fn(3)977 1346 y Fo(\031)1036 1317 y Fn(3)1102 1142 y Fi(Z)1158 1367 y Fj(D)1215 1348 y Fh(3)1269 1142 y Fi(Z)1325 1367 y Fk(r)r Fj(2)p Fk(@)t(D)r Fn(\(0)p Fk(;r)1621 1376 y Fh(1)1655 1367 y Fn(\))1687 1277 y Fq(\()p Fo(f)11 b Fq(\()p Fo(\025)1879 1292 y Fn(1)p Fk(;)p Fn(0)1973 1277 y Fq(\()p Fo(x)p Fq(\))22 b(+)g Fo(\030)2272 1236 y Fn(2)2311 1277 y Fo(;)17 b(\025)p Fq(\))22 b Fm(\000)g Fo(f)11 b Fq(\()p Fo(\025)2725 1292 y Fn(2)p Fk(;)p Fn(0)2819 1277 y Fq(\()p Fo(x)p Fq(\))23 b(+)f Fo(\030)3119 1236 y Fn(2)3158 1277 y Fo(;)17 b(\025)p Fq(\)\))826 1460 y Fk(n)776 1490 y Fi(X)786 1700 y Fk(j)t Fn(=1)936 1584 y Fq(2)p Fo(\030)1033 1543 y Fn(2)1028 1609 y Fk(j)1072 1584 y Fq(\()p Fo(d)1161 1543 y Fn(0)1161 1609 y Fk(i)1200 1584 y Fo(b)1241 1543 y Fn(0)1241 1609 y Fk(i)1281 1584 y Fq(\)\()p Fo(x)p Fq(\))1450 1504 y Fi(\002)1491 1584 y Fq(\()p Fo(\025)1586 1599 y Fn(1)p Fk(;)p Fn(0)1703 1584 y Fm(\000)22 b Fo(\025)1859 1599 y Fn(2)p Fk(;)p Fn(0)1954 1584 y Fq(\)\()p Fo(x)p Fq(\))2123 1504 y Fi(\003)2164 1526 y Fj(\000)p Fn(4)2259 1584 y Fo(d)p Fm(L)2379 1543 y Fn(3)2418 1584 y Fo(:)386 1852 y Fq(Then)34 b(the)f(residue)g(theorem)f(giv)m(es)410 2068 y Fm(h)p Fo(\027)q(;)17 b(f)11 b Fm(i)27 b Fq(=)837 2001 y Fo(i)p 780 2045 148 4 v 780 2138 a Fq(4)p Fo(\031)888 2100 y Fn(2)954 1933 y Fi(Z)1009 2158 y Fj(D)1066 2139 y Fh(3)1121 2068 y Fm(f)p Fo(@)1222 2083 y Fk(r)1260 2068 y Fq(\()p Fo(\025)1355 2083 y Fn(1)p Fk(;)p Fn(0)1472 2068 y Fm(\000)22 b Fo(\025)1628 2083 y Fn(2)p Fk(;)p Fn(0)1723 2068 y Fq(\)\(0)p Fo(;)g Fq(^)-54 b Fo(x)o Fq(\)\))p Fm(g)2072 2020 y Fj(\000)p Fn(3)2166 2068 y Fo(L)p Fq(\()p Fo(d)6 b Fq(^)-55 b Fo(x)q Fq(\))p Fo(L)p Fq(\()p Fo(d)2580 2042 y Fq(^)2570 2068 y Fo(\030)t Fq(\))p Fo(\032)2705 2027 y Fn(2)2745 2068 y Fo(d\032)432 2310 y Fm(\002)532 2199 y Fi(h)579 2310 y Fq(\()617 2215 y Fi(X)677 2425 y Fk(i)777 2310 y Fo(d)828 2269 y Fn(0)828 2335 y Fk(i)867 2310 y Fo(b)908 2269 y Fn(0)908 2335 y Fk(i)948 2310 y Fq(\)\(0)p Fo(;)23 b Fq(^)-55 b Fo(x)p Fq(\))p Fo(f)11 b Fq(\()p Fo(\032)1357 2269 y Fn(2)1419 2310 y