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begin H.S end 691 2658 a 691 2658 a SDict begin 12 H.A end 691 2658 a 691 2658 a SDict begin [/View [/XYZ H.V]/Dest (equation.1.20) cvn /DEST pdfmark end 691 2658 a FF(G)766 2624 y Fh(a)p Fq(;\014)861 2658 y Fu(\()p FE(v)s Fu(\))23 b(:=)g FF(G)1177 2618 y Fh(a)p Fq(;\014)1177 2680 y Fr(0)1271 2658 y Fu(\()p FE(v)s Fu(\))p FE(;)98 b Fu(where)1808 2566 y Fp(\020)1858 2658 y FF(G)1933 2618 y Fh(a)p Fq(;\014)1933 2682 y(M)2027 2658 y Fu(\()p FE(v)s Fu(\))2134 2566 y Fp(\021)2184 2583 y Fr(2)2245 2658 y Fu(:=)2397 2554 y FD(1)2370 2579 y Fp(X)2355 2758 y Fq(`)p Fr(=)p Fq(M)2528 2596 y Ft(a)2581 2537 y FC(2)p 2577 2546 35 3 v 2577 2579 a Fl(\014)2622 2559 y Fq(`)2668 2529 y Fp(\000)2706 2596 y Fs(h)p FE(v)s Fs(i)2813 2566 y Fq(\014)2859 2529 y Fp(\001)2910 2524 y FC(2)p 2907 2533 V 2907 2566 a Fl(\014)2952 2546 y Fq(`)p 2528 2639 457 4 v 2665 2735 a Fu(\()p FE(`)p Fu(!\))2800 2675 y FC(2)p 2797 2684 35 3 v 2797 2718 a Fl(\014)2994 2658 y FE(:)p 0 TeXcolorgray 645 w Fu(\(1.20\))p 0 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FE(C)1914 3375 y FD(j)p Fq(\013)p FD(j)2015 3409 y Fu(\()p Fs(j)p FE(\013)p Fs(j)p Fu(!\))2202 3368 y Fq(\027)2257 3409 y FE(:)p 0 TeXcolorgray 1382 w Fu(\(1.22\))p 0 TeXcolorgray 0 3575 a(Equila)n(v)n(en)n(tly)-7 b(,)26 b(there)h(exists)h FE(c)23 b(>)f Fu(0)27 b(suc)n(h)h(that)g FE(e)1561 3545 y Fq(c)p Fr(\()p FD(\000)p Fr(\001\))1774 3500 y FC(1)p 1758 3507 61 3 v 1758 3538 a(2)p Fl(\027)1837 3575 y FE(f)k Fs(2)23 b FE(L)2045 3545 y Fr(2)2082 3575 y Fu(\()p Fo(R)2174 3538 y Fr(3)2211 3575 y Fu(\).)0 3646 y SDict begin H.S end 0 3646 a 0 3646 a SDict begin 12 H.A end 0 3646 a 0 3646 a SDict begin [/View [/XYZ H.V]/Dest (subsubsection.1.4.3) cvn /DEST pdfmark end 0 3646 a 79 x Fu(1.4.3.)40 b Fj(Main)34 b(r)l(esults.)41 b Fu(Our)31 b(\014rst)g(result)g(is)g(to)g(pro)n(v)n(e)e(the)j(existence)f(and)g (uniqueness)g(results)f(of)h(the)h(equation)e(\()p 1 0 0 TeXcolorrgb 3735 3725 a SDict begin H.S end 3735 3725 a 1 0 0 TeXcolorrgb Fu(1.1)p 1 0 0 TeXcolorrgb 3842 3671 a SDict begin H.R end 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(subsubsection.2.2.1) cvn /DEST pdfmark end 0 1915 a 71 x Fu(2.2.1.)40 b Fj(Dyadic)31 b(de)l(c)l(omp)l(osition)h (in)d(the)h(phase)h(sp)l(ac)l(e.)43 b Fu(The)27 b(dy)n(adic)g(op)r (erator)f(in)h(the)h(phase)f(space)g Fs(P)3289 1998 y Fq(j)3351 1986 y Fu(can)g(b)r(e)h(de\014ned)0 2086 y(as)f(follo)n(ws:) 874 2246 y SDict begin H.S end 874 2246 a 874 2246 a SDict begin 12 H.A end 874 2246 a 874 2246 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.4) cvn /DEST pdfmark end 874 2246 a Fs(P)932 2258 y FD(\000)p Fr(1)1021 2246 y FE(f)9 b Fu(\()p FE(x)p Fu(\))23 b(:=)g FE( )s Fu(\()p FE(x)p Fu(\))p FE(f)9 b Fu(\()p FE(x)p Fu(\))p FE(;)43 b Fs(P)1769 2258 y Fq(j)1804 2246 y FE(f)9 b Fu(\()p FE(x)p Fu(\))24 b(:=)e FE(')p Fu(\(2)2227 2211 y FD(\000)p Fq(j)2314 2246 y FE(x)p Fu(\))p FE(f)9 b Fu(\()p FE(x)p Fu(\))p FE(;)43 b(j)28 b Fs(\025)23 b Fu(0)p FE(:)p 0 TeXcolorgray 868 w Fu(\(2.4\))p 0 TeXcolorgray 0 2406 a(By)j(the)g(de\014nition,)h(there)e(exists)h(a)f(in)n(teger)g FE(N)1506 2418 y Fr(0)1566 2406 y Fs(\025)e Fu(2)j(suc)n(h)f(that)h Fs(P)2143 2418 y Fq(j)2178 2406 y Fs(P)2236 2418 y Fq(k)2300 2406 y Fu(=)c(0)k(if)g Fs(j)p FE(j)20 b Fs(\000)15 b FE(k)s Fs(j)23 b(\025)g FE(N)2933 2418 y Fr(0)2970 2406 y Fu(.)36 b(In)26 b(applications,)g(w)n(e)f(also)0 2514 y(in)n(tro)r(duce)388 2493 y(~)369 2514 y Fs(P)427 2526 y Fq(l)452 2514 y FE(f)9 b Fu(\()p FE(x)p Fu(\))24 b(:=)847 2452 y Fp(P)748 2593 y FD(j)p Fq(k)q FD(\000)p Fq(l)p FD(j\024)p Fq(N)1002 2601 y FC(0)1048 2514 y Fs(P)1106 2526 y Fq(k)1146 2514 y FE(f)9 b Fu(\()p FE(x)p Fu(\))29 b(and)e Ff(U)1551 2526 y Fq(j)1586 2514 y FE(f)9 b Fu(\()p FE(x)p Fu(\))23 b(:=)1897 2452 y Fp(P)1881 2589 y Fq(k)q FD(\024)p Fq(j)2014 2514 y Fs(P)2072 2526 y Fq(k)2113 2514 y FE(f)9 b Fu(\()p FE(x)p Fu(\).)38 b(F)-7 b(or)27 b(an)n(y)f(smo)r(oth)i(function)g FE(f)9 b Fu(,)27 b(w)n(e)g(ha)n(v)n (e)1561 2771 y FE(f)k Fu(=)23 b Fs(P)1779 2783 y FD(\000)p Fr(1)1868 2771 y FE(f)k Fu(+)2019 2693 y Fp(X)2021 2871 y Fq(j)s FD(\025)p Fr(0)2153 2771 y Fs(P)2211 2783 y Fq(j)2245 2771 y FE(f)t(:)0 2991 y SDict begin H.S end 0 2991 a 0 2991 a SDict begin 12 H.A end 0 2991 a 0 2991 a SDict begin [/View [/XYZ H.V]/Dest (subsubsection.2.2.2) cvn /DEST pdfmark end 0 2991 a 76 x Fu(2.2.2.)40 b Fj(Dyadic)30 b(de)l(c)l(omp)l(osition)g(in)e(the)g(fr)l(e)l(quency)h(sp)l(ac)l(e.)43 b Fu(W)-7 b(e)26 b(denote)41 b(~)-57 b FE(m)23 b Fu(:=)f Fs(F)2582 3037 y FD(\000)p Fr(1)2671 3067 y FE( )29 b Fu(and)40 b(~)-56 b FE(')24 b Fu(:=)e Fs(F)3170 3037 y FD(\000)p Fr(1)3259 3067 y FE(')27 b Fu(where)e(they)h(are)0 3167 y(the)34 b(in)n(v)n(erse)e(F)-7 b(ourier)33 b(transform)g(of)h FE(')g Fu(and)f FE( )s Fu(.)56 b(If)34 b(w)n(e)g(set)47 b(~)-56 b FE(')2011 3179 y Fq(j)2047 3167 y Fu(\()p FE(x)p Fu(\))34 b(=)f(2)2332 3137 y Fr(3)p Fq(j)2413 3167 y Fu(~)-55 b FE(')p Fu(\(2)2528 3137 y Fq(j)2563 3167 y FE(x)p Fu(\),)36 b(then)e(the)h(dy)n(adic)e(op)r(erator)f(in)i(the)0 3267 y(frequency)27 b(space)g Ff(F)649 3279 y Fq(j)711 3267 y Fu(can)h(b)r(e)g(de\014ned)g(as)e(follo)n(ws)553 3472 y SDict begin H.S end 553 3472 a 553 3472 a SDict begin 12 H.A end 553 3472 a 553 3472 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.5) cvn /DEST pdfmark end 553 3472 a Ff(F)604 3484 y FD(\000)p Fr(1)693 3472 y FE(f)9 b Fu(\()p FE(x)p Fu(\))23 b(:=)988 3359 y Fp(Z)1034 3548 y Fn(R)1076 3528 y FC(3)1142 3472 y Fu(~)-58 b FE(m)p Fu(\()p FE(x)19 b Fs(\000)f FE(y)s Fu(\))p FE(f)9 b Fu(\()p FE(y)s Fu(\))p FE(dy)s(;)42 b Ff(F)1817 3484 y Fq(j)1851 3472 y FE(f)9 b Fu(\()p FE(x)p Fu(\))24 b(:=)2147 3359 y Fp(Z)2193 3548 y Fn(R)2235 3528 y FC(3)2299 3472 y Fu(~)-56 b FE(')2339 3484 y Fq(j)2375 3472 y Fu(\()p FE(x)19 b Fs(\000)f FE(y)s Fu(\))p FE(f)9 b Fu(\()p FE(y)s Fu(\))p FE(dy)s(;)41 b(j)28 b Fs(\025)23 b Fu(0)p FE(:)p 0 TeXcolorgray 547 w Fu(\(2.5\))p 0 TeXcolorgray 0 3694 a(W)-7 b(e)32 b(also)e(in)n(tro)r(duce)695 3672 y(~)690 3694 y Ff(F)741 3706 y Fq(j)776 3694 y FE(f)9 b Fu(\()p FE(x)p Fu(\))30 b(=)1061 3631 y Fp(P)1149 3719 y FD(j)p Fq(k)q FD(\000)p Fq(j)s FD(j\024)p Fr(3)p Fq(N)1445 3727 y FC(0)1496 3694 y Ff(F)1547 3706 y Fq(k)1587 3694 y FE(f)9 b Fu(\()p FE(x)p Fu(\))33 b(and)e Fs(S)1996 3706 y Fq(j)2031 3694 y FE(f)9 b Fu(\()p FE(x)p Fu(\))30 b(=)2316 3631 y Fp(P)2404 3719 y Fq(k)q FD(\024)p Fq(j)2541 3694 y Ff(F)2592 3706 y Fq(k)2633 3694 y FE(f)9 b Fu(.)48 b(Then)32 b(for)f(an)n(y)g FE(f)38 b Fs(2)30 b Fi(S)3519 3664 y FD(0)3542 3694 y Fu(,)j(it)e(holds)0 3799 y(that)1568 3924 y FE(f)g Fu(=)23 b Ff(F)1779 3936 y FD(\000)p Fr(1)1868 3924 y FE(f)k Fu(+)2019 3846 y Fp(X)2021 4024 y Fq(j)s FD(\025)p Fr(0)2153 3924 y Ff(F)2204 3936 y Fq(j)2238 3924 y FE(f)t(:)0 4144 y SDict begin H.S end 0 4144 a 0 4144 a SDict begin 12 H.A end 0 4144 a 0 4144 a SDict begin [/View [/XYZ H.V]/Dest (subsubsection.2.2.3) cvn /DEST pdfmark end 0 4144 a 71 x Fu(2.2.3.)40 b Fj(Applic)l(ations)32 b(to)d(the)h(c)l(ol)t(lision)i(op)l(er)l(ator.)43 b Fu(Recall)27 b(that)h FE(a)p Fu(\()p FE(z)t Fu(\))g(de\014ned)f(in)h(\()p 1 0 0 TeXcolorrgb 2662 4215 a SDict begin H.S end 2662 4215 a 1 0 0 TeXcolorrgb Fu(1.3)p 1 0 0 TeXcolorrgb 2769 4161 a SDict begin H.R end 2769 4161 a 2769 4215 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.1.3) cvn H.B /ANN pdfmark end 2769 4215 a 0 TeXcolorgray Fu(\).)37 b(By)27 b(in)n(tro)r (ducing)1138 4450 y SDict begin H.S end 1138 4450 a 1138 4450 a SDict begin 12 H.A end 1138 4450 a 1138 4450 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.6) cvn /DEST pdfmark end 1138 4450 a FE(a)1182 4462 y Fq(k)1223 4450 y Fu(\()p FE(z)t Fu(\))c(=)1441 4308 y Fp(\()1507 4385 y FE(a)p Fu(\()p FE(z)t Fu(\))p FE(')p Fu(\(2)1786 4350 y FD(\000)p Fq(k)1879 4385 y FE(z)t Fu(\))p FE(;)230 b Fu(if)84 b FE(k)26 b Fs(\025)c Fu(0;)1507 4514 y FE(a)p Fu(\()p FE(z)t Fu(\))p FE( )s Fu(\()p FE(z)t Fu(\))p FE(;)298 b Fu(if)83 b FE(k)26 b Fu(=)d Fs(\000)p Fu(1)p FE(;)p 0 TeXcolorgray 3703 4450 a Fu(\(2.6\))p 0 TeXcolorgray 0 4685 a(w)n(e)k(set)h FE(b)288 4697 y Fq(k)328 4685 y Fu(\()p FE(z)t Fu(\))23 b(:=)g Fs(r)c(\001)f FE(a)742 4697 y Fq(k)783 4685 y Fu(\()p FE(z)t Fu(\))p FE(;)97 b(c)1046 4697 y Fq(k)1086 4685 y Fu(\()p FE(z)t Fu(\))23 b(:=)g Fs(r)c(\001)f FE(b)1492 4697 y Fq(k)1533 4685 y Fu(\()p FE(z)t Fu(\))p FE(;)96 b(Q)1825 4697 y Fq(k)1866 4685 y Fu(\()p FE(g)s(;)14 b(h)p Fu(\))23 b(:=)f Fs(r)d(\001)g Fu(\([)p FE(a)2420 4697 y Fq(k)2479 4685 y Fs(\003)f FE(g)s Fu(])p Fs(r)p FE(h)g Fs(\000)g Fu([)p FE(a)2890 4697 y Fq(k)2949 4685 y Fs(\003)g(r)p FE(g)s Fu(])p FE(h)p Fu(\).)37 b(Then)28 b(w)n(e)f(ha)n(v)n(e)802 4887 y SDict begin H.S end 802 4887 a 802 4887 a SDict begin 12 H.A end 802 4887 a 802 4887 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.7) cvn /DEST pdfmark end 802 4887 a Fs(h)p FE(Q)900 4899 y Fq(k)941 4887 y Fu(\()p FE(g)s(;)14 b(h)p Fu(\))p FE(;)g(f)9 b Fs(i)1252 4899 y Fq(v)1374 4887 y Fu(=)1521 4774 y Fp(Z)1604 4887 y Fu(\()p FE(a)1680 4899 y Fq(k)1740 4887 y Fs(\003)18 b FE(g)s Fu(\))23 b(:)g Fs(r)2013 4853 y Fr(2)2050 4887 y FE(hf)9 b(dv)22 b Fs(\000)2336 4774 y Fp(Z)2419 4887 y Fu(\()p FE(c)2487 4899 y Fq(k)2546 4887 y Fs(\003)c FE(g)s Fu(\))p FE(hf)9 b(dv)1374 5105 y Fu(=)1521 4992 y Fp(Z)1604 5105 y Fu(\()p FE(a)1680 5117 y Fq(k)1740 5105 y Fs(\003)18 b FE(g)s Fu(\))23 b(:)g FE(h)p Fs(r)2061 5071 y Fr(2)2098 5105 y FE(f)9 b(dv)22 b Fu(+)c(2)2392 4992 y Fp(Z)2474 5105 y Fu(\()p FE(b)2542 5117 y Fq(k)2601 5105 y Fs(\003)g FE(g)s Fu(\))g Fs(\001)h FE(h)p Fs(r)p FE(f)9 b(dv)s(:)p 0 TeXcolorgray 631 w Fu(\(2.7\))p 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 10 10 TeXDict begin HPSdict begin 10 9 bop 0 TeXcolorgray 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 0 TeXcolorgray -8 -245 a SDict begin H.S end -8 -245 a -8 -245 a SDict begin H.R end -8 -245 a -8 -245 a SDict begin [/View [/XYZ H.V]/Dest (page.10) cvn /DEST pdfmark end -8 -245 a 0 TeXcolorgray 0 TeXcolorgray 0 -170 a Fr(10)1263 b(LING-BING)24 b(HE,)e(JIE)g(JI,)g(AND)g(YUE)g(LUO) p 0 TeXcolorgray 0 29 a Fu(On)27 b(one)h(hand,)f(in)h(phase)f(space,)g (applying)g(the)h(dy)n(adic)f(decomp)r(osition)g(to)g(the)h(op)r (erator,)e(w)n(e)i(deduce)f(that)538 177 y SDict begin H.S end 538 177 a 538 177 a SDict begin 12 H.A end 538 177 a 538 177 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.8) cvn /DEST pdfmark end 538 177 a Fs(h)p FE(Q)p Fu(\()p FE(g)s(;)14 b(h)p Fu(\))p FE(;)g(f)9 b Fs(i)947 189 y Fq(v)1009 177 y Fu(=)1144 98 y Fp(X)1097 277 y Fq(k)q(;)p Fe(l)p FD(\025\000)p Fr(1)1311 177 y Fs(h)p FE(Q)1409 189 y Fq(k)1450 177 y Fu(\()p Fs(P)1540 189 y Fe(l)1565 177 y FE(g)s(;)14 b(h)p Fu(\))p FE(;)g(f)9 b Fs(i)1844 189 y Fq(v)1906 177 y Fu(=)2063 98 y Fp(X)1994 277 y Fq(k)q FD(\025)p Fq(N)2135 285 y FC(0)2167 277 y FD(\000)p Fr(1)2252 177 y Fs(h)p FE(Q)2350 189 y Fq(k)2391 177 y Fu(\()p Ff(U)2477 189 y Fq(k)q FD(\000)p Fq(N)2618 197 y FC(0)2655 177 y FE(g)s(;)2753 156 y Fu(~)2735 177 y Fs(P)2793 189 y Fq(k)2833 177 y FE(h)p Fu(\))p FE(;)2968 156 y Fu(~)2950 177 y Fs(P)3008 189 y Fq(k)3048 177 y FE(f)g Fs(i)3130 189 y Fq(v)860 417 y Fu(+)1001 338 y Fp(X)939 517 y Fe(l)p FD(\025)p Fq(k)q Fr(+)p Fq(N)1152 525 y FC(0)1184 417 y Fs(h)p FE(Q)1282 429 y Fq(k)1323 417 y Fu(\()p Fs(P)1413 429 y Fe(l)1438 417 y FE(g)s(;)1536 396 y Fu(~)1518 417 y Fs(P)1576 429 y Fe(l)1600 417 y FE(h)p Fu(\))p FE(;)1736 396 y Fu(~)1717 417 y Fs(P)1775 429 y Fe(l)1800 417 y FE(f)g Fs(i)1882 429 y Fq(v)1940 417 y Fu(+)2118 338 y Fp(X)2023 517 y Fe(m)p FD(\024)p Fq(k)q Fr(+2)p Fq(N)2300 525 y FC(0)2332 417 y Fs(h)p FE(Q)2430 429 y Fq(k)2471 417 y Fu(\()2522 396 y(~)2503 417 y Fs(P)2561 429 y Fq(k)2602 417 y FE(g)s(;)14 b Fs(P)2740 429 y Fe(m)2795 417 y FE(h)p Fu(\))p FE(;)2930 396 y Fu(~)2912 417 y Fs(P)2970 429 y Fe(m)3025 417 y FE(f)9 b Fs(i)3107 429 y Fq(v)3147 417 y FE(:)p 0 TeXcolorgray 533 w Fu(\(2.8\))p 0 TeXcolorgray 0 637 a(On)27 b(the)h(other)f(hand,)h (o)n(wing)f(to)g(the)h(Bob)n(ylev's)e(equalit)n(y)660 811 y Fs(h)p FE(Q)758 823 y Fq(k)799 811 y Fu(\()p FE(g)s(;)14 b(h)p Fu(\))p FE(;)g(f)9 b Fs(i)1110 823 y Fq(v)1172 811 y Fu(=)1260 698 y Fp(Z)g(Z)1413 811 y(c)-84 b FE(a)1456 823 y Fq(k)1497 811 y Fu(\()p FE(\030)23 b Fs(\000)18 b FE(\021)s Fu(\))23 b(:)g FE(\030)g Fs(\012)18 b Fu(\()p FE(\030)23 b Fs(\000)18 b Fu(2)p FE(\021)s Fu(\))s(^)-45 b FE(g)s Fu(\()p FE(\030)22 b Fs(\000)c FE(\021)s Fu(\))2543 789 y(^)2542 811 y FE(h)q Fu(\()p FE(\021)s Fu(\))p 2699 721 155 4 v 2717 789 a(^)2699 811 y FE(f)9 b Fu(\()p FE(\030)t Fu(\))q FE(d\021)s(d\030)t(;)0 984 y Fu(w)n(e)27 b(infer)h(that)212 1119 y SDict begin H.S end 212 1119 a 212 1119 a SDict begin 12 H.A end 212 1119 a 212 1119 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.9) cvn /DEST pdfmark end 212 1119 a 166 w Fs(h)p FE(Q)476 1131 y Fq(k)516 1119 y Fu(\()p FE(g)s(;)14 b(h)p Fu(\))p FE(;)g(f)9 b Fs(i)827 1131 y Fq(v)890 1119 y Fu(=)1024 1041 y Fp(X)977 1219 y Fe(p)p Fq(;)p Fe(l)p FD(\025\000)p Fr(1)1190 1119 y Fs(h)p FE(Q)1288 1131 y Fq(k)1328 1119 y Fu(\()p Ff(F)1411 1131 y Fe(p)1450 1119 y FE(g)s(;)14 b Ff(F)1581 1131 y Fe(l)1606 1119 y FE(h)p Fu(\))p FE(;)g(f)9 b Fs(i)1805 1131 y Fq(v)1867 1119 y Fu(=)2017 1041 y Fp(X)1955 1219 y Fe(l)p FD(\024)p Fe(p)p FD(\000)p Fq(N)2168 1227 y FC(0)2199 1119 y Fs(h)p FE(Q)2297 1131 y Fq(k)2338 1119 y Fu(\()p Ff(F)2421 1131 y Fe(p)2460 1119 y FE(g)s(;)14 b Ff(F)2591 1131 y Fe(l)2616 1119 y FE(h)p Fu(\))p FE(;)2737 1098 y Fu(~)2733 1119 y Ff(F)2784 1131 y Fe(p)2822 1119 y FE(f)9 b Fs(i)2904 1131 y Fq(v)378 1359 y Fu(+)475 1280 y Fp(X)457 1459 y Fe(l)p FD(\025\000)p Fr(1)614 1359 y Fs(h)p FE(Q)712 1371 y Fq(k)753 1359 y Fu(\()p Fs(S)835 1371 y Fe(l)p FD(\000)p Fq(N)961 1379 y FC(0)997 1359 y FE(g)s(;)14 b Ff(F)1128 1371 y Fe(l)1153 1359 y FE(h)p Fu(\))p FE(;)1274 1338 y Fu(~)1270 1359 y Ff(F)1321 1371 y Fe(l)1345 1359 y FE(f)9 b Fs(i)1427 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SDict begin 12 H.A end 289 718 a 289 718 a SDict begin [/View [/XYZ H.V]/Dest (prop.2.2) cvn /DEST pdfmark end 289 718 a 0 TeXcolorgray 0 818 a FF(Prop)s(osition)j(2.2.)p 0 TeXcolorgray 0 TeXcolorgray 177 w Fu(\(i\).)p 0 TeXcolorgray 42 w Fj(L)l(et)e FE(a)1181 830 y Fr(2)1218 818 y FE(;)14 b(b)1291 830 y Fr(2)1351 818 y Fs(2)23 b Fu([0)p FE(;)14 b Fu(1])p FE(;)g(!)1685 830 y Fq(i)1734 818 y Fs(2)24 b Fo(R)6 b FE(;)14 b(i)22 b Fu(=)h(1)14 b Fs(\001)g(\001)g(\001)27 b FE(;)14 b Fu(4)p FE(;)g Ft(s)22 b FE(>)2497 785 y Fr(1)p 2497 799 34 4 v 2497 846 a(2)2568 818 y Fj(verifying)30 b(that)e FE(a)3121 830 y Fr(2)3173 818 y Fu(+)14 b FE(b)3288 830 y Fr(2)3348 818 y Fu(=)23 b(1)p FE(;)14 b(!)3567 830 y Fr(1)3618 818 y Fu(+)g FE(!)3749 830 y Fr(2)3809 818 y Fu(=)289 917 y Fs(\000)p Fu(3)p FE(;)g(!)485 929 y Fr(3)539 917 y Fu(+)k FE(!)674 929 y Fr(4)734 917 y Fu(=)23 b Fs(\000)p Fu(2)p Fj(.)37 b(Then)97 1103 y SDict begin H.S end 97 1103 a 97 1103 a SDict begin 12 H.A end 97 1103 a 97 1103 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.12) cvn /DEST pdfmark end 97 1103 a Fs(jh)p FE(Q)p Fu(\()p FE(g)s(;)14 b(h)p Fu(\))p FE(;)g(f)9 b Fs(i)529 1115 y Fq(v)568 1103 y Fs(j)83 b Fk(.)739 1115 y Fh(s)850 986 y Fp(\022)911 1103 y Fs(k)p FE(g)s Fs(k)1038 1121 y Fq(L)1084 1101 y FC(1)1084 1139 y(3)1137 1103 y Fu(+)18 b Fs(k)p FE(g)s Fs(k)1347 1160 y Fq(H)1405 1117 y Fm(\000)1459 1101 y FC(1)p 1459 1108 29 3 v 1459 1139 a(2)1498 1117 y Fl(;)p Fg(s)1401 1178 y FC(3)1542 986 y Fp(\023)1617 1011 y(\020)1666 1103 y Fs(k)c Fu(\()p Fs(\000)p Fu(\001)1888 1118 y Fn(S)1926 1101 y FC(2)1957 1103 y Fu(\))1999 1039 y FC(1)p 1999 1048 V 1999 1081 a(2)2055 1103 y FE(h)p Fs(k)2145 1118 y Fq(L)2191 1101 y FC(2)2191 1135 y Fl(!)2228 1147 y FC(1)2286 1103 y Fu(+)k Fs(k)p FE(h)p Fs(k)2501 1118 y Fq(H)2559 1101 y FC(1)2555 1135 y Fl(!)2592 1147 y FC(1)2632 1011 y Fp(\021)13 b(\020)2745 1103 y Fs(k)h Fu(\()o Fs(\000)p Fu(\001)2966 1118 y Fn(S)3004 1101 y FC(2)3035 1103 y Fu(\))3077 1039 y FC(1)p 3077 1048 V 3077 1081 a(2)3133 1103 y FE(f)9 b Fs(k)3225 1118 y Fq(L)3271 1101 y FC(2)3271 1135 y Fl(!)3308 1147 y FC(2)3366 1103 y Fu(+)18 b Fs(k)p FE(f)9 b Fs(k)3583 1118 y Fq(H)3641 1101 y FC(1)3637 1135 y Fl(!)3674 1147 y FC(2)3713 1011 y Fp(\021)850 1278 y Fu(+)p Fs(k)p FE(g)s Fs(k)1042 1293 y Fq(L)1088 1276 y FC(1)1122 1278 y Fs(k)p FE(h)p Fs(k)1254 1299 y Fq(H)1312 1267 y Fl(a)1344 1279 y FC(2)1308 1308 y Fl(!)1345 1320 y FC(3)1385 1278 y Fs(k)p FE(f)g Fs(k)1519 1310 y Fq(H)1577 1278 y Fl(b)1603 1290 y FC(2)1573 1319 y Fl(!)1610 1331 y FC(4)1649 1278 y FE(:)p 0 TeXcolorgray 1990 w Fu(\(2.12\))p 0 TeXcolorgray 0 TeXcolorgray 113 1451 a(\(ii\).)p 0 TeXcolorgray 43 w Fj(L)l(et)29 b FE(a)476 1463 y Fr(1)513 1451 y FE(;)14 b(b)586 1463 y Fr(1)646 1451 y Fs(2)23 b Fu([)757 1418 y Fr(1)p 757 1432 34 4 v 757 1480 a(2)800 1451 y FE(;)847 1418 y Fr(3)p 847 1432 V 847 1480 a(2)890 1451 y Fu(])p FE(;)14 b(!)1002 1463 y Fr(1)1039 1451 y FE(;)g(!)1128 1463 y Fr(2)1188 1451 y Fs(2)24 b Fo(R)6 b FE(;)14 b Ft(s)22 b FE(>)1516 