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y Fp(\()p Fh(T)s Fp(\)\(1)c Fr(<)d(p)f(<)g Fo(1)p Fp(\))40 b Fj(by)1015 4084 y Fr(H)9 b Fp(\()p Fr(b)p Fp(\))k(:)38 b Fr(H)1453 4034 y Fq(p)1533 4084 y Fo(!)33 b Fr(H)1792 4034 y Fq(p)1851 4084 y Fp(:)39 b Fr(f)46 b Fo(7!)33 b Fr(P)16 b(M)c Fp(\()p Fr(b)p Fp(\)\()p Fr(I)35 b Fo(\000)27 b Fr(P)16 b Fp(\))p Fr(J)11 b(f)6 b(;)-9 4322 y Fj(wher)-6 b(e)1380 4549 y Fr(P)50 b Fp(:)1660 4399 y Fq(N)1612 4435 y Fi(X)1571 4691 y Fq(n)p Fn(=)p Fl(\000)p Fq(N)1846 4549 y Fr(f)1904 4567 y Fq(n)1957 4549 y Fr(\037)2030 4567 y Fq(n)2118 4549 y Fo(7!)2319 4399 y Fq(N)2271 4435 y Fi(X)2279 4688 y Fq(n)p Fn(=0)2463 4549 y Fr(f)2521 4567 y Fq(n)2575 4549 y Fr(\037)2648 4567 y Fq(n)2702 4549 y Fr(;)1235 4858 y(M)12 b Fp(\()p Fr(b)p Fp(\))32 b(:)h Fr(L)1680 4809 y Fq(p)1726 4858 y Fp(\()p Fh(T)r Fp(\))39 b Fo(!)32 b Fr(L)2162 4809 y Fq(p)2208 4858 y Fp(\()p Fh(T)r Fp(\))39 b(:)33 b Fr(f)45 b Fo(7!)33 b Fr(bf)6 b(;)964 5105 y(J)44 b Fp(:)33 b Fr(f)13 b Fp(\()p Fr(t)p Fp(\))32 b Fo(7!)1539 5024 y Fp(1)p 1539 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Fr(p)33 b Fo(2)g Fp(\(1)p Fr(;)20 b Fo(1)p Fp(\))36 b Fs(and)i Fr(z)h Fo(2)33 b Fh(D)25 b Fs(.)166 2568 y(Next,)38 b(since)g(for)f(all)i Fr(z)g Fo(2)33 b Fh(D)466 2913 y SDict begin H.S end 466 2913 a 466 2913 a SDict begin 17 H.A end 466 2913 a 466 2913 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.5) cvn /DEST pdfmark end 466 2913 a Fp(\()p Fr(S)7 b(\037)664 2931 y Fq(n)718 2913 y Fp(\)\()p Fr(z)e Fp(\))32 b(=)1071 2750 y Fi(Z)1191 2782 y Fn(1)1138 3020 y(0)1372 2832 y Fr(t)1414 2789 y Fq(n)p 1266 2885 307 5 v 1266 2995 a Fp(1)27 b Fo(\000)f Fr(tz)1583 2913 y(dt)33 b Fp(=)1892 2763 y Fl(1)1847 2799 y Fi(X)1844 3052 y Fq(m)p Fn(=0)2043 2913 y Fr(z)2102 2863 y Fq(m)2199 2750 y Fi(Z)2319 2782 y Fn(1)2266 3020 y(0)2384 2913 y Fr(t)2426 2863 y Fq(n)p Fn(+)p Fq(m)2618 2913 y Fr(dt)g Fp(=)2927 2763 y Fl(1)2882 2799 y Fi(X)2879 3052 y Fq(m)p Fn(=0)3197 2832 y Fr(\037)3270 2850 y Fq(m)3347 2832 y Fp(\()p Fr(z)5 b Fp(\))p 3087 2885 520 5 v 3087 2995 a Fr(n)27 b Fp(+)g Fr(m)g Fp(+)f(1)3617 2913 y Fr(;)329 b Fs(\(5\))-9 3278 y(op)s(erator)39 b Fr(S)45 b Fs(is)40 b(Hank)m(el)f(and)g(has)g(Hilb)s(ert)h(matrix)g Fp(\000)35 b(=)g(\(1)p Fr(=n)25 b Fp(+)i Fr(m)h Fp(+)f(1\))3219 3235 y Fl(1)3219 3308 y Fq(m;n)p Fn(=0)3513 3278 y Fs(with)40 b(resp)s(ect)-9 3419 y(to)51 b(the)h(standard)g(basis)g Fp(\()p Fr(\037)1253 3437 y Fq(n)1307 3419 y Fp(\))1353 3437 y Fq(n)p Fl(2)p Ff(Z)1519 3449 y Fc(+)1624 3419 y Fs(of)f Fr(H)1873 3376 y Fq(p)1919 3419 y Fp(\()p Fh(D)25 b Fp(\))p Fs(.)57 b(Indeed,)51 b(b)m(y)g(form)m(ula)h(\(5\))g(and)g(P)m (arcev)-6 b(al's)-9 3560 y(form)m(ula,)675 3622 y Fi(\015)675 3694 y(\015)675 3766 y(\015)675 3838 y(\015)675 3909 y(\015)742 3832 y Fr(S)7 b(\037)894 3849 y Fq(n)975 3832 y Fo(\000)1139 3682 y Fq(M)1097 3718 y Fi(X)1094 3971 y Fq(m)p Fn(=0)1487 3751 y Fr(\037)1560 3769 y Fq(m)p 1303 3804 V 1303 3914 a Fr(n)26 b Fp(+)h Fr(m)g Fp(+)g(1)1832 3622 y Fi(\015)1832 3694 y(\015)1832 3766 y(\015)1832 3838 y(\015)1832 3909 y(\015)1899 3645 y Fn(2)1899 3987 y Fq(H)1975 3963 y Fc(2)2049 3832 y Fp(=)2173 3622 y Fi(\015)2173 3694 y(\015)2173 3766 y(\015)2173 3838 y(\015)2173 3909 y(\015)2365 3682 y Fl(1)2320 3718 y Fi(X)2240 3974 y Fq(m)p Fn(=)p Fq(M)9 b Fn(+1)2787 3751 y Fr(\037)2860 3769 y Fq(m)p 2603 3804 V 2603 3914 a Fr(n)26 b Fp(+)h Fr(m)g Fp(+)f(1)3132 3622 y Fi(\015)3132 3694 y(\015)3132 3766 y(\015)3132 3838 y(\015)3132 3909 y(\015)3198 3654 y Fn(2)3198 3997 y Fq(H)3274 3973 y Fc(2)3349 3832 y Fp(=)1249 4111 y Fl(1)1205 4147 y Fi(X)1124 