Fq(+)22 b Fo(E)1589 2325 y Fn(0)1628 2310 y Fo(;)17 b(\025)p Fq(\))22 b(+)1887 2215 y Fi(X)1947 2425 y Fk(i)2047 2229 y Fi(\000)2093 2310 y Fq(\()p Fo(d)2182 2269 y Fn(0)2182 2335 y Fk(i)2221 2310 y Fo(b)2262 2269 y Fn(0)2262 2335 y Fk(i)2302 2310 y Fq(\)\(0)p Fo(;)h Fq(^)-55 b Fo(x)p Fq(\))p Fo(\030)2612 2269 y Fn(2)2607 2335 y Fk(i)2651 2229 y Fi(\001)2696 2310 y Fo(f)2755 2269 y Fj(0)2778 2310 y Fq(\()p Fo(\032)2866 2269 y Fn(2)2928 2310 y Fq(+)22 b Fo(E)3098 2325 y Fn(0)3138 2310 y Fo(;)17 b(\025)p Fq(\))3277 2199 y Fi(i)3323 2310 y Fo(:)386 2565 y Fq(A)33 b(deformation)d(of)i(the)h(con)m(tour)g(in)f Fo(\032)h Fq(giv)m(es)428 2781 y Fm(h)p Fo(\027)q(;)17 b(f)11 b Fm(i)27 b Fq(=)855 2713 y Fo(i)p 798 2758 V 798 2851 a Fq(4)p Fo(\031)906 2813 y Fn(2)972 2645 y Fi(Z)12 b(Z)1098 2871 y Fn(^)-39 b Fk(x;)1161 2853 y Fn(^)1154 2871 y Fk(\030)r Fj(2)p Fn(S)1273 2852 y Fh(2)1328 2645 y Fi(Z)1384 2871 y Fk(\032)p Fj(2)p Fc(R)p Fk(;)p Fj(j)p Fk(\032)p Fj(j\024)1666 2823 y(p)p 1724 2823 38 3 v 1724 2871 a Fk(a)1782 2781 y Fm(f)p Fo(@)1883 2796 y Fk(r)1921 2781 y Fq(\()p Fo(\025)2016 2796 y Fn(1)p Fk(;)p Fn(0)2132 2781 y Fm(\000)23 b Fo(\025)2289 2796 y Fn(2)p Fk(;)p Fn(0)2383 2781 y Fq(\)\(0)p Fo(;)g Fq(^)-55 b Fo(x)p Fq(\)\))p Fm(g)2733 2732 y Fj(\000)p Fn(3)2844 2781 y Fo(L)p Fq(\()p Fo(d)6 b Fq(^)-55 b Fo(x)p Fq(\))p Fo(L)p Fq(\()p Fo(d)3257 2755 y Fq(^)3247 2781 y Fo(\030)t Fq(\))2097 2929 y Fi(h)2144 3039 y Fq(\()2182 2945 y Fi(X)2242 3155 y Fk(i)2343 3039 y Fo(d)2394 2998 y Fn(0)2394 3064 y Fk(i)2433 3039 y Fo(b)2474 2998 y Fn(0)2474 3064 y Fk(i)2514 3039 y Fq(\)\(0)p Fo(;)22 b Fq(^)-54 b Fo(x)p Fq(\))p Fo(f)11 b Fq(\()p Fo(\032)2923 2998 y Fn(2)2984 3039 y Fq(+)22 b Fo(E)3154 3054 y Fn(0)3194 3039 y Fo(;)17 b(\025)p Fq(\))1602 3317 y(+)1678 3236 y Fi(\000)1740 3222 y(X)1800 3432 y Fk(i)1884 3317 y Fq(\()p Fo(d)1973 3275 y Fn(0)1973 3341 y Fk(i)2012 3317 y Fo(b)2053 3275 y Fn(0)2053 3341 y Fk(i)2093 3317 y Fq(\)\(0)p Fo(;)23 b Fq(^)-55 b Fo(x)p Fq(\))p Fo(\030)2403 3275 y Fn(2)2398 3341 y Fk(i)2442 3236 y Fi(\001)2487 3317 y Fo(f)2546 