1418 y Fr(1)p 1516 1432 V 1516 1480 a(2)1589 1451 y Fj(verifying)31 b(that)f FE(a)2145 1463 y Fr(1)2201 1451 y Fu(+)18 b FE(b)2320 1463 y Fr(1)2380 1451 y Fu(=)k(2)p FE(;)14 b(!)2598 1463 y Fr(1)2653 1451 y Fu(+)k FE(!)2788 1463 y Fr(2)2848 1451 y Fu(=)23 b Fs(\000)p Fu(1)p Fj(.)37 b(Then)879 1649 y SDict begin H.S end 879 1649 a 879 1649 a SDict begin 12 H.A end 879 1649 a 879 1649 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.13) cvn /DEST pdfmark end 879 1649 a Fs(jh)p FE(Q)p Fu(\()p FE(g)s(;)14 b(h)p Fu(\))p FE(;)g(f)9 b Fs(i)1311 1661 y Fq(v)1350 1649 y Fs(j)23 b Fk(.)1461 1661 y Fh(s)1512 1532 y Fp(\022)1573 1649 y Fs(k)p FE(g)s Fs(k)1700 1668 y Fq(L)1746 1648 y FC(1)1746 1686 y(1)1799 1649 y Fu(+)18 b Fs(k)p FE(g)s Fs(k)2009 1706 y Fq(H)2067 1664 y Fm(\000)2121 1648 y FC(1)p 2121 1655 29 3 v 2121 1686 a(2)2160 1664 y Fl(;)p Fg(s)2063 1724 y FC(1)2204 1532 y Fp(\023)2279 1649 y Fs(k)p FE(h)p Fs(k)2411 1671 y Fq(H)2469 1639 y Fl(a)2501 1651 y FC(1)2465 1680 y Fl(!)2502 1692 y FC(1)2541 1649 y Fs(k)p FE(f)9 b Fs(k)2675 1682 y Fq(H)2733 1650 y Fl(b)2759 1662 y FC(1)2729 1691 y Fl(!)2766 1703 y FC(2)2806 1649 y FE(:)p 0 TeXcolorgray 833 w Fu(\(2.13\))p 0 TeXcolorgray 0 TeXcolorgray 0 1852 a Fj(Pr)l(o)l(of.)p 0 TeXcolorgray 43 w Fu(Due)28 b(to)g(the)g(decomp)r(osition)f(\()p 1 0 0 TeXcolorrgb 1258 1852 a SDict begin H.S end 1258 1852 a 1 0 0 TeXcolorrgb Fu(2.8)p 1 0 0 TeXcolorrgb 1364 1799 a SDict begin H.R end 1364 1799 a 1364 1852 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.8) cvn H.B /ANN pdfmark end 1364 1852 a 0 TeXcolorgray Fu(\))h(and)f(\()p 1 0 0 TeXcolorrgb 1617 1852 a SDict begin H.S end 1617 1852 a 1 0 0 TeXcolorrgb Fu(2.9)p 1 0 0 TeXcolorrgb 1724 1799 a SDict begin H.R end 1724 1799 a 1724 1852 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.9) cvn H.B /ANN pdfmark end 1724 1852 a 0 TeXcolorgray Fu(\),)h(w)n(e)f(split)h(it)g(in)n (to)f(t)n(w)n(o)g(steps.)100 1952 y(Step)h(1:)36 b(Preliminary)26 b(estimates)h(of)h Fs(h)p FE(Q)1404 1964 y Fq(k)1445 1952 y Fu(\()p FE(g)s(;)14 b(h)p Fu(\))p FE(;)g(f)9 b Fs(i)1756 1964 y Fq(v)1823 1952 y Fu(with)28 b FE(k)e Fs(\025)c Fu(0.)p 100 1986 2134 4 v 27 w(Thanks)27 b(to)h(\()p 1 0 0 TeXcolorrgb 2692 1952 a SDict begin H.S end 2692 1952 a 1 0 0 TeXcolorrgb Fu(2.6)p 1 0 0 TeXcolorrgb 2798 1898 a SDict begin H.R end 2798 1898 a 2798 1952 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.6) cvn H.B /ANN pdfmark end 2798 1952 a 0 TeXcolorgray Fu(\),)g(one)f(ma)n(y)g(ha)n(v)n(e)g FE(a)3449 1964 y Fr(0)3509 1952 y Fs(2)c Fi(S)44 b Fu(and)651 2113 y FE(a)695 2125 y Fq(k)735 2113 y Fu(\()p FE(z)t Fu(\))23 b(=)g(2)995 2079 y FD(\000)p Fq(k)1087 2113 y FE(a)1131 2125 y Fr(0)1168 2113 y Fu(\(2)1242 2079 y FD(\000)p Fq(k)1335 2113 y FE(z)t Fu(\))p FE(;)96 b(b)1565 2125 y Fq(k)1606 2113 y Fu(\()p FE(z)t Fu(\))23 b(=)f(2)1865 2079 y FD(\000)p Fr(2)p Fq(k)1991 2113 y FE(b)2027 2125 y Fr(0)2063 2113 y Fu(\(2)2137 2079 y FD(\000)p Fq(k)2230 2113 y FE(z)t Fu(\))p FE(;)96 b(c)2460 2125 y Fq(k)2501 2113 y Fu(\()p FE(z)t Fu(\))23 b(=)g(2)2761 2079 y FD(\000)p Fr(3)p Fq(k)2886 2113 y FE(c)2922 2125 y Fr(0)2959 2113 y Fu(\(2)3033 2079 y FD(\000)p Fq(k)3126 2113 y FE(z)t Fu(\))p FE(;)651 2255 y Fp(c)-83 b FE(a)695 2267 y Fq(k)735 2255 y Fu(\()p FE(\030)t Fu(\))24 b(=)f(2)993 2221 y Fr(2)p Fq(k)1084 2255 y Fp(b)-64 b FE(a)1110 2267 y Fr(0)1147 2255 y Fu(\(2)1221 2221 y Fq(k)1262 2255 y FE(\030)t Fu(\))p FE(;)97 b Ff(F)1505 2267 y Fq(p)1543 2255 y FE(a)1587 2267 y Fq(k)1628 2255 y Fu(\()p FE(z)t Fu(\))23 b(=)g(2)1888 2221 y FD(\000)p Fq(k)1980 2255 y Fu(\()p Ff(F)2063 2267 y Fq(p)p Fr(+)p Fq(k)2189 2255 y FE(a)2233 2267 y Fr(0)2270 2255 y Fu(\)\(2)2376 2221 y FD(\000)p Fq(k)2469 2255 y FE(z)t Fu(\))p FE(;)96 b Fu(for)27 b FE(p)c Fs(\025)g Fu(0)p FE(:)0 2396 y Fu(Then)28 b(for)f(an)n(y)g FE(N)k Fs(2)24 b Fo(N)10 b Fu(,)28 b(it)g(holds)f(that)385 2536 y SDict begin H.S end 385 2536 a 385 2536 a SDict begin 12 H.A end 385 2536 a 385 2536 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.14) cvn /DEST pdfmark end 385 2536 a 166 w Fs(k)p FE(a)637 2548 y Fq(k)678 2536 y Fs(k)720 2548 y Fq(L)766 2532 y Fm(1)852 2536 y Fk(.)c Fu(2)982 2502 y FD(\000)p Fq(k)1074 2536 y FE(;)97 b Fs(k)p Ff(F)1287 2548 y Fq(p)1325 2536 y FE(a)1369 2548 y Fq(k)1410 2536 y Fs(k)1452 2548 y Fq(L)1498 2532 y Fm(1)1585 2536 y Fs(\024)22 b Fu(2)1714 2502 y FD(\000)p Fq(k)1806 2536 y Fs(k)p Ff(F)1899 2548 y Fq(p)p Fr(+)p Fq(k)2025 2536 y FE(a)2069 2548 y Fr(0)2106 2536 y Fs(k)2148 2548 y Fq(L)2194 2532 y Fm(1)2281 2536 y Fk(.)2346 2548 y Fq(N)2431 2536 y Fu(2)2473 2502 y FD(\000)p Fq(k)2566 2536 y Fu(2)2608 2502 y FD(\000)p Fq(pN)2756 2536 y Fu(2)2798 2502 y FD(\000)p Fq(k)q(N)2949 2536 y Fu(;)551 2668 y Fs(k)p FE(b)629 2680 y Fq(k)669 2668 y Fs(k)711 2680 y Fq(L)757 2664 y Fm(1)844 2668 y Fk(.)h Fu(2)974 2634 y FD(\000)p Fr(2)p Fq(k)1099 2668 y FE(;)97 b Fs(k)p Ff(F)1312 2680 y Fq(p)1350 2668 y FE(b)1386 2680 y Fq(k)1426 2668 y Fs(k)1468 2680 y Fq(L)1514 2664 y Fm(1)1601 2668 y Fs(\024)23 b Fu(2)1731 2634 y FD(\000)p Fr(2)p Fq(k)1856 2668 y Fs(k)p Ff(F)1949 2680 y Fq(p)p Fr(+)p Fq(k)2074 2668 y FE(a)2118 2680 y Fr(0)2156 2668 y Fs(k)2198 2680 y Fq(L)2244 2664 y Fm(1)2330 2668 y Fk(.)2395 2680 y Fq(N)2481 2668 y Fu(2)2523 2634 y FD(\000)p Fq(k)2615 2668 y Fu(2)2657 2634 y FD(\000)p Fq(pN)2806 2668 y Fu(2)2848 2634 y FD(\000)p Fq(k)q(N)2999 2668 y Fu(;)551 2800 y Fs(k)p FE(c)629 2812 y Fq(k)670 2800 y Fs(k)712 2812 y Fq(L)758 2795 y Fm(1)845 2800 y Fk(.)f Fu(2)974 2765 y FD(\000)p Fr(3)p Fq(k)1099 2800 y FE(;)97 b Fs(k)p Ff(F)1312 2812 y Fq(p)1350 2800 y FE(c)1386 2812 y Fq(k)1427 2800 y Fs(k)1469 2812 y Fq(L)1515 2795 y Fm(1)1602 2800 y Fs(\024)22 b Fu(2)1731 2765 y FD(\000)p Fr(3)p Fq(k)1857 2800 y Fs(k)p Ff(F)1950 2812 y Fq(p)p Fr(+)p Fq(k)2075 2800 y FE(c)2111 2812 y Fr(0)2148 2800 y Fs(k)2190 2812 y Fq(L)2236 2795 y Fm(1)2323 2800 y Fk(.)2388 2812 y Fq(N)2474 2800 y Fu(2)2516 2765 y FD(\000)p Fr(3)p Fq(k)2641 2800 y Fu(2)2683 2765 y FD(\000)p Fq(pN)2832 2800 y Fu(2)2874 2765 y FD(\000)p Fq(k)q(N)3025 2800 y FE(;)97 b Fu(for)27 b FE(p)c Fs(\025)f Fu(0)p FE(:)p 0 TeXcolorgray 173 w Fu(\(2.14\))p 0 TeXcolorgray 0 2940 a Fs(\017)27 b Fj(Estimate)j(of)h Ff(M)603 2910 y Fr(1)640 2940 y Fj(.)38 b Fu(Thanks)27 b(to)g(\()p 1 0 0 TeXcolorrgb 1134 2940 a SDict begin H.S end 1134 2940 a 1 0 0 TeXcolorrgb Fu(2.10)p 1 0 0 TeXcolorrgb 1282 2887 a SDict begin H.R end 1282 2887 a 1282 2940 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.10) cvn H.B /ANN pdfmark end 1282 2940 a 0 TeXcolorgray Fu(\),)h(w)n(e)f(\014rst)h(ha)n(v)n (e)358 3123 y Ff(M)445 3089 y Fr(1)445 3144 y Fq(k)q(;)p Fe(p)p Fq(;)p Fe(l)581 3123 y Fu(\()p FE(g)s(;)14 b(h;)g(f)9 b Fu(\))23 b(=)970 3010 y Fp(Z)1053 3123 y Fu(\(\()1122 3101 y(~)1117 3123 y Ff(F)1168 3135 y Fe(p)1208 3123 y FE(a)1252 3135 y Fq(k)1292 3123 y Fu(\))c Fs(\003)f Fu(\()p Ff(F)1486 3135 y Fe(p)1525 3123 y FE(g)s Fu(\)\))23 b(:)g Fs(r)1770 3089 y Fr(2)1808 3123 y Fu(\()p Ff(F)1891 3135 y Fe(l)1916 3123 y FE(h)p Fu(\))2001 3101 y(~)1996 3123 y Ff(F)2047 3135 y Fe(p)2086 3123 y FE(f)9 b(dv)21 b Fs(\000)2323 3010 y Fp(Z)2406 3123 y Fu(\(\()2475 3101 y(~)2470 3123 y Ff(F)2521 3135 y Fe(p)2561 3123 y FE(c)2597 3135 y Fq(k)2638 3123 y Fu(\))d Fs(\003)g Fu(\()p Ff(F)2831 3135 y Fe(p)2870 3123 y FE(g)s Fu(\)\))p Ff(F)3028 3135 y Fe(l)3053 3123 y FE(h)3106 3101 y Fu(~)3101 3123 y Ff(F)3152 3135 y Fe(p)3191 3123 y FE(f)9 b(dv)s(:)0 3306 y Fu(Th)n(us)29 b(b)n(y)h(Cauc)n(h)n(y)f(inequalit)n(y)-7 b(,)30 b(Y)-7 b(oung's)29 b(inequalit)n(y)g(and)h(\()p 1 0 0 TeXcolorrgb 1933 3306 a SDict begin H.S end 1933 3306 a 1 0 0 TeXcolorrgb Fu(2.14)p 1 0 0 TeXcolorrgb 2081 3253 a SDict begin H.R end 2081 3253 a 2081 3306 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.14) cvn H.B /ANN pdfmark end 2081 3306 a 0 TeXcolorgray Fu(\),)g(together)f(with)i(the)f (fact)g(that)g Ff(l)d Fs(\024)f Ff(p)19 b Fs(\000)h FE(N)3535 3318 y Fr(0)3572 3306 y Fu(,)30 b(w)n(e)f(can)0 3406 y(deriv)n(e)e(that)68 3603 y SDict begin H.S end 68 3603 a 68 3603 a SDict begin 12 H.A end 68 3603 a 68 3603 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.15) cvn /DEST pdfmark end 68 3603 a 209 3461 a Fp(\014)209 3511 y(\014)237 3532 y Ff(M)324 3498 y Fr(1)324 3553 y Fq(k)q(;)p Fe(p)p Fq(;)p Fe(l)460 3532 y Fu(\()p FE(g)s(;)14 b(h;)g(f)9 b Fu(\))739 3461 y Fp(\014)739 3511 y(\014)789 3532 y Fs(\024)23 b(k)924 3510 y Fu(~)919 3532 y Ff(F)970 3544 y Fe(p)1008 3532 y FE(a)1052 3544 y Fq(k)1093 3532 y Fs(k)1135 3544 y Fq(L)1181 3528 y Fm(1)1245 3532 y Fs(k)p Ff(F)1338 3544 y Fe(p)1376 3532 y FE(g)s Fs(k)1461 3547 y Fq(L)1507 3530 y FC(1)1542 3532 y Fs(kr)1653 3498 y Fr(2)1690 3532 y Ff(F)1741 3544 y Fe(l)1766 3532 y FE(h)p Fs(k)1856 3547 y Fq(L)1902 3530 y FC(2)1937 3532 y Fs(k)1984 3510 y Fu(~)1979 3532 y Ff(F)2030 3544 y Fe(p)2069 3532 y FE(f)9 b Fs(k)2161 3547 y Fq(L)2207 3530 y FC(2)2260 3532 y Fu(+)18 b Fs(k)2390 3510 y Fu(~)2385 3532 y Ff(F)2436 3544 y Fe(p)2475 3532 y FE(c)2511 3544 y Fq(k)2551 3532 y Fs(k)2593 3544 y Fq(L)2639 3528 y Fm(1)2703 3532 y Fs(k)p Ff(F)2796 3544 y Fe(p)2835 3532 y FE(g)s Fs(k)2920 3547 y Fq(L)2966 3530 y FC(1)3001 3532 y Fs(k)p Ff(F)3094 3544 y Fe(l)3118 3532 y FE(h)p Fs(k)3208 3547 y Fq(L)3254 3530 y FC(2)3290 3532 y Fs(k)3337 3510 y Fu(~)3332 3532 y Ff(F)3383 3544 y Fe(p)3421 3532 y FE(f)9 b Fs(k)3513 3547 y Fq(L)3559 3530 y FC(2)68 3676 y Fk(.)133 3688 y Fq(N)195 3676 y Fu(2)237 3642 y FD(\000)p Fr(2)p Fq(k)q(N)422 3676 y Fu(2)464 3642 y FD(\000)p Fr(2)p Fe(p)p Fq(N)646 3676 y Fs(k)p Ff(F)739 3688 y Fe(p)777 3676 y FE(g)s Fs(k)862 3691 y Fq(L)908 3674 y FC(1)943 3676 y Fs(k)p Ff(F)1036 3688 y Fe(l)1061 3676 y FE(h)p Fs(k)1151 3691 y Fq(L)1197 3674 y FC(2)1232 3676 y Fs(k)1279 3655 y Fu(~)1274 3676 y Ff(F)1325 3688 y Fe(p)1363 3676 y FE(f)g Fs(k)1455 3691 y Fq(L)1501 3674 y FC(2)1537 3676 y Fu(2)1579 3642 y Fr(2)p Fe(l)1659 3676 y Fk(.)1724 3688 y Fq(N)1810 3676 y Fu(2)1852 3642 y FD(\000)p Fq(k)q(N)2003 3676 y Fu(2)2045 3642 y FD(\000)p Fe(p)p Fq(N)2194 3676 y Fu(2)2236 3642 y FD(\000)p Fe(l)p Fq(N)2372 3676 y Fs(k)p FE(g)s Fs(k)2499 3691 y Fq(L)2545 3674 y FC(1)2579 3676 y Fs(k)p Ff(F)2672 3688 y Fe(l)2697 3676 y FE(h)p Fs(k)2787 3691 y Fq(L)2833 3674 y FC(2)2868 3676 y Fs(k)2915 3655 y Fu(~)2910 3676 y Ff(F)2961 3688 y Fe(p)2999 3676 y FE(f)g Fs(k)3091 3691 y Fq(L)3137 3674 y FC(2)3173 3676 y FE(:)3662 3603 y Fu(\(2.15\))0 3817 y(Th)n(us)27 b(w)n(e)h(can)f (obtain)g(that)577 3958 y SDict begin H.S end 577 3958 a 577 3958 a SDict begin 12 H.A end 577 3958 a 577 3958 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.16) cvn /DEST pdfmark end 577 3958 a 639 3879 a Fp(X)577 4058 y Fe(l)p FD(\024)p Fe(p)p FD(\000)p Fq(N)790 4066 y FC(0)835 3958 y Fs(j)p Ff(M)945 3924 y Fr(1)945 3978 y Fq(k)q(;)p Fe(p)p Fq(;)p Fe(l)1081 3958 y Fu(\()p FE(g)s(;)14 b(h;)g(f)9 b Fu(\))p Fs(j)83 b Fk(.)1531 3970 y Fq(N)1739 3879 y Fp(X)1676 4058 y Fe(l)p FD(\024)p Fe(p)p FD(\000)p Fq(N)1889 4066 y FC(0)1935 3958 y Fu(2)1977 3924 y FD(\000)p Fq(k)q(N)2128 3958 y Fu(2)2170 3924 y FD(\000)p Fe(p)p Fq(N)2319 3958 y Fu(2)2361 3924 y FD(\000)p Fe(l)p Fq(N)2496 3958 y Fs(k)p FE(g)s Fs(k)2623 3973 y Fq(L)2669 3956 y FC(1)2704 3958 y Fs(k)p Ff(F)2797 3970 y Fe(l)2821 3958 y FE(h)p Fs(k)2911 3973 y Fq(L)2957 3956 y FC(2)2993 3958 y Fs(k)3040 3936 y Fu(~)3035 3958 y Ff(F)3086 3970 y Fe(p)3124 3958 y FE(f)9 b Fs(k)3216 3973 y Fq(L)3262 3956 y FC(2)1466 4185 y Fk(.)1531 4197 y Fq(N)1676 4185 y Fu(2)1718 4150 y FD(\000)p Fq(k)q(N)1869 4185 y Fs(k)p FE(g)s Fs(k)1996 4200 y Fq(L)2042 4183 y FC(1)2077 4185 y Fs(k)p FE(h)p Fs(k)2209 4200 y Fq(L)2255 4183 y FC(2)2290 4185 y Fs(k)p FE(f)g Fs(k)2424 4200 y Fq(L)2470 4183 y FC(2)2505 4185 y FE(:)p 0 TeXcolorgray 1134 w Fu(\(2.16\))p 0 TeXcolorgray 0 4325 a Fs(\017)27 b Fj(Estimate)j(of)h Ff(M)603 4295 y Fr(4)640 4325 y Fj(.)38 b Fu(By)27 b(the)h(similar)f(argumen)n(t)f (as)h Ff(M)1809 4295 y Fr(1)1847 4325 y Fu(,)g(w)n(e)h(can)f(get)g (that)750 4468 y SDict begin H.S end 750 4468 a 750 4468 a SDict begin 12 H.A end 750 4468 a 750 4468 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.17) cvn /DEST pdfmark end 750 4468 a -71 x Fp(\014)750 4447 y(\014)778 4468 y Ff(M)865 4433 y Fr(4)865 4488 y Fq(k)q(;)p Fe(p)p Fq(;)p Fe(m)1031 4468 y Fu(\()p FE(g)s(;)14 b(h;)g(f)9 b Fu(\))1310 4397 y Fp(\014)1310 4447 y(\014)1361 4468 y Fk(.)1426 4480 y Fq(N)1511 4468 y Fu(2)1553 4433 y FD(\000)p Fq(k)q(N)1704 4468 y Fu(2)1746 4433 y FD(\000)p Fe(p)p Fq(N)1895 4468 y Fu(2)1937 4433 y FD(\000)p Fe(m)p Fq(N)2103 4468 y Fs(k)p FE(g)s Fs(k)2230 4483 y Fq(L)2276 4466 y FC(1)2311 4468 y Fs(k)2358 4446 y Fu(~)2353 4468 y Ff(F)2404 4480 y Fe(p)2442 4468 y FE(h)p Fs(k)2532 4483 y Fq(L)2578 4466 y FC(2)2614 4468 y Fs(k)p Ff(F)2707 4480 y Fe(m)2762 4468 y FE(f)g Fs(k)2854 4483 y Fq(L)2900 4466 y FC(2)2935 4468 y FE(:)p 0 TeXcolorgray 704 w Fu(\(2.17\))p 0 TeXcolorgray 0 4608 a(Moreo)n(v)n(er,)25 b(w)n(e)i(ha)n(v)n(e)908 4761 y SDict begin H.S end 908 4761 a 908 4761 a SDict begin 12 H.A end 908 4761 a 908 4761 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.18) cvn /DEST pdfmark end 908 4761 a 985 4682 a Fp(X)908 4860 y Fe(m)p FD(\024)p Fe(p)p FD(\000)p Fq(N)1152 4868 y FC(0)1197 4761 y Fs(j)p Ff(M)1307 4726 y Fr(4)1307 4781 y Fq(k)q(;)p Fe(p)p Fq(;)p Fe(m)1474 4761 y Fu(\()p FE(g)s(;)14 b(h;)g(f)9 b Fu(\))p Fs(j)22 b Fk(.)1863 4773 y Fq(N)1949 4761 y Fu(2)1991 4726 y FD(\000)p Fq(k)q(N)2142 4761 y Fs(k)p FE(g)s Fs(k)2269 4776 y Fq(L)2315 4759 y FC(1)2350 4761 y Fs(k)p FE(h)p Fs(k)2482 4776 y Fq(L)2528 4759 y FC(2)2562 4761 y Fs(k)p FE(f)9 b Fs(k)2696 4776 y Fq(L)2742 4759 y FC(2)2777 4761 y FE(:)p 0 TeXcolorgray 862 w Fu(\(2.18\))p 0 TeXcolorgray 0 4996 a Fs(\017)27 b Fj(Estimates)j(of)h Ff(M)637 4966 y Fr(2)704 4996 y Fj(and)f Ff(M)952 4966 y Fr(3)990 4996 y Fj(.)37 b Fu(F)-7 b(rom)27 b(\()p 1 0 0 TeXcolorrgb 1302 4996 a SDict begin H.S end 1302 4996 a 1 0 0 TeXcolorrgb Fu(2.10)p 1 0 0 TeXcolorrgb 1450 4943 a SDict begin H.R end 1450 4943 a 1450 4996 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.10) cvn H.B /ANN pdfmark end 1450 4996 a 0 TeXcolorgray Fu(\))h(and)f(\()p 1 0 0 TeXcolorrgb 1703 4996 a SDict begin H.S end 1703 4996 a 1 0 0 TeXcolorrgb Fu(2.14)p 1 0 0 TeXcolorrgb 1851 4943 a SDict begin H.R end 1851 4943 a 1851 4996 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.14) cvn H.B /ANN pdfmark end 1851 4996 a 0 TeXcolorgray Fu(\))q(,)g(w)n(e)g(can)h(deduce)f(that)1030 5143 y SDict begin H.S end 1030 5143 a 1030 5143 a SDict begin 12 H.A end 1030 5143 a 1030 5143 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.19) cvn /DEST pdfmark end 1030 5143 a Fs(j)p Ff(M)1140 5109 y Fr(2)1140 5164 y Fq(k)q(;)p Fe(l)1222 5143 y Fu(\()p FE(g)s(;)14 b(h;)g(f)9 b Fu(\))p Fs(j)22 b Fk(.)h Fu(2)1676 5109 y FD(\000)p Fq(k)1768 5143 y Fs(k)p FE(g)s Fs(k)1895 5158 y Fq(L)1941 5141 y FC(1)1976 5143 y Fs(k)p Ff(F)2069 5155 y Fe(l)2093 5143 y FE(h)p Fs(k)2183 5158 y Fq(L)2229 5141 y FC(2)2265 5143 y Fs(k)2312 5122 y Fu(~)2307 5143 y Ff(F)2358 5155 y Fe(l)2382 5143 y FE(f)9 b Fs(k)2474 5158 y Fq(L)2520 5141 y FC(2)2555 5143 y Fu(2)2597 5109 y Fr(2)p Fe(l)2655 5143 y FE(:)p 0 TeXcolorgray 984 w Fu(\(2.19\))p 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 12 12 TeXDict begin HPSdict begin 12 11 bop 0 TeXcolorgray 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 0 TeXcolorgray -8 -245 a SDict begin H.S end -8 -245 a -8 -245 a SDict begin H.R end -8 -245 a -8 -245 a SDict begin [/View [/XYZ H.V]/Dest (page.12) cvn /DEST pdfmark end -8 -245 a 0 TeXcolorgray 0 TeXcolorgray 0 -170 a Fr(12)1263 b(LING-BING)24 b(HE,)e(JIE)g(JI,)g(AND)g(YUE)g(LUO) p 0 TeXcolorgray 0 29 a Fu(Similarly)-7 b(,)27 b(for)g Ff(M)583 -1 y Fr(3)621 29 y Fu(,)g(one)h(has)749 255 y SDict begin H.S end 749 255 a 749 255 a SDict begin 12 H.A end 749 255 a 749 255 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.20) cvn /DEST pdfmark end 749 255 a -71 x Fp(\014)749 234 y(\014)776 255 y Ff(M)863 220 y Fr(3)863 275 y Fq(k)q(;)p Fe(p)959 255 y Fu(\()p FE(g)s(;)14 b(h;)g(f)9 b Fu(\))1238 184 y Fp(\014)1238 234 y(\014)1348 255 y Fk(.)1413 267 y Fq(N)1559 113 y Fp(\()1626 183 y Fu(2)1668 149 y FD(\000)p Fq(k)1760 183 y Fs(k)p FE(g)s Fs(k)1887 198 y Fq(L)1933 181 y FC(1)1967 183 y Fs(k)2014 162 y Fu(~)2009 183 y Ff(F)2060 195 y FD(\000)p Fr(1)2149 183 y FE(h)p Fs(k)2239 198 y Fq(L)2285 181 y FC(2)2320 183 y Fs(k)2367 162 y Fu(~)2362 183 y Ff(F)2413 195 y FD(\000)p Fr(1)2502 183 y FE(f)g Fs(k)2594 198 y Fq(L)2640 181 y FC(2)2758 183 y Ff(p)23 b Fu(=)f Fs(\000)p Fu(1)1626 328 y(2)1668 293 y FD(\000)p Fq(k)q(N)1819 328 y Fu(2)1861 293 y FD(\000)p Fe(p)p Fq(N)2010 328 y Fs(k)p FE(g)s Fs(k)2137 343 y Fq(L)2183 326 y FC(1)2218 328 y Fs(k)2265 306 y Fu(~)2260 328 y Ff(F)2311 340 y Fe(p)2349 328 y FE(h)p Fs(k)2439 343 y Fq(L)2485 326 y FC(2)2520 328 y Fs(k)2567 306 y Fu(~)2562 328 y Ff(F)2613 340 y Fe(p)2651 328 y FE(f)9 b Fs(k)2743 343 y Fq(L)2789 326 y FC(2)2908 328 y Ff(p)22 b Fs(\025)h Fu(0)p 0 TeXcolorgray 3662 255 