4403 y Fq(m)p Fn(=)p Fq(M)9 b Fn(+1)1786 4180 y Fp(1)p 1487 4233 656 5 v 1487 4343 a(\()p Fr(n)26 b Fp(+)h Fr(m)g Fp(+)g(1\))2099 4308 y Fn(2)2187 4261 y Fo(!)32 b Fp(0)38 b(\()p Fr(M)45 b Fo(!)33 b(1)p Fp(\))p Fr(:)-9 4585 y Fs(This)44 b(implies)h(that)f Fr(S)7 b(\037)1052 4603 y Fq(n)1149 4585 y Fp(=)1283 4495 y Fi(P)1409 4527 y Fl(1)1409 4619 y Fq(m)p Fn(=0)1612 4585 y Fr(\037)1685 4603 y Fq(m)1763 4585 y Fr(=)p Fp(\()p Fr(n)29 b Fp(+)h Fr(m)h Fp(+)g(1\))42 b Fs(in)j(the)e(sense)g(of)g Fr(H)3369 4542 y Fn(2)3458 4585 y Fs(and)h(therefore)-9 4726 y Fo(h)p Fr(S)7 b(\037)189 4744 y Fq(j)232 4726 y Fr(;)20 b(\037)358 4744 y Fq(k)408 4726 y Fo(i)38 b Fp(=)g(1)p Fr(=)p Fp(\()p Fr(j)c Fp(+)28 b Fr(k)k Fp(+)c(1\))40 b Fs(for)g(all)h Fr(j;)20 b(k)42 b Fo(\025)37 b Fp(0)j Fs(where)g Fo(h\001)p Fr(;)20 b Fo(\001i)41 b Fs(denotes)f(the)g(inner) h(pro)s(duct)h(in)-9 4867 y Fr(H)97 4824 y Fn(2)142 4867 y Fs(.)166 5009 y(No)m(w)49 b(w)m(e)f(use)f(the)i(follo)m(wing)h (remark)e(to)g(the)g(Nehari)h(Theorem.)f(If)g(the)g Fr(n)p Fs(-th)h(F)-9 b(ourier)-9 5150 y(co)s(e\036cien)m(t)40 b(of)i(a)f(function)h Fr(a)f Fo(2)e Fr(H)1533 5106 y Fq(p)1579 5150 y Fp(\()p Fh(D)25 b Fp(\))47 b Fs(equals)42 b(to)g Fr(a)2351 5168 y Fq(n)2447 5150 y Fs(for)g Fr(n)d Fo(2)h Fh(N)15 b Fs(,)47 b(and)42 b(the)g(op)s(erator)g Fr(A)g Fs(in)-9 5291 y Fr(H)97 5247 y Fq(p)143 5291 y Fp(\()p Fh(T)r Fp(\))47 b Fs(satis\034es)41 b Fo(h)p Fr(A\037)984 5309 y Fq(j)1028 5291 y Fr(;)20 b(\037)1154 5309 y Fq(k)1203 5291 y Fo(i)40 b Fp(=)f Fr(a)1480 5309 y Fq(j)5 b Fn(+)p Fq(k)s Fn(+1)1781 5291 y Fs(for)41 b(all)h Fr(j;)20 b(k)43 b Fo(\025)c Fp(0)i Fs(then)g Fr(A)g Fs(is)h(b)s(ounded)g(on)f Fr(H)3800 5247 y Fq(p)3846 5291 y Fp(\()p Fh(T)r Fp(\))47 b Fs(if)-9 5432 y(\(and)c(only)g(if)8 b(\))43 b Fr(a)f Fo(2)g Fr(B)6 b(M)12 b(O)s Fs(,)43 b(the)g(space)f(of) g(functions)h(of)g(b)s(ounded)g(mean)f(oscillation)j(on)e Fh(T)-9 5573 y Fs([)23 5573 y SDict begin H.S end 23 5573 a Fs(15)136 5497 y SDict begin H.R end 136 5497 a 136 5573 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Bott) cvn H.B /ANN pdfmark end 136 5573 a Fs(,)37 b(p.)h(55].)f(But)1372 5808 y Fo(\000)20 b Fp(log)q(\(1)25 b Fo(\000)i Fr(z)5 b Fp(\))33 b(=)2194 5659 y Fl(1)2149 5695 y Fi(X)2158 5947 y Fq(n)p Fn(=1)2358 5727 y Fp(1)p 2352 5781 70 5 v 2352 5890 a Fr(n)2432 5808 y(\037)2505 5826 y Fq(n)2559 5808 y Fp(\()p Fr(z)5 b Fp(\))p Fr(;)2029 6192 y Fs(5)p eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -18 -696 a SDict begin H.S end -18 -696 a -18 -696 a SDict begin H.R end -18 -696 a -18 -696 a SDict begin [/View [/XYZ H.V]/Dest (page.6) cvn /DEST pdfmark end -18 -696 a -9 -248 a Fs(and)39 b(the)g(function)h Fr(a)p Fp(\()p Fr(z)5 b Fp(\))36 b(:=)f Fo(\000)20 b Fp(log\(1)26 b Fo(\000)i Fr(z)5 b Fp(\))39 b Fs(b)s(elongs)h(to)f Fr(B)6 b(M)12 b(O)44 b Fs(\()p Fr(a)c Fs(is)f(analytic)i(in)f Fh(D)70 b Fs(and)40 b(its)-9 -106 y(imaginary)27 b(part)f(b)s(elongs)h (to)f Fr(L)1351 -150 y Fl(1)1438 -106 y Fp(\()p Fh(T)r Fp(\))p Fs(,)31 b(th)m(us,)26 b Fr(a)h Fs(has)f(the)g(form)f Fr(f)14 b Fp(+)6 b(~)-63 b Fr(g)t Fs(,)26 b(where)g Fr(f)6 b(;)20 b(g)37 b Fo(2)c Fr(L)3658 -150 y Fl(1)3744 -106 y Fp(\()p Fh(T)s 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w(H)1445 504 y Fl(1)1531 539 y Fp(\))i Fo(\024)g(k)p Fr(S)7 b Fo(k)1935 573 y Fq(H)2011 549 y Fe(p)2049 573 y Fl(!)p Fq(H)2208 549 y Fe(p)2283 539 y Fo(\024)33 b Fr(c)2459 557 y Fq(p)2506 539 y Fp(dist)2695 557 y Fq(L)2752 533 y Fd(1)2826 539 y Fp(\()p Fr(b;)p 2975 443 V 20 w(H)3081 504 y Fl(1)3167 539 y Fp(\))p Fr(;)-9 761 y Fs(where)537 927 y Fp(dist)726 945 y Fq(L)783 