3275 y Fj(0)2569 3317 y Fq(\()p Fo(\032)2657 3275 y Fn(2)2719 3317 y Fq(+)22 b Fo(E)2889 3332 y Fn(0)2929 3317 y Fo(;)17 b(\025)p Fq(\))3068 3206 y Fi(i)3114 3317 y Fo(\032)3164 3275 y Fn(2)3204 3317 y Fo(d\032:)386 3577 y Fq(W)-8 b(e)33 b(get,)g(for)f(all)e Fo(j)k Fm(2)28 b(f)p Fq(1)p Fo(;)17 b(n)p Fm(g)p Fq(,)32 b(that)1304 3674 y Fi(Z)1366 3882 y Fn(^)1359 3899 y Fk(\030)s Fj(2)p Fn(S)1479 3880 y Fh(2)1535 3809 y Fo(\030)1583 3768 y Fn(2)1578 3834 y Fk(j)1638 3809 y Fo(L)p Fq(\()p Fo(d)1803 3783 y Fq(^)1793 3809 y Fo(\030)t Fq(\))c(=)2020 3742 y(V)-8 b(ol)n(\(S)2252 3706 y Fn(2)2292 3742 y Fq(\))p 2020 3786 311 4 v 2150 3877 a(3)2340 3809 y Fo(\032)2390 3768 y Fn(2)2430 3809 y Fo(;)504 4025 y Fi(Z)560 4251 y Fk(\032)p Fj(2)p Fc(R)p Fk(;)p Fj(j)p Fk(\032)p Fj(j\024)842 4203 y(p)p 899 4203 38 3 v 899 4251 a Fk(a)958 4161 y Fo(f)1017 4120 y Fj(0)1040 4161 y Fq(\()p Fo(\032)1128 4120 y Fn(2)1189 4161 y Fq(+)22 b Fo(E)1359 4176 y Fn(0)1399 4161 y Fo(;)17 b(\025)p Fq(\))p Fo(\032)1588 4120 y Fn(4)1627 4161 y Fo(d\032)28 b Fq(=)22 b Fm(\000)1964 4093 y Fq(3)p 1964 4138 49 4 v 1964 4229 a(2)2039 4025 y Fi(Z)2094 4251 y Fk(\032)p Fj(2)p Fc(R)p Fk(;)p Fj(j)p Fk(\032)p Fj(j\024)2376 4203 y(p)p 2434 4203 38 3 v 2434 4251 a Fk(a)2492 4161 y Fo(f)11 b Fq(\()p Fo(\032)2639 4120 y Fn(2)2701 4161 y Fq(+)22 b Fo(E)2871 4176 y Fn(0)2911 4161 y Fo(;)17 b(\025)p Fq(\))p Fo(\032)3100 4120 y Fn(2)3155 4161 y Fo(d\032)1854 4392 y Fq(+)22 b Fm(O)s Fq(\()p Fo(e)2117 4351 y Fj(\000)p Fk(C)2222 4360 y Fh(2)2257 4351 y Fk(\025)2302 4392 y Fq(\))p Fo(:)386 4566 y Fq(Up)33 b(to)f(a)g(remainder,)g(included)g(in) g(\(2.2\),)g(w)m(e)i(get)554 4787 y Fm(h)p Fo(\027)q(;)17 b(f)11 b Fm(i)26 b Fq(=)1016 4720 y Fo(i)p 896 4765 272 4 v 896 4868 a Fq(2\(4)p Fo(\031)t Fq(\))1129 4819 y Fn(2)1178 4787 y Fq(V)-8 b(ol)o(\(S)1411 4746 y Fn(2)1450 4787 y Fq(\))1505 4652 y Fi(Z)1564 4877 y Fn(^)-39 b Fk(x)p Fj(2)p Fn(S)1686 4859 y Fh(2)1742 4652 y Fi(Z)1797 4877 y Fk(\032)p Fj(2)p Fc(R)p Fk(;)p Fj(j)p Fk(\032)p Fj(j\024)2079 4830 y(p)p 2137 4830 38 3 v 2137 4877 a