a(\(2.20\))p 0 TeXcolorgray 1348 472 a Fk(.)1413 484 y Fq(N)1559 472 y Fu(2)1601 438 y FD(\000)p Fq(k)1693 472 y Fs(k)p FE(g)s Fs(k)1820 487 y Fq(L)1866 470 y FC(1)1901 472 y Fs(k)1948 450 y Fu(~)1943 472 y Ff(F)1994 484 y Fe(p)2032 472 y FE(h)p Fs(k)2122 487 y Fq(L)2168 470 y FC(2)2203 472 y Fs(k)2250 450 y Fu(~)2245 472 y Ff(F)2296 484 y Fe(p)2335 472 y FE(f)9 b Fs(k)2427 487 y Fq(L)2473 470 y FC(2)2508 472 y Fu(2)2550 438 y FD(\000)p Fe(p)p Fq(N)2699 472 y FE(:)0 624 y Fu(Th)n(us)27 b(com)n(bining)h(\()p 1 0 0 TeXcolorrgb 642 624 a SDict begin H.S end 642 624 a 1 0 0 TeXcolorrgb Fu(2.19)p 1 0 0 TeXcolorrgb 790 570 a SDict begin H.R end 790 570 a 790 624 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.19) cvn H.B /ANN pdfmark end 790 624 a 0 TeXcolorgray Fu(\))g(and)g(\()p 1 0 0 TeXcolorrgb 1044 624 a SDict begin H.S end 1044 624 a 1 0 0 TeXcolorrgb Fu(2.20)p 1 0 0 TeXcolorrgb 1192 570 a SDict begin H.R end 1192 570 a 1192 624 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.20) cvn H.B /ANN pdfmark end 1192 624 a 0 TeXcolorgray Fu(\),)g(w)n(e)f(ha)n(v)n(e)638 792 y SDict begin H.S end 638 792 a 638 792 a SDict begin 12 H.A end 638 792 a 638 792 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.21) cvn /DEST pdfmark end 638 792 a 657 713 a Fp(X)638 892 y Fe(l)p FD(\025\000)p Fr(1)809 792 y Fs(j)p Ff(M)919 758 y Fr(2)919 813 y Fq(k)q(;)p Fe(l)1001 792 y Fu(\()p 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Fu(,)g(w)n(e)h(observ)n(e)d(that) 358 1323 y Ff(M)445 1289 y Fr(2)445 1344 y Fq(k)q(;)p Fe(l)526 1323 y Fu(\()p FE(g)s(;)14 b(h;)g(f)9 b Fu(\))23 b(=)f Fs(\000)994 1210 y Fp(Z)1077 1323 y Fu(\()p FE(a)1153 1335 y Fq(k)1212 1323 y Fs(\003)c(S)1322 1335 y Fe(l)p FD(\000)p Fq(N)1448 1343 y FC(0)1484 1323 y FE(g)s Fu(\))23 b(:)g Fs(r)p Ff(F)1748 1335 y Fe(l)1773 1323 y FE(h)c Fs(\012)f(r)1997 1301 y Fu(~)1992 1323 y Ff(F)2043 1335 y Fe(l)2068 1323 y FE(f)9 b(dv)21 b Fu(+)2305 1210 y Fp(Z)2388 1323 y Fu(\()p FE(b)2456 1335 y Fq(k)2516 1323 y Fs(\003)d(S)2626 1335 y Fe(l)p FD(\000)p Fq(N)2752 1343 y FC(0)2788 1323 y FE(g)s Fu(\))g Fs(\001)h Ff(F)2974 1335 y Fe(l)2998 1323 y FE(h)p Fs(r)3120 1301 y Fu(~)3115 1323 y Ff(F)3166 1335 y Fe(l)3191 1323 y FE(f)9 b(dv)s(:)0 1513 y Fu(Then)28 b(one)f(can)g(infer)h(that)197 1573 y Fp(\014)197 1623 y(\014)197 1673 y(\014)197 1723 y(\014)224 1581 y(Z)307 1694 y Fu(\()p FE(a)383 1706 y Fq(k)443 1694 y Fs(\003)18 b(S)553 1706 y Fe(l)p FD(\000)p Fq(N)679 1714 y FC(0)715 1694 y FE(g)s Fu(\))23 b(:)g Fs(r)p Ff(F)979 1706 y Fe(l)1004 1694 y FE(h)18 b Fs(\012)g(r)1227 1672 y Fu(~)1222 1694 y Ff(F)1273 1706 y Fe(l)1298 1694 y FE(f)9 b(dv)1434 1573 y Fp(\014)1434 1623 y(\014)1434 1673 y(\014)1434 1723 y(\014)118 1926 y Fu(=)197 1806 y Fp(\014)197 1856 y(\014)197 1905 y(\014)197 1955 y(\014)224 1813 y(Z)h(Z)376 1926 y Fs(j)p FE(v)22 b Fs(\000)c FE(v)584 1938 y FD(\003)623 1926 y Fs(j)646 1892 y FD(\000)p Fr(3)735 1926 y FE(')p Fu(\(2)863 1892 y FD(\000)p Fq(k)956 1926 y Fu(\()p FE(v)k Fs(\000)c FE(v)1173 1938 y FD(\003)1211 1926 y Fu(\)\))c(\()q Fs(S)1372 1938 y Fe(l)p FD(\000)p Fq(N)1498 1946 y FC(0)1534 1926 y FE(g)s Fu(\))1609 1951 y FD(\003)1661 1926 y Fu(\(\()p FE(v)22 b Fs(\000)c FE(v)1910 1938 y FD(\003)1948 1926 y Fu(\))h Fs(\002)f(r)p Ff(F)2202 1938 y Fe(l)2227 1926 y FE(h)p Fu(\))h Fs(\001)2367 1834 y Fp(\020)2417 1926 y Fu(\()p FE(v)j Fs(\000)c FE(v)2634 1938 y FD(\003)2672 1926 y Fu(\))h Fs(\002)f(r)2880 1905 y Fu(~)2875 1926 y Ff(F)2926 1938 y Fe(l)2951 1926 y FE(f)3001 1834 y Fp(\021)3064 1926 y FE(dv)3147 1938 y FD(\003)3186 1926 y FE(dv)3272 1806 y Fp(\014)3272 1856 y(\014)3272 1905 y(\014)3272 1955 y(\014)118 2182 y Fk(.)p Fu(2)225 2147 y FD(\000)p Fr(3)p Fq(k)364 2065 y Fp(\022)425 2069 y(Z)9 b(Z)577 2182 y Fs(j)p Fu(\()q Fs(S)683 2194 y Fe(l)p FD(\000)p Fq(N)809 2202 y FC(0)845 2182 y FE(g)s Fu(\))920 2207 y FD(\003)958 2182 y Fs(j)14 b(j)p Fu(\()p FE(v)22 b Fs(\000)c FE(v)1235 2194 y FD(\003)1273 2182 y Fu(\))h Fs(\002)f(r)p Ff(F)1527 2194 y Fe(l)1552 2182 y FE(h)p Fs(j)1623 2140 y Fr(2)1674 2182 y FE(dv)1757 2194 y FD(\003)1796 2182 y FE(dv)1882 2065 y Fp(\023)1943 2082 y Fr(1)p Fq(=)p Fr(2)2061 2065 y Fp(\022)2122 2069 y(Z)10 b(Z)2275 2182 y Fs(j)p Fu(\()p Fs(S)2380 2194 y Fe(l)p FD(\000)p Fq(N)2506 2202 y FC(0)2542 2182 y FE(g)s Fu(\))2617 2207 y FD(\003)2655 2182 y Fs(j)2692 2086 y Fp(\014)2692 2136 y(\014)2692 2186 y(\014)2720 2182 y Fu(\()p FE(v)22 b Fs(\000)c FE(v)2937 2194 y FD(\003)2975 2182 y Fu(\))h Fs(\002)f(r)3183 2160 y Fu(~)3178 2182 y Ff(F)3229 2194 y Fe(l)3254 2182 y FE(f)3304 2086 y Fp(\014)3304 2136 y(\014)3304 2186 y(\014)3331 2107 y Fr(2)3382 2182 y FE(dv)3465 2194 y FD(\003)3504 2182 y FE(dv)3590 2065 y Fp(\023)3652 2082 y Fr(1)p Fq(=)p Fr(2)118 2389 y Fk(.)p Fu(2)225 2355 y FD(\000)p Fr(3)p Fq(k)350 2389 y Fs(k)p FE(g)s Fs(k)477 2407 y Fq(L)523 2387 y FC(1)523 2425 y(2)572 2297 y Fp(\020)621 2389 y Fs(k)c Fu(\()p Fs(\000)p Fu(\001)843 2404 y Fn(S)881 2387 y FC(2)911 2389 y Fu(\))954 2325 y FC(1)p 954 2334 29 3 v 954 2368 a(2)1010 2389 y Ff(F)1061 2401 y Fe(l)1086 2389 y FE(h)p Fs(k)1176 2404 y Fq(L)1222 2387 y FC(2)1276 2389 y Fu(+)k Fs(k)p Ff(F)1452 2401 y Fe(l)1476 2389 y FE(h)p Fs(k)1566 2404 y Fq(L)1612 2387 y FC(2)1647 2389 y Fu(2)1689 2355 y Fe(l)1714 2297 y Fp(\021)13 b(\020)1827 2389 y Fs(k)h Fu(\()p Fs(\000)p Fu(\001)2049 2404 y Fn(S)2087 2387 y FC(2)2117 2389 y Fu(\))2160 2325 y FC(1)p 2160 2334 V 2160 2368 a(2)32 b Fu(~)2216 2389 y Ff(F)2267 2401 y Fe(l)2292 2389 y FE(f)9 b Fs(k)2384 2404 y Fq(L)2430 2387 y FC(2)2483 2389 y Fu(+)18 b Fs(k)2613 2368 y Fu(~)2608 2389 y Ff(F)2659 2401 y Fe(l)2684 2389 y FE(f)9 b Fs(k)2776 2404 y Fq(L)2822 2387 y FC(2)2857 2389 y Fu(2)2899 2355 y Fe(l)2923 2297 y Fp(\021)0 2561 y Fu(and)738 2622 y Fp(\014)738 2672 y(\014)738 2722 y(\014)738 2771 y(\014)766 2629 y(Z)849 2742 y Fu(\()p FE(b)917 2754 y Fq(k)976 2742 y Fs(\003)18 b(S)1086 2754 y Fe(l)p FD(\000)p Fq(N)1212 2762 y FC(0)1248 2742 y FE(g)s Fu(\))g Fs(\001)h Ff(F)1434 2754 y Fe(l)1459 2742 y FE(h)p Fs(r)1581 2721 y Fu(~)1576 2742 y Ff(F)1627 2754 y Fe(l)1652 2742 y FE(f)9 b(dv)1788 2622 y Fp(\014)1788 2672 y(\014)1788 2722 y(\014)1788 2771 y(\014)1838 2742 y Fk(.)23 b Fu(2)1968 2708 y FD(\000)p Fr(2)p Fq(k)2093 2742 y Fs(k)p FE(g)s Fs(k)2220 2757 y Fq(L)2266 2741 y FC(1)2301 2742 y Fs(k)p Ff(F)2394 2754 y Fe(l)2418 2742 y FE(h)p Fs(k)2508 2757 y Fq(L)2554 2741 y FC(2)2590 2742 y Fs(k)2637 2721 y Fu(~)2632 2742 y Ff(F)2683 2754 y Fe(l)2707 2742 y FE(f)9 b Fs(k)2799 2757 y Fq(L)2845 2741 y FC(2)2880 2742 y Fu(2)2922 2708 y Fe(l)2947 2742 y FE(:)0 2939 y Fu(These)27 b(imply)h(that)583 3112 y SDict begin H.S end 583 3112 a 583 3112 a SDict begin 12 H.A end 583 3112 a 583 3112 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.22) cvn /DEST pdfmark end 583 3112 a 602 3033 a Fp(X)583 3211 y Fe(l)p FD(\025\000)p Fr(1)755 3112 y Fs(j)p Ff(M)865 3077 y Fr(2)865 3132 y Fq(k)q(;)p Fe(l)946 3112 y Fu(\()p FE(g)s(;)14 b(h;)g(f)9 b Fu(\))p Fs(j)23 b Fk(.)1377 3033 y Fp(X)1359 3211 y Fe(l)p FD(\025\000)p Fr(1)1530 3112 y Fu(2)1572 3077 y FD(\000)p Fr(3)p Fq(k)1698 3112 y Fs(k)p FE(g)s Fs(k)1825 3130 y Fq(L)1871 3110 y FC(1)1871 3148 y(2)1919 3019 y Fp(\020)1969 3112 y Fs(k)14 b Fu(\()o Fs(\000)p Fu(\001)2190 3127 y Fn(S)2228 3110 y FC(2)2259 3112 y Fu(\))2301 3048 y FC(1)p 2301 3057 29 3 v 2301 3090 a(2)2357 3112 y Ff(F)2408 3124 y Fe(l)2433 3112 y FE(h)p Fs(k)2523 3127 y Fq(L)2569 3110 y FC(2)2623 3112 y Fu(+)k Fs(k)p Ff(F)2799 3124 y Fe(l)2823 3112 y FE(h)p Fs(k)2913 3127 y Fq(L)2959 3110 y FC(2)2995 3112 y Fu(2)3037 3077 y Fe(l)3062 3019 y Fp(\021)728 3359 y Fs(\002)807 3267 y Fp(\020)856 3359 y Fs(k)c Fu(\()p Fs(\000)p Fu(\001)1078 3374 y Fn(S)1116 3357 y FC(2)1147 3359 y Fu(\))1189 3295 y FC(1)p 1189 3304 V 1189 3337 a(2)33 b Fu(~)1245 3359 y Ff(F)1296 3371 y Fe(l)1321 3359 y FE(f)9 b Fs(k)1413 3374 y Fq(L)1459 3357 y FC(2)1513 3359 y Fu(+)18 b Fs(k)1643 3337 y Fu(~)1638 3359 y Ff(F)1689 3371 y Fe(l)1713 3359 y FE(f)9 b Fs(k)1805 3374 y Fq(L)1851 3357 y FC(2)1886 3359 y Fu(2)1928 3325 y Fe(l)1953 3267 y Fp(\021)2021 3359 y Fu(+)18 b(2)2146 3325 y FD(\000)p Fr(2)p Fq(k)2271 3359 y Fs(k)p FE(g)s Fs(k)2398 3374 y Fq(L)2444 3357 y FC(1)2479 3359 y Fs(k)p Ff(F)2572 3371 y Fe(l)2596 3359 y FE(h)p Fs(k)2686 3374 y Fq(L)2732 3357 y FC(2)2768 3359 y Fs(k)2815 3337 y Fu(~)2810 3359 y Ff(F)2861 3371 y Fe(l)2885 3359 y FE(f)9 b Fs(k)2977 3374 y Fq(L)3023 3357 y FC(2)3058 3359 y Fu(2)3100 3325 y Fe(l)620 3542 y Fk(.)23 b Fu(2)750 3507 y FD(\000)p Fr(3)p Fq(k)875 3542 y Fs(k)p FE(g)s Fs(k)1002 3560 y Fq(L)1048 3540 y FC(1)1048 3578 y(2)1097 3450 y Fp(\020)1146 3542 y Fs(k)14 b Fu(\()p Fs(\000)p Fu(\001)1368 3557 y Fn(S)1406 3540 y FC(2)1437 3542 y Fu(\))1479 3478 y FC(1)p 1479 3487 V 1479 3520 a(2)1535 3542 y FE(h)p Fs(k)1625 3557 y Fq(L)1671 3540 y FC(2)1725 3542 y Fu(+)k Fs(k)p FE(h)p Fs(k)1940 3557 y Fq(H)1998 3540 y FC(1)2034 3450 y Fp(\021)c(\020)2147 3542 y Fs(k)g Fu(\()p Fs(\000)p Fu(\001)2369 3557 y Fn(S)2407 3540 y FC(2)2437 3542 y Fu(\))2480 3478 y FC(1)p 2480 3487 V 2480 3520 a(2)2536 3542 y FE(f)9 b Fs(k)2628 3557 y Fq(L)2674 3540 y FC(2)2727 3542 y Fu(+)19 b Fs(k)p FE(f)9 b Fs(k)2945 3557 y Fq(H)3003 3540 y FC(1)3038 3450 y Fp(\021)3102 3542 y FE(:)p 0 TeXcolorgray 537 w Fu(\(2.22\))p 0 TeXcolorgray 0 3714 a(Similarly)-7 b(,)27 b(for)g Ff(p)c Fu(=)g Fs(\000)p Fu(1,)j(w)n(e)i(can)f(get)583 3886 y SDict begin H.S end 583 3886 a 583 3886 a SDict begin 12 H.A end 583 3886 a 583 3886 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.23) cvn /DEST pdfmark end 583 3886 a 749 3815 a Fp(\014)749 3865 y(\014)777 3886 y Ff(M)864 3852 y Fr(3)864 3907 y Fq(k)q(;)p FD(\000)p Fr(1)1009 3886 y Fu(\()p FE(g)s(;)14 b(h;)g(f)9 b Fu(\))1288 3815 y Fp(\014)1288 3865 y(\014)1338 3886 y Fk(.)23 b Fu(2)1468 3852 y FD(\000)p Fr(3)p Fq(k)1593 3886 y Fs(k)p FE(g)s Fs(k)1720 3904 y Fq(L)1766 3884 y FC(1)1766 3922 y(2)1815 3794 y Fp(\020)1864 3886 y Fs(k)14 b Fu(\()p Fs(\000)p Fu(\001)2086 3901 y Fn(S)2124 3884 y FC(2)2155 3886 y Fu(\))2197 3822 y FC(1)p 2197 3831 29 3 v 2197 3864 a(2)33 b Fu(~)2253 3886 y Ff(F)2304 3898 y FD(\000)p Fr(1)2393 3886 y FE(h)p Fs(k)2483 3901 y Fq(L)2529 3884 y FC(2)2583 3886 y Fu(+)18 b Fs(k)2713 3864 y Fu(~)2708 3886 y Ff(F)2759 3898 y FD(\000)p Fr(1)2847 3886 y FE(h)p Fs(k)2937 3901 y Fq(L)2983 3884 y FC(2)3019 3794 y Fp(\021)749 4069 y Fs(\002)828 3977 y Fp(\020)877 4069 y Fs(k)c Fu(\()o Fs(\000)p Fu(\001)1098 4084 y Fn(S)1137 4067 y FC(2)1167 4069 y Fu(\))1209 4005 y FC(1)p 1209 4014 V 1209 4047 a(2)33 b Fu(~)1266 4069 y Ff(F)1317 4081 y FD(\000)p Fr(1)1405 4069 y FE(f)9 b Fs(k)1497 4084 y Fq(L)1543 4067 y FC(2)1597 4069 y Fu(+)18 b Fs(k)1727 4047 y Fu(~)1722 4069 y Ff(F)1773 4081 y FD(\000)p Fr(1)1861 4069 y FE(f)9 b Fs(k)1953 4084 y Fq(L)1999 4067 y FC(2)2035 3977 y Fp(\021)2103 4069 y Fu(+)18 b(2)2228 4034 y FD(\000)p Fr(2)p Fq(k)2353 4069 y Fs(k)p FE(g)s Fs(k)2480 4084 y Fq(L)2526 4067 y FC(1)2561 4069 y Fs(k)2608 4047 y Fu(~)2603 4069 y Ff(F)2654 4081 y FD(\000)p Fr(1)2742 4069 y FE(h)p Fs(k)2832 4084 y Fq(L)2878 4067 y FC(2)2914 4069 y Fs(k)2961 4047 y Fu(~)2956 4069 y Ff(F)3007 4081 y FD(\000)p Fr(1)3095 4069 y FE(f)9 b Fs(k)3187 4084 y Fq(L)3233 4067 y FC(2)3268 4069 y FE(:)p 0 TeXcolorgray 371 w Fu(\(2.23\))p 0 TeXcolorgray 100 4245 a(No)n(w)27 b(patc)n(hing)g(together)f(\()p 1 0 0 TeXcolorrgb 987 4245 a SDict begin H.S end 987 4245 a 1 0 0 TeXcolorrgb Fu(2.16)p 1 0 0 TeXcolorrgb 1135 4192 a SDict begin H.R end 1135 4192 a 1135 4245 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.16) cvn H.B /ANN pdfmark end 1135 4245 a 0 TeXcolorgray Fu(\))q(\()p 1 0 0 TeXcolorrgb 1200 4245 a SDict begin H.S end 1200 4245 a 1 0 0 TeXcolorrgb Fu(2.18)p 1 0 0 TeXcolorrgb 1347 4192 a SDict begin H.R end 1347 4192 a 1347 4245 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.18) cvn H.B /ANN pdfmark end 1347 4245 a 0 TeXcolorgray Fu(\))i(and)g(\()p 1 0 0 TeXcolorrgb 1601 4245 a SDict begin H.S end 1601 4245 a 1 0 0 TeXcolorrgb Fu(2.21)p 1 0 0 TeXcolorrgb 1749 4192 a SDict begin H.R end 1749 4192 a 1749 4245 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.21) cvn H.B /ANN pdfmark end 1749 4245 a 0 TeXcolorgray Fu(\),)g(w)n(e)f(can)g(get)1100 4401 y SDict begin H.S end 1100 4401 a 1100 4401 a SDict begin 12 H.A end 1100 4401 a 1100 4401 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.24) cvn /DEST pdfmark end 1100 4401 a Fs(jh)p FE(Q)1221 4413 y Fq(k)1262 4401 y Fu(\()p FE(g)s(;)14 b(h)p Fu(\))p FE(;)g(f)9 b Fs(i)1573 4413 y Fq(v)1612 4401 y Fs(j)23 b Fk(.)f Fu(2)1787 4367 y FD(\000)p Fq(k)1880 4401 y Fs(k)p FE(g)s Fs(k)2007 4416 y Fq(L)2053 4400 y FC(1)2087 4401 y Fs(k)p FE(h)p Fs(k)2219 4413 y Fq(H)2277 4395 y Fl(a)2309 4407 y FC(1)2350 4401 y Fs(k)p FE(f)9 b Fs(k)2484 4420 y Fq(H)2542 4402 y Fl(b)2568 4414 y FC(1)p 0 TeXcolorgray 3662 4401 a Fu(\(2.24\))p 0 TeXcolorgray 0 4553 a(for)27 b FE(a)171 4565 y Fr(1)208 4553 y FE(;)14 b(b)281 4565 y Fr(1)341 4553 y Fs(2)23 b Fu([0)p FE(;)14 b Fu(2])27 b(and)h FE(a)819 4565 y Fr(1)874 4553 y Fu(+)18 b FE(b)993 4565 y Fr(1)1053 4553 y Fu(=)23 b(2.)100 4653 y(On)k(the)h(other)f (hand,)h(com)n(bining)g(\()p 1 0 0 TeXcolorrgb 1261 4653 a SDict begin H.S end 1261 4653 a 1 0 0 TeXcolorrgb Fu(2.16)p 1 0 0 TeXcolorrgb 1409 4599 a SDict begin H.R end 1409 4599 a 1409 4653 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.16) cvn H.B /ANN pdfmark end 1409 4653 a 0 TeXcolorgray Fu(\)\()p 1 0 0 TeXcolorrgb 1473 4653 a SDict begin H.S end 1473 4653 a 1 0 0 TeXcolorrgb Fu(2.18)p 1 0 0 TeXcolorrgb 1621 4599 a SDict begin H.R end 1621 4599 a 1621 4653 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.18) cvn H.B /ANN pdfmark end 1621 4653 a 0 TeXcolorgray Fu(\)\()p 1 0 0 TeXcolorrgb 1685 4653 a SDict begin H.S end 1685 4653 a 1 0 0 TeXcolorrgb Fu(2.22)p 1 0 0 TeXcolorrgb 1833 4599 a SDict begin H.R end 1833 4599 a 1833 4653 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.22) cvn H.B /ANN pdfmark end 1833 4653 a 0 TeXcolorgray Fu(\))g(\()p 1 0 0 TeXcolorrgb 1925 4653 a SDict begin H.S end 1925 4653 a 1 0 0 TeXcolorrgb Fu(2.23)p 1 0 0 TeXcolorrgb 2073 4599 a SDict begin H.R end 2073 4599 a 2073 4653 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.23) cvn H.B /ANN pdfmark end 2073 4653 a 0 TeXcolorgray Fu(\))g(and)f(\()p 1 0 0 TeXcolorrgb 2326 4653 a SDict begin H.S end 2326 4653 a 1 0 0 TeXcolorrgb Fu(2.20)p 1 0 0 TeXcolorrgb 2474 4599 a SDict begin H.R end 2474 4599 a 2474 4653 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.20) cvn H.B /ANN pdfmark end 2474 4653 a 0 TeXcolorgray Fu(\)\()p Ff(p)c Fs(\025)g Fu(0\),)k(w)n(e)h(ha)n(v)n(e)357 4830 y SDict begin H.S end 357 4830 a 357 4830 a SDict begin 12 H.A end 357 4830 a 357 4830 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.25) cvn /DEST pdfmark end 357 4830 a Fs(jh)p FE(Q)478 4842 y Fq(k)519 4830 y Fu(\()p FE(g)s(;)14 b(h)p Fu(\))p FE(;)g(f)9 b Fs(i)830 4842 y Fq(v)869 4830 y Fs(j)83 b Fk(.)g Fu(2)1165 4795 y FD(\000)p Fr(3)p Fq(k)1290 4830 y Fs(k)p FE(g)s Fs(k)1417 4848 y Fq(L)1463 4828 y FC(1)1463 4866 y(2)1512 4737 y Fp(\020)1561 4830 y Fs(k)14 b Fu(\()p Fs(\000)p 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2057 y(~)1335 2079 y Ff(F)1386 2091 y Fe(p)1425 2079 y FE(a)1469 2091 y FD(\000)p Fr(1)1577 2079 y Fs(\003)18 b Ff(F)1688 2091 y Fe(p)1727 2079 y FE(g)s Fu(\))23 b(:)1875 2057 y(~)1871 2079 y Ff(F)1922 2091 y Fe(p)1960 2079 y FE(h)p Fs(r)2077 2045 y Fr(2)2115 2079 y Ff(F)2166 2091 y Fe(m)2221 2079 y FE(f)9 b(dv)22 b Fu(+)c(2)2515 1966 y Fp(Z)2597 2079 y Fu(\()p FE(b)2665 2091 y FD(\000)p Fr(1)2772 2079 y Fs(\003)g Ff(F)2883 2091 y Fe(p)2922 2079 y FE(g)s Fu(\))h Fs(\001)3062 2057 y Fu(~)3057 2079 y Ff(F)3108 2091 y Fe(p)3147 2079 y FE(h)p Fs(r)p Ff(F)3315 2091 y Fe(m)3370 2079 y FE(f)9 b(dv)3506 1959 y Fp(\014)3506 2008 y(\014)3506 2058 y(\014)3506 2108 y(\014)1045 2274 y Fk(.)83 b Fs(k)p Ff(F)1286 2286 y Fe(p)1324 2274 y FE(g)s Fs(k)1409 2289 y Fq(L)1455 2273 y FC(2)1490 2274 y Fs(k)1537 2253 y Fu(~)1532 2274 y Ff(F)1583 2286 y Fe(p)1621 2274 y FE(h)p Fs(k)1711 2289 y Fq(L)1757 2273 y FC(2)1793 2274 y Fs(k)p Ff(F)1886 2286 y Fe(m)1941 2274 y FE(f)9 b Fs(k)2033 2289 y Fq(L)2079 2273 y FC(2)2114 2274 y Fu(2)2166 2218 y FC(3)p 2166 2227 29 3 v 2166 2260 a(2)2204 2240 y Fe(m)2259 2274 y FE(:)p 0 TeXcolorgray 1380 w Fu(\(2.30\))p 0 TeXcolorgray 0 2427 a(Th)n(us)27 b(for)g(an)n(y)g FE(a)538 2439 y Fr(1)598 2427 y Fs(2)d Fu([0)p FE(;)14 b Fu(2])p FE(;)g(b)917 2439 y Fr(1)976 2427 y Fs(2)23 b Fu([0)p FE(;)1166 2394 y Fr(3)p 1166 2408 34 4 v 1166 2456 a(2)1209 2427 y Fu(])k(v)n(erifying)g FE(a)1645 2439 y Fr(1)1700 2427 y Fu(+)18 b FE(b)1819 2439 y Fr(1)1879 2427 y Fu(=)23 b(2)k(and)h Ft(s)23 b FE(>)2350 2394 y Fr(1)p 2350 2408 V 2350 2456 a(2)2393 2427 y Fu(,)28 b(w)n(e)f(ha)n(v)n(e)g(that)123 2523 y Fp(X)46 2701 y Fe(m)p FD(\024)p Fe(p)p FD(\000)p Fq(N)290 2709 y FC(0)335 2531 y Fp(\014)335 2581 y(\014)362 2602 y Ff(M)449 2568 y Fr(4)449 2622 y FD(\000)p Fr(1)p Fq(;)p Fe(p)p Fq(;)p Fe(m)665 2602 y Fu(\()p