921 y Fd(1)857 927 y Fp(\()p Fr(b;)p 1006 831 V 20 w(H)1112 893 y Fl(1)1198 927 y Fp(\))g(=)g(inf)8 b Fo(fk)p Fr(b)26 b Fo(\000)g Fr(f)13 b Fo(k)1989 945 y Fq(L)2046 921 y Fd(1)2153 927 y Fp(:)32 b Fr(f)46 b Fo(2)p 2435 831 V 33 w Fr(H)2541 893 y Fl(1)2628 927 y Fo(g)p Fr(;)58 b(c)2829 945 y Fq(p)2908 927 y Fp(=)3276 846 y(1)p 3042 900 526 5 v 3042 1009 a(sin)3362 962 y Fq(\031)p 3216 982 343 5 v 3216 1050 a Fn(max)n Fl(f)p Fq(p;q)s Fl(g)-9 1212 y Fs(\(see,)47 b(e.g.,)f([)543 1212 y SDict begin H.S end 543 1212 a Fs(15)656 1135 y SDict begin H.R end 656 1135 a 656 1212 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 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begin H.R end 2531 2144 a 2531 2220 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Pell) cvn H.B /ANN pdfmark end 2531 2220 a Fs(,)i(p.)f(575,)h(Theorem)f(1.7]\).)166 2361 y(T)-9 b(o)48 b(pro)m(v)m(e)f(1\))h(it)h(remains)f(to)g(sho)m(w)g (that)h Fr(S)54 b Fs(is)49 b(non)f(compact)f(on)h Fr(H)3286 2318 y Fq(p)3380 2361 y Fp(\(1)i Fr(<)h(p)f(<)h Fo(1)p Fp(\))p Fs(.)-9 2503 y(F)-9 b(or)37 b(this)i(w)m(e)f(recall)g(that)g (the)g(sym)m(b)s(ol)h(of)e(the)h(Hank)m(el)h(op)s(erator)f(on)g(the)g (space)f Fr(H)3649 2459 y Fn(2)3695 2503 y Fp(\()p Fh(T)r Fp(\))43 b Fs(with)-9 2644 y(Hilb)s(ert)32 b(matrix)g(with)f(resp)s (ect)f(to)h(the)g(standard)g(basis)g(is)g Fr(b)p Fp(\()p Fr(e)2688 2600 y Fq(it)2751 2644 y Fp(\))h(=)i Fr(ie)3048 2600 y Fl(\000)p Fq(it)3175 2644 y Fp(\()p Fr(\031)15 b Fo(\000)c Fr(t)p Fp(\))p Fr(;)51 b Fp(0)32 b Fo(\024)h Fr(t)h(<)f Fp(2)p Fr(\031)-9 2785 y Fs(\(see,)j(e.g.,)g([)522 2785 y SDict begin H.S end 522 2785 a Fs(16)635 2708 y SDict begin H.R end 635 2708 a 635 2785 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Pell) cvn H.B /ANN pdfmark end 635 2785 a Fs(,)i(p.)f(6]\).)g(A)m(ccording)h(to)g(the)g(Hartman)g (Theorem)f([)2906 2785 y SDict begin H.S end 2906 2785 a Fs(15)3020 2708 y SDict begin H.R end 3020 2708 a 3020 2785 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Bott) cvn H.B /ANN pdfmark end 3020 2785 a Fs(,)g(p.)g(80])g(if)h(the)g(op)s (erator)-9 2926 y Fr(S)65 b Fp(=)59 b Fr(H)9 b Fp(\()p Fr(b)p Fp(\))51 b Fs(is)i(compact)f(then)g Fr(b)58 b Fo(2)h Fr(C)9 b Fp(\()p Fh(T)q Fp(\))42 b(+)p 2123 2830 194 5 v 37 w Fr(H)2229 2891 y Fl(1)2317 2926 y Fs(.)52 b(But)g(this)h(inclusion)i(con)m(tradicts)d(the)-9 3067 y(Lindel)p Fp(\177)-59 b(o)p Fs(f)26 b(theorem)f(on)h(one-sided)f (limits)j(of)d Fr(H)2012 3024 y Fl(1)2099 3067 y Fs(-functions)h (\(see,)f(e.g.,)e([)3122 3067 y SDict begin H.S end 3122 3067 a Fs(17)3236 2991 y SDict begin H.R end 3236 2991 a 3236 3067 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Nik) cvn H.B /ANN pdfmark end 3236 3067 a Fs(,)i(Corollary)i (5.3.5]\).)-9 3208 y(This)38 b(completes)g(the)g(pro)s(of)g(of)f(the)h (\034rst)h(part)f(of)g(the)g(theorem.)166 3349 y(2\))f(As)g(it)g(w)m (as)g(sho)m(wn)g(ab)s(o)m(v)m(e)f Fr(S)7 b(f)45 b Fp(=)34 b Fr(P)16 b(M)c Fp(\()p Fr(b)p Fp(\)\()p Fr(I)32 b Fo(\000)24 b Fr(P)16 b Fp(\))p Fr(J)11 b(f)49 b Fs(for)37 b Fr(f)45 b Fo(2)33 b Fr(H)3193 3306 y Fq(p)3240 3349 y Fs(.)j(Since)h Fr(J)44 b Fp(:)32 b Fr(H)3884 3306 y Fl(1)4005 3349 y Fo(!)