Fk(a)2195 4787 y Fo(f)11 b Fq(\()p Fo(\032)2342 4746 y Fn(2)2404 4787 y Fq(+)22 b Fo(E)2574 4802 y Fn(0)2613 4787 y Fo(;)17 b(\025)p Fq(\))908 5046 y Fm(\002)1008 4935 y Fi(h)1055 5046 y Fq(\()1093 4951 y Fi(X)1153 5161 y Fk(i)1254 5046 y Fo(d)1305 5005 y Fn(0)1305 5071 y Fk(i)1344 5046 y Fo(b)1385 5005 y Fn(0)1385 5071 y Fk(i)1425 5046 y Fq(\)\(0)p Fo(;)22 b Fq(^)-54 b Fo(x)o Fq(\))p Fm(f)p Fo(@)1787 5061 y Fk(r)1826 5046 y Fq(\()p Fo(\025)1921 5061 y Fn(1)p Fk(;)p Fn(0)2037 5046 y Fm(\000)23 b Fo(\025)2194 5061 y Fn(2)p Fk(;)p Fn(0)2288 5046 y Fq(\)\(0)p Fo(;)f Fq(^)-54 b Fo(x)p Fq(\))p Fm(g)2599 4998 y Fj(\000)p Fn(3)2694 4935 y Fi(i)2741 5046 y Fo(L)p Fq(\()p Fo(d)6 b Fq(^)-55 b Fo(x)p Fq(\))p Fo(\032)3039 5005 y Fn(2)3079 5046 y Fo(d\032;)386 5301 y Fq(with)649 5497 y Fo(C)7 b Fq(\()p Fo(V)821 5512 y Fn(0)860 5497 y Fq(\))27 b(=)1029 5362 y Fi(Z)1088 5587 y Fn(^)-39 b Fk(x)p Fj(2)p Fn(S)1210 5568 y Fh(2)1266 5417 y Fi(\000)1328 5403 y(X)1388 5613 y Fk(i)1488 5497 y Fo(d)1539 5456 y Fn(0)1539 5522 y Fk(i)1579 5497 y Fo(b)1620 5456 y Fn(0)1620 5522 y Fk(i)1659 5417 y Fi(\001)1705 5497 y Fq(\(0)p Fo(;)23 b Fq(^)-55 b Fo(x)p Fq(\))p Fo(@)1980 5512 y Fk(r)2018 5497 y Fq(\()p Fo(\025)2113 5512 y Fn(1)p Fk(;)p Fn(0)2230 5497 y Fm(\000)22 b Fo(\025)2386 5512 y Fn(2)p Fk(;)p Fn(0)2481 5497 y Fq(\)\(0)p Fo(;)g Fq(^)-54 b Fo(x)o Fq(\))2742 5449 y Fj(\000)p Fn(3)2837 5497 y Fo(L)p Fq(\()p Fo(d)6 b Fq(^)-55 b Fo(x)p Fq(\))p Fo(:)1842 5722 y Fg(21)p eop %%Page: 22 22 22 21 bop 386 459 a Fq(Denote)37 b(b)m(y)i Fo(a)917 474 y Fn(1)956 459 y Fo(;)17 b(a)1051 474 y Fn(2)1091 459 y Fo(;)g(a)1186 474 y Fn(3)1261 459 y Fm(2)37 b Fl(R)1430 423 y Fn(3)1512 459 y Fq(the)h(v)m(ectors)h(suc)m(h)g(that)f(the)f (\014rst)h(order)g(T)-8 b(a)m(ylor)386 575 y(expansion)33 b(of)f Fo(V)1007 590 y Fn(0)1046 575 y Fq(\()p Fo(x)p Fq(\))23 b Fm(\000)f Fo(b)p Fq(\()p Fo(x)p Fq(\))p Fo(I)1514 590 y Fn(3)1587 575 y Fq(is)1113 784 y(~)1098 809 y Fo(V)1155 824 y Fn(0)1194 809 y Fq(\()p Fo(x)p Fq(\))28 b(=)1457 669 y Fi(\022)1572 749 y Fo(a)1623 764 y Fn(1)1679 749 