FE(g)s(;)14 b(h;)g(f)9 b Fu(\))944 2531 y Fp(\014)944 2581 y(\014)1054 2602 y Fk(.)82 b Fu(2)1243 2568 y FD(\000)1305 2545 y FC(1)p 1305 2554 29 3 v 1305 2588 a(2)1343 2568 y Fe(p)1382 2602 y Fu(\(2)18 b(+)g Ff(p)p Fu(\))1631 2568 y Fh(s)1659 2602 y Fs(k)p Ff(F)1752 2614 y Fe(p)1790 2602 y FE(g)s Fs(k)1875 2617 y Fq(L)1921 2600 y FC(2)1956 2602 y Fu(2)1998 2568 y Fq(a)2034 2576 y FC(1)2066 2568 y Fe(p)2105 2602 y Fs(k)2152 2580 y Fu(~)2147 2602 y Ff(F)2198 2614 y Fe(p)2236 2602 y FE(h)p Fs(k)2326 2617 y Fq(L)2372 2600 y FC(2)2408 2602 y Fu(2)2450 2568 y Fq(b)2479 2576 y FC(1)2511 2568 y Fe(m)2567 2602 y Fs(k)p Ff(F)2660 2614 y Fe(m)2715 2602 y FE(f)9 b Fs(k)2807 2617 y Fq(L)2853 2600 y FC(2)2888 2602 y Fu(2)2930 2568 y Fr(\()2965 2545 y FC(3)p 2965 2554 V 2965 2588 a(2)3003 2568 y FD(\000)p Fq(b)3084 2576 y FC(1)3117 2568 y Fr(\))p Fe(m)3199 2602 y Fu(2)3241 2568 y Fr(\()3276 2545 y FC(1)p 3276 2554 V 3276 2588 a(2)3314 2568 y FD(\000)p Fq(a)3402 2576 y FC(1)3435 2568 y Fr(\))p Fe(p)3500 2602 y Fu(\(2)18 b(+)g Ff(p)p Fu(\))3749 2568 y FD(\000)p Fh(s)1054 2816 y Fk(.)82 b Fs(k)p FE(g)s Fs(k)1328 2859 y Fq(H)1386 2830 y Fm(\000)1441 2814 y FC(1)p 1441 2821 V 1441 2852 a(2)1479 2830 y Fl(;)p Fg(s)1523 2816 y Fs(k)p FE(h)p Fs(k)1655 2828 y Fq(H)1713 2810 y Fl(a)1745 2822 y FC(1)1786 2816 y Fs(k)p FE(f)9 b Fs(k)1920 2835 y Fq(H)1978 2816 y Fl(b)2004 2828 y FC(1)2044 2816 y FE(;)0 2988 y Fu(where)27 b(w)n(e)g(use)h(the)g(fact)f(that)h(\()1034 2955 y Fr(3)p 1035 2969 34 4 v 1035 3016 a(2)1096 2988 y Fs(\000)18 b FE(b)1215 3000 y Fr(1)1252 2988 y Fu(\))p Ff(m)g Fu(+)g(\()1491 2955 y Fr(1)p 1492 2969 V 1492 3016 a(2)1553 2988 y Fs(\000)g FE(a)1680 3000 y Fr(1)1717 2988 y Fu(\))p Ff(p)23 b Fu(=)g Fs(\000)p Fu(\()2009 2955 y Fr(3)p 2009 2969 V 2009 3016 a(2)2070 2988 y Fs(\000)18 b FE(b)2189 3000 y Fr(1)2226 2988 y Fu(\)\()p Ff(m)h Fs(\000)f Ff(p)p Fu(\))p FE(;)c(b)2603 3000 y Fr(1)2662 2988 y Fs(2)24 b Fu([0)p FE(;)2852 2955 y Fr(3)p 2852 2969 V 2852 3016 a(2)2895 2988 y Fu(])k(and)g Ft(s)22 b FE(>)3260 2955 y Fr(1)p 3260 2969 V 3260 3016 a(2)3303 2988 y Fu(.)100 3092 y(F)-7 b(or)27 b Ff(M)336 3062 y Fr(2)401 3092 y Fu(and)g Ff(M)649 3062 y Fr(3)686 3092 y Fu(,)h(w)n(e)f(ha)n(v)n(e)838 3255 y SDict begin H.S end 838 3255 a 838 3255 a SDict begin 12 H.A end 838 3255 a 838 3255 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.31) cvn /DEST pdfmark end 838 3255 a -71 x Fp(\014)838 3234 y(\014)865 3255 y Ff(M)952 3220 y Fr(2)952 3275 y FD(\000)p Fr(1)p Fq(;)p Fe(l)1082 3255 y Fu(\()p FE(g)s(;)14 b(h;)g(f)9 b Fu(\))1361 3184 y Fp(\014)1361 3234 y(\014)1471 3255 y Fk(.)1681 3176 y Fp(X)1619 3354 y Fe(p)p FD(\024)p Fe(l)p FD(\000)p Fq(N)1832 3362 y FC(0)1877 3255 y Fu(2)1919 3220 y FD(\000)1981 3194 y Fd(p)p 1981 3208 33 3 v 1983 3240 a FC(2)2028 3255 y Fs(k)p Ff(F)2121 3267 y Fe(p)2159 3255 y FE(g)s Fs(k)2244 3270 y Fq(L)2290 3253 y FC(2)2325 3255 y Fs(k)p Ff(F)2418 3267 y Fe(l)2442 3255 y FE(h)p Fs(k)2532 3270 y Fq(L)2578 3253 y FC(2)2614 3255 y Fs(k)2661 3233 y Fu(~)2656 3255 y Ff(F)2707 3267 y Fe(l)2731 3255 y FE(f)g Fs(k)2823 3270 y Fq(L)2869 3253 y FC(2)2904 3255 y Fu(2)2946 3220 y Fr(2)p Fe(l)3004 3255 y FE(;)p 0 TeXcolorgray 635 w Fu(\(2.31\))p 0 TeXcolorgray 833 3490 a Fs(j)p Ff(M)943 3456 y Fr(3)943 3511 y FD(\000)p Fr(1)p Fq(;)p Fe(p)1087 3490 y Fu(\()p FE(g)s(;)14 b(h;)g(f)9 b Fu(\))p Fs(j)82 b Fk(.)h Fs(k)p Ff(F)1712 3502 y Fe(p)1750 3490 y FE(g)s Fs(k)1835 3505 y Fq(L)1881 3488 y FC(2)1916 3490 y Fs(k)1963 3469 y Fu(~)1958 3490 y Ff(F)2009 3502 y Fe(p)2048 3490 y FE(h)p Fs(k)2138 3505 y Fq(L)2184 3488 y FC(2)2219 3490 y Fs(k)2266 3469 y Fu(~)2261 3490 y Ff(F)2312 3502 y Fe(p)2350 3490 y FE(f)9 b Fs(k)2442 3505 y Fq(L)2488 3488 y FC(2)2523 3490 y Fu(2)2575 3434 y FC(3)p 2575 3443 29 3 v 2575 3476 a(2)2613 3456 y Fq(p)2651 3490 y FE(:)p 0 TeXcolorgray 988 w Fu(\(2.32\))p 0 TeXcolorgray 0 3641 a(Then)28 b(it)g(is)f(easy)g(to)g(deriv)n(e)g(that)609 3724 y Fp(X)590 3903 y Fe(l)p FD(\025\000)p Fr(1)762 3732 y Fp(\014)762 3782 y(\014)790 3803 y Ff(M)877 3769 y Fr(2)877 3823 y FD(\000)p Fr(1)p Fq(;)p Fe(l)1007 3803 y Fu(\()p FE(g)s(;)14 b(h;)g(f)9 b Fu(\))1286 3732 y Fp(\014)1286 3782 y(\014)1331 3803 y Fu(+)1440 3724 y Fp(X)1414 3902 y Fe(p)p FD(\025\000)p Fr(1)1600 3732 y Fp(\014)1600 3782 y(\014)1627 3803 y Ff(M)1714 3769 y Fr(3)1714 3823 y FD(\000)p Fr(1)p Fq(;)p Fe(p)1858 3803 y Fu(\()p FE(g)s(;)14 b(h;)g(f)9 b Fu(\))2137 3732 y Fp(\014)2137 3782 y(\014)2187 3803 y Fk(.)23 b Fs(k)p FE(g)s Fs(k)2402 3845 y Fq(H)2460 3817 y Fm(\000)2514 3801 y FC(1)p 2514 3808 V 2514 3839 a(2)2552 3817 y Fl(;)p Fg(s)2597 3803 y Fs(k)p FE(h)p Fs(k)2729 3815 y Fq(H)2787 3797 y Fl(a)2819 3809 y FC(1)2859 3803 y Fs(k)p FE(f)9 b Fs(k)2993 3822 y Fq(H)3051 3803 y Fl(b)3077 3815 y FC(1)0 4043 y Fu(with)28 b FE(a)233 4055 y Fr(1)270 4043 y FE(;)14 b(b)343 4055 y Fr(1)403 4043 y Fs(2)23 b Fu([0)p FE(;)14 b Fu(2])27 b(and)h FE(a)881 4055 y Fr(1)936 4043 y Fu(+)18 b FE(b)1055 4055 y Fr(1)1115 4043 y Fu(=)23 b(2.)100 4143 y(P)n(atc)n(hing)j(together)g(all)i(the)g(ab)r(o)n(v)n(e) e(estimates,)h(if)h FE(a)1776 4155 y Fr(1)1814 4143 y FE(;)14 b(b)1887 4155 y Fr(1)1946 4143 y Fs(2)24 b Fu([)2058 4110 y Fr(1)p 2058 4124 34 4 v 2058 4171 a(2)2101 4143 y FE(;)2148 4110 y Fr(3)p 2148 4124 V 2148 4171 a(2)2191 4143 y Fu(])k(and)f FE(a)2447 4155 y Fr(1)2503 4143 y Fu(+)18 b FE(b)2622 4155 y Fr(1)2681 4143 y Fu(=)23 b(2,)k(w)n(e)h(ha)n (v)n(e)1056 4293 y SDict begin H.S end 1056 4293 a 1056 4293 a SDict begin 12 H.A end 1056 4293 a 1056 4293 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.33) cvn /DEST pdfmark end 1056 4293 a Fs(jh)p FE(Q)1177 4305 y FD(\000)p Fr(1)1266 4293 y Fu(\()p FE(g)s(;)14 b(h)p Fu(\))p FE(;)g(f)9 b Fs(i)1577 4305 y Fq(v)1617 4293 y Fs(j)23 b Fk(.)1728 4305 y Fq(")1786 4293 y Fs(k)p FE(g)s Fs(k)1913 4336 y Fq(H)1971 4307 y Fm(\000)2025 4291 y FC(1)p 2025 4298 29 3 v 2025 4329 a(2)2063 4307 y Fl(;)p Fg(s)2108 4293 y Fs(k)p FE(h)p Fs(k)2240 4305 y Fq(H)2298 4287 y Fl(a)2330 4299 y FC(1)2370 4293 y Fs(k)p FE(f)9 b Fs(k)2504 4312 y Fq(H)2562 4293 y Fl(b)2588 4305 y FC(1)2629 4293 y FE(:)p 0 TeXcolorgray 1010 w Fu(\(2.33\))p 0 TeXcolorgray 100 4457 a(Observing)28 b(that)j(for)f FE(Q)876 4469 y FD(\000)p Fr(1)965 4457 y Fu(,)h(it)g(holds)f(that)g Fs(j)p FE(v)24 b Fs(\000)c FE(v)1719 4469 y FD(\003)1757 4457 y Fs(j)28 b Fk(.)f Fu(1)j(whic)n(h)g(implies)h Fs(h)p FE(v)s Fs(i)d(\030)g(h)p FE(v)2797 4469 y FD(\003)2835 4457 y Fs(i)p Fu(,)k(tigether)e(with)h(\()p 1 0 0 TeXcolorrgb 3458 4457 a SDict begin H.S end 3458 4457 a 1 0 0 TeXcolorrgb Fu(2.33)p 1 0 0 TeXcolorrgb 3606 4403 a SDict begin H.R end 3606 4403 a 3606 4457 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.33) cvn H.B /ANN pdfmark end 3606 4457 a 0 TeXcolorgray Fu(\))f(yields)0 4556 y(that)397 4707 y SDict begin H.S end 397 4707 a 397 4707 a SDict begin 12 H.A end 397 4707 a 397 4707 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.34) cvn /DEST pdfmark end 397 4707 a Fs(jh)p FE(Q)518 4719 y FD(\000)p Fr(1)607 4707 y Fu(\()p FE(g)s(;)14 b(h)p Fu(\))p FE(;)g(f)9 b Fs(i)918 4719 y Fq(v)957 4707 y Fs(j)23 b Fk(.)1068 4719 y Fq(")1126 4628 y Fp(X)1176 4807 y Fe(l)1260 4707 y Fs(kP)1360 4719 y Fe(l)1384 4707 y FE(g)s Fs(k)1469 4749 y Fq(H)1527 4721 y Fm(\000)1582 4705 y FC(1)p 1582 4712 29 3 v 1582 4743 a(2)1621 4721 y Fl(;)p Fg(s)1665 4707 y Fs(k)1725 4686 y Fu(~)1707 4707 y Fs(P)1765 4719 y Fe(l)1789 4707 y FE(h)p Fs(k)1879 4719 y Fq(H)1937 4701 y Fl(a)1969 4713 y FC(1)2010 4707 y Fs(k)2070 4686 y Fu(~)2052 4707 y Fs(P)2110 4719 y Fe(l)2135 4707 y FE(f)9 b Fs(k)2227 4726 y Fq(H)2285 4707 y Fl(b)2311 4719 y FC(1)2374 4707 y Fk(.)23 b Fs(k)p FE(g)s Fs(k)2589 4763 y Fq(H)2647 4721 y Fm(\000)2701 4705 y FC(1)p 2701 4712 V 2701 4743 a(2)2740 4721 y Fl(;)p Fg(s)2643 4772 y Fl(!)2680 4784 y FC(1)2784 4707 y Fs(k)p FE(h)p Fs(k)2916 4729 y Fq(H)2974 4696 y Fl(a)3006 4708 y FC(1)2970 4738 y Fl(!)3007 4750 y FC(2)3047 4707 y Fs(k)p FE(f)9 b Fs(k)3181 4740 y Fq(H)3239 4707 y Fl(b)3265 4719 y FC(1)3235 4749 y Fl(!)3272 4761 y FC(3)p 0 TeXcolorgray 3662 4707 a Fu(\(2.34\))p 0 TeXcolorgray 0 4944 a(for)27 b FE(a)171 4956 y Fr(1)208 4944 y FE(;)14 b(b)281 4956 y Fr(1)341 4944 y Fs(2)23 b Fu([)452 4911 y Fr(1)p 452 4925 34 4 v 452 4973 a(2)496 4944 y FE(;)542 4911 y Fr(3)p 542 4925 V 542 4973 a(2)585 4944 y Fu(])p FE(;)14 b(a)689 4956 y Fr(1)745 4944 y Fu(+)k FE(b)864 4956 y Fr(1)924 4944 y Fu(=)23 b(2)k(and)g FE(!)1294 4956 y Fr(1)1350 4944 y Fu(+)18 b FE(!)1485 4956 y Fr(2)1540 4944 y Fu(+)g FE(!)1675 4956 y Fr(3)1735 4944 y Fu(=)23 b(0.)100 5044 y(One)37 b(ma)n(y)h(easy)f(to)h(c)n(hec)n(k)f(that)h(\()p 1 0 0 TeXcolorrgb 1237 5044 a SDict begin H.S end 1237 5044 a 1 0 0 TeXcolorrgb Fu(2.27)p 1 0 0 TeXcolorrgb 1385 4990 a SDict begin H.R end 1385 4990 a 1385 5044 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.27) cvn H.B /ANN pdfmark end 1385 5044 a 0 TeXcolorgray Fu(\))h(and)f(\()p 1 0 0 TeXcolorrgb 1660 5044 a SDict begin H.S end 1660 5044 a 1 0 0 TeXcolorrgb Fu(2.34)p 1 0 0 TeXcolorrgb 1807 4990 a SDict begin H.R end 1807 4990 a 1807 5044 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.34) cvn H.B /ANN pdfmark end 1807 5044 a 0 TeXcolorgray Fu(\))h(yield)f(\()p 1 0 0 TeXcolorrgb 2121 5044 a SDict begin H.S end 2121 5044 a 1 0 0 TeXcolorrgb Fu(2.13)p 1 0 0 TeXcolorrgb 2269 4990 a SDict begin H.R end 2269 4990 a 2269 5044 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.13) cvn H.B /ANN pdfmark end 2269 5044 a 0 TeXcolorgray Fu(\),)j(\()p 1 0 0 TeXcolorrgb 2397 5044 a SDict begin H.S end 2397 5044 a 1 0 0 TeXcolorrgb Fu(2.28)p 1 0 0 TeXcolorrgb 2545 4990 a SDict begin H.R end 2545 4990 a 2545 5044 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.28) cvn H.B /ANN pdfmark end 2545 5044 a 0 TeXcolorgray Fu(\))d(and)g(\()p 1 0 0 TeXcolorrgb 2819 5044 a SDict begin H.S end 2819 5044 a 1 0 0 TeXcolorrgb Fu(2.34)p 1 0 0 TeXcolorrgb 2967 4990 a SDict begin H.R end 2967 4990 a 2967 5044 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.34) cvn H.B /ANN pdfmark end 2967 5044 a 0 TeXcolorgray Fu(\))h(yield)f(\()p 1 0 0 TeXcolorrgb 3281 5044 a SDict begin H.S end 3281 5044 a 1 0 0 TeXcolorrgb Fu(2.12)p 1 0 0 TeXcolorrgb 3429 4990 a SDict begin H.R end 3429 4990 a 3429 5044 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.12) cvn H.B /ANN pdfmark end 3429 5044 a 0 TeXcolorgray Fu(\).)68 b(Then)38 b(w)n(e)0 5143 y(complete)28 b(the)g(pro)r(of)f(of)g(this)h(prop)r (osition.)2409 b Fk(\003)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 15 15 TeXDict begin HPSdict begin 15 14 bop 0 TeXcolorgray 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 0 TeXcolorgray -8 -245 a SDict begin H.S end -8 -245 a -8 -245 a SDict begin H.R end -8 -245 a -8 -245 a SDict begin [/View [/XYZ H.V]/Dest (page.15) cvn /DEST pdfmark end -8 -245 a 0 TeXcolorgray 0 TeXcolorgray 1335 -165 a Fr(LAND)n(A)n(U)30 b(EQUA)-5 b(TION)28 b(IN)h Fq(H)2196 -193 y Fm(\000)2252 -209 y FC(1)p 2252 -202 29 3 v 2252 -171 a(2)2324 -165 y Fr(SP)-5 b(A)n(CE)1268 b(15)p 0 TeXcolorgray 0 -54 a SDict begin H.S end 0 -54 a 0 -54 a SDict begin 12 H.A end 0 -54 a 0 -54 a SDict begin [/View [/XYZ H.V]/Dest (subsection.2.4) cvn /DEST pdfmark end 0 -54 a 83 x Fu(2.4.)41 b FF(Comm)m(utator)27 b(b)s(et)m(w)m(een)i Ff(F)1155 41 y Fq(j)1220 29 y FF(and)h FE(Q)p FF(.)40 b Fu(No)n(w)25 b(w)n(e)h(estimate)f(the)h(comm)n(utator) e(b)r(et)n(w)n(een)i Ff(F)3150 41 y Fq(j)3210 29 y Fu(and)g FE(Q)p Fu(.)36 b(W)-7 b(e)26 b(split)g(it)0 129 y(in)n(to)h(t)n(w)n(o)g (parts:)30 353 y Fs(h)p Ff(F)113 365 y Fq(j)148 353 y FE(Q)p Fu(\()p FE(g)s(;)14 b(h)p Fu(\))k Fs(\000)g FE(Q)p Fu(\()p FE(g)s(;)c Ff(F)736 365 y Fq(j)770 353 y FE(h)p Fu(\))p 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1530 y Fs(k)p FE(f)9 b Fs(k)3196 1556 y Fq(H)3254 1530 y Fm(\000)p Fl(N)3250 1576 y Fm(\000)p Fl(N)3356 1530 y FE(;)p 0 TeXcolorgray 283 w Fu(\(2.35\))p 0 TeXcolorgray 0 1827 a Fj(wher)l(e)30 b FE(N)i Fs(2)24 b Fo(N)39 b Fj(c)l(an)30 b(b)l(e)g(suitably)g(lar)l(ge,)h(and)g FF(F)1507 1839 y Fq(j)1542 1827 y FE(;)14 b FF(P)1644 1839 y Fq(k)1714 1827 y Fj(ar)l(e)30 b(de\014ne)l(d)g(in)g(De\014nition)p 1 0 0 TeXcolorrgb 2629 1827 a SDict begin H.S end 2629 1827 a 1 0 0 TeXcolorrgb Fj(2.2)p 1 0 0 TeXcolorrgb 2739 1773 a SDict begin H.R end 2739 1773 a 2739 1827 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (defi.2.2) cvn H.B /ANN pdfmark end 2739 1827 a 0 TeXcolorgray Fj(.)p 0 TeXcolorgray 0 2021 a(Pr)l(o)l(of.)p 0 TeXcolorgray 43 w Fu(W)-7 b(e)28 b(divide)g(the)g(pro)r(of)f(in)n(to)g(sev)n(eral)f(steps.)100 2120 y(Step)i(1:)36 b(Preliminary)26 b(estimate)i(of)f Fs(h)p Ff(F)1356 2132 y Fq(j)1391 2120 y FE(Q)1457 2132 y Fq(k)1498 2120 y Fu(\()p 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Ff(F)912 2380 y Fq(j)946 2368 y FE(h)p Fu(\))p FE(;)g Ff(F)1114 2380 y Fq(j)1149 2368 y FE(f)9 b Fs(i)1231 2380 y Fq(v)1294 2368 y Fu(=)1448 2289 y Fp(X)1381 2468 y Fe(p)p FD(\025)p Fq(j)s Fr(+)p Fq(N)1602 2476 y FC(0)1635 2368 y Fs(h)p FE(Q)1733 2380 y Fq(k)1773 2368 y Fu(\()p Ff(F)1856 2380 y Fe(p)1896 2368 y FE(g)s(;)1980 2347 y Fu(~)1976 2368 y Ff(F)2027 2380 y Fe(p)2065 2368 y FE(h)p Fu(\))p FE(;)14 b Ff(F)2233 2334 y Fr(2)2233 2389 y Fq(j)2270 2368 y FE(f)9 b Fs(i)2352 2380 y Fq(v)2410 2368 y Fu(+)2560 2289 y Fp(X)2493 2468 y Fe(p)p FD(\024)p Fq(j)s FD(\000)p Fq(N)2715 2476 y FC(0)2747 2368 y Fs(h)p Ff(F)2830 2380 y Fq(j)2865 2368 y FE(Q)2931 2380 y Fq(k)2971 2368 y Fu(\()p Ff(F)3054 2380 y Fe(p)3094 2368 y FE(g)s(;)3178 2347 y Fu(~)3174 2368 y Ff(F)3225 2380 y Fq(j)3259 2368 y FE(h)p Fu(\))143 2608 y Fs(\000)18 b FE(Q)292 2620 y Fq(k)333 2608 y Fu(\()p Ff(F)416 2620 y Fe(p)455 2608 y FE(g)s(;)c Ff(F)586 2620 y Fq(j)620 2608 y FE(h)p Fu(\))p FE(;)g Ff(F)788 2620 y Fq(j)823 2608 y FE(f)9 b Fs(i)905 2620 y Fq(v)962 2608 y Fu(+)1113 2529 y Fp(X)1045 2707 y Fe(a)p FD(\024)p Fq(j)s FD(\000)p Fq(N)1267 2715 y FC(0)1300 2608 y Fs(h)p FE(Q)1398 2620 y Fq(k)1439 2608 y Fu(\()1476 2586 y(~)1471 2608 y Ff(F)1522 2620 y Fq(j)1557 2608 y FE(g)s(;)14 b Ff(F)1688 2620 y Fe(a)1726 2608 y FE(h)p Fu(\))p FE(;)g Ff(F)1894 2573 y Fr(2)1894 2628 y Fq(j)1931 2608 y FE(f)9 b Fs(i)2013 2620 y Fq(v)2071 2608 y Fu(+)18 b Fs(h)p FE(Q)2252 2620 y Fq(k)2292 2608 y Fu(\()2329 2586 y(~)2324 2608 y Ff(F)2375 2620 y Fq(j)2410 2608 y FE(g)s(;)2494 2586 y Fu(~)2490 2608 y Ff(F)2541 2620 y Fq(j)2576 2608 y FE(h)p Fu(\))p FE(;)c Ff(F)2744 2573 y Fr(2)2744 2628 y Fq(j)2781 2608 y FE(f)9 b Fs(i)2863 2620 y Fq(v)2920 2608 y Fs(\000)18 b(h)p FE(Q)3101 2620 y Fq(k)3142 2608 y Fu(\()3179 2586 y(~)3174 2608 y Ff(F)3225 2620 y Fq(j)3260 2608 y FE(g)s(;)c Ff(F)3391 2620 y Fq(j)3425 2608 y FE(h)p Fu(\))p FE(;)g Ff(F)3593 2620 y Fq(j)3628 2608 y FE(f)9 b Fs(i)3710 2620 y Fq(v)148 2818 y Fu(:=)22 b Fi(B)333 2830 y Fr(1)389 2818 y Fu(+)c Fi(B)547 2830 y Fr(2)603 2818 y Fu(+)h Fi(B)762 2830 y Fr(3)818 2818 y Fu(+)f Fi(B)976 2830 y Fr(4)1013 2818 y FE(:)3662 2923 y Fu(\(2.36\))100 3022 y(Observing)25 b(that)i(the)g Fi(B)890 3034 y Fr(1)928 3022 y Fu(,)g Fi(B)1053 3034 y Fr(3)1118 3022 y Fu(and)f Fi(B)1353 3034 y Fr(4)1418 3022 y Fu(enjo)n(y)g(the)i(similar)e(structure)g(as)g Ff(M)2594 2992 y Fr(4)2594 3046 y Fq(k)q(;j;)p Fe(a)2737 3022 y Fu(,)h Ff(M)2874 2992 y Fr(1)2874 3046 y Fq(k)q(;j;)p Fe(a)3043 3022 y Fu(and)g Ff(M)3291 2992 y Fr(3)3291 3046 y Fq(k)q(;j)3383 3022 y Fu(,)g(resp)r(ectiv)n(ely)-7 b(.)0 3128 y(Th)n(us)27 b(w)n(e)h(can)f(deriv)n(e)f(from)i(\()p 1 0 0 TeXcolorrgb 958 3128 a SDict begin H.S end 958 3128 a 1 0 0 TeXcolorrgb Fu(2.15)p 1 0 0 TeXcolorrgb 1106 3074 a SDict begin H.R end 1106 3074 a 1106 3128 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.15) cvn H.B /ANN pdfmark end 1106 3128 a 0 TeXcolorgray Fu(\)\()p 1 0 0 TeXcolorrgb 1170 3128 a SDict begin H.S end 1170 3128 a 1 0 0 TeXcolorrgb Fu(2.17)p 1 0 0 TeXcolorrgb 1318 3074 a SDict begin H.R end 1318 3074 a 1318 3128 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.17) cvn H.B /ANN pdfmark end 1318 3128 a 0 TeXcolorgray Fu(\))g(and)f(\()p 1 0 0 TeXcolorrgb 1571 3128 a SDict begin H.S end 1571 3128 a 1 0 0 TeXcolorrgb Fu(2.20)p 1 0 0 TeXcolorrgb 1719 3074 a SDict begin H.R end 1719 3074 a 1719 3128 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.20) cvn H.B /ANN pdfmark end 1719 3128 a 0 TeXcolorgray Fu(\))h(that)871 3344 y Fs(j)p Fi(B)969 3356 y Fr(1)1006 3344 y Fs(j)19 b Fu(+)f Fs(j)p Fi(B)1229 3356 y Fr(3)1266 3344 y Fs(j)h Fu(+)f Fs(j)p Fi(B)1489 3356 y Fr(4)1527 3344 y Fs(j)23 b Fk(.)1638 3356 y Fq(N)1723 3344 y Fu(2)1765 3309 y FD(\000)p Fq(k)q(N)1916 3344 y Fu(2)1958 3309 y FD(\000)p Fq(j)s(N)2104 3344 y Fs(k)p FE(g)s Fs(k)2231 3359 y Fq(L)2277 3342 y FC(1)2311 3344 y Fs(k)p FE(h)p Fs(k)2443 3360 y Fq(H)2501 3343 y Fm(\000)p Fl(N)2603 3344 y Fs(k)p FE(f)9 b Fs(k)2737 3360 y Fq(H)2795 3343 y Fm(\000)p Fl(N)2897 3344 y FE(:)0 3556 y Fu(F)-7 b(or)27 b Fi(B)224 3568 y Fr(2)262 3556 y Fu(,)g(b)n(y)h(F)-7 b(ourier)26 b(transform,)143 3806 y Fi(B)218 3818 y Fr(2)279 3806 y Fu(=)c(2)408 3772 y Fr(2)p Fq(k)496 