-9 3491 y Fp(\()p Fr(H)143 3447 y Fn(2)188 3491 y Fp(\))234 3447 y Fl(?)302 3491 y Fs(,)38 b(it)g(follo)m(ws)h(that)g Fr(S)7 b(f)45 b Fp(=)33 b Fr(P)16 b(M)c Fp(\()p Fr(b)p Fp(\))p Fr(J)f(f)51 b Fs(for)38 b Fr(f)45 b Fo(2)33 b Fr(H)2451 3447 y Fl(1)2539 3491 y Fs(.)k(Moreo)m(v)m(er,)1269 3712 y Fr(J)44 b Fp(:)33 b Fr(H)1550 3663 y Fl(1)1670 3712 y Fo(!)g Fr(L)1903 3663 y Fl(1)1990 3712 y Fr(;)20 b Fo(k)p Fr(J)11 b Fo(k)2239 3730 y Fq(H)2315 3706 y Fd(1)2384 3730 y Fl(!)p Fq(L)2524 3706 y Fd(1)2631 3712 y Fp(=)33 b(1)p Fr(;)-9 3934 y Fs(and)866 4076 y Fr(M)12 b Fp(\()p Fr(b)p Fp(\))33 b(:)f Fr(L)1311 4026 y Fl(1)1432 4076 y Fo(!)g Fr(L)1664 4026 y Fl(1)1751 4076 y Fr(;)20 b Fo(k)p Fr(M)12 b Fp(\()p Fr(b)p Fp(\))p Fo(k)2191 4094 y Fq(L)2248 4070 y Fd(1)2316 4094 y Fl(!)p Fq(L)2456 4070 y Fd(1)2564 4076 y Fp(=)33 b Fo(k)p Fr(b)p Fo(k)2858 4094 y Fq(L)2915 4070 y Fd(1)3021 4076 y Fp(=)h Fr(\031)t(:)-9 4260 y Fs(It)i(is)g(also)g(kno)m(wn)g(\(see,)f(e.g.,)f([)1336 4260 y SDict begin H.S end 1336 4260 a Fs(18)1450 4183 y SDict begin H.R end 1450 4183 a 1450 4260 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Zh) cvn H.B /ANN pdfmark end 1450 4260 a Fs(,)h(Theorem)g(8.3.10]\))g(that)h Fr(P)53 b Fs(is)36 b(a)g(b)s(ounded)g(op)s(erator)h(from)-9 4401 y Fr(L)71 4357 y Fl(1)190 4401 y Fs(to)32 b Fr(B)6 b(M)12 b(O)s(A)p Fs(.)33 b(This)h(implies)f(that)g Fr(S)39 b Fs(is)33 b(a)f(b)s(ounded)i(op)s(erator)e(from)g Fr(H)3254 4357 y Fl(1)3374 4401 y Fs(to)h Fr(B)6 b(M)12 b(O)s(A)33 b Fs(and)138 4623 y Fo(k)p Fr(S)7 b Fo(k)337 4641 y Fq(H)413 4617 y Fd(1)483 4641 y Fl(!)p Fq(B)t(M)i(O)r(A)888 4623 y Fo(\024)33 b(k)p Fr(P)16 b Fo(k)1225 4641 y Fq(L)1282 4617 y Fd(1)1352 4641 y Fl(!)p Fq(B)t(M)9 b(O)r(A)1724 4623 y Fo(k)p Fr(M)j Fp(\()p Fr(b)p Fp(\))p Fo(k)2111 4641 y Fq(L)2168 4617 y Fd(1)2237 4641 y Fl(!)p Fq(L)2377 4617 y Fd(1)2451 4623 y Fo(k)p Fr(J)f Fo(k)2647 4641 y Fq(H)2723 4617 y Fd(1)2792 4641 y Fl(!)p Fq(L)2932 4617 y Fd(1)3039 4623 y Fp(=)34 b Fr(\031)t Fo(k)p Fr(P)16 b Fo(k)3445 4641 y Fq(L)3502 4617 y Fd(1)3573 4641 y Fl(!)p Fq(B)t(M)9 b(O)r(A)3944 4623 y Fr(:)166 4845 y Fs(Finally)-9 b(,)45 b(since)f Fr(b)58 b(=)-73 b Fo(2)43 b Fr(C)9 b Fp(\()p Fh(T)r Fp(\))36 b(+)p 1499 4749 194 5 v 31 w Fr(H)1605 4810 y Fl(1)1692 4845 y Fs(,)44 b(the)g(op)s(erator) g Fr(S)51 b Fp(:)43 b Fr(H)2739 4801 y Fl(1)2827 4845 y Fp(\()p Fh(T)r Fp(\))49 b Fo(!)44 b Fr(B)6 b(M)12 b(O)s(A)p Fp(\()p Fh(T)s Fp(\))50 b Fs(is)44 b(non)-9 4986 y(compact)37 b(b)m(y)h(the)g(main)g(result)h(of)e(the)h(pap)s(er)h([)2078 4986 y SDict begin H.S end 2078 4986 a Fs(19)2191 4909 y SDict begin H.R end 2191 4909 a 2191 4986 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.CJY) cvn H.B /ANN pdfmark end 2191 4986 a Fs(].)166 5127 y(Results)26 b(lik)m(e)g(the)g (previous)g(theorem)f(ma)m(y)h(ha)m(v)m(e)f(applications)i(to)f(appro)m (ximation)h(theory)-9 b(.)-9 5268 y(Let)43 b Fb(r)255 5286 y Fq(n)352 5268 y Fs(b)s(e)g(the)g(set)g(of)g(rational)h (functions)g(of)f(order)g(at)h(most)f Fr(n)g Fs(whose)g(p)s(oles)g (lies)h(outside)-9 5409 y(of)g Fh(D)25 b Fs(,)51 b Fr(X)j Fs(a)45 b(Banac)m(h)f(space)g(of)g(functions)i(de\034ned)e(on)h(some)f (set)h Fr(E)52 b Fo(\022)45 b Fh(D)76 b Fs(and)45 b Fr(X)9 b(R)3772 5427 y Fq(n)3827 5409 y Fp(\()p Fr(f)k Fp(\))43 b(=)-9 5550 y(inf)132 5568 y Fq(r)r Fl(2)p Fa(r)265 5580 y Fe(n)336 5550 y Fo(k)p Fr(f)f Fo(\000)30 b Fr(r)s Fo(k)735 5568 y Fq(X)856 5550 y Fs(the)43 b(degree)e(of)h(appro)m(ximation)j(of) d(a)g(function)h Fr(f)55 b Fs(from)42 b Fr(X)52 b Fs(b)m(y)43 b(rationals)-9 5691 y(from)37 b Fb(r)314 5709 y Fq(n)368 5691 y Fs(.)h(By)f(the)h(triangle)h(inequalit)m(y)-9 b(,)932 5913 y SDict begin H.S end 932 5913 a 932 5913 a SDict begin 17 H.A end 932 5913 a 932 5913 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.6) cvn /DEST pdfmark end 932 5913 a Fo(j)p Fr(X)9 b(R)1160 5931 y Fq(n)1215 5913 y Fp(\()p Fr(f)k Fp(\))25 b Fo(\000)h Fr(X)9 b(R)1717 5931 y Fq(n)1772 5913 y Fp(\()p Fr(g)t Fp(\))p Fo(j)33 b(\024)g(k)p Fr(f)39 b Fo(\000)26 b Fr(g)t Fo(k)2512 