y Fq(_)-44 b Fo(x)385 b(a)2153 764 y Fn(2)2210 749 y Fq(_)-44 b Fo(x)22 b Fq(+)g Fo(ia)2452 764 y Fn(3)2509 749 y Fq(_)-44 b Fo(x)1572 866 y(a)1623 881 y Fn(2)1679 866 y Fq(_)g Fo(x)23 b Fm(\000)g Fo(ia)1924 881 y Fn(3)1980 866 y Fq(_)-43 b Fo(x)83 b Fm(\000)p Fo(a)2230 881 y Fn(1)2287 866 y Fq(_)-44 b Fo(x)2589 669 y Fi(\023)386 1060 y Fq(W)-8 b(e)33 b(get)f Fo(C)7 b Fq(\()p Fo(V)888 1075 y Fn(0)928 1060 y Fq(\))27 b(=)h Fo(C)7 b Fq(\()1227 1035 y(~)1212 1060 y Fo(V)1269 1075 y Fn(0)1308 1060 y Fq(\).)43 b(Using)33 b(the)g(c)m(hange)g(of)f(co)s(ordinates)905 1234 y Fo(y)953 1249 y Fn(1)1019 1234 y Fq(=)c Fo(a)1174 1249 y Fn(1)1230 1234 y Fq(_)-44 b Fo(x)q(;)44 b(y)1388 1249 y Fn(2)1455 1234 y Fq(=)27 b Fo(a)1609 1249 y Fn(2)1666 1234 y Fq(_)-44 b Fo(x)q(;)44 b(y)1824 1249 y Fn(3)1891 1234 y Fq(=)27 b Fo(a)2045 1249 y Fn(3)2102 1234 y Fq(_)-44 b Fo(x;)45 b(y)30 b Fq(=)e Fo(s)7 b Fq(^)-56 b Fo(y)s(;)51 b Fq(^)-56 b Fo(y)31 b Fm(2)d Fq(S)2789 1193 y Fn(2)2829 1234 y Fo(;)386 1409 y Fq(w)m(e)34 b(obtain)d(:)596 1565 y Fo(C)7 b Fq(\()726 1540 y(~)711 1565 y Fo(V)768 1580 y Fn(0)807 1565 y Fq(\))27 b(=det)q(\()p Fo(a)1173 1580 y Fn(1)1212 1565 y Fo(;)17 b(a)1307 1580 y Fn(2)1347 1565 y Fo(;)g(a)1442 1580 y Fn(3)1481 1565 y Fq(\))1519 1524 y Fj(\000)p Fn(1)970 1777 y Fm(\002)1070 1642 y Fi(Z)1130 1867 y Fn(^)-40 b Fk(y)s Fj(2)p Fn(S)1249 1848 y Fh(2)1288 1777 y Fq(\()1326 1683 y Fi(X)1386 1893 y Fk(i)1486 1777 y Fo(d)1537 1736 y Fn(0)1537 1802 y Fk(i)1576 1777 y Fo(b)1617 1736 y Fn(0)1617 1802 y Fk(i)1657 1777 y Fq(\)\(0)p Fo(;)23 b Fq(^)-55 b Fo(y)s Fq(\)\()p 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y Fq(:)740 2851 y Fm(j)p Fq(T)-8 b(r[\()p Fo(\037f)11 b Fq(\)\()p Fo(P)1192 2866 y Fj(\017)1231 2851 y Fo(;)17 b(\025)p Fq(\)])1397 2810 y Fj(1)1397 2875 y Fn(0)1471 2851 y Fm(j)28 b(\025)g Fo(C)7 b(\025)1766 2810 y Fj(\000)1831 2782 y Fh(3)p 1831 2794 V 1831 2836 a(2)1875 2851 y Fo(h)1931 2810 y Fj(\000)p Fn(1)2048 2851 y Fq(+)22 b Fm(O)s Fq(\()p Fo(h)p Fq(\))p Fo(e)2405 2810 y Fk(C)2455 2819 y Fh(1)2490 2810 y Fk(\025)2557 