3693 y Fp(Z)9 b(Z)665 3806 y(b)-63 b FE(a)692 3818 y Fr(0)729 3806 y Fu(\(2)803 3772 y Fq(k)843 3806 y Fu(\()p FE(\030)23 b Fs(\000)18 b FE(\021)s Fu(\)\))24 b(:)f FE(\030)g Fs(\012)18 b Fu(\()p FE(\030)23 b Fs(\000)18 b Fu(2)p FE(\021)s Fu(\))1664 3788 y Fo(\\)1629 3806 y Fs(S)1679 3818 y Fq(j)s FD(\000)p Fq(N)1814 3826 y FC(0)1851 3806 y FE(g)s Fu(\()p FE(\030)k Fs(\000)c FE(\021)s Fu(\))2150 3765 y Fp(d)2148 3784 y Fu(~)2143 3806 y Ff(F)2194 3818 y Fq(j)2230 3806 y FE(h)p Fu(\()p FE(\021)s Fu(\))p 2386 3712 241 4 v 2394 3784 a Fp(d)2386 3806 y Ff(F)2437 3818 y Fq(j)2472 3806 y FE(f)9 b Fu(\()p FE(\030)t Fu(\)\()p FE(')p 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4340 y Fq(j)s FD(\000)p Fq(N)1713 4348 y FC(0)1750 4328 y FE(g)s Fs(k)1835 4343 y Fq(L)1881 4326 y FC(1)1916 4328 y Fs(k)1963 4307 y Fu(~)1958 4328 y Ff(F)2009 4340 y Fq(j)2043 4328 y FE(h)p Fs(k)2133 4343 y Fq(L)2179 4326 y FC(2)2215 4328 y Fs(k)p Ff(F)2308 4340 y Fq(j)2342 4328 y FE(f)9 b Fs(k)2434 4343 y Fq(L)2480 4326 y FC(2)2538 4328 y Fk(.)23 b Fu(2)2668 4294 y FD(\000)p Fr(2)p Fq(k)2793 4328 y Fs(k)p FE(g)s Fs(k)2920 4343 y Fq(L)2966 4326 y FC(1)3001 4328 y Fs(k)3048 4307 y Fu(~)3043 4328 y Ff(F)3094 4340 y Fq(j)3128 4328 y FE(h)p Fs(k)3218 4343 y Fq(L)3264 4326 y FC(2)3300 4328 y Fs(k)p Ff(F)3393 4340 y Fq(j)3427 4328 y FE(f)9 b Fs(k)3519 4343 y Fq(L)3565 4326 y FC(2)3600 4328 y Fu(2)3642 4294 y Fq(j)3677 4328 y FE(:)0 4585 y Fu(These)27 b(lead)h(to)f(that)569 4797 y SDict begin H.S end 569 4797 a 569 4797 a SDict begin 12 H.A end 569 4797 a 569 4797 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.37) cvn /DEST pdfmark end 569 4797 a 231 w Fs(jh)p Ff(F)906 4809 y Fq(j)941 4797 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Fu(\()p Ff(U)727 229 y Fq(k)q FD(\000)p Fq(N)868 237 y FC(0)905 217 y FE(g)s(;)1003 196 y Fu(~)985 217 y Fs(P)1043 229 y Fq(k)1083 217 y FE(h;)14 b Fu([)1209 196 y(~)1191 217 y Fs(P)1249 229 y Fq(k)1289 217 y FE(;)g Ff(F)1377 182 y Fr(2)1377 237 y Fq(j)1414 217 y Fu(])p FE(f)9 b Fu(\))p Fs(j)23 b Fk(.)1679 138 y Fp(X)1653 316 y Fe(a)p FD(\025\000)p Fr(1)1838 217 y Fu(2)1880 182 y FD(\000)p Fq(k)1973 217 y Fs(k)p FE(g)s Fs(k)2100 232 y Fq(L)2146 215 y FC(1)2180 217 y Fs(k)p Ff(F)2273 229 y Fe(a)2330 196 y Fu(~)2312 217 y Fs(P)2370 229 y Fq(k)2410 217 y FE(h)p Fs(k)2500 232 y Fq(L)2546 215 y FC(2)2582 217 y Fs(k)2629 195 y Fu(~)2624 217 y Ff(F)2675 229 y Fe(a)2713 217 y Fu([)2754 196 y(~)2736 217 y Fs(P)2794 229 y Fq(k)2834 217 y FE(;)14 b Ff(F)2922 182 y Fr(2)2922 237 y Fq(j)2959 217 y Fu(])p FE(f)9 b Fs(k)3074 232 y Fq(L)3120 215 y FC(2)3156 217 y Fu(2)3198 182 y Fr(2)p Fe(a)172 464 y Fk(.)o Fu(2)278 429 y FD(\000)p Fq(k)370 464 y Fs(k)p FE(g)s Fs(k)497 479 y Fq(L)543 462 y FC(1)578 372 y Fp(\020)747 385 y(X)641 567 y FD(j)p Fe(a)p FD(\000)p Fq(j)s FD(j\024)p Fe(n)p Fq(N)939 575 y FC(0)986 464 y Fs(k)p Ff(F)1079 476 y Fe(a)1135 443 y Fu(~)1117 464 y Fs(P)1175 476 y Fq(k)1216 464 y FE(h)p Fs(k)1306 479 y Fq(L)1352 462 y FC(2)1387 464 y Fs(k)1434 442 y Fu(~)1429 464 y Ff(F)1480 476 y Fe(a)1518 464 y Fu([)1560 443 y(~)1541 464 y Fs(P)1599 476 y Fq(k)1640 464 y FE(;)14 b Ff(F)1728 429 y Fr(2)1728 484 y Fq(j)1765 464 y Fu(])p FE(f)9 b Fs(k)1880 479 y Fq(L)1926 462 y FC(2)1961 464 y Fu(2)2003 429 y Fr(2)p Fq(j)2089 464 y Fu(+)2277 385 y Fp(X)2172 567 y FD(j)p Fe(a)p FD(\000)p Fq(j)s FD(j)p Fq(>)p Fe(n)p Fq(N)2470 575 y FC(0)2516 464 y Fs(k)p Ff(F)2609 476 y Fe(a)2666 443 y Fu(~)2648 464 y Fs(P)2706 476 y Fq(k)2746 464 y FE(h)p Fs(k)2836 479 y Fq(L)2882 462 y FC(2)2918 464 y Fs(k)2965 442 y Fu(~)2960 464 y Ff(F)3011 476 y Fe(a)3049 464 y Fu([)3090 443 y(~)3072 464 y Fs(P)3130 476 y Fq(k)3171 464 y FE(;)14 b Ff(F)3259 429 y Fr(2)3259 484 y Fq(j)3295 464 y Fu(])p FE(f)9 b Fs(k)3410 479 y Fq(L)3456 462 y FC(2)3492 464 y Fu(2)3534 429 y Fr(2)p Fe(a)3605 372 y Fp(\021)109 775 y Fk(.)174 787 y Fq(N)236 775 y Fu(2)278 740 y FD(\000)p Fr(2)p Fq(k)403 775 y Fs(k)p FE(g)s Fs(k)530 790 y Fq(L)576 773 y FC(1)611 682 y Fp(\020)780 696 y(X)675 878 y FD(j)p Fe(a)p FD(\000)p Fq(j)s FD(j\024)p Fe(n)p Fq(N)973 886 y FC(0)1019 775 y Fs(k)p Ff(F)1112 787 y Fe(a)1168 754 y Fu(~)1150 775 y Fs(P)1208 787 y Fq(k)1249 775 y FE(h)p Fs(k)1339 790 y Fq(L)1385 773 y FC(2)1420 682 y Fp(\021\020)1570 671 y Fr(2)p Fq(N)1556 696 y Fp(X)1533 878 y FD(j)p Fq(\013)p FD(j)p Fr(=1)1714 775 y Fs(k)1761 753 y Fu(^)1756 775 y Ff(F)1807 787 y Fq(j;\013)1919 754 y Fu(~)1900 775 y Fs(P)1958 787 y Fq(k)q(;\013)2062 775 y FE(f)g Fs(k)2154 790 y Fq(L)2200 773 y FC(2)2235 682 y Fp(\021)2284 775 y Fu(2)2326 740 y Fq(j)2379 775 y Fu(+)18 b(2)2504 740 y FD(\000)p Fq(k)q(N)2655 775 y Fu(2)2697 740 y FD(\000)p Fq(j)s(N)2843 775 y Fs(k)p FE(g)s Fs(k)2970 790 y Fq(L)3016 773 y FC(1)3050 775 y Fs(k)p FF(P)3157 787 y Fq(k)3198 775 y FE(h)p Fs(k)3288 791 y Fq(H)3346 774 y Fm(\000)p Fl(N)3449 775 y Fs(k)p FE(f)9 b Fs(k)3583 801 y Fq(H)3641 774 y Fm(\000)p Fl(N)3637 821 y Fm(\000)p Fl(N)3742 775 y FE(:)0 1023 y Fu(Similarly)27 b(it)h(holds)f(that)276 1113 y Fp(X)250 1291 y Fe(p)p FD(\025\000)p Fr(1)435 1192 y Fs(j)p Ff(M)545 1158 y Fr(3)545 1212 y Fq(k)q(;)p Fe(p)641 1192 y Fu(\()p Ff(U)727 1204 y Fq(k)q FD(\000)p Fq(N)868 1212 y FC(0)905 1192 y FE(g)s(;)1003 1171 y Fu(~)985 1192 y Fs(P)1043 1204 y Fq(k)1083 1192 y FE(h;)14 b Fu([)1209 1171 y(~)1191 1192 y Fs(P)1249 1204 y Fq(k)1289 1192 y FE(;)g Ff(F)1377 1158 y Fr(2)1377 1212 y Fq(j)1414 1192 y Fu(])p FE(f)9 b Fu(\))p Fs(j)109 1495 y Fk(.)174 1507 y Fq(N)236 1495 y Fu(2)278 1461 y FD(\000)p Fr(2)p Fq(k)403 1495 y Fs(k)p FE(g)s Fs(k)530 1510 y Fq(L)576 1494 y FC(1)611 1403 y Fp(\020)780 1417 y(X)675 1599 y FD(j)p Fe(p)p FD(\000)p Fq(j)s FD(j\024)p Fe(n)p Fq(N)973 1607 y FC(0)1019 1495 y Fs(k)p Ff(F)1112 1507 y Fe(p)1168 1474 y 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begin H.R end 842 1691 a 842 1744 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (defi.2.2) cvn H.B /ANN pdfmark end 842 1744 a 0 TeXcolorgray 28 w Fu(for)g FF(F)1057 1756 y Fq(j)1120 1744 y Fu(and)g FF(P)1346 1756 y Fq(k)1415 1744 y Fu(and)g(patc)n(h)h(together)e(all)i(the)g(estimates,)f(w)n(e)g (ha)n(v)n(e)g(that)487 1920 y Fs(j)p Ff(D)579 1886 y Fr(2)579 1941 y(1)p Fq(;)p Fr(1)669 1920 y Fs(j)c Fk(.)780 1932 y Fq(N)866 1841 y Fp(X)908 2020 y Fq(k)1000 1920 y Fu(2)1042 1886 y FD(\000)p Fr(2)p Fq(k)1167 1920 y Fs(k)p FE(g)s Fs(k)1294 1935 y Fq(L)1340 1918 y FC(1)1375 1920 y Fs(k)p FF(F)1477 1932 y Fq(j)1511 1920 y FF(P)1576 1932 y Fq(k)1618 1920 y FE(h)p Fs(k)1708 1935 y Fq(L)1754 1918 y FC(2)1789 1920 y Fs(k)p FF(F)1891 1932 y Fq(j)1926 1920 y FF(P)1991 1932 y Fq(k)2032 1920 y FE(f)9 b Fs(k)2124 1935 y Fq(L)2170 1918 y FC(2)2205 1920 y Fu(2)2247 1886 y Fq(j)2300 1920 y Fu(+)18 b(2)2425 1886 y FD(\000)p Fq(j)s(N)2570 1920 y Fs(k)p FE(g)s Fs(k)2697 1935 y Fq(L)2743 1918 y FC(1)2778 1920 y Fs(k)p FE(h)p Fs(k)2910 1947 y Fq(H)2968 1920 y Fm(\000)p Fl(N)2964 1967 y Fm(\000)p Fl(N)3070 1920 y Fs(k)p FE(f)9 b Fs(k)3204 1947 y Fq(H)3262 1920 y Fm(\000)p Fl(N)3258 1967 y Fm(\000)p Fl(N)3364 1920 y FE(;)0 2151 y Fu(whic)n(h)28 b(is)f(enough)g(to)g(conclude)h (our)f(desired)g(result.)2102 b Fk(\003)0 2249 y SDict begin H.S end 0 2249 a 0 2249 a SDict begin 12 H.A end 0 2249 a 0 2249 a SDict begin [/View [/XYZ H.V]/Dest (subsubsection.2.4.2) cvn /DEST pdfmark end 0 2249 a 83 x Fu(2.4.2.)40 b Fj(Estimate)30 b(of)h Ff(D)751 2344 y Fr(2)788 2332 y Fj(.)42 b Fu(Then)28 b(w)n(e)f(giv)n(e)g(the)h(upp)r(er)g(b)r(ound)g (of)f Ff(D)2165 2344 y Fr(2)2202 2332 y Fu(.)0 2394 y SDict begin H.S end 0 2394 a 0 2394 a SDict begin 12 H.A end 0 2394 a 0 2394 a SDict begin [/View [/XYZ H.V]/Dest (prop.2.4) cvn /DEST pdfmark end 0 2394 a 0 TeXcolorgray 100 x FF(Prop)s(osition)j(2.4.)p 0 TeXcolorgray 40 w Fj(It)g(holds)h(that)0 2663 y SDict begin H.S end 0 2663 a 0 2663 a SDict begin 12 H.A end 0 2663 a 0 2663 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.38) cvn /DEST pdfmark end 0 2663 a 231 w Fs(j)p Ff(D)323 2675 y Fr(2)360 2663 y Fs(j)23 b Fk(.)471 2675 y Fq(N)576 2584 y Fp(X)557 2762 y Fe(l)p FD(\025\000)p Fr(1)795 2584 y Fp(X)728 2762 y Fe(p)p FD(\025)p Fq(j)s Fr(+)p Fq(N)949 2770 y FC(0)996 2663 y Fs(k)p FF(F)1098 2675 y Fe(p)1136 2663 y FF(P)1201 2675 y Fe(l)1227 2663 y FE(g)s Fs(k)1312 2678 y Fq(L)1358 2661 y FC(2)1393 2663 y Fs(k)p FF(F)1495 2675 y Fe(p)1533 2663 y FF(P)1598 2675 y Fe(l)1624 2663 y FE(h)p Fs(k)1714 2678 y Fq(L)1760 2661 y FC(2)1795 2663 y Fs(k)p FF(F)1897 2675 y Fq(j)1932 2663 y FF(P)1997 2675 y Fe(l)2022 2663 y FE(f)9 b Fs(k)2114 2678 y Fq(L)2160 2661 y FC(2)2195 2663 y Fu(2)2247 2606 y FC(3)p 2247 2615 29 3 v 2247 2648 a(2)2285 2628 y Fq(j)83 2902 y Fu(+)250 2823 y Fp(X)231 3002 y Fe(l)p FD(\025\000)p Fr(1)470 2823 y Fp(X)402 3001 y Fe(p)p FD(\024)p Fq(j)s FD(\000)p Fq(N)624 3009 y FC(0)671 2902 y Fu(2)723 2841 y Fd(p)p 722 2855 33 3 v 724 2888 a FC(2)769 2902 y Fs(k)p FF(F)871 2914 y Fe(p)910 2902 y FF(P)975 2914 y Fe(l)1000 2902 y FE(g)s Fs(k)1085 2917 y Fq(L)1131 2900 y FC(2)1166 2902 y Fs(k)p FF(F)1268 2914 y Fq(j)1303 2902 y FF(P)1368 2914 y Fe(l)1393 2902 y FE(h)p Fs(k)1483 2917 y Fq(L)1529 2900 y FC(2)1565 2902 y Fs(k)p FF(F)1667 2914 y Fq(j)1701 2902 y FF(P)1766 2914 y Fe(l)1792 2902 y FE(f)g Fs(k)1884 2917 y Fq(L)1930 2900 y FC(2)1965 2902 y Fu(2)2007 2868 y Fq(j)2060 2902 y Fu(+)2162 2823 y Fp(X)2143 3002 y Fe(l)p FD(\025\000)p Fr(1)2382 2823 y Fp(X)2314 3001 y Fe(a)p FD(\024)p Fq(j)s FD(\000)p Fq(N)2536 3009 y FC(0)2583 2902 y Fs(k)p FF(F)2685 2914 y Fq(j)2719 2902 y FF(P)2784 2914 y Fe(l)2810 2902 y FE(g)s Fs(k)2895 2917 y Fq(L)2941 2900 y FC(2)2976 2902 y Fs(k)p FF(F)3078 2914 y Fe(a)3116 2902 y FF(P)3181 2914 y Fe(l)3207 2902 y FE(h)p Fs(k)3297 2917 y Fq(L)3343 2900 y FC(2)3378 2902 y Fs(k)p FF(F)3480 2914 y Fq(j)3515 2902 y FF(P)3580 2914 y Fe(l)3605 2902 y FE(f)g Fs(k)3697 2917 y Fq(L)3743 2900 y FC(2)3778 2902 y Fu(2)3830 2845 y FC(3)p 3830 2854 29 3 v 3830 2888 a(2)3868 2868 y Fe(a)83 3150 y Fu(+)250 3071 y Fp(X)231 3249 y Fe(l)p FD(\025\000)p Fr(1)402 3150 y Fs(k)p FF(F)504 3162 y Fq(j)539 3150 y FF(P)604 3162 y Fe(l)629 3150 y FE(g)s Fs(k)714 3165 y Fq(L)760 3148 y FC(2)795 3150 y Fs(k)p FF(F)897 3162 y Fq(j)932 3150 y FF(P)997 3162 y Fe(l)1022 3150 y FE(h)p Fs(k)1112 3165 y Fq(L)1158 3148 y FC(2)1194 3150 y Fs(k)p FF(F)1296 3162 y Fq(j)1330 3150 y FF(P)1395 3162 y Fe(l)1421 3150 y FE(f)g Fs(k)1513 3165 y Fq(L)1559 3148 y FC(2)1594 3150 y Fu(2)1646 3093 y FC(3)p 1645 3102 V 1645 3135 a(2)1684 3115 y Fq(j)1737 3150 y Fu(+)1820 3057 y Fp(\020)1902 3071 y(X)1884 3249 y Fe(l)p FD(\025\000)p Fr(1)2122 3071 y Fp(X)2055 3249 y Fe(p)p FD(\025)p Fq(j)s Fr(+)p Fq(N)2276 3257 y FC(0)2323 3150 y Fs(k)p FF(F)2425 3162 y Fe(p)2463 3150 y FF(P)2528 3162 y Fe(l)2554 3150 y FE(g)s Fs(k)2639 3165 y Fq(L)2685 3148 y FC(2)2720 3150 y Fs(k)p FF(F)2822 3162 y 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1446 y FC(2)3129 1448 y Fu(2)3181 1391 y FC(3)p 3180 1400 V 3180 1433 a(2)3218 1413 y Fq(j)3253 1448 y FE(:)p 0 TeXcolorgray 386 w Fu(\(2.39\))p 0 TeXcolorgray 100 1688 a(Step)28 b(2:)36 b(Dy)n(adic)27 b(decomp)r(osition)g(in)h(phase)f(space.)p 100 1718 1663 4 v 33 w(Noticing)34 b(that)g(for)f FE(Q)2522 1700 y FD(\000)p Fr(1)2611 1688 y Fu(,)i(w)n(e)e(ha)n(v)n(e)g(that)h Fs(j)p FE(v)25 b Fs(\000)e FE(v)3397 1700 y FD(\003)3435 1688 y Fs(j)33 b Fk(.)g Fu(1)g(whic)n(h)0 1792 y(implies)28 b(that)g Fs(h)p FE(v)s Fs(i)c(\030)e(h)p FE(v)752 1804 y FD(\003)791 1792 y Fs(i)p Fu(.)37 b(Th)n(us)27 b(w)n(e)h(ha)n(v)n(e)e (that)287 1959 y Ff(D)356 1971 y Fr(2)417 1959 y Fu(=)523 1880 y Fp(X)504 2059 y Fe(l)p FD(\025\000)p Fr(1)662 1959 y Fs(h)p Ff(F)745 1971 y Fq(j)780 1959 y FE(Q)846 1971 y FD(\000)p Fr(1)935 1959 y Fu(\()p Fs(P)1025 1971 y Fe(l)1050 1959 y FE(g)s(;)1148 1938 y Fu(~)1130 1959 y Fs(P)1188 1971 y Fe(l)1212 1959 y FE(h)p Fu(\))19 b Fs(\000)f FE(Q)1460 1971 y FD(\000)p Fr(1)1549 1959 y Fu(\()p Fs(P)1639 1971 y Fe(l)1664 1959 y FE(g)s(;)c Ff(F)1795 1971 y Fq(j)1847 1938 y Fu(~)1829 1959 y Fs(P)1887 1971 y Fe(l)1912 1959 y FE(h)p Fu(\))p FE(;)g Ff(F)2080 1971 y Fq(j)2133 1938 y Fu(~)2115 1959 y Fs(P)2173 1971 y Fe(l)2198 1959 y FE(f)9 b Fs(i)2280 1971 y Fq(v)2337 1959 y Fu(+)2439 1880 y Fp(X)2420 2059 y Fe(l)p FD(\025\000)p Fr(1)2592 1892 y Fp(\000)2630 1959 y Fs(h)p FE(Q)2728 1971 y FD(\000)p Fr(1)2817 1959 y Fu(\()p Fs(P)2907 1971 y Fe(l)2932 1959 y FE(g)s(;)3030 1938 y Fu(~)3012 1959 y Fs(P)3070 1971 y Fe(l)3095 1959 y FE(h)p Fu(\))p FE(;)14 b Fu([)3253 1938 y(~)3235 1959 y Fs(P)3293 1971 y Fe(l)3318 1959 y FE(;)g Ff(F)3406 1925 y Fr(2)3406 1980 y Fq(j)3442 1959 y Fu(])p FE(f)9 b Fs(i)3547 1971 y Fq(v)329 2188 y Fu(+)p Fs(h)p FE(Q)492 2200 y FD(\000)p Fr(1)581 2188 y Fu(\()p Fs(P)671 2200 y Fe(l)696 2188 y FE(g)s(;)14 b Fu([)p Ff(F)850 2200 y Fq(j)884 2188 y FE(;)939 2167 y Fu(~)921 2188 y Fs(P)979 2200 y Fe(l)1004 2188 y Fu(])p FE(h)p Fu(\))p FE(;)g Ff(F)1195 2200 y Fq(j)1248 2167 y Fu(~)1230 2188 y Fs(P)1288 2200 y Fe(l)1312 2188 y FE(f)9 b Fs(i)1394 2200 y Fq(v)1452 2188 y Fu(+)18 b Fs(h)p FE(Q)1633 2200 y FD(\000)p Fr(1)1722 2188 y Fu(\()p Fs(P)1812 2200 y Fe(l)1837 2188 y FE(g)s(;)1935 2167 y Fu(~)1917 2188 y Fs(P)1975 2200 y Fe(l)2000 2188 y Ff(F)2051 2200 y Fq(j)2085 2188 y FE(h)p Fu(\))p FE(;)c Fu([)p Ff(F)2276 2200 y Fq(j)2311 2188 y FE(;)2367 2167 y Fu(~)2348 2188 y Fs(P)2406 2200 y Fe(l)2431 2188 y Fu(])p FE(f)9 b Fs(i)2536 2200 y Fq(v)2575 2120 y Fp(\001)2637 2188 y Fu(:=)22 b Ff(D)2816 2153 y Fr(1)2816 2208 y(2)2872 2188 y Fu(+)c Ff(D)3024 2153 y Fr(2)3024 2208 y(2)3061 2188 y FE(:)100 2343 y Fu(Thanks)27 b(to)g(\()p 1 0 0 TeXcolorrgb 531 2343 a SDict begin H.S end 531 2343 a 1 0 0 TeXcolorrgb Fu(2.39)p 1 0 0 TeXcolorrgb 679 2290 a SDict begin H.R end 679 2290 a 679 2343 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.39) cvn H.B /ANN pdfmark end 679 2343 a 0 TeXcolorgray 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TeXcolorgray Fu(\),)g(w)n(e)f (further)g(ha)n(v)n(e)g(the)h(decomp)r(osition)f(as)g(follo)n(ws:)62 1892 y Fi(C)117 1904 y Fr(1)178 1892 y Fu(=)335 1814 y Fp(X)266 1992 y Fe(a)p FD(\024)p Fe(p)p FD(\000)p Fq(N)493 2000 y FC(0)524 1892 y Fu(\(\003)614 1904 y Fr(1)651 1892 y Fu(\()688 1871 y(~)683 1892 y Ff(F)734 1904 y Fe(p)773 1892 y FE(a)817 1904 y Fe(l)842 1892 y FE(;)884 1871 y Fu(~)879 1892 y Ff(F)930 1904 y Fe(p)969 1892 y FE(c)1005 1904 y Fe(l)1030 1892 y FE(;)14 b Ff(F)1118 1904 y Fe(p)1157 1892 y FE(g)s(;)g Ff(F)1288 1904 y Fe(a)1326 1892 y FE(h;)g Fs(P)1469 1904 y Fq(k)1514 1871 y Fu(~)1509 1892 y Ff(F)1560 1904 y Fe(p)1599 1892 y FE(f)9 b Fu(\))18 b Fs(\000)g Fu(\003)1840 1904 y Fr(1)1877 1892 y Fu(\()1914 1871 y(~)1909 1892 y Ff(F)1960 1904 y Fe(p)1999 1892 y FE(a)2043 1904 y Fe(l)2068 1892 y FE(;)2110 1871 y Fu(~)2105 1892 y Ff(F)2156 1904 y Fe(p)2195 1892 y FE(c)2231 1904 y Fe(l)2256 1892 y FE(;)c Ff(F)2344 1904 y Fe(p)2382 1892 y FE(g)s(;)g Fs(P)2520 1904 y Fq(k)2561 1892 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Fs(k)p FE(f)d Fs(k)2883 1762 y Fq(H)2941 1746 y Fm(\000)p Fl(N)j FE(:)0 2072 y Fu(These)24 b(are)e(enough)i(to)f(conclude)h(\()p 1 0 0 TeXcolorrgb 1121 2072 a SDict begin H.S end 1121 2072 a 1 0 0 TeXcolorrgb Fu(2.41)p 1 0 0 TeXcolorrgb 1269 2019 a SDict begin H.R end 1269 2019 a 1269 2072 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.41) cvn H.B /ANN pdfmark end 1269 2072 a 0 TeXcolorgray Fu(\).)36 b(W)-7 b(e)24 b(remark)f(that)h(b)n(y)f (the)i(similar)e(argumen)n(t)g(\()p 1 0 0 TeXcolorrgb 2876 2072 a SDict begin H.S end 2876 2072 a 1 0 0 TeXcolorrgb Fu(2.41)p 1 0 0 TeXcolorrgb 3023 2019 a SDict begin H.R end 3023 2019 a 3023 2072 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.41) cvn H.B /ANN pdfmark end 3023 2072 a 0 TeXcolorgray Fu(\))i(holds)e(true)h(for)f FE(k)j Fu(=)d Fs(\000)p Fu(1.)100 2172 y(Step)28 b(2.)p 100 2201 254 4 v 27 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FC(2)668 660 y Fu(2)720 603 y Fd(b)p 719 612 34 3 v 722 645 a FC(2)785 660 y Fu(+)894 581 y Fp(X)868 759 y Fe(p)p FD(\025\000)p Fr(1)1054 660 y Fu(2)1096 625 y FD(\000)p Fq(k)q(N)1247 660 y Fs(k)p FF(F)1349 672 y Fe(p)1387 660 y FF(P)1452 672 y Fq(k)1493 660 y FE(g)s Fs(k)1578 675 y Fq(L)1624 658 y FC(2)1660 660 y Fs(k)p FF(F)1762 672 y Fe(p)1800 660 y FF(P)1865 672 y Fq(k)1906 660 y FE(f)g Fs(k)1998 675 y Fq(L)2044 658 y FC(2)2080 660 y Fs(k)p FF(P)2187 672 y Fq(k)2227 660 y FE(h)p Fs(k)2317 686 y Fq(H)2375 659 y Fm(\000)p Fl(N)2371 706 y Fm(\000)p Fl(N)2496 660 y Fu(+)2605 581 y Fp(X)2579 759 y Fe(p)p FD(\025\000)p Fr(1)2765 660 y Fu(2)2807 625 y FD(\000)p Fq(k)q(N)2958 660 y Fs(k)p FF(F)3060 672 y Fe(p)3098 660 y FF(P)3163 672 y Fq(k)3205 660 y FE(g)s Fs(k)3290 675 y Fq(L)3336 658 y FC(2)182 886 y Fs(\002k)p FF(F)349 898 y Fe(p)387 886 y FF(P)452 898 y Fq(k)494 886 y FE(h)p Fs(k)584 901 y Fq(L)630 884 y FC(2)665 886 y Fs(k)p FF(P)772 898 y Fq(k)813 886 y FE(f)g Fs(k)905 913 y Fq(H)963 886 y Fm(\000)p Fl(N)959 933 y Fm(\000)p Fl(N)1083 886 y Fu(+)18 b(2)1208 852 y FD(\000)p Fq(k)q(N)1359 886 y Fs(k)p FF(P)1466 898 y Fq(k)1507 886 y FE(g)s Fs(k)1592 903 