5931 y Fq(X)2630 5913 y Fp(\()p Fr(f)6 b(;)20 b(g)36 b Fo(2)d Fr(X)9 b Fp(\))p Fr(:)796 b Fs(\(6\))-9 6024 y SDict begin H.S end -9 6024 a -9 6024 a SDict begin 17 H.A end -9 6024 a -9 6024 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.3) cvn /DEST pdfmark end -9 6024 a 2029 6192 a Fs(6)p eop end end %%Page: 7 7 TeXDict begin HPSdict begin 7 6 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -18 -696 a SDict begin H.S end -18 -696 a -18 -696 a SDict begin H.R end -18 -696 a -18 -696 a SDict begin [/View [/XYZ H.V]/Dest (page.7) cvn /DEST pdfmark end -18 -696 a -9 -248 a Fk(Corollary)45 b(4.3)40 b Fj(The)h(map)h Fr(f)j Fo(7!)33 b Fr(H)1633 -291 y Fq(p)1679 -248 y Fr(R)1768 -230 y Fq(n)1822 -248 y Fp(\()p Fr(S)7 b(f)13 b Fp(\))40 b Fj(is)g(c)-6 b(ontinuous)43 b(on)f Fr(H)3072 -291 y Fq(p)3118 -248 y Fj(.)166 26 y Fs(Pro)s(of.)c(Indeed,)f(the)h(map)g Fr(g)g Fo(7!)33 b Fr(H)1712 -17 y Fq(p)1758 26 y Fr(R)1847 44 y Fq(n)1901 26 y Fr(g)42 b Fs(is)c(con)m(tin)m(uous)h(on)f Fr(H)2947 -17 y Fq(p)3031 26 y Fs(b)m(y)g(the)g(inequalit)m(y)i(\(6\).) 166 168 y(The)d(follo)m(wing)h(corollary)g(is)f(a)g(generalization)h (\(for)e(a)h(Hausdor\033)h(momen)m(t)e(problem\))i(of)-9 309 y(a)f(result)i(due)f(to)g(K.)f(Zh)m(u)h([)1177 309 y SDict begin H.S end 1177 309 a Fs(21)1290 232 y SDict begin H.R end 1290 232 a 1290 309 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Li) cvn H.B /ANN pdfmark end 1290 309 a Fs(,)g(p.)f(372,)h(Prop)s(osition)i(9].)-9 339 y SDict begin H.S end -9 339 a -9 339 a SDict begin 17 H.A end -9 339 a -9 339 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.4) cvn /DEST pdfmark end -9 339 a 244 x Fk(Corollary)45 b(4.4)40 b Fj(L)-6 b(et)43 b Fp(1)36 b Fr(<)i(p)f Fo(\024)g Fp(2)p Fj(.)42 b(F)-9 b(or)44 b Fr(f)50 b Fo(2)36 b Fr(H)2117 539 y Fq(p)2207 583 y Fj(we)42 b(let)h Fr(T)16 b(f)56 b Fj(b)-6 b(e)42 b(the)h(se)-6 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Fj(The)e(Markov-Stieltjes)f(tr)-6 b(ansform)38 b(is)f(an)h(unb)-6 b(ounde)g(d)40 b(op)-6 b(er)g(ator)40 b(on)e Fr(L)3819 -291 y Fn(1)3864 -248 y Fp(\(0)p Fr(;)20 b Fp(1\))-9 -106 y Fj(and)32 b Fr(L)283 -150 y Fl(1)370 -106 y Fp(\(0)p Fr(;)20 b Fp(1\))p Fj(,)29 b(but)j(it)g(is)f(b)-6 b(ounde)g(d)33 b(as)f(an)h(op)-6 b(er)g(ator)33 b(fr)-6 b(om)32 b Fr(L)2598 -150 y Fq(p)2644 -106 y Fp(\(0)p Fr(;)20 b Fp(1\))29 b(\(1)j Fr(<)i(p)f(<)g Fo(1)p Fp(\))e Fj(to)h Fr(L)3819 -150 y Fn(1)3864 -106 y Fp(\(0)p Fr(;)20 b Fp(1\))-9 35 y Fj([)26 35 y SDict begin H.S end 26 35 a Fj(8)84 -42 y SDict begin H.R end 84 -42 a 84 35 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Mir) cvn H.B /ANN pdfmark end 84 35 a Fj(,)50 b(p.)h(187].)h(It)f (is)f(also)h(a)g(c)-6 b(ontinuous)52 b(map)f(fr)-6 b(om)51 b Fr(L)2432 -9 y Fn(1)2477 35 y Fp(\(0)p Fr(;)20 b Fp(1\))48 b Fj(to)i Fr(L)3014 -9 y Fq(p)3060 35 y Fp(\(0)p Fr(;)20 b Fp(1\))48 b Fj(for)j Fp(0)f Fr(<)h(p)g(<)g Fp(1)-9 176 y Fj(b)-6 b(e)g(c)g(ause)40 b(if)h Fr(f)6 b(;)20 b(g)37 b Fo(2)c Fr(L)905 132 y Fq(p)951 176 y Fp(\(0)p Fr(;)20 b Fp(1\))38 b Fj(then)67 507 y Fo(k)p Fr(S)7 b Fp(\()p Fr(f)38 b Fo(\000)27 b Fr(g)t Fp(\))p Fo(k)634 450 y Fq(p)634 548 y(L)691 524 y Fe(p)728 548 y Fn(\(0)p Fq(;)p Fn(1\))936 507 y Fp(=)1060 345 y Fi(Z)1180 377 y Fn(1)1127 614 y(0)1226 298 y Fi(\014)1226 370 y(\014)1226 442 y(\014)1226 513 y(\014)1226 585 y(\014)1265 345 y(Z)1385 377 y Fn(1)1332 614 y(0)1461 426 y Fr(f)13 b Fp(\()p Fr(t)p Fp(\))25 b Fo(\000)h Fr(g)t Fp(\()p Fr(t)p Fp(\))p 1461 480 544 5 v 1579 589 a(1)g Fo(\000)h Fr(tz)2014 507 y(dt)2117 298 y Fi(\014)2117 370 y(\014)2117 442 y(\014)2117 513 y(\014)2117 585 y(\014)2157 