2851 y Fq(+)g Fm(O)s Fq(\()p Fo(h)2831 2810 y Fj(\000)p Fn(3)2926 2851 y Fq(\))p Fo(e)3009 2810 y Fj(\000)p Fk(C)3114 2819 y Fh(2)3149 2810 y Fk(\025)3194 2851 y Fo(:)386 2845 y Fq(\(7.2\))386 3030 y(The)33 b(comparison)f(of)g(\(7.2\))g(and)h (\(7.1\))f(giv)m(es)h(that)753 3130 y Fi(X)584 3346 y Fk(z)617 3356 y Fe(j)649 3346 y Fj(2)p Fn(Res)q(\()p Fk(P)880 3355 y Fh(0)914 3346 y Fn(\))p Fj(\\)p Fk(W)1082 3225 y Fm(j)p Fo(f)11 b Fq(\()p Fo(z)1252 3240 y Fk(j)1288 3225 y Fo(;)17 b(\025)p Fq(\))p Fm(j)27 b(\025)h Fo(C)7 b(h)1720 3184 y Fj(\000)p Fn(1)1815 3225 y 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b(\025)p Fq(.)43 b(There)34 b(exist)f Fo(h)1505 3795 y Fn(0)1572 3780 y Fm(\024)28 b Fo(h)1733 3795 y Fn(1)1806 3780 y Fq(suc)m(h)34 b(that)e(for)g Fo(h)c(<)f(h)2629 3795 y Fn(0)2701 3780 y Fq(w)m(e)34 b(obtain)e(:)739 3961 y Fo(C)7 b(h)872 3919 y Fj(\000)p Fn(1)966 3961 y Fo(\025)1023 3919 y Fj(\000)1088 3892 y Fh(3)p 1088 3904 V 1088 3945 a(2)1155 3961 y Fq(+)22 b Fm(O)s Fq(\()p Fo(h)p Fq(\))p Fo(e)1512 3919 y Fk(C)1562 3928 y Fh(1)1597 3919 y Fk(\025)1664 3961 y Fq(+)g Fm(O)s Fq(\()p Fo(h)1938 3919 y Fj(\000)p Fn(3)2033 3961 y Fq(\))p Fo(e)2116 3919 y Fj(\000)p Fk(C)2221 3928 y Fh(2)2256 3919 y Fk(\025)1160 4139 y Fq(=)28 b Fo(C)7 b(e)1402 4081 y Fn(~)1386 4098 y Fk(C)e(\025)1486 4139 y Fo(\025)1543 4098 y Fj(\000)1608 4071 y Fh(3)p 1608 4083 V 1608 4124 a(2)1675 4139 y Fq(+)22 b Fm(O)s Fq(\(1\))p Fo(e)2025 4098 y Fn(\()p Fk(C)2102 4107 y Fh(1)2137 4098 y Fj(\000)2208 4081 y Fn(~)2192 4098 y Fk(C)5 b Fn(\))p Fk(\025)2342 4139 y Fq(+)22 b Fm(O)s Fq(\(1\))p Fo(e)2692 4098 y Fn(\(3)2770 4081 y(~)2754 4098 y Fk(C)5 b Fj(\000)p Fk(C)2914 4107 y Fh(2)2949 4098 y Fn(\))p Fk(\025)1160 4318 y Fm(\030)28 b Fo(C)7 b(e)1403 4260 y Fn(~)1387 4277 y Fk(C)e(\025)1488 4318 y Fo(\025)1545 4277 y Fj(\000)1610 4250 y Fh(3)p 1610 4262 V 1610 4303 a(2)1654 4318 y Fo(:)386 4491 y Fq(Hence)730 4670 y(2)p Fo(C)i(h)912 4629 y Fj(\000)p Fn(1)1006 4670 y Fo(\025)1063 4629 y Fj(\000)1128 4602 y Fh(3)p 1128 4614 V 1128 4655 a(2)1195 4670 y Fq(+)22 