y Fq(H)1650 886 y Fm(\000)p Fl(N)12 b Fs(k)p FF(P)1859 898 y Fq(k)1900 886 y FE(h)p Fs(k)1990 903 y Fq(H)2048 886 y Fm(\000)p Fl(N)h Fs(k)p FF(P)2258 898 y Fq(k)2298 886 y FE(f)c Fs(k)2390 903 y Fq(H)2448 886 y Fm(\000)p Fl(N)k FE(:)p 0 TeXcolorgray 1088 w Fu(\(2.42\))p 0 TeXcolorgray 0 TeXcolorgray 0 1058 a Fj(Pr)l(o)l(of.)p 0 TeXcolorgray 43 w Fu(Step)28 b(1.)p 260 1088 254 4 v 27 w(W)-7 b(e)28 b(\014rst)g(apply)f(the)h(decomp)r(osition)f(to)h Ff(E)1931 1070 y Fr(2)1996 1058 y Fu(in)f(frequency)h(space.)45 1227 y Ff(E)100 1239 y Fr(2)160 1227 y Fu(=)317 1148 y Fp(X)248 1326 y Fe(a)p FD(\024)p Fe(p)p FD(\000)p Fq(N)475 1334 y FC(0)506 1227 y Fu(\()p Fs(h)p FE(Q)636 1239 y FD(\000)p Fr(1)726 1227 y Fu(\()p Ff(F)809 1239 y Fe(p)848 1227 y FE(g)s(;)14 b Ff(F)979 1239 y Fe(a)1017 1227 y FE(h)p Fu(\))p FE(;)1139 1205 y Fu(~)1134 1227 y Ff(F)1185 1239 y Fe(p)1224 1227 y Fs(P)1282 1239 y Fq(k)1322 1227 y FE(f)9 b Fs(i)1404 1239 y Fq(v)1462 1227 y Fs(\000)18 b(h)p FE(Q)1643 1239 y FD(\000)p Fr(1)1732 1227 y Fu(\()p Ff(F)1815 1239 y Fe(p)1854 1227 y FE(g)s(;)c Ff(F)1985 1239 y Fe(a)2023 1227 y Fs(P)2081 1239 y Fq(k)2122 1227 y FE(h)p Fu(\))p FE(;)2244 1205 y Fu(~)2239 1227 y Ff(F)2290 1239 y Fe(p)2329 1227 y FE(f)9 b Fs(i)2411 1239 y Fq(v)2450 1227 y Fu(\))19 b(+)2610 1148 y Fp(X)2584 1326 y Fe(a)p FD(\025\000)p Fr(1)2755 1227 y Fu(\()p Fs(h)p FE(Q)2885 1239 y FD(\000)p Fr(1)2975 1227 y Fu(\()p Fs(S)3057 1239 y Fe(a)p FD(\000)p Fq(N)3197 1247 y FC(0)3233 1227 y FE(g)s(;)14 b Ff(F)3364 1239 y Fe(a)3402 1227 y FE(h)p Fu(\))p FE(;)3524 1205 y Fu(~)3519 1227 y Ff(F)3570 1239 y Fe(a)3609 1227 y Fs(P)3667 1239 y Fq(k)3708 1227 y FE(f)9 b Fs(i)3790 1239 y Fq(v)156 1466 y Fs(\000)18 b(h)p FE(Q)337 1478 y FD(\000)p Fr(1)426 1466 y Fu(\()p Ff(F)509 1478 y Fe(p)548 1466 y FE(g)s(;)c Ff(F)679 1478 y Fe(a)717 1466 y Fs(P)775 1478 y Fq(k)816 1466 y FE(h)p Fu(\))p FE(;)937 1444 y Fu(~)933 1466 y Ff(F)984 1478 y Fe(a)1022 1466 y FE(f)9 b Fs(i)1104 1478 y Fq(v)1144 1466 y Fu(\))18 b(+)1303 1387 y Fp(X)1277 1565 y Fe(p)p FD(\025\000)p Fr(1)1449 1466 y Fu(\()p Fs(h)p FE(Q)1579 1478 y FD(\000)p Fr(1)1668 1466 y Fu(\()p Ff(F)1751 1478 y Fe(p)1790 1466 y FE(g)s(;)1875 1444 y Fu(~)1870 1466 y Ff(F)1921 1478 y Fe(p)1960 1466 y FE(h)p Fu(\))p FE(;)2081 1444 y Fu(~)2077 1466 y Ff(F)2128 1478 y Fe(p)2166 1466 y Fs(P)2224 1478 y Fq(k)2265 1466 y FE(f)9 b Fs(i)2347 1478 y Fq(v)2405 1466 y Fs(\000)18 b(h)p FE(Q)2586 1478 y FD(\000)p Fr(1)2675 1466 y Fu(\()p Ff(F)2758 1478 y Fe(p)2797 1466 y FE(g)s(;)2881 1444 y Fu(~)2877 1466 y Ff(F)2928 1478 y Fe(p)2966 1466 y Fs(P)3024 1478 y Fq(k)3065 1466 y FE(h)p Fu(\))p FE(;)3186 1444 y Fu(~)3182 1466 y Ff(F)3233 1478 y Fe(p)3271 1466 y FE(f)9 b Fs(i)3353 1478 y Fq(v)3393 1466 y Fu(\))156 1705 y(+)308 1626 y Fp(X)239 1805 y Fe(b)p FD(\024)p Fe(p)p FD(\000)p Fq(N)467 1813 y FC(0)498 1705 y Fs(h)p FE(Q)596 1717 y FD(\000)p Fr(1)685 1705 y Fu(\()p Ff(F)768 1717 y Fe(p)807 1705 y FE(g)s(;)891 1683 y Fu(~)887 1705 y Ff(F)938 1717 y Fe(p)976 1705 y FE(h)p Fu(\))p FE(;)14 b Ff(F)1144 1717 y Fe(b)1184 1705 y Fs(P)1242 1717 y Fq(k)1282 1705 y FE(f)9 b Fs(i)1364 1717 y Fq(v)1422 1705 y Fs(\000)18 b(h)p FE(Q)1603 1717 y FD(\000)p Fr(1)1692 1705 y Fu(\()p Ff(F)1775 1717 y Fe(p)1814 1705 y FE(g)s(;)1898 1683 y Fu(~)1894 1705 y Ff(F)1945 1717 y Fe(p)1983 1705 y Fs(P)2041 1717 y Fq(k)2082 1705 y FE(h)p Fu(\))p FE(;)c Ff(F)2250 1717 y Fe(b)2290 1705 y FE(f)9 b Fs(i)2372 1717 y Fq(v)2411 1705 y Fu(\))23 b(:=)g Fi(C)2646 1671 y FD(0)2670 1717 y Fr(1)2725 1705 y Fu(+)18 b Fi(C)2878 1671 y FD(0)2901 1717 y Fr(2)2957 1705 y Fu(+)g Fi(C)3109 1671 y FD(0)3132 1717 y Fr(3)3188 1705 y Fu(+)g Fi(C)3340 1671 y FD(0)3364 1717 y Fr(4)3401 1705 y FE(;)0 1930 y Fu(where)80 2066 y Fi(C)149 2031 y FD(0)173 2078 y Fr(1)233 2066 y Fu(=)390 1987 y Fp(X)321 2165 y Fe(a)p FD(\024)p Fe(p)p FD(\000)p Fq(N)548 2173 y FC(0)579 2066 y Fs(hP)669 2078 y Fq(k)710 2066 y FE(Q)776 2078 y FD(\000)p Fr(1)865 2066 y Fu(\()p Ff(F)948 2078 y Fe(p)987 2066 y FE(g)s(;)c Ff(F)1118 2078 y Fe(a)1156 2066 y FE(h)p Fu(\))19 b Fs(\000)f FE(Q)1404 2078 y FD(\000)p Fr(1)1492 2066 y Fu(\()p Ff(F)1575 2078 y Fe(p)1614 2066 y FE(g)s(;)c Fs(P)1752 2078 y Fq(k)1792 2066 y Ff(F)1843 2078 y Fe(a)1882 2066 y FE(h)p Fu(\))p FE(;)2004 2044 y Fu(~)1999 2066 y Ff(F)2050 2078 y Fe(p)2089 2066 y FE(f)9 b Fs(i)2171 2078 y Fq(v)2229 2066 y Fu(+)2381 1987 y Fp(X)2312 2165 y Fe(a)p FD(\024)p Fe(p)p FD(\000)p Fq(N)2539 2173 y FC(0)2570 2066 y Fs(h)p FE(Q)2668 2078 y FD(\000)p Fr(1)2757 2066 y Fu(\()p Ff(F)2840 2078 y Fe(p)2879 2066 y FE(g)s(;)14 b Ff(F)3010 2078 y Fe(a)3049 2066 y FE(h)p Fu(\))p FE(;)g Fu([)3194 2044 y(~)3189 2066 y Ff(F)3240 2078 y Fe(p)3278 2066 y FE(;)g Fs(P)3373 2078 y Fq(k)3414 2066 y Fu(])p FE(f)9 b Fs(i)3519 2078 y Fq(v)228 2305 y Fu(+)381 2226 y Fp(X)311 2404 y Fe(a)p FD(\024)p Fe(p)p FD(\000)p Fq(N)538 2412 y FC(0)570 2305 y Fs(h)p FE(Q)668 2317 y FD(\000)p Fr(1)757 2305 y Fu(\()p Ff(F)840 2317 y Fe(p)879 2305 y FE(g)s(;)14 b Fu([)p Fs(P)1040 2317 y Fq(k)1080 2305 y FE(;)g Ff(F)1168 2317 y Fe(a)1207 2305 y Fu(])p FE(h)p Fu(\))p FE(;)1351 2283 y Fu(~)1347 2305 y Ff(F)1398 2317 y Fe(p)1437 2305 y FE(f)9 b Fs(i)1519 2317 y Fq(v)1581 2305 y Fu(:=)23 b Fi(C)1761 2270 y FD(0)1784 2259 y Fr(1)1784 2325 y(1)1840 2305 y Fu(+)18 b Fi(C)1992 2270 y FD(0)2016 2259 y Fr(2)2016 2325 y(1)2071 2305 y Fu(+)g Fi(C)2224 2270 y FD(0)2247 2259 y Fr(3)2247 2325 y(1)2284 2305 y Fu(;)80 2544 y Fi(C)149 2509 y FD(0)173 2556 y Fr(2)233 2544 y Fu(=)347 2465 y Fp(X)321 2643 y Fe(a)p FD(\025\000)p Fr(1)492 2544 y Fs(hP)582 2556 y Fq(k)623 2544 y FE(Q)689 2556 y FD(\000)p Fr(1)778 2544 y Fu(\()p Fs(S)860 2556 y Fe(a)p FD(\000)p Fq(N)1000 2564 y FC(0)1036 2544 y FE(g)s(;)c Ff(F)1167 2556 y Fe(a)1206 2544 y FE(h)p Fu(\))k Fs(\000)g FE(Q)1453 2556 y FD(\000)p Fr(1)1542 2544 y Fu(\()p Fs(S)1624 2556 y Fe(a)p FD(\000)p Fq(N)1764 2564 y FC(0)1800 2544 y FE(g)s(;)c Fs(P)1938 2556 y Fq(k)1979 2544 y Ff(F)2030 2556 y Fe(a)2068 2544 y FE(h)p Fu(\))p FE(;)2190 2522 y Fu(~)2185 2544 y Ff(F)2236 2556 y Fe(a)2275 2544 y FE(f)9 b Fs(i)2357 2556 y Fq(v)2415 2544 y Fu(+)2524 2465 y Fp(X)2498 2643 y Fe(a)p FD(\025\000)p Fr(1)2670 2544 y Fs(h)p FE(Q)2768 2556 y FD(\000)p Fr(1)2857 2544 y Fu(\()p Fs(S)2939 2556 y Fe(a)p FD(\000)p Fq(N)3079 2564 y FC(0)3115 2544 y FE(g)s(;)14 b Ff(F)3246 2556 y Fe(a)3284 2544 y FE(h)p Fu(\))p FE(;)g Fu([)3429 2522 y(~)3424 2544 y Ff(F)3475 2556 y Fe(a)3514 2544 y FE(;)g Fs(P)3609 2556 y Fq(k)3650 2544 y Fu(])p FE(f)9 b Fs(i)3755 2556 y Fq(v)228 2783 y Fu(+)337 2704 y Fp(X)311 2882 y Fe(a)p FD(\025\000)p Fr(1)483 2783 y Fs(h)p FE(Q)581 2795 y FD(\000)p Fr(1)670 2783 y Fu(\()p Fs(S)752 2795 y Fe(a)p FD(\000)p Fq(N)892 2803 y FC(0)929 2783 y FE(g)s(;)14 b Fu([)p Fs(P)1090 2795 y Fq(k)1130 2783 y FE(;)g Ff(F)1218 2795 y Fe(a)1256 2783 y Fu(])p FE(h)p Fu(\))p FE(;)1401 2761 y Fu(~)1396 2783 y Ff(F)1447 2795 y Fe(a)1486 2783 y FE(f)9 b Fs(i)1568 2795 y Fq(v)1631 2783 y Fu(:=)22 b Fi(C)1811 2748 y FD(0)1834 2736 y Fr(1)1834 2803 y(2)1890 2783 y Fu(+)c Fi(C)2042 2748 y FD(0)2065 2736 y Fr(2)2065 2803 y(2)2121 2783 y Fu(+)g Fi(C)2273 2748 y FD(0)2297 2736 y Fr(3)2297 2803 y(2)2334 2783 y Fu(;)196 3042 y Fi(C)266 3008 y FD(0)289 3054 y Fr(3)349 3042 y Fu(=)463 2963 y Fp(X)437 3141 y Fe(p)p FD(\025\000)p Fr(1)609 3042 y Fs(hP)699 3054 y Fq(k)740 3042 y FE(Q)806 3054 y FD(\000)p Fr(1)894 3042 y Fu(\()p Ff(F)977 3054 y Fe(p)1016 3042 y FE(g)s(;)1100 3020 y Fu(~)1096 3042 y Ff(F)1147 3054 y Fe(p)1186 3042 y FE(h)p Fu(\))g Fs(\000)g FE(Q)1433 3054 y FD(\000)p Fr(1)1522 3042 y Fu(\()p Ff(F)1605 3054 y Fe(p)1644 3042 y FE(g)s(;)c Fs(P)1782 3054 y Fq(k)1827 3020 y Fu(~)1822 3042 y Ff(F)1873 3054 y Fe(p)1912 3042 y FE(h)p Fu(\))p FE(;)2033 3020 y Fu(~)2029 3042 y Ff(F)2080 3054 y Fe(p)2118 3042 y FE(f)9 b Fs(i)2200 3054 y Fq(v)2258 3042 y Fu(+)2367 2963 y Fp(X)2341 3141 y Fe(p)p FD(\025\000)p Fr(1)2513 3042 y Fs(h)p FE(Q)2611 3054 y FD(\000)p Fr(1)2700 3042 y Fu(\()p Ff(F)2783 3054 y Fe(p)2822 3042 y FE(g)s(;)2906 3020 y Fu(~)2902 3042 y Ff(F)2953 3054 y Fe(p)2991 3042 y FE(h)p Fu(\))p FE(;)14 b Fu([)3136 3020 y(~)3131 3042 y Ff(F)3182 3054 y Fe(p)3221 3042 y FE(;)g Fs(P)3316 3054 y Fq(k)3357 3042 y Fu(])p FE(f)9 b Fs(i)3462 3054 y Fq(v)345 3281 y Fu(+)454 3202 y Fp(X)428 3380 y Fe(p)p FD(\025\000)p Fr(1)599 3281 y Fs(h)p FE(Q)697 3293 y FD(\000)p Fr(1)786 3281 y Fu(\()p Ff(F)869 3293 y Fe(p)908 3281 y FE(g)s(;)14 b Fu([)p Fs(P)1069 3293 y Fq(k)1110 3281 y FE(;)1151 3259 y Fu(~)1147 3281 y Ff(F)1198 3293 y Fe(p)1236 3281 y Fu(])p FE(h)p Fu(\))p FE(;)1381 3259 y Fu(~)1376 3281 y Ff(F)1427 3293 y Fe(p)1466 3281 y FE(f)9 b Fs(i)1548 3293 y Fq(v)1610 3281 y Fu(:=)23 b Fi(C)1790 3247 y FD(0)1814 3235 y Fr(1)1814 3302 y(3)1869 3281 y Fu(+)18 b Fi(C)2022 3247 y FD(0)2045 3235 y Fr(2)2045 3302 y(3)2101 3281 y Fu(+)g Fi(C)2253 3247 y FD(0)2276 3235 y Fr(3)2276 3302 y(3)2314 3281 y Fu(;)196 3520 y Fi(C)266 3486 y FD(0)289 3532 y Fr(4)349 3520 y Fu(=)507 3441 y Fp(X)437 3620 y Fe(b)p FD(\024)p Fe(p)p FD(\000)p Fq(N)665 3628 y FC(0)696 3520 y Fs(hP)786 3532 y Fq(k)827 3520 y FE(Q)893 3532 y FD(\000)p Fr(1)982 3520 y Fu(\()p Ff(F)1065 3532 y Fe(p)1104 3520 y FE(g)s(;)1188 3498 y Fu(~)1184 3520 y Ff(F)1235 3532 y Fe(p)1273 3520 y FE(h)p Fu(\))h Fs(\000)f FE(Q)1521 3532 y FD(\000)p Fr(1)1609 3520 y Fu(\()p Ff(F)1692 3532 y Fe(p)1731 3520 y FE(g)s(;)c Fs(P)1869 3532 y Fq(k)1914 3498 y Fu(~)1910 3520 y Ff(F)1961 3532 y Fe(p)1999 3520 y FE(h)p Fu(\))p FE(;)g Ff(F)2167 3532 y Fe(b)2207 3520 y FE(f)9 b Fs(i)2289 3532 y Fq(v)2347 3520 y Fu(+)2499 3441 y Fp(X)2430 3620 y Fe(b)p FD(\024)p Fe(p)p FD(\000)p Fq(N)2658 3628 y FC(0)2689 3520 y Fs(h)p FE(Q)2787 3532 y FD(\000)p Fr(1)2876 3520 y Fu(\()p Ff(F)2959 3532 y Fe(p)2998 3520 y FE(g)s(;)3082 3498 y Fu(~)3078 3520 y Ff(F)3129 3532 y Fe(p)3167 3520 y FE(h)p Fu(\))p FE(;)14 b Fu([)p Ff(F)3358 3532 y Fe(b)3398 3520 y FE(;)g Fs(P)3493 3532 y Fq(k)3533 3520 y Fu(])p FE(f)9 b Fs(i)3638 3532 y Fq(v)345 3759 y Fu(+)497 3681 y Fp(X)428 3859 y Fe(b)p FD(\024)p Fe(p)p FD(\000)p Fq(N)656 3867 y FC(0)687 3759 y Fs(h)p FE(Q)785 3771 y FD(\000)p Fr(1)874 3759 y Fu(\()p Ff(F)957 3771 y Fe(p)996 3759 y FE(g)s(;)14 b Fu([)p Fs(P)1157 3771 y Fq(k)1197 3759 y FE(;)1239 3738 y Fu(~)1234 3759 y Ff(F)1285 3771 y Fe(p)1324 3759 y Fu(])p FE(h)p Fu(\))p FE(;)g Ff(F)1515 3771 y Fe(b)1554 3759 y FE(f)9 b Fs(i)1636 3771 y Fq(v)1699 3759 y Fu(:=)22 b Fi(C)1879 3725 y FD(0)1902 3713 y Fr(1)1902 3780 y(4)1958 3759 y Fu(+)c Fi(C)2110 3725 y FD(0)2133 3713 y Fr(2)2133 3780 y(4)2189 3759 y Fu(+)g Fi(C)2342 3725 y FD(0)2365 3713 y Fr(3)2365 3780 y(4)2402 3759 y FE(:)100 3989 y Fu(In)36 b(the)h(sequel,)i(w)n(e)d(will)h(c)n(ho)r(ose)e Fi(C)1284 3959 y FD(0)1308 3947 y Fr(1)1308 4009 y(1)1345 3989 y FE(;)14 b Fi(C)1451 3959 y FD(0)1474 3947 y Fr(1)1474 4009 y(4)1512 3989 y Fu(,)39 b Fi(C)1643 3959 y FD(0)1666 3947 y Fr(2)1666 4009 y(4)1740 3989 y Fu(and)e Fi(C)1980 3959 y FD(0)2003 3947 y Fr(3)2003 4009 y(1)2077 3989 y Fu(as)f(t)n(ypical)g(terms)g(since)h(the)g(others)e(can)i(b)r(e)f (treated)0 4089 y(similarly)-7 b(.)36 b(F)-7 b(or)27 b FE(k)f Fs(\025)d Fu(0,)k(w)n(e)g(ha)n(v)-5 b(a)27 b(that)99 4266 y SDict begin H.S end 99 4266 a 99 4266 a SDict begin 12 H.A end 99 4266 a 99 4266 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.43) cvn /DEST pdfmark end 99 4266 a 166 w Fs(hP)355 4278 y Fq(k)396 4266 y FE(Q)462 4278 y FD(\000)p Fr(1)551 4266 y Fu(\()p FE(g)s(;)14 b(h)p Fu(\))k Fs(\000)g FE(Q)910 4278 y FD(\000)p Fr(1)999 4266 y Fu(\()p FE(g)s(;)c Fs(P)1169 4278 y Fq(k)1209 4266 y FE(h)p Fu(\))p FE(;)g(f)9 b Fs(i)1408 4278 y Fq(v)1470 4266 y Fu(=)1558 4153 y Fp(Z)1641 4266 y Fu(\()p FE(a)1717 4278 y FD(\000)p Fr(1)1825 4266 y Fs(\003)18 b FE(g)s Fu(\))23 b(:)2029 4199 y Fp(\000)2067 4266 y Fu(2)2109 4232 y FD(\000)p Fr(2)p Fq(k)2234 4266 y Fs(r)2303 4232 y Fr(2)2341 4266 y FE(')p Fu(\(2)2469 4232 y FD(\000)p Fq(k)2561 4266 y FE(v)s Fu(\))p FE(hf)28 b Fu(+)18 b(2)2878 4232 y Fr(1)p FD(\000)p Fq(k)3003 4266 y Fs(r)p FE(')p Fu(\(2)3200 4232 y FD(\000)p Fq(k)3293 4266 y FE(v)s Fu(\))h Fs(\012)f(r)p FE(f)9 b(h)3637 4199 y Fp(\001)3689 4266 y FE(dv)1759 4484 y Fu(+2)1866 4450 y Fr(1)p FD(\000)p Fq(k)2005 4371 y Fp(Z)2088 4484 y Fu(\()p FE(b)2156 4496 y FD(\000)p Fr(1)2264 4484 y Fs(\003)18 b FE(g)s Fu(\))g Fs(\001)g(r)p FE(')p Fu(\(2)2655 4450 y FD(\000)p Fq(k)2748 4484 y FE(v)s Fu(\))p FE(hf)9 b(dv)p 0 TeXcolorgray 658 w Fu(\(2.43\))p 0 TeXcolorgray 1095 4702 a(=)1183 4589 y Fp(Z)1266 4702 y Fu(\()p FE(a)1342 4714 y FD(\000)p Fr(1)1450 4702 y Fs(\003)17 b FE(g)s Fu(\))23 b(:)1654 4634 y Fp(\000)1692 4702 y Fu(2)1734 4667 y FD(\000)p Fr(2)p Fq(k)1859 4702 y Fs(r)1928 4667 y Fr(2)1965 4702 y FE(')p Fu(\(2)2093 4667 y FD(\000)p Fq(k)2186 4702 y FE(v)s Fu(\))p FE(hf)28 b Fu(+)18 b(2)2503 4667 y Fr(1)p FD(\000)p Fq(k)2628 4702 y Fs(r)p FE(')p 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SDict begin H.R end 2861 2984 a 2861 3037 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (lem.9.5) cvn H.B /ANN pdfmark end 2861 3037 a 0 TeXcolorgray Fu(,)h(one)f(can)h(c)n(hec)n(k)e(that)46 3131 y Fp(Z)129 3244 y Fu(\()p FE(a)19 b Fs(\003)f FE(f)9 b Fu(\))23 b(:)g Fs(r)p Ff(F)555 3256 y Fq(j)590 3244 y Fs(P)648 3256 y Fq(k)688 3244 y FE(f)k Fs(\012)18 b(r)p Ff(F)959 3256 y Fq(j)994 3244 y Fs(P)1052 3256 y Fq(k)1093 3244 y FE(f)9 b(dv)26 b Fs(\025)c FE(\024)p Fs(k)p Ff(F)1480 3256 y Fq(j)1514 3244 y Fs(P)1572 3256 y Fq(k)1613 3244 y FE(f)9 b Fs(k)1705 3210 y Fr(2)1705 3266 y Fq(L)1751 3250 y FC(2)1786 3244 y Fu(2)1828 3210 y FD(\000)p Fr(3)p Fq(k)1954 3244 y Fu(2)1996 3210 y Fr(2)p Fq(j)2082 3244 y Fs(\000)18 b FE(C)2224 3256 y Fq(N)2301 3152 y Fp(\020)2350 3244 y Fs(k)p FF(F)2452 3256 y Fq(j)2487 3244 y FF(P)2552 3256 y Fq(k)2593 3244 y FE(f)9 b Fs(k)2685 3210 y Fr(2)2685 3266 y Fq(L)2731 3250 y FC(2)2766 3244 y Fu(2)2808 3210 y FD(\000)p Fr(3)p 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4232 y Fq(k)2307 4220 y FE(Q)p Fu(\()p FE(f)t(;)c(f)9 b Fu(\))18 b Fs(\000)g FE(Q)p Fu(\()p FE(f)t(;)c Fs(P)2908 4232 y Fq(k)2949 4220 y FE(f)9 b Fu(\))p FE(;)14 b Ff(F)3119 4185 y Fr(2)3119 4240 y Fq(j)3155 4220 y Fs(P)3213 4232 y Fq(k)3254 4220 y FE(f)9 b Fs(i)3336 4232 y Fq(v)3375 4220 y FE(:)p 0 TeXcolorgray 305 w Fu(\(3.6\))p 0 TeXcolorgray 0 4392 a(Th)n(us)27 b(b)n(y)h(virtue)f(of)34 b(\()p 1 0 0 TeXcolorrgb 701 4392 a SDict begin H.S end 701 4392 a 1 0 0 TeXcolorrgb Fu(2.35)p 1 0 0 TeXcolorrgb 849 4338 a SDict begin H.R end 849 4338 a 849 4392 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.35) cvn H.B /ANN pdfmark end 849 4392 a 0 TeXcolorgray Fu(\)\()p 1 0 0 TeXcolorrgb 913 4392 a SDict begin H.S end 913 4392 a 1 0 0 TeXcolorrgb Fu(2.38)p 1 0 0 TeXcolorrgb 1061 4338 a SDict begin H.R end 1061 4338 a 1061 4392 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.38) cvn H.B /ANN pdfmark 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FE(f)g Fs(k)3113 5023 y Fr(2)3113 5080 y Fq(L)3159 5063 y FC(2)3213 5058 y Fu(+)18 b(2)3338 5023 y FD(\000)p Fq(j)s(N)3483 5058 y Fs(k)p FE(f)9 b Fs(k)3617 5023 y Fr(2)3617 5083 y Fq(L)3663 5063 y FC(2)3663 5103 y Fm(\000)p Fl(N)3764 5058 y Fu(\))p FE(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 24 24 TeXDict begin HPSdict begin 24 23 bop 0 TeXcolorgray 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 0 TeXcolorgray -8 -245 a SDict begin H.S end -8 -245 a -8 -245 a SDict begin H.R end -8 -245 a -8 -245 a SDict begin [/View [/XYZ H.V]/Dest (page.24) cvn /DEST pdfmark end -8 -245 a 0 TeXcolorgray 0 TeXcolorgray 0 -170 a Fr(24)1263 b(LING-BING)24 b(HE,)e(JIE)g(JI,)g(AND)g(YUE)g(LUO) p 0 TeXcolorgray 0 29 a Fu(Com)n(bining)27 b(the)h(ab)r(o)n(v)n(e)e 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1796 1364 29 3 v 1796 1397 a(2)1835 1377 y Fq(;)p Fh(s)1729 1448 y Fr(3)1882 1426 y Fu(\))19 b Fs(\\)g FE(L)2064 1392 y Fr(2)2100 1426 y Fu(\([0)p FE(;)14 b(T)e Fu(];)i FE(H)2440 1355 y FC(1)p 2440 1364 V 2440 1397 a(2)2478 1377 y Fq(;)p Fh(s)2433 1437 y FC(3)p 2433 1446 V 2433 1480 a(2)2526 1426 y Fu(\))p FE(:)p 0 TeXcolorgray 1122 w Fu(\(3.9\))p 0 TeXcolorgray 100 1641 a(Step)28 b(3:)36 b(Pro)r(of)27 b(of)g(the)h(con)n(tin)n(uit)n(y)-7 b(.)p 100 1671 1137 4 v 27 w(No)n(w,)27 b(w)n(e)g(will)g(address)f(the)i(issue)f(of)g FE(f)k Fs(2)24 b FE(C)6 b Fu(\([0)p FE(;)14 b(T)e Fu(];)i FE(H)3032 1593 y FD(\000)3092 1570 y FC(1)p 3092 1579 29 3 v 3092 1613 a(2)3131 1593 y Fq(;)p Fh(s)3025 1663 y Fr(3)3178 1641 y Fu(\).)37 b(Since)28 b(w)n(e)f(already)0 1750 y(ha)n(v)n(e)c(that)i(con)n(tin)n(uit)n(y)f(of)g(function)h FE(X)7 b Fu(\()p FE(t)p Fu(\))23 b Fs(\030)1425 1762 y Fh(s)1476 1750 y Fs(k)p FE(f)9 b Fu(\()p FE(t)p Fu(\))p Fs(k)1704 1720 y Fr(2)1704 1818 y Fq(H)1762 1776 y Fm(\000)1817 1760 y FC(1)p 1817 1767 V 1817 1798 a(2)1855 1776 y Fl(;)p Fg(s)1758 1836 y FC(3)1924 1750 y Fu(for)24 b FE(t)f Fs(2)g Fu([0)p FE(;)14 b(T)e Fu(],)24 b(th)n(us)g(b)n(y)h(Randon-Riesz) e(Theorem,)h(w)n(e)h(only)0 1942 y(need)j(to)f(sho)n(w)g(the)h(con)n (tin)n(uit)n(y)f(of)g(function)i Fs(h)p