330 y Fq(p)2203 507 y Fr(dz)39 b Fo(\024)2483 345 y Fi(Z)2602 377 y Fn(1)2549 614 y(0)2648 303 y Fi( )2743 345 y(Z)2862 377 y Fn(1)2809 614 y(0)2938 426 y Fo(j)p Fr(f)13 b Fp(\()p Fr(t)p Fp(\))25 b Fo(\000)i Fr(g)t Fp(\()p Fr(t)p Fp(\))p Fo(j)p 2938 480 610 5 v 3090 589 a Fp(1)f Fo(\000)g Fr(tz)3558 507 y(dt)3661 303 y Fi(!)3756 330 y Fq(p)3801 507 y Fr(dz)39 b Fo(\024)-9 958 y(\024)117 796 y Fi(Z)237 828 y Fn(1)184 1065 y(0)283 754 y Fi( )377 796 y(Z)497 828 y Fn(1)444 1065 y(0)572 877 y Fo(j)p Fr(f)13 b Fp(\()p Fr(t)p Fp(\))26 b Fo(\000)g Fr(g)t Fp(\()p Fr(t)p Fp(\))p Fo(j)p 572 931 V 745 1040 a Fp(1)g Fo(\000)h Fr(z)1192 958 y(dt)1295 754 y Fi(!)1390 781 y Fq(p)1436 958 y Fr(dz)39 b Fp(=)1714 796 y Fi(Z)1834 828 y Fn(1)1780 1065 y(0)2049 877 y Fr(dz)p 1909 931 402 5 v 1909 1040 a Fp(\(1)26 b Fo(\000)g Fr(z)5 b Fp(\))2264 1006 y Fq(p)2320 958 y Fo(k)p Fr(f)39 b Fo(\000)27 b Fr(g)t Fo(k)2717 901 y Fq(p)2717 998 y(L)2774 975 y Fc(1)2811 998 y Fn(\(0)p Fq(;)p Fn(1\))3019 958 y Fp(=)3255 877 y(1)p 3153 931 264 5 v 3153 1040 a(1)f Fo(\000)g Fr(p)3426 958 y Fo(k)p Fr(f)39 b Fo(\000)27 b Fr(g)t Fo(k)3823 901 y Fq(p)3823 998 y(L)3880 975 y Fc(1)3916 998 y Fn(\(0)p Fq(;)p Fn(1\))4091 958 y Fr(:)166 1296 y Fs(Recall)42 b(that)h(the)f(Banac)m(h)g(space)f Fr(`)1732 1238 y Fq(p)1732 1330 y(A)1841 1296 y Fp(\(1)e Fr(<)i(p)g Fo(\024)f(1)p Fp(\))h Fs(consists)i(of)f(functions)g Fr(f)55 b Fs(that)43 b(are)-9 1437 y(holomorphic)h(on)g(the)g(unit)h (disc,)e Fr(f)13 b Fp(\()p Fr(z)5 b Fp(\))43 b(=)1923 1347 y Fi(P)2049 1379 y Fl(1)2049 1471 y Fq(n)p Fn(=0)2229 1437 y Fr(f)2287 1455 y Fq(n)2341 1437 y Fr(z)2400 1394 y Fq(n)2455 1437 y Fp(\()p Fr(z)48 b Fo(2)43 b Fh(D)25 b Fp(\))p Fs(,)49 b(and)44 b(suc)m(h)g(that)g Fo(k)p Fr(f)13 b Fo(k)3866 1380 y Fq(p)3866 1480 y(`)3901 1440 y Fe(p)3901 1504 y(A)4001 1437 y Fp(:=)-9 1516 y Fi(P)117 1548 y Fl(1)117 1640 y Fq(n)p Fn(=0)297 1606 y Fo(j)p Fr(f)388 1624 y Fq(n)442 1606 y Fo(j)475 1562 y Fq(p)569 1606 y Fr(<)48 b Fo(1)p Fs(.)e(Ob)m(viously)-9 b(,)47 b(the)f(space)g Fr(`)2038 1549 y Fq(p)2038 1641 y(A)2151 1606 y Fs(can)g(b)s(e)h(iden)m(ti\034ed)g(with)h Fr(`)3353 1562 y Fq(p)3399 1606 y Fs(.)e(A)m(ccording)h(to)-9 1747 y([)23 1747 y SDict begin H.S end 23 1747 a Fs(15)136 1670 y SDict begin H.R end 136 1670 a 136 1747 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Bott) cvn H.B /ANN pdfmark end 136 1747 a Fs(,)d(p.)h(53],)f(a)g(Hank)m(el)i(op)s(erator)f Fr(H)9 b Fp(\()p Fr(a)p Fp(\))45 b Fs(on)f Fr(`)2047 1690 y Fq(p)2047 1782 y(A)2159 1747 y Fs(asso)s(ciated)g(with)i(a)f (sequence)f Fr(a)h Fp(=)h(\()p Fr(a)3904 1765 y Fq(n)3958 1747 y Fp(\))e Fs(is)-9 1888 y(de\034ned)37 b(b)m(y)1223 2117 y Fr(H)9 b Fp(\()p Fr(a)p Fp(\))p Fr(f)k Fp(\()p Fr(z)5 b Fp(\))32 b(=)1905 1967 y Fl(1)1860 2003 y Fi(X)1874 2257 y Fq(j)5 b Fn(=0)2053 2117 y Fr(b)2103 2135 y Fq(j)2146 2117 y Fr(z)2205 2067 y Fq(j)2476 2117 y Fp(\()p Fr(z)39 b Fo(2)33 b Fh(D)25 b Fp(\))p Fr(;)-9 2419 y Fs(where)979 2637 y Fr(b)1029 2655 y Fq(j)1105 2637 y Fp(=)1274 2487 y Fl(1)1229 2523 y Fi(X)1239 2780 y Fq(k)s Fn(=0)1422 2637 y Fr(a)1483 2655 y Fq(k)s Fn(+)p Fq(j)5 b Fn(+1)1743 2637 y Fr(f)1801 2655 y Fq(k)2098 2432 y Fi( )2193 2637 y Fr(f)13 b Fp(\()p Fr(z)5 b Fp(\))32 b(=)2616 2487 y Fl(1)2571 2523 y Fi(X)2580 2776 y Fq(n)p Fn(=0)2764 2637 y Fr(f)2822 2655 y Fq(n)2875 2637 y Fr(z)2934 2588 y Fq(n)2989 2432 y Fi(!)3104 2637 y Fr(:)166 2936 y Fs(In)k(the)f(follo)m (wing)i(theorem)e(w)m(e)g(consider)h Fr(S)42 b Fs(as)35 b(an)h(op)s(erator)g(on)f Fr(`)3090 2879 y Fq(p)3090 2971 y(A)3192 2936 y Fs(\(this)i(means)d(that)i Fr(S)-9 3077 y Fs(is)i(de\034ned)g(b)m(y)g(the)g(form)m(ula)g(\(1\),)g(where)g Fr(f)45 b Fo(2)33 b Fr(`)2077 3020 y Fq(p)2077 3112 y(A)2144 3077 y Fr(;)20 b(z)38 b Fo(2)33 b Fh(D)25 b Fs(\))q(.)