b Fm(O)s Fq(\()p Fo(h)p Fq(\))p Fo(e)1552 4629 y Fk(C)1602 4638 y Fh(1)1637 4629 y Fk(\025)1704 4670 y Fq(+)g Fm(O)s Fq(\()p Fo(h)1978 4629 y Fj(\000)p Fn(3)2073 4670 y Fq(\))p Fo(e)2156 4629 y Fj(\000)p Fk(C)2261 4638 y Fh(2)2296 4629 y Fk(\025)2369 4670 y Fm(\025)28 b Fo(C)7 b Fm(j)p Fq(ln)o Fo(h)p Fm(j)2744 4629 y Fj(\000)2809 4602 y Fh(3)p 2808 4614 V 2808 4655 a(2)2853 4670 y Fo(h)2909 4629 y Fj(\000)p Fn(1)3004 4670 y Fo(:)386 4849 y Fq(F)-8 b(or)32 b Fo(f)43 b Fq(w)m(e)34 b(see)f(that)g Fm(j)p Fo(f)11 b Fq(\()p Fo(z)t(;)17 b(\025)p Fq(\))p Fm(j)27 b(\024)h Fq(1)k(for)g(Im\()p Fo(z)t Fq(\))c Fm(\024)g Fq(0.)43 b(W)-8 b(e)33 b(conclude)g(that)637 5044 y(card)p Fm(f)p Fq(Res\()p Fo(P)1125 5059 y Fn(0)1165 5044 y Fq(\))22 b Fm(\\)g Fo(W)14 b Fm(g)27 b Fq(=)1745 4949 y Fi(X)1589 5165 y Fk(z)1622 5175 y Fe(j)1654 5165 y Fj(2)p Fn(Res\()p Fk(P)10 b Fn(\))p Fj(\\)p Fk(W)2062 5044 y Fq(1)1495 5337 y Fm(\025)1745 5242 y Fi(X)1589 5458 y Fk(z)1622 5468 y Fe(j)1654 5458 y Fj(2)p Fn(Res\()p Fk(P)g Fn(\))p Fj(\\)p Fk(W)2062 5337 y Fm(j)p Fo(f)h Fq(\()p Fo(z)2232 5352 y Fk(j)2268 5337 y Fo(;)17 b(\025)p Fq(\))p Fm(j)27 b(\025)h Fo(C)7 b(h)2700 5296 y Fj(\000)p Fn(1)2795 5337 y Fm(j)p Fq(ln)o Fo(h)p Fm(j)2988 5296 y Fj(\000)3053 5269 y Fh(3)p 3052 5281 V 3052 5322 a(2)3097 5337 y Fo(:)386 5623 y Fq(This)33 b(\014nishes)g(the)g(pro)s(of)f(of)g (Theorem)h(2.3.)1842 5722 y Fg(23)p eop %%Page: 24 24 24 23 bop 1174 459 a Fp(Appendix)37 b Fq(A.)49 b Fp(Weyl)37 b(calculus)486 633 y Fq(W)-8 b(e)37 b(gather)g(in)f(this)g(app)s(endix) h(some)g(results)h(on)e(the)i(W)-8 b(eyl)37 b(calculus)f(of)h(h)386 749 y(sp)s(eudo)i(di\013eren)m(tial)d(op)s(erators)i(needed)h(v)-5 b(arious)37 b(places)h(in)f(the)i(pap)s(er.)59 b(Let)386 866 y Fo(m)p Fq(\()p Fo(x;)17 b(\030)5 b Fq(\))39 b(b)s(e)h(a)g(temp)s (ered)g(function)f(on)g Fl(R)1997 829 y Fn(2)q Fk(n)2125 866 y Fq(\(i.e.)64 b(there)41 b(exists)f Fo(N)51 b(>)39 b Fq(0)h(and)386 982 y Fo(C)35 b(>)27 b Fq(0)32 b(suc)m(h)i(that)f(:) 392 1146 y Fm(8)p Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(u;)g(v)t