FE(f)9 b Fu(\()p FE(t)p Fu(\))p FE(;)14 b(\036)p Fs(i)28 b Fu(with)h FE(\036)23 b Fs(2)g FE(H)2204 1871 y FC(1)p 2204 1880 V 2204 1913 a(2)2242 1893 y Fq(;)p FD(\000)p Fh(s)2187 1964 y FD(\000)p Fr(3)2369 1942 y Fu(whic)n(h)28 b(is)g(the)f(dual)h(space)f(of)g FE(H)3410 1893 y FD(\000)3472 1871 y FC(1)p 3472 1880 V 3472 1913 a(2)3510 1893 y Fq(;)p Fh(s)3403 1964 y Fr(3)3558 1942 y Fu(.)100 2077 y(Indeed,)h(b)n(y)g(the)h(same)f(argumen)n(t)f(as) h(\()p 1 0 0 TeXcolorrgb 1369 2077 a SDict begin H.S end 1369 2077 a 1 0 0 TeXcolorrgb Fu(3.4)p 1 0 0 TeXcolorrgb 1475 2023 a SDict begin H.R end 1475 2023 a 1475 2077 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.4) cvn H.B /ANN pdfmark end 1475 2077 a 0 TeXcolorgray Fu(\),)h(the)g(con)n(tin) n(uit)n(y)f(holds)g(true)g(for)g(an)n(y)f FE(\036)e Fs(2)f FE(C)2993 2047 y FD(1)2987 2097 y Fq(c)3064 2077 y Fu(.)39 b(If)29 b FE(\036)c Fs(2)f FE(H)3449 2006 y FC(1)p 3449 2015 29 3 v 3449 2048 a(2)3487 2028 y Fq(;)p FD(\000)p Fh(s)3432 2099 y FD(\000)p Fr(3)3587 2077 y Fu(,)k(c)n(ho)r(ose)0 2212 y Fs(f)p FE(\036)91 2224 y Fq(n)136 2212 y Fs(g)23 b(\032)f FE(C)353 2182 y FD(1)347 2232 y Fq(c)452 2212 y Fu(suc)n(h)27 b(that)h FE(\036)868 2224 y Fq(n)937 2212 y Fs(!)23 b FE(\036)28 b Fu(in)g FE(H)1302 2141 y FC(1)p 1302 2150 V 1302 2183 a(2)1341 2163 y Fq(;)p FD(\000)p Fh(s)1286 2234 y FD(\000)p Fr(3)1440 2212 y Fu(,)g(then)455 2367 y(lim)452 2417 y Fq(t)p FD(!)p Fq(s)574 2367 y Fu(\()p FE(f)9 b Fu(\()p FE(t)p Fu(\))19 b Fs(\000)f FE(f)9 b Fu(\()p FE(s)p Fu(\))p FE(;)14 b(\036)p Fu(\))84 b(=)i(lim)1355 2417 y Fq(t)p FD(!)p Fq(s)1477 2367 y Fu(\()p FE(f)9 b Fu(\()p FE(t)p Fu(\))19 b Fs(\000)f FE(f)9 b Fu(\()p FE(s)p Fu(\))p FE(;)14 b(\036)1994 2379 y Fq(n)2040 2367 y Fu(\))k(+)k(lim)2173 2417 y Fq(t)p FD(!)p Fq(s)2296 2367 y Fu(\()p FE(f)9 b Fu(\()p FE(t)p Fu(\))19 b Fs(\000)f FE(f)9 b Fu(\()p FE(s)p Fu(\))p FE(;)14 b(\036)19 b Fs(\000)f FE(\036)2964 2379 y Fq(n)3010 2367 y Fu(\))1207 2521 y Fs(\024)86 b Fu(lim)1355 2571 y Fq(t)p FD(!)p Fq(s)1477 2521 y Fu(\()p FE(f)9 b Fu(\()p FE(t)p Fu(\))19 b Fs(\000)f FE(f)9 b Fu(\()p FE(s)p Fu(\))p FE(;)14 b(\036)1994 2533 y Fq(n)2040 2521 y Fu(\))k(+)g(2)56 b(sup)2229 2594 y Fq(t)p FD(2)p Fr([0)p Fq(;T)9 b Fr(])2451 2521 y Fs(k)p FE(f)g Fu(\()p FE(t)p Fu(\))p Fs(k)2679 2577 y Fq(H)2737 2535 y Fm(\000)2792 2519 y FC(1)p 2792 2526 V 2792 2557 a(2)2830 2535 y Fl(;)p Fg(s)2733 2595 y FC(3)2874 2521 y Fs(k)p FE(\036)19 b Fs(\000)f FE(\036)3116 2533 y Fq(n)3161 2521 y Fs(k)3203 2577 y Fq(H)3272 2519 y FC(1)p 3272 2526 V 3272 2557 a(2)3310 2535 y Fl(;)p Fm(\000)p Fg(s)3257 2596 y Fm(\000)p FC(3)3399 2521 y FE(:)0 2743 y Fu(Therefore)26 b(from)i(\()p 1 0 0 TeXcolorrgb 605 2743 a SDict begin H.S end 605 2743 a 1 0 0 TeXcolorrgb Fu(3.9)p 1 0 0 TeXcolorrgb 711 2690 a SDict begin H.R end 711 2690 a 711 2743 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.9) cvn H.B /ANN pdfmark end 711 2743 a 0 TeXcolorgray Fu(\),)g(the)g(ab)r(o)n(v)n(e)e(limit)j(is)e(0)g(if)h (w)n(e)g(tak)n(e)e(the)i(su\016cien)n(tly)g(large)e FE(n)i Fu(and)f FE(t)h Fu(approac)n(hes)d FE(s)p Fu(.)100 2843 y(Step)j(4:)36 b(Exten)n(tion)27 b(to)h(the)g FE(H)1095 2813 y Fh(r)1148 2843 y Fu(space.)p 100 2872 1266 4 v 30 w(F)-7 b(or)30 b(the)h(case)e FE(f)1913 2855 y Fr(0)1978 2843 y Fs(2)f FE(H)2137 2813 y Fh(r)2130 2867 y Fq(l)2193 2843 y Fu(with)j Ft(r)e Fs(2)g Fu(\()p Fs(\000)2633 2810 y Fr(1)p 2632 2824 34 4 v 2632 2872 a(2)2675 2843 y FE(;)14 b Fu(0])30 b(and)h FE(l)e Fs(\025)3128 2810 y Fr(9)p FD(\000)p Fr(6)p Fh(r)p 3128 2824 141 4 v 3182 2872 a Fr(4)3278 2843 y Fu(,)i(w)n(e)f(can)g(use)h(the)0 2943 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3786 y Fp(Z)1794 3807 y Fq(T)1757 3975 y Fr(0)1861 3899 y FE(Y)1909 3911 y Fh(r)q Fq(;l)1976 3899 y Fu(\()p FE(\034)9 b Fu(\))p FE(d\034)34 b Fk(.)23 b FE(X)2355 3911 y Fh(r)q Fq(;l)2422 3899 y Fu(\(0\))p FE(:)0 4122 y Fu(The)31 b(con)n(tin)n(uit)n(y)f(can)g (b)r(e)i(established)e(as)g(previously)g(demonstrated.)45 b(Then)31 b(w)n(e)g(end)g(the)g(pro)r(of)f(of)h(Theorem)p 1 0 0 TeXcolorrgb 3662 4122 a SDict begin H.S end 3662 4122 a 1 0 0 TeXcolorrgb Fu(1.1)p 1 0 0 TeXcolorrgb 3768 4068 a SDict begin H.R end 3768 4068 a 3768 4122 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.1.1) cvn H.B /ANN pdfmark end 3768 4122 a 0 TeXcolorgray Fu(\(1\))0 4221 y(for)c Ft(s)c FE(>)g Fu(0)k(as)g(w)n(ell)g(as)g(the)h(lo)r(cal)f(existence)g (for)g(Corollary)p 1 0 0 TeXcolorrgb 1898 4221 a SDict begin H.S end 1898 4221 a 1 0 0 TeXcolorrgb Fu(1.1)p 1 0 0 TeXcolorrgb 2004 4168 a SDict begin H.R end 2004 4168 a 2004 4221 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cor.1.1) cvn H.B /ANN pdfmark end 2004 4221 a 0 TeXcolorgray Fu(.)1782 b Fk(\003)100 4456 y Fu(No)n(w)27 b(w)n(e)g(b)r(egin)h(to)f (pro)n(v)n(e)f(Theorem)p 1 0 0 TeXcolorrgb 1311 4456 a SDict begin H.S end 1311 4456 a 1 0 0 TeXcolorrgb Fu(1.2)p 1 0 0 TeXcolorrgb 1417 4403 a SDict begin H.R end 1417 4403 a 1417 4456 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.1.2) cvn H.B /ANN pdfmark end 1417 4456 a 0 TeXcolorgray Fu(\(1.1\))h(and)h(w)n(e)f(\014rst)g(handle)h(with)g(the)g(case)f Ft(r)d FE(>)f Fs(\000)3061 4423 y Fr(1)p 3060 4437 34 4 v 3060 4485 a(2)3103 4456 y Fu(.)p 0 TeXcolorgray 0 4646 a Fj(Pr)l(o)l(of)31 b(of)f(The)l(or)l(em)p 1 0 0 TeXcolorrgb 665 4646 a SDict begin H.S end 665 4646 a 1 0 0 TeXcolorrgb Fj(1.2)p 1 0 0 TeXcolorrgb 776 4592 a SDict begin H.R end 776 4592 a 776 4646 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.1.2) cvn H.B /ANN pdfmark end 776 4646 a 0 TeXcolorgray Fj(\(1.1\))g(for)h Ft(r)25 b FE(>)d Fs(\000)1331 4613 y Fr(1)p 1331 4627 34 4 v 1331 4674 a(2)1374 4646 y Fj(.)p 0 TeXcolorgray 42 w Fu(Recall)29 b(that)g FE(f)34 b Fs(2)26 b FE(C)6 b Fu(\([0)p FE(;)14 b Fs(1)p Fu(\);)g FE(H)2457 4615 y Fh(r)2450 4669 y Fq(l)2483 4646 y Fu(\))20 b Fs(\\)g FE(L)2667 4615 y Fr(2)2667 4669 y Fq(loc)2754 4646 y Fu(\([0)p FE(;)14 b Fs(1)p Fu(\);)g FE(H)3116 4610 y Fh(r)q Fr(+1)3109 4679 y Fq(l)p FD(\000)3192 4657 y FC(3)p 3192 4666 29 3 v 3192 4699 a(2)3235 4646 y Fu(\))29 b(with)h Ft(r)d Fs(2)f Fu(\()p Fs(\000)3730 4613 y Fr(1)p 3729 4627 34 4 v 3729 4674 a(2)3773 4646 y FE(;)14 b Fu(0])0 4778 y(and)32 b FE(l)g Fs(\025)329 4745 y Fr(9)p 329 4759 V 329 4806 a(4)393 4778 y Fs(\000)489 4745 y Fr(3)p 489 4759 V 489 4806 a(2)532 4778 y Ft(r)q Fu(,)i(it)f(follo)n(ws)e(that)i(\()p 1 0 0 TeXcolorrgb 1200 4778 a SDict begin H.S end 1200 4778 a 1 0 0 TeXcolorrgb Fu(3.5)p 1 0 0 TeXcolorrgb 1306 4724 a SDict begin H.R end 1306 4724 a 1306 4778 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.5) cvn H.B /ANN pdfmark end 1306 4778 a 0 TeXcolorgray Fu(\))g(and)f(\()p 1 0 0 TeXcolorrgb 1569 4778 a SDict begin H.S end 1569 4778 a 1 0 0 TeXcolorrgb Fu(3.7)p 1 0 0 TeXcolorrgb 1675 4724 a SDict begin H.R end 1675 4724 a 1675 4778 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.7) cvn H.B /ANN pdfmark end 1675 4778 a 0 TeXcolorgray Fu(\))h(are)e(satis\014ed.)50 b(F)-7 b(or)32 b(n)f Fs(\025)f Ft(r)k Fu(and)e FE(`)e Fs(\024)h FE(l)23 b Fs(\000)3107 4745 y Fr(3)p 3107 4759 34 4 v 3107 4806 a(2)3150 4778 y Fu(n)f(+)3314 4745 y Fr(3)p 3314 4759 V 3314 4806 a(2)3357 4778 y Ft(r)q Fu(,)34 b(w)n(e)e(m)n(ultiply)0 4885 y(2)42 4855 y Fr(2)p Fq(j)s Fr(n)146 4885 y Fu(2)188 4855 y Fr(2)p Fq(k)q(`)317 4885 y Fu(on)27 b(b)r(oth)i(sides)e(of)34 b(\()p 1 0 0 TeXcolorrgb 961 4885 a SDict begin H.S end 961 4885 a 1 0 0 TeXcolorrgb Fu(3.7)p 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1210 y FE(f)862 1222 y Fr(2)899 1210 y Fs(k)941 1225 y Fq(L)987 1208 y FC(2)1022 1210 y Fs(k)p FF(F)1124 1222 y Fq(j)1159 1210 y FF(P)1224 1222 y Fq(k)1265 1210 y FE(f)1306 1222 y Fr(1)1343 1210 y Fs(k)1385 1225 y Fq(L)1431 1208 y FC(2)1467 1210 y Fs(k)p FF(F)1569 1222 y Fq(j)1603 1210 y FF(P)1668 1222 y Fq(k)1709 1210 y FE(f)1750 1222 y Fr(2)1787 1210 y Fs(k)1829 1225 y Fq(L)1875 1208 y FC(2)1911 1210 y Fu(2)1963 1153 y FC(3)p 1962 1162 29 3 v 1962 1195 a(2)2000 1175 y Fe(p)2058 1210 y Fu(+)18 b(2)2183 1175 y FD(\000)p Fq(j)s(N)2328 1210 y Fu(2)2370 1175 y FD(\000)p Fq(k)q(N)2521 1210 y Fs(k)p FE(f)2604 1222 y Fr(1)2641 1210 y Fs(k)2683 1228 y Fq(L)2729 1208 y FC(2)2729 1247 y Fm(\000)p Fl(N)2830 1210 y Fs(k)p FE(f)2913 1222 y Fr(2)2949 1210 y Fs(k)2991 1228 y Fq(L)3037 1208 y FC(2)3037 1247 y Fm(\000)p Fl(N)3139 1210 y Fs(k)p FE(f)3222 1222 y Fr(2)3258 1210 y Fs(k)3300 1236 y Fq(H)3358 1209 y Fm(\000)p Fl(N)3354 1256 y Fm(\000)p Fl(N)3480 1210 y Fu(+)g(2)3605 1175 y FD(\000)p Fr(2)p Fq(k)307 1436 y Fs(\002k)p FE(f)455 1448 y Fr(1)491 1436 y Fs(k)533 1455 y Fq(L)579 1434 y FC(1)579 1473 y(2)614 1436 y Fs(k)p FF(F)716 1448 y Fq(j)751 1436 y FF(P)816 1448 y Fq(k)857 1436 y FE(f)898 1448 y Fr(2)935 1436 y Fs(k)977 1402 y Fr(2)977 1459 y Fq(L)1023 1442 y FC(2)1059 1436 y Fu(2)1101 1402 y Fq(j)1153 1436 y Fu(+)h(2)1279 1402 y FD(\000)p Fr(2)p Fq(k)1404 1436 y Fu(2)1446 1402 y FD(\000)p Fq(j)s(N)1591 1436 y Fs(k)p FE(f)1674 1448 y Fr(1)1710 1436 y Fs(k)1752 1455 y Fq(L)1798 1434 y FC(1)1798 1473 y(2)1834 1436 y Fs(k)p FF(P)1941 1448 y Fq(k)1982 1436 y FE(f)2023 1448 y Fr(2)2060 1436 y Fs(k)2102 1402 y Fr(2)2102 1460 y Fq(H)2160 1444 y Fm(\000)p Fl(N)2281 1436 y Fu(+)f(2)2406 1402 y FD(\000)p Fq(j)s(N)2551 1436 y Fu(2)2593 1402 y FD(\000)p Fq(k)q(N)2744 1436 y Fs(k)p FE(f)2827 1448 y Fr(1)2863 1436 y Fs(k)2905 1455 y Fq(L)2951 1434 y FC(1)2951 1473 y(2)2987 1436 y Fs(k)p FE(f)3070 1448 y Fr(2)3107 1436 y Fs(k)3149 1402 y Fr(2)3149 1471 y Fq(H)3207 1444 y Fm(\000)p Fl(N)3203 1490 y Fm(\000)p Fl(N)3309 1436 y FE(:)p 0 TeXcolorgray 371 w Fu(\(4.3\))p 0 TeXcolorgray 0 1647 a(and)262 1830 y SDict begin H.S end 262 1830 a 262 1830 a SDict begin 12 H.A end 262 1830 a 262 1830 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.4) cvn /DEST pdfmark end 262 1830 a 231 w Fs(j)p Ff(G)581 1795 y Fr(2)581 1850 y(2)618 1830 y Fs(j)23 b Fk(.)729 1842 y Fq(N)882 1751 y Fp(X)815 1929 y Fe(p)p FD(\025)p Fq(j)s Fr(+)p Fq(N)1036 1937 y FC(0)1082 1830 y Fs(k)p FF(F)1184 1842 y Fe(p)1223 1830 y FF(P)1288 1842 y Fq(k)1329 1830 y FE(f)1370 1842 y Fr(2)1407 1830 y Fs(k)1449 1795 y Fr(2)1449 1852 y Fq(L)1495 1836 y FC(2)1530 1830 y Fs(k)p FF(F)1632 1842 y Fq(j)1667 1830 y FF(P)1732 1842 y Fq(k)1773 1830 y FE(f)1814 1842 y Fr(2)1851 1830 y Fs(k)1893 1845 y Fq(L)1939 1828 y FC(2)1975 1830 y Fu(2)2027 1773 y FC(3)p 2026 1782 29 3 v 2026 1815 a(2)2064 1795 y Fq(j)2118 1830 y Fu(+)2268 1751 y Fp(X)2201 1929 y Fe(p)p FD(\024)p Fq(j)s Fr(+)p Fq(N)2422 1937 y FC(0)2468 1830 y Fs(k)p FF(F)2570 1842 y Fe(p)2609 1830 y FF(P)2674 1842 y Fq(k)2715 1830 y FE(f)2756 1842 y Fr(2)2793 1830 y Fs(k)2835 1845 y Fq(L)2881 1828 y FC(2)2916 1830 y Fu(2)2968 1769 y Fd(p)p 2968 1783 33 3 v 2970 1815 a FC(2)3015 1830 y Fs(k)p FF(F)3117 1842 y Fq(j)3151 1830 y FF(P)3216 1842 y Fq(k)3258 1830 y FE(f)3299 1842 y Fr(2)3335 1830 y Fs(k)3377 1795 y Fr(2)3377 1852 y Fq(L)3423 1836 y FC(2)3459 1830 y Fu(2)3501 1795 y Fq(j)345 2057 y Fu(+)83 b(2)535 2022 y FD(\000)p Fr(2)p Fq(k)660 2057 y Fs(k)p FE(f)743 2069 y Fr(2)779 2057 y Fs(k)821 2075 y Fq(L)867 2055 y FC(1)867 2093 y(2)903 2057 y Fs(k)p FF(F)1005 2069 y Fq(j)1040 2057 y FF(P)1105 2069 y Fq(k)1146 2057 y FE(f)1187 2069 y Fr(2)1224 2057 y Fs(k)1266 2022 y Fr(2)1266 2079 y Fq(L)1312 2062 y FC(2)1347 2057 y Fu(2)1389 2022 y Fq(j)1442 2057 y Fu(+)18 b(2)1567 2022 y FD(\000)p Fr(2)p Fq(k)1693 2057 y Fu(2)1735 2022 y FD(\000)p Fq(j)s(N)1880 2057 y Fs(k)p FE(f)1963 2069 y Fr(2)1999 2057 y Fs(k)2041 2075 y Fq(L)2087 2055 y FC(1)2087 2093 y(2)2123 2057 y Fs(k)p FF(P)2230 2069 y Fq(k)2270 2057 y FE(f)2311 2069 y Fr(2)2348 2057 y Fs(k)2390 2022 y Fr(2)2390 2081 y Fq(H)2448 2064 y Fm(\000)p Fl(N)2569 2057 y Fu(+)h(2)2695 2022 y FD(\000)p Fq(k)q(N)2846 2057 y Fs(k)p FF(F)2948 2069 y Fq(j)2982 2057 y FF(P)3047 2069 y Fq(k)3088 2057 y FE(f)3129 2069 y Fr(2)3166 2057 y Fs(k)3208 2022 y Fr(2)3208 2079 y Fq(L)3254 2062 y FC(2)3290 2057 y Fs(k)p FE(f)3373 2069 y Fr(2)3409 2057 y Fs(k)3451 2083 y Fq(H)3509 2056 y Fm(\000)p Fl(N)3505 2103 y Fm(\000)p Fl(N)345 2218 y Fu(+)83 b(2)535 2183 y FD(\000)p Fq(j)s(N)680 2218 y Fu(2)722 2183 y FD(\000)p Fq(k)q(N)873 2218 y Fs(k)p FE(f)956 2230 y Fr(2)992 2218 y Fs(k)1034 2183 y Fr(2)1034 2243 y Fq(L)1080 2223 y FC(2)1080 2263 y Fm(\000)p Fl(N)1182 2218 y Fs(k)p FE(f)1265 2230 y Fr(2)1301 2218 y Fs(k)1343 2244 y Fq(H)1401 2217 y Fm(\000)p Fl(N)1397 2264 y Fm(\000)p Fl(N)1504 2218 y FE(:)p 0 TeXcolorgray 2176 w Fu(\(4.4\))p 0 TeXcolorgray 100 2430 a(No)n(w)27 b(w)n(e)g(fo)r(cus)h(on)f(the)h(estimate)f(of)h Ff(G)1374 2400 y Fr(3)1374 2451 y(2)1411 2430 y Fu(.)37 b(W)-7 b(e)28 b(split)g(it)g(in)n(to)g(t)n(w)n(o)f(parts:)610 2630 y Ff(G)675 2596 y Fr(3)675 2650 y(2)735 2630 y Fu(=)823 2551 y Fp(X)830 2730 y Fe(l)p FD(\025)p Fr(0)943 2630 y Fs(h)p FE(Q)1041 2642 y Fe(l)1066 2630 y Fu(\()p FE(f)1139 2642 y Fr(2)1176 2630 y FE(;)14 b(f)1254 2642 y Fr(1)1291 2630 y Fu(\))p FE(;)g Fs(P)1418 2642 y Fq(k)1458 2630 y Ff(F)1509 2596 y Fr(2)1509 2650 y Fq(j)1546 2630 y Fs(P)1604 2642 y Fq(k)1645 2630 y FE(f)1686 2642 y Fr(2)1723 2630 y Fs(i)1755 2642 y Fq(v)1813 2630 y Fu(+)k Fs(h)p FE(Q)1994 2642 y FD(\000)p Fr(1)2083 2630 y Fu(\()p FE(f)2156 2642 y Fr(2)2193 2630 y FE(;)c(f)2271 2642 y Fr(1)2308 2630 y Fu(\))p FE(;)g Fs(P)2435 2642 y Fq(k)2476 2630 y Ff(F)2527 2596 y Fr(2)2527 2650 y Fq(j)2564 2630 y Fs(P)2622 2642 y Fq(k)2662 2630 y FE(f)2703 2642 y Fr(2)2740 2630 y Fs(i)2772 2642 y Fq(v)2835 2630 y Fu(:=)23 b Ff(H)3006 2642 y Fr(1)3061 2630 y Fu(+)18 b Ff(H)3204 2642 y Fr(2)3241 2630 y FE(:)100 2898 y Fu(Estimate)27 b(of)g Ff(H)606 2910 y Fr(1)643 2898 y Fu(.)p 100 2924 567 4 v 35 w(First)34 b(w)n(e)f(p)r(erform)h(dy)n(adic)g(decomp)r(osition)f(in)i(frequency)e (space.)56 b(Thanks)34 b(to)g(decomp)r(osition)0 2998 y(\()p 1 0 0 TeXcolorrgb 32 2998 a SDict begin H.S end 32 2998 a 1 0 0 TeXcolorrgb Fu(2.9)p 1 0 0 TeXcolorrgb 138 2944 a SDict begin H.R end 138 2944 a 138 2998 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.9) cvn H.B /ANN pdfmark end 138 2998 a 0 TeXcolorgray Fu(\))28 b(together)f(with)h(the)g (estimates)f(\()p 1 0 0 TeXcolorrgb 1255 2998 a SDict begin H.S end 1255 2998 a 1 0 0 TeXcolorrgb Fu(2.15)p 1 0 0 TeXcolorrgb 1403 2944 a SDict begin H.R end 1403 2944 a 1403 2998 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.15) cvn H.B /ANN pdfmark end 1403 2998 a 0 TeXcolorgray Fu(\))q(\()p 1 0 0 TeXcolorrgb 1468 2998 a SDict begin H.S end 1468 2998 a 1 0 0 TeXcolorrgb Fu(2.17)p 1 0 0 TeXcolorrgb 1615 2944 a SDict begin H.R end 1615 2944 a 1615 2998 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.17) cvn H.B /ANN pdfmark end 1615 2998 a 0 TeXcolorgray Fu(\))q(\()p 1 0 0 TeXcolorrgb 1680 2998 a SDict begin H.S end 1680 2998 a 1 0 0 TeXcolorrgb Fu(2.19)p 1 0 0 TeXcolorrgb 1828 2944 a SDict begin H.R end 1828 2944 a 1828 2998 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.19) cvn H.B /ANN pdfmark end 1828 2998 a 0 TeXcolorgray Fu(\))h(and)f(\()p 1 0 0 TeXcolorrgb 2081 2998 a SDict begin H.S end 2081 2998 a 1 0 0 TeXcolorrgb Fu(2.20)p 1 0 0 TeXcolorrgb 2229 2944 a SDict begin H.R end 2229 2944 a 2229 2998 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.20) cvn H.B /ANN pdfmark end 2229 2998 a 0 TeXcolorgray Fu(\),)h(w)n(e)f(can)g(deduce)h(that)0 3197 y SDict begin H.S end 0 3197 a 0 3197 a SDict begin 12 H.A end 0 3197 a 0 3197 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.5) cvn /DEST pdfmark end 0 3197 a 166 w Fs(h)p FE(Q)264 3209 y Fe(l)289 3197 y Fu(\()p FE(f)362 3209 y Fr(2)399 3197 y FE(;)14 b(f)477 3209 y Fr(1)514 3197 y Fu(\))p FE(;)g Fs(P)641 3209 y Fq(k)682 3197 y Ff(F)733 3163 y Fr(2)733 3218 y Fq(j)770 3197 y Fs(P)828 3209 y Fq(k)868 3197 y FE(f)909 3209 y Fr(2)946 3197 y Fs(i)978 3209 y Fq(v)1041 3197 y Fu(=)1198 3119 y Fp(X)1129 3297 y Fe(a)p FD(\024)p Fe(p)p FD(\000)p Fq(N)1356 3305 y FC(0)1387 3197 y Fs(h)p FE(Q)1485 3209 y Fe(l)1510 3197 y Fu(\()p Ff(F)1593 3209 y Fe(p)1632 3197 y FE(f)1673 3209 y Fr(2)1710 3197 y FE(;)g Ff(F)1798 3209 y Fe(a)1837 3197 y FE(f)1878 3209 y Fr(1)1915 3197 y Fu(\))p FE(;)1988 3176 y Fu(~)1984 3197 y Ff(F)2035 3209 y Fe(p)2074 3197 y Fs(P)2132 3209 y Fq(k)2172 3197 y Ff(F)2223 3163 y Fr(2)2223 3218 y Fq(j)2260 3197 y Fs(P)2318 3209 y Fq(k)2359 3197 y FE(f)2400 3209 y Fr(2)2437 3197 y Fs(i)2469 3209 y Fq(v)2527 3197 y Fu(+)2636 3119 y Fp(X)2610 3297 y Fe(a)p FD(\025\000)p Fr(1)2782 3197 y Fs(h)p FE(Q)2880 3209 y Fe(l)2905 3197 y Fu(\()p Fs(S)2987 3209 y Fe(a)p FD(\000)p Fq(N)3127 3217 y FC(0)3163 3197 y FE(f)3204 3209 y Fr(2)3241 3197 y FE(;)g Ff(F)3329 3209 y Fe(a)3368 3197 y FE(f)3409 3209 y Fr(1)3446 3197 y Fu(\))p FE(;)p 0 TeXcolorgray 204 w Fu(\(4.5\))p 0 TeXcolorgray 171 3415 a(~)166 3436 y Ff(F)217 3448 y Fe(a)256 3436 y Fs(P)314 3448 y Fq(k)354 3436 y Ff(F)405 3402 y Fr(2)405 3457 y Fq(j)442 3436 y Fs(P)500 3448 y Fq(k)541 3436 y FE(f)582 3448 y Fr(2)619 3436 y Fs(i)651 3448 y Fq(v)709 3436 y Fu(+)869 3358 y Fp(X)792 3536 y Fe(m)p FD(\024)p Fe(p)p Fr(+)p Fq(N)1035 3544 y FC(0)1066 3436 y Fs(h)p