-9 3112 y SDict begin H.S end -9 3112 a -9 3112 a SDict begin 17 H.A end -9 3112 a -9 3112 a SDict begin [/View [/XYZ H.V]/Dest (theorem.5.5) cvn /DEST pdfmark end -9 3112 a 193 x Fk(Theorem)43 b(5.5)e Fj(The)i(Markov-Stieltjes)h(tr)-6 b(ansform)45 b Fr(S)51 b Fj(is)44 b(a)h(b)-6 b(ounde)g(d)45 b(Hankel)f(op)-6 b(er)g(ator)47 b(on)-9 3446 y Fr(`)40 3389 y Fq(p)40 3481 y(A)139 3446 y Fp(\(1)32 b Fr(<)h(p)g(<)h Fo(1)p Fp(\))d Fj(and)j(has)g(Hilb)-6 b(ert)33 b(matrix)f(with)i(r)-6 b(esp)g(e)g(ct)32 b(to)h(the)g(standar)-6 b(d)35 b(b)-6 b(asis.)33 b(Mor)-6 b(e)g(over,)1604 3702 y Fo(k)p Fr(S)7 b Fo(k)1803 3726 y Fq(`)1838 3686 y Fe(p)1838 3750 y(A)1891 3726 y Fl(!)p Fq(`)2009 3686 y Fe(p)2009 3750 y(A)2100 3702 y Fp(=)2316 3621 y Fr(\031)p 2235 3675 234 5 v 2235 3784 a Fp(sin)2408 3737 y Fq(\031)p 2408 3757 51 5 v 2412 3825 a(p)2478 3702 y Fr(:)166 4029 y Fs(Pro)s(of.)46 b(First)i(note,)d(that)i(the)f(Mark)m(o)m(v-Stieltjes)h(transform)g (exists)f(for)g Fr(f)60 b Fo(2)47 b Fr(`)3739 3972 y Fq(p)3739 4064 y(A)3806 4029 y Fs(,)f(since)-9 4171 y Fr(f)13 b Fo(j)p Fp(\(0)p Fr(;)20 b Fp(1\))30 b Fo(2)j Fr(L)581 4127 y Fn(1)626 4171 y Fp(\(0)p Fr(;)20 b Fp(1\))p Fs(.)35 b(In)j(fact,)f(if)i Fr(f)13 b Fp(\()p Fr(t)p Fp(\))32 b(=)1820 4081 y Fi(P)1946 4113 y Fl(1)1946 4205 y Fq(n)p Fn(=0)2126 4171 y Fr(f)2184 4189 y Fq(n)2238 4171 y Fr(t)2280 4127 y Fq(n)2372 4171 y Fs(then)983 4328 y Fi(Z)1103 4360 y Fn(1)1050 4597 y(0)1169 4490 y Fo(j)p Fr(f)13 b Fp(\()p Fr(t)p Fp(\))p Fo(j)p Fr(dt)32 b Fo(\024)1701 4328 y Fi(Z)1821 4360 y Fn(1)1768 4597 y(0)1931 4341 y Fl(1)1886 4377 y Fi(X)1895 4630 y Fq(n)p Fn(=0)2079 4490 y Fo(j)p Fr(f)2170 4508 y Fq(n)2224 4490 y Fo(j)p Fr(t)2299 4441 y Fq(n)2353 4490 y Fr(dt)i Fp(=)2659 4341 y Fl(1)2614 4377 y Fi(X)2622 4630 y Fq(n)p Fn(=0)2864 4410 y Fo(j)p Fr(f)2955 4427 y Fq(n)3009 4410 y Fo(j)p 2816 4463 273 5 v 2816 4572 a Fr(n)27 b Fp(+)g(1)3099 4490 y Fr(;)-9 4813 y Fs(the)38 b(series)f(con)m(v)m(erges)g(b)m(y)h (the)g(H)p Fp(\177)-59 b(o)p Fs(lder)38 b(inequalit)m(y)-9 b(.)166 4954 y(Arguing)27 b(as)e(in)h(the)f(pro)s(of)h(of)f(Theorem)f (4.2)h(w)m(e)g(get)g(that)h(the)f(Mark)m(o)m(v-Stieltjes)h(transform)-9 5095 y Fr(S)48 b Fs(is)41 b(a)g(Hank)m(el)h(op)s(erator)g(on)f Fr(`)1384 5038 y Fq(p)1384 5130 y(A)1492 5095 y Fs(and)g(has)h(Hilb)s (ert)h(matrix)f(with)g(resp)s(ect)f(to)g(the)h(standard)-9 5237 y(basis)c Fo(f)p Fr(e)384 5255 y Fq(n)471 5237 y Fp(:)33 b Fr(n)g Fo(2)g Fh(Z)830 5255 y Fn(+)901 5237 y Fo(g)p Fr(;)57 b(e)1105 5255 y Fq(n)1160 5237 y Fp(\()p Fr(z)5 b Fp(\))33 b(:=)f Fr(z)1559 5193 y Fq(n)1652 5237 y Fs(of)38 b Fr(`)1831 5179 y Fq(p)1831 5271 y(A)1897 5237 y Fs(.)166 5378 y(T)-9 b(o)25 b(compute)g(the)g(norm)h(of)f Fr(S)7 b Fs(,)24 b(note)i(that)f(the)h(Hardy-Littlew)m(o)s(o)s(d-P)m (oly)m(a-Sh)m(ur)j(inequalit)m(y)-9 5519 y([)23 5519 y SDict begin H.S end 23 5519 a Fs(22)136 5442 y SDict begin H.R end 136 5442 a 136 5519 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.HLP) cvn H.B /ANN pdfmark end 136 5519 a Fs(,)37 b(Theorem)h(318])f(applied)i(to)f(the)g(function)h Fr(K)9 b Fp(\()p Fr(x;)20 b(y)t Fp(\))31 b(=)i(1)p Fr(=)p Fp(\()p Fr(x)25 b Fp(+)h Fr(y)t Fp(\))38 b Fs(implies,)h(that)977 5850 y Fo(k)p Fr(S)7 b(f)13 b Fo(k)1247 5793 y Fq(p)1247 5893 y(`)1282 5853 y Fe(p)1282 5918 y(A)1372 5850 y Fp(=)1541 5701 y Fl(1)1496 5737 y Fi(X)1505 5989 y Fq(n)p Fn(=0)1689 5641 y Fi(\014)1689 5713 y(\014)1689 5785 y(\014)1689 5856 y(\014)1689 