Fq(\))26 b Fm(2)i Fl(R)1051 1105 y Fn(4)p Fk(n)1161 1146 y Fm(\002)23 b Fl(C)d Fo(;)50 b Fm(j)p Fo(m)p Fq(\()p Fo(x)23 b Fq(+)f Fo(u;)17 b(\030)25 b Fq(+)d Fo(v)t Fq(\))p Fm(j)27 b(\024)i Fo(C)7 b Fm(j)p Fo(m)p Fq(\()p Fo(x;)17 b(\030)5 b Fq(\))p Fm(j)p Fq(\(1)21 b(+)h Fo(u)2950 1105 y Fn(2)3011 1146 y Fq(+)g Fo(v)3160 1105 y Fn(2)3199 1146 y Fq(\))3237 1105 y Fk(N)3304 1146 y Fq(\))p Fo(:)386 1318 y Fq(Let)33 b(\012)f(b)s(e)h(a)f(compact)g(set.)44 b(W)-8 b(e)33 b(denote)g(b)m(y)h(S\()p Fo(m)p Fq(\()p Fo(x;)17 b(\030)5 b Fq(\))p Fo(;)17 b Fq(1)p Fo(;)g Fq(1\))31 b(the)i(set)g(of)f(func-)386 1435 y(tions)g Fo(a)d Fm(2)g(C)858 1399 y Fj(1)933 1435 y Fq(\()p Fl(R)1037 1399 y Fn(2)p Fk(n)1125 1435 y Fq(\),)k(dep)s(ending)g(on)g Fo(h)g Fq(and)g Fo(z)t Fq(,)h(called)e(sym)m(b)s(ols,)h(with)g(v)-5 b(alues)386 1551 y(in)32 b Fm(M)620 1566 y Fk(r)657 1551 y Fq(\()p Fl(C)20 b Fq(\))39 b(suc)m(h)34 b(that,)e(for)g(all)f(\()p Fo(\013)q(;)17 b(\014)6 b Fq(\))26 b Fm(2)i Fl(N)2011 1515 y Fn(2)q Fk(n)2099 1551 y Fq(,)33 b(there)g(exist)g Fo(C)2709 1566 y Fk(\013;\014)2854 1551 y Fq(suc)m(h)h(that)e(:)510 1729 y Fm(8)p Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(h;)g(z)t Fq(\))28 b Fm(2)g Fl(R)1169 1689 y Fn(2)p Fk(n)1257 1729 y Fm(\002)p Fq(]0)p Fo(;)17 b(h)1510 1744 y Fn(0)1550 1729 y Fq([)p Fm(\002)p Fq(\012)56 b(:)g Fm(k)p Fo(@)1969 1689 y Fk(\013)1964 1754 y(x)2019 1729 y Fo(@)2075 1682 y Fk(\014)2070 1757 y(\030)2123 1729 y Fo(a)p Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(h)p Fq(\))p Fm(k)27 b(\024)h Fo(C)2749 1744 y Fk(\013;\014)2878 1729 y Fo(m)p Fq(\()p Fo(x;)17 b(\030)5 b Fq(\)\))p Fo(:)386 1897 y Fq(A)44 b(sym)m(b)s(ol)e Fo(a)i Fq(is)f Fo(h)p Fq(-admissible)f(if)g(and)i (only)f(if)f(there)i(exists)h(sym)m(b)s(ols)e Fo(a)3225 1912 y Fk(j)3308 1897 y Fm(2)386 2013 y Fq(S\()p Fo(m)p Fq(\()p Fo(x;)17 b(\030)5 b Fq(\))p Fo(;)17 b Fq(1)p Fo(;)g Fq(1\))31 b(suc)m(h)j(that)f(for)f(all)e Fo(N)43 b Fq(:)847 2200 y Fo(a)p Fq(\()p Fo(x;)17 b(\030)5 b(;)17 b(h;)g(z)t Fq(\))28 b(=)1449 2106 y Fi(X)1445 2317 y Fk(j)t(