5928 y(\014)1777 5701 y Fl(1)1732 5737 y Fi(X)1729 5989 y Fq(m)p Fn(=0)2130 5769 y Fr(f)2188 5787 y Fq(m)p 1938 5823 520 5 v 1938 5932 a Fr(n)26 b Fp(+)h Fr(m)g Fp(+)f(1)2467 5641 y Fi(\014)2467 5713 y(\014)2467 5785 y(\014)2467 5856 y(\014)2467 5928 y(\014)2507 5673 y Fq(p)2586 5850 y Fo(\024)33 b Fr(k)2777 5801 y Fq(p)2823 5850 y Fo(k)p Fr(f)13 b Fo(k)3014 5793 y Fq(p)3014 5893 y(`)3049 5853 y Fe(p)3049 5918 y(A)3105 5850 y Fr(;)2001 6192 y Fs(11)p eop end end %%Page: 12 12 TeXDict begin HPSdict begin 12 11 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -18 -696 a SDict begin H.S end -18 -696 a -18 -696 a SDict begin H.R end -18 -696 a -18 -696 a SDict begin [/View [/XYZ H.V]/Dest (page.12) cvn /DEST pdfmark end -18 -696 a -9 -248 a Fs(where)37 b(the)h(b)s(est)g(p)s(ossible)h(constan)m (t)f Fr(k)k Fs(is)c([)1895 -248 y SDict begin H.S end 1895 -248 a Fs(22)2009 -324 y SDict begin H.R end 2009 -324 a 2009 -248 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.HLP) cvn H.B /ANN pdfmark end 2009 -248 a Fs(,)f(p.)h(229])1340 61 y Fr(k)f Fp(=)1562 -101 y Fi(Z)1681 -69 y Fl(1)1628 169 y Fn(0)2013 -19 y Fr(dx)p 1798 34 556 5 v 1798 145 a(x)1865 111 y Fn(1)p Fq(=p)1994 145 y Fp(\(1)26 b(+)g Fr(x)p Fp(\))2397 61 y(=)2613 -19 y Fr(\031)p 2532 34 234 5 v 2532 143 a Fp(sin)2705 96 y Fq(\031)p 2705 116 51 5 v 2709 185 a(p)-9 403 y Fs(\(for)38 b(the)g(last)g(equalit)m(y)h (see,)e(e.g.,)f([)1541 403 y SDict begin H.S end 1541 403 a Fs(23)1654 327 y SDict begin H.R end 1654 327 a 1654 403 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.BMP) cvn H.B /ANN pdfmark end 1654 403 a Fs(,)i(Section)g(2.2.4,)e(form)m (ula)i(25]\).)-9 433 y SDict begin H.S end -9 433 a -9 433 a SDict begin 17 H.A end -9 433 a -9 433 a SDict begin [/View [/XYZ H.V]/Dest (theorem.5.6) cvn /DEST pdfmark end -9 433 a 229 x Fk(Corollary)45 b(5.6)40 b Fj(L)-6 b(et)55 b Fp(1)61 b Fr(<)g(p)g Fo(\024)g Fp(2)p Fj(.)56 b(The)f(Markov-Stieltjes)g(tr)-6 b(ansform)57 b Fr(S)63 b Fj(is)56 b(a)g(b)-6 b(ounde)g(d)-9 803 y(op)g(er)g(ator)43 b(fr)-6 b(om)41 b Fr(`)760 746 y Fq(p)760 838 y(A)868 803 y Fj(to)g Fr(H)1111 760 y Fq(q)1196 803 y Fp(\(1)p Fr(=p)25 b Fp(+)i(1)p Fr(=q)36 b Fp(=)e(1\))40 b Fj(and)1591 1083 y Fo(k)p Fr(S)7 b Fo(k)1790 1107 y Fq(`)1825 1067 y Fe(p)1825 1131 y(A)1878 1107 y Fl(!)p Fq(H)2037 1083 y Fe(q)44 b Fo(\024)2329 1002 y Fr(\031)p 2247 1055 234 5 v 2247 1165 a Fp(sin)2421 1118 y Fq(\031)p 2421 1137 51 5 v 2425 1206 a(p)2491 1083 y Fr(:)166 1438 y Fs(Pro)s(of.)h(It)g (follo)m(ws)h(from)e(the)h(ab)s(o)m(v)m(e)f(theorem,)g(b)s(ecause)g(b)m (y)h([)2940 1438 y SDict begin H.S end 2940 1438 a Fs(14)3054 1361 y SDict begin H.R end 3054 1361 a 3054 1438 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Dur) cvn H.B /ANN pdfmark end 3054 1438 a Fs(,)f(Theorem)h(6.1,)e(p.)i(94])-9 1579 y Fr(`)40 1522 y Fq(p)40 1614 y(A)158 1579 y Fo(\032)53 b Fr(H)410 1536 y Fq(q)454 1579 y Fp(\()p Fh(D)25 b Fp(\))54 b Fs(and)49 b(the)g(norm)g(of)g(the)f(natural)i(em)m(b)s(edding)g(of)e Fr(`)2959 1522 y Fq(p)2959 1614 y(A)3074 1579 y Fs(in)m(to)i Fr(H)3422 1536 y Fq(q)3466 1579 y Fp(\()p Fh(D)25 b Fp(\))55 b Fs(do)s(es)48 b(not)-9 1720 y(exceed)36 b Fp(1)p Fs(.)-9 1720 y SDict begin H.S end -9 1720 a -9 1720 a SDict begin 17 H.A end -9 1720 a -9 1720 a SDict begin [/View [/XYZ H.V]/Dest (theorem.5.7) cvn /DEST pdfmark end -9 1720 a 259 x Fk(Corollary)45 b(5.7)40 b Fj(The)f(Markov-Stieltjes)g(tr)-6 b(ansform)40 b Fr(S)46 b Fj(is)39 b(a)h(b)-6 b(ounde)g(d)40 b(op)-6 b(er)g(ator)42 b(fr)-6 b(om)39 b Fr(`)3922 1922 y Fq(p)3922 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