FE(Q)1164 3448 y Fe(l)1189 3436 y Fu(\()p Ff(F)1272 3448 y Fe(p)1311 3436 y FE(f)1352 3448 y Fr(2)1389 3436 y FE(;)1431 3415 y Fu(~)1426 3436 y Ff(F)1477 3448 y Fe(p)1516 3436 y FE(f)1557 3448 y Fr(1)1594 3436 y Fu(\))p FE(;)g Ff(F)1714 3448 y Fe(m)1769 3436 y Fs(P)1827 3448 y Fq(k)1868 3436 y Ff(F)1919 3402 y Fr(2)1919 3457 y Fq(j)1956 3436 y Fs(P)2014 3448 y Fq(k)2055 3436 y FE(f)2096 3448 y Fr(2)2133 3436 y Fs(i)2165 3448 y Fq(v)2227 3436 y Fk(.)2292 3448 y Fq(N)2404 3358 y Fp(X)2378 3536 y Fe(p)p FD(\025\000)p Fr(1)2564 3436 y Fu(2)2606 3402 y FD(\000)p Fe(l)2682 3436 y Fs(k)p FE(f)2765 3448 y Fr(2)2801 3436 y Fs(k)2843 3451 y Fq(L)2889 3435 y FC(1)2925 3436 y Fs(k)2972 3415 y Fu(~)2967 3436 y Ff(F)3018 3448 y Fe(p)3056 3436 y FE(f)3097 3448 y Fr(1)3134 3436 y Fs(k)3176 3451 y Fq(L)3222 3435 y FC(2)3258 3436 y Fs(k)3305 3415 y Fu(~)3300 3436 y Ff(F)3351 3448 y Fe(p)3389 3436 y Fs(P)3447 3448 y Fq(k)3488 3436 y Ff(F)3539 3402 y Fr(2)3539 3457 y Fq(j)3576 3436 y Fs(P)3634 3448 y Fq(k)3674 3436 y FE(f)3715 3448 y Fr(2)3752 3436 y Fs(k)3794 3451 y Fq(L)3840 3435 y FC(2)166 3684 y Fs(\002)p Fu(2)273 3649 y Fr(2)p Fe(p)363 3684 y Fu(+)523 3605 y Fp(X)446 3783 y Fe(m)p FD(\024)p Fe(p)p Fr(+)p Fq(N)689 3791 y FC(0)734 3684 y Fu(2)776 3649 y FD(\000)p Fe(l)p Fq(N)911 3684 y Fu(2)953 3649 y FD(\000)p Fr(2)p Fe(p)p Fq(N)1135 3684 y Fu(2)1177 3649 y FD(\000)p Fr(2)p Fe(m)p Fq(N)1377 3684 y Fs(k)p FE(f)1460 3696 y Fr(2)1496 3684 y Fs(k)1538 3699 y Fq(L)1584 3682 y FC(1)1633 3592 y Fp(\020)1683 3684 y Fs(k)p Ff(F)1776 3696 y Fe(m)1831 3684 y FE(f)1872 3696 y Fr(1)1909 3684 y Fs(k)1951 3699 y Fq(L)1997 3682 y FC(2)2032 3684 y Fs(k)2079 3662 y Fu(~)2074 3684 y Ff(F)2125 3696 y Fe(p)2164 3684 y Fs(P)2222 3696 y Fq(k)2262 3684 y Ff(F)2313 3649 y Fr(2)2313 3704 y Fq(j)2350 3684 y Fs(P)2408 3696 y Fq(k)2449 3684 y FE(f)2490 3696 y Fr(2)2527 3684 y Fs(k)2569 3699 y Fq(L)2615 3682 y FC(2)2669 3684 y Fu(+)k Fs(k)2799 3662 y Fu(~)2794 3684 y Ff(F)2845 3696 y Fe(p)2883 3684 y FE(f)2924 3696 y Fr(1)2961 3684 y Fs(k)3003 3699 y Fq(L)3049 3682 y FC(2)3085 3684 y Fs(k)p Ff(F)3178 3696 y Fe(m)3233 3684 y Fs(P)3291 3696 y Fq(k)3331 3684 y Ff(F)3382 3649 y Fr(2)3382 3704 y Fq(j)3419 3684 y Fs(P)3477 3696 y Fq(k)3518 3684 y FE(f)3559 3696 y Fr(2)3596 3684 y Fs(k)3638 3699 y Fq(L)3684 3682 y FC(2)3719 3592 y Fp(\021)3783 3684 y FE(:)0 3956 y Fu(Then)28 b(w)n(e)f(p)r(erform)g(dy)n(adic)g(decomp)r (osition)g(in)h(phase)f(space\(see)g(\()p 1 0 0 TeXcolorrgb 2184 3956 a SDict begin H.S end 2184 3956 a 1 0 0 TeXcolorrgb Fu(2.8)p 1 0 0 TeXcolorrgb 2290 3903 a SDict begin H.R end 2290 3903 a 2290 3956 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.8) cvn H.B /ANN pdfmark end 2290 3956 a 0 TeXcolorgray Fu(\)\))h(to)g(get)f(that)393 4156 y SDict begin H.S end 393 4156 a 393 4156 a SDict begin 12 H.A end 393 4156 a 393 4156 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.6) cvn /DEST pdfmark end 393 4156 a Ff(H)453 4168 y Fr(1)573 4156 y Fu(=)826 4077 y Fp(X)720 4256 y Fr(0)p FD(\024)p Fe(l)p FD(\024)p Fq(k)q FD(\000)p Fq(N)1019 4264 y FC(0)1052 4156 y Fs(h)p FE(Q)1150 4168 y Fe(l)1175 4156 y Fu(\()1226 4135 y(~)1207 4156 y Fs(P)1265 4168 y Fq(k)1306 4156 y FE(f)1347 4168 y 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5123 y Fs(k)730 5141 y Fq(L)776 5121 y FC(1)776 5159 y(1)826 5030 y Fp(\020)875 5123 y Fs(k)p Ff(F)968 5135 y Fe(m)1042 5102 y Fu(~)1023 5123 y Fs(P)1081 5135 y Fq(k)1122 5123 y FE(f)1163 5135 y Fr(1)1200 5123 y Fs(k)1242 5138 y Fq(L)1288 5121 y FC(2)1323 5123 y Fs(k)1370 5101 y Fu(~)1365 5123 y Ff(F)1416 5135 y Fe(p)1455 5123 y Fs(P)1513 5135 y Fq(k)1553 5123 y Ff(F)1604 5088 y Fr(2)1604 5143 y Fq(j)1641 5123 y Fs(P)1699 5135 y Fq(k)1740 5123 y FE(f)1781 5135 y Fr(2)1818 5123 y Fs(k)1860 5138 y Fq(L)1906 5121 y FC(2)1960 5123 y Fu(+)18 b Fs(k)2090 5101 y Fu(~)2085 5123 y Ff(F)2136 5135 y Fe(p)2192 5102 y Fu(~)2174 5123 y Fs(P)2232 5135 y Fq(k)2273 5123 y FE(f)2314 5135 y Fr(1)2350 5123 y Fs(k)2392 5138 y Fq(L)2438 5121 y FC(2)2474 5123 y Fs(k)p Ff(F)2567 5135 y Fe(m)2622 5123 y Fs(P)2680 5135 y Fq(k)2721 5123 y Ff(F)2772 5088 y Fr(2)2772 5143 y Fq(j)2809 5123 y Fs(P)2867 5135 y Fq(k)2907 5123 y FE(f)2948 5135 y Fr(2)2985 5123 y Fs(k)3027 5138 y Fq(L)3073 5121 y FC(2)3109 5030 y Fp(\021)3172 5123 y FE(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 29 29 TeXDict begin HPSdict begin 29 28 bop 0 TeXcolorgray 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 0 TeXcolorgray -8 -245 a SDict begin H.S end -8 -245 a -8 -245 a SDict begin H.R end -8 -245 a -8 -245 a SDict begin [/View [/XYZ H.V]/Dest (page.29) cvn /DEST pdfmark end -8 -245 a 0 TeXcolorgray 0 TeXcolorgray 1335 -165 a Fr(LAND)n(A)n(U)30 b(EQUA)-5 b(TION)28 b(IN)h Fq(H)2196 -193 y Fm(\000)2252 -209 y FC(1)p 2252 -202 29 3 v 2252 -171 a(2)2324 -165 y Fr(SP)-5 b(A)n(CE)1268 b(29)p 0 TeXcolorgray 0 29 a Fu(F)-7 b(uthermore,)27 b(due)h(to)f(Prop)r(osition)p 1 0 0 TeXcolorrgb 1177 29 a SDict begin H.S end 1177 29 a 1 0 0 TeXcolorrgb Fu(9.4)p 1 0 0 TeXcolorrgb 1283 -24 a SDict begin H.R end 1283 -24 a 1283 29 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (lem.9.4) cvn H.B /ANN pdfmark end 1283 29 a 0 TeXcolorgray Fu(,)h(w)n(e)f(can)h(deriv)n(e)e(that)0 199 y SDict begin H.S end 0 199 a 0 199 a SDict begin 12 H.A end 0 199 a 0 199 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.7) cvn /DEST pdfmark end 0 199 a 231 w Fs(j)p Ff(H)314 211 y Fr(1)351 199 y Fs(j)d Fk(.)571 120 y Fp(X)484 302 y FD(j)p Fe(p)p FD(\000)p Fq(j)s FD(j\024)p Fq(N)746 310 y FC(0)792 199 y Fu(2)834 165 y FD(\000)p Fq(k)926 199 y Fs(k)p FE(f)1009 211 y Fr(2)1046 199 y Fs(k)1088 217 y Fq(L)1134 197 y FC(1)1134 235 y(1)1169 199 y Fs(k)1216 178 y Fu(~)1211 199 y Ff(F)1262 211 y Fe(p)1319 178 y Fu(~)1301 199 y Fs(P)1359 211 y Fq(k)1399 199 y FE(f)1440 211 y Fr(1)1477 199 y Fs(k)1519 214 y Fq(L)1565 197 y FC(2)1601 199 y Fs(k)p Ff(F)1694 165 y Fr(2)1694 220 y Fq(j)1730 199 y Fs(P)1788 211 y Fq(k)1829 199 y FE(f)1870 211 y Fr(2)1907 199 y Fs(k)1949 214 y Fq(L)1995 197 y 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Fq(j)s(N)2216 1189 y Fu(2)2258 1154 y FD(\000)p Fq(k)q(N)2409 1189 y Fs(k)p FE(f)2492 1201 y Fr(2)2528 1189 y Fs(k)2570 1207 y Fq(L)2616 1187 y FC(1)2616 1225 y(1)2652 1189 y Fs(k)p FE(f)2735 1201 y Fr(1)2771 1189 y Fs(k)2813 1215 y Fq(H)2871 1188 y Fm(\000)p Fl(N)2867 1235 y Fm(\000)p Fl(N)2974 1189 y Fs(k)p FE(f)3057 1201 y Fr(2)3094 1189 y Fs(k)3136 1215 y Fq(H)3194 1188 y Fm(\000)p Fl(N)3190 1235 y Fm(\000)p Fl(N)3296 1189 y FE(:)p 0 TeXcolorgray 618 w Fu(\(4.7\))p 0 TeXcolorgray 0 1371 a(Estimate)27 b(of)h Ff(H)507 1383 y Fr(2)544 1371 y Fu(.)p 0 1396 567 4 v 26 w(First)f(w)n(e)f(p)r(erform)g(dy)n(adic)f(decomp)r(osition) h(in)h(frequency)f(space.)35 b(Thanks)26 b(to)h(decomp)r(osition)e(\()p 1 0 0 TeXcolorrgb 3735 1371 a SDict begin H.S end 3735 1371 a 1 0 0 TeXcolorrgb Fu(2.9)p 1 0 0 TeXcolorrgb 3842 1317 a SDict begin H.R end 3842 1317 a 3842 1371 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.9) cvn H.B /ANN pdfmark end 3842 1371 a 0 TeXcolorgray Fu(\))0 1470 y(together)i(with)h(the) g(estimates)f(\()p 1 0 0 TeXcolorrgb 1057 1470 a SDict begin H.S end 1057 1470 a 1 0 0 TeXcolorrgb Fu(2.29)p 1 0 0 TeXcolorrgb 1205 1417 a SDict begin H.R end 1205 1417 a 1205 1470 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.29) cvn H.B /ANN pdfmark end 1205 1470 a 0 TeXcolorgray Fu(\)\()p 1 0 0 TeXcolorrgb 1269 1470 a SDict begin H.S end 1269 1470 a 1 0 0 TeXcolorrgb Fu(2.30)p 1 0 0 TeXcolorrgb 1417 1417 a SDict begin H.R end 1417 1417 a 1417 1470 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.30) cvn H.B /ANN pdfmark end 1417 1470 a 0 TeXcolorgray Fu(\)\()p 1 0 0 TeXcolorrgb 1481 1470 a SDict begin H.S end 1481 1470 a 1 0 0 TeXcolorrgb Fu(2.31)p 1 0 0 TeXcolorrgb 1629 1417 a SDict begin H.R end 1629 1417 a 1629 1470 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.31) cvn H.B /ANN pdfmark end 1629 1470 a 0 TeXcolorgray Fu(\))h(and)g(\()p 1 0 0 TeXcolorrgb 1883 1470 a SDict begin H.S end 1883 1470 a 1 0 0 TeXcolorrgb Fu(2.32)p 1 0 0 TeXcolorrgb 2031 1417 a SDict begin H.R end 2031 1417 a 2031 1470 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.31) cvn H.B /ANN pdfmark end 2031 1470 a 0 TeXcolorgray Fu(\),)g(w)n(e)f(ha)n(v)n(e)0 1645 y SDict begin H.S end 0 1645 a 0 1645 a SDict begin 12 H.A end 0 1645 a 0 1645 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.8) cvn /DEST pdfmark end 0 1645 a 231 w Fs(j)p Ff(H)314 1657 y Fr(2)351 1645 y Fs(j)c(\024)554 1566 y Fp(X)484 1744 y Fe(a)p FD(\024)p Fe(p)p FD(\000)p Fq(N)711 1752 y FC(0)757 1645 y Fs(jh)p FE(Q)878 1657 y FD(\000)p Fr(1)967 1645 y Fu(\()p Ff(F)1050 1657 y Fe(p)1089 1645 y FE(f)1130 1657 y Fr(2)1167 1645 y FE(;)14 b Ff(F)1255 1657 y Fe(a)1293 1645 y FE(f)1334 1657 y Fr(1)1371 1645 y Fu(\))p FE(;)1445 1623 y Fu(~)1440 1645 y Ff(F)1491 1657 y Fe(p)1530 1645 y Fs(P)1588 1657 y Fq(k)1629 1645 y Ff(F)1680 1610 y Fr(2)1680 1665 y Fq(j)1717 1645 y Fs(P)1775 1657 y Fq(k)1815 1645 y FE(f)1856 1657 y Fr(2)1893 1645 y Fs(i)1925 1657 y Fq(v)1965 1645 y Fs(j)19 b Fu(+)2116 1566 y Fp(X)2090 1744 y Fe(a)p FD(\025\000)p Fr(1)2275 1645 y Fs(jh)p FE(Q)2396 1657 y FD(\000)p Fr(1)2485 1645 y Fu(\()p Fs(S)2567 1657 y Fe(a)p FD(\000)p Fq(N)2707 1665 y FC(0)2744 1645 y FE(f)2785 1657 y Fr(2)2822 1645 y FE(;)14 b Ff(F)2910 1657 y Fe(a)2948 1645 y FE(f)2989 1657 y Fr(1)3026 1645 y Fu(\))p FE(;)3100 1623 y Fu(~)3095 1645 y Ff(F)3146 1657 y Fe(a)3185 1645 y Fs(P)3243 1657 y Fq(k)3284 1645 y Ff(F)3335 1610 y Fr(2)3335 1665 y Fq(j)3372 1645 y Fs(P)3430 1657 y Fq(k)3470 1645 y FE(f)3511 1657 y Fr(2)3548 1645 y Fs(i)3580 1657 y Fq(v)3620 1645 y Fs(j)p 0 TeXcolorgray 95 w Fu(\(4.8\))p 0 TeXcolorgray 231 1884 a(+)386 1805 y Fp(X)310 1983 y Fe(m)p FD(\024)p Fe(p)p Fr(+)p Fq(N)553 1991 y FC(0)597 1884 y Fs(jh)p FE(Q)718 1896 y FD(\000)p Fr(1)807 1884 y Fu(\()p Ff(F)890 1896 y Fe(p)929 1884 y FE(f)970 1896 y Fr(2)1007 1884 y FE(;)1049 1862 y Fu(~)1044 1884 y Ff(F)1095 1896 y Fe(p)1134 1884 y FE(f)1175 1896 y Fr(1)1212 1884 y Fu(\))p FE(;)g Ff(F)1332 1896 y Fe(m)1387 1884 y Fs(P)1445 1896 y Fq(k)1486 1884 y Ff(F)1537 1849 y Fr(2)1537 1904 y Fq(j)1574 1884 y Fs(P)1632 1896 y Fq(k)1673 1884 y FE(f)1714 1896 y Fr(2)1750 1884 y Fs(i)1782 1896 y Fq(v)1822 1884 y Fs(j)23 b Fk(.)2025 1805 y Fp(X)1956 1983 y Fe(a)p FD(\024)p Fe(p)p FD(\000)p Fq(N)2183 1991 y FC(0)2228 1884 y Fs(k)p Ff(F)2321 1896 y Fe(p)2359 1884 y FE(f)2400 1896 y Fr(2)2437 1884 y Fs(k)2479 1899 y Fq(L)2525 1882 y FC(2)2561 1884 y Fs(k)2608 1862 y Fu(~)2603 1884 y Ff(F)2654 1896 y Fe(p)2692 1884 y Fs(P)2750 1896 y Fq(k)2791 1884 y Ff(F)2842 1849 y Fr(2)2842 1904 y Fq(j)2879 1884 y Fs(P)2937 1896 y Fq(k)2977 1884 y FE(f)3018 1896 y Fr(2)3055 1884 y Fs(k)3097 1899 y Fq(L)3143 1882 y FC(2)3179 1884 y Fs(k)p Ff(F)3272 1896 y Fe(a)3310 1884 y FE(f)3351 1896 y Fr(1)3388 1884 y Fs(k)3430 1899 y Fq(L)3476 1882 y FC(2)3512 1884 y Fu(2)3564 1827 y FC(3)p 3563 1836 29 3 v 3563 1869 a(2)3601 1849 y Fe(a)83 2123 y Fu(+)300 2044 y Fp(X)231 2222 y Fe(p)p FD(\024)p Fe(a)p FD(\000)p Fq(N)458 2230 y FC(0)503 2123 y Fs(k)p Ff(F)596 2135 y Fe(p)634 2123 y FE(f)675 2135 y Fr(2)712 2123 y Fs(k)754 2138 y Fq(L)800 2121 y FC(2)836 2123 y Fu(2)878 2088 y FD(\000)939 2062 y Fd(p)p 939 2076 33 3 v 941 2108 a FC(2)986 2123 y Fs(k)p Ff(F)1079 2135 y Fe(a)1117 2123 y FE(f)1158 2135 y Fr(1)1195 2123 y Fs(k)1237 2138 y Fq(L)1283 2121 y FC(2)1319 2123 y Fs(k)1366 2101 y Fu(~)1361 2123 y Ff(F)1412 2135 y Fe(a)1450 2123 y Fs(P)1508 2135 y Fq(k)1549 2123 y Ff(F)1600 2088 y Fr(2)1600 2143 y Fq(j)1637 2123 y Fs(P)1695 2135 y Fq(k)1735 2123 y FE(f)1776 2135 y Fr(2)1813 2123 y Fs(k)1855 2138 y Fq(L)1901 2121 y FC(2)1937 2123 y Fu(2)1979 2088 y Fr(2)p Fe(a)2069 2123 y Fu(+)2229 2044 y Fp(X)2152 2222 y Fe(m)p FD(\024)p Fe(p)p Fr(+)p Fq(N)2395 2230 y FC(0)2440 2123 y Fu(2)2492 2066 y FC(3)p 2491 2075 29 3 v 2491 2108 a(2)2530 2088 y Fe(m)2585 2123 y Fs(k)p Ff(F)2678 2135 y Fe(p)2717 2123 y FE(f)2758 2135 y Fr(2)2795 2123 y Fs(k)2837 2138 y Fq(L)2883 2121 y FC(2)2918 2123 y Fs(k)2965 2101 y Fu(~)2960 2123 y Ff(F)3011 2135 y Fe(p)3049 2123 y FE(f)3090 2135 y Fr(1)3127 2123 y Fs(k)3169 2138 y Fq(L)3215 2121 y FC(2)3251 2123 y Fs(k)p Ff(F)3344 2135 y Fe(m)3399 2123 y Fs(P)3457 2135 y Fq(k)3498 2123 y Ff(F)3549 2088 y Fr(2)3549 2143 y Fq(j)3585 2123 y Fs(P)3643 2135 y Fq(k)3684 2123 y FE(f)3725 2135 y Fr(2)3762 2123 y Fs(k)3804 2138 y Fq(L)3850 2121 y FC(2)3886 2123 y FE(:)0 2370 y Fu(Then)j(w)n(e)g(p)r(erform)f(the)h(decomp)r(osition)g(in)g(phase)f (space.)36 b(Note)26 b(that)g Fs(h)p FE(v)s Fs(i)e(\030)f(h)p FE(v)2581 2382 y FD(\003)2619 2370 y Fs(i)k Fu(for)e FE(Q)2869 2382 y FD(\000)p Fr(1)2958 2370 y Fu(,)h(th)n(us)g(b)n(y)h (\()p 1 0 0 TeXcolorrgb 3335 2370 a SDict begin H.S end 3335 2370 a 1 0 0 TeXcolorrgb Fu(4.8)p 1 0 0 TeXcolorrgb 3441 2316 a SDict begin H.R end 3441 2316 a 3441 2370 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.8) cvn H.B /ANN pdfmark end 3441 2370 a 0 TeXcolorgray Fu(\),)g(w)n(e)e(obtain)0 2470 y(that)259 2627 y Fs(j)p Ff(H)342 2639 y Fr(2)379 2627 y Fs(j)e Fk(.)582 2549 y Fp(X)513 2727 y Fe(a)p FD(\024)p Fe(p)p FD(\000)p Fq(N)740 2735 y FC(0)785 2627 y Fs(k)p Ff(F)878 2639 y Fe(p)935 2606 y Fu(~)916 2627 y Fs(P)974 2639 y Fq(k)1015 2627 y FE(f)1056 2639 y Fr(2)1093 2627 y Fs(k)1135 2642 y Fq(L)1181 2626 y FC(2)1216 2627 y Fs(k)1263 2606 y Fu(~)1258 2627 y Ff(F)1309 2639 y Fe(p)1348 2627 y Fs(P)1406 2639 y Fq(k)1446 2627 y Ff(F)1497 2593 y Fr(2)1497 2648 y Fq(j)1534 2627 y Fs(P)1592 2639 y Fq(k)1633 2627 y FE(f)1674 2639 y Fr(2)1711 2627 y Fs(k)1753 2642 y Fq(L)1799 2626 y FC(2)1834 2627 y Fs(k)p Ff(F)1927 2639 y Fe(a)1984 2606 y Fu(~)1966 2627 y Fs(P)2024 2639 y Fq(k)2064 2627 y FE(f)2105 2639 y Fr(1)2142 2627 y Fs(k)2184 2642 y Fq(L)2230 2626 y FC(2)2266 2627 y Fu(2)2318 2571 y FC(3)p 2317 2580 29 3 v 2317 2613 a(2)2355 2593 y Fe(a)2413 2627 y Fu(+)2565 2549 y Fp(X)2496 2727 y Fe(p)p FD(\024)p Fe(a)p FD(\000)p Fq(N)2723 2735 y FC(0)2768 2627 y Fs(k)p Ff(F)2861 2639 y Fe(p)2918 2606 y Fu(~)2899 2627 y Fs(P)2957 2639 y Fq(k)2998 2627 y FE(f)3039 2639 y Fr(2)3076 2627 y Fs(k)3118 2642 y Fq(L)3164 2626 y FC(2)3199 2627 y Fu(2)3241 2593 y FD(\000)3303 2566 y Fd(p)p 3303 2580 33 3 v 3305 2613 a FC(2)3350 2627 y Fs(k)p Ff(F)3443 2639 y Fe(a)3500 2606 y Fu(~)3481 2627 y Fs(P)3539 2639 y Fq(k)3580 2627 y FE(f)3621 2639 y Fr(1)3658 2627 y Fs(k)3700 2642 y Fq(L)3746 2626 y FC(2)259 2866 y Fs(\002k)371 2845 y Fu(~)366 2866 y Ff(F)417 2878 y Fe(a)455 2866 y Fs(P)513 2878 y Fq(k)553 2866 y Ff(F)604 2832 y Fr(2)604 2887 y Fq(j)641 2866 y Fs(P)699 2878 y Fq(k)740 2866 y FE(f)781 2878 y Fr(2)818 2866 y Fs(k)860 2881 y Fq(L)906 2865 y FC(2)941 2866 y Fu(2)983 2832 y Fr(2)p Fe(a)1073 2866 y Fu(+)1234 2788 y Fp(X)1156 2966 y Fe(m)p FD(\024)p Fe(p)p Fr(+)p Fq(N)1399 2974 y FC(0)1445 2866 y Fs(k)p Ff(F)1538 2878 y Fe(p)1594 2845 y Fu(~)1576 2866 y Fs(P)1634 2878 y Fq(k)1674 2866 y FE(f)1715 2878 y Fr(2)1752 2866 y Fs(k)1794 2881 y Fq(L)1840 2865 y FC(2)1876 2866 y Fs(k)1923 2845 y Fu(~)1918 2866 y Ff(F)1969 2878 y Fe(p)2026 2845 y Fu(~)2007 2866 y Fs(P)2065 2878 y Fq(k)2106 2866 y FE(f)2147 2878 y Fr(1)2184 2866 y Fs(k)2226 2881 y Fq(L)2272 2865 y FC(2)2307 2866 y Fs(k)p Ff(F)2400 2878 y Fe(m)2455 2866 y Fs(P)2513 2878 y Fq(k)2554 2866 y Ff(F)2605 2832 y Fr(2)2605 2887 y Fq(j)2642 2866 y Fs(P)2700 2878 y Fq(k)2740 2866 y FE(f)2781 2878 y Fr(2)2818 2866 y Fs(k)2860 2881 y Fq(L)2906 2865 y FC(2)2942 2866 y Fu(2)2994 2810 y FC(3)p 2993 2819 29 3 v 2993 2852 a(2)3032 2832 y Fe(m)3087 2866 y FE(:)0 3114 y Fu(F)-7 b(uthermore,)27 b(due)h(to)f(Prop)r(osition)p 1 0 0 TeXcolorrgb 1177 3114 a SDict begin H.S end 1177 3114 a 1 0 0 TeXcolorrgb Fu(9.4)p 1 0 0 TeXcolorrgb 1283 3060 a SDict begin H.R end 1283 3060 a 1283 3114 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (lem.9.4) cvn H.B /ANN pdfmark end 1283 3114 a 0 TeXcolorgray 28 w Fu(and)g(similar)g(argumen)n(t)g (as)g(\()p 1 0 0 TeXcolorrgb 2251 3114 a SDict begin H.S end 2251 3114 a 1 0 0 TeXcolorrgb Fu(4.7)p 1 0 0 TeXcolorrgb 2357 3060 a SDict begin H.R end 2357 3060 a 2357 3114 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.7) cvn H.B /ANN pdfmark end 2357 3114 a 0 TeXcolorgray Fu(\),)h(w)n(e)f(can)h(deriv)n(e)e(that)454 3288 y SDict begin H.S end 454 3288 a 454 3288 a SDict begin 12 H.A end 454 3288 a 454 3288 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.9) cvn /DEST pdfmark end 454 3288 a 166 w Fs(j)p Ff(H)703 3300 y Fr(2)740 3288 y Fs(j)d Fk(.)851 3300 y Fq(N)937 3209 y Fp(X)938 3388 y Fe(a)p FD(\024)p Fq(j)1070 3288 y Fs(k)p FF(F)1172 3300 y Fe(a)1211 3288 y FF(P)1276 3300 y Fq(k)1317 3288 y FE(f)1358 3300 y Fr(1)1395 3288 y Fs(k)1437 3303 y Fq(L)1483 3286 y FC(2)1519 3288 y Fu(2)1571 3232 y FC(3)p 1570 3241 29 3 v 1570 3274 a(2)1608 3254 y Fe(a)1647 3288 y Fs(k)p FF(F)1749 3300 y Fq(j)1784 3288 y FF(P)1849 3300 y Fq(k)1890 3288 y FE(f)1931 3300 y Fr(2)1968 3288 y Fs(k)2010 3254 y Fr(2)2010 3311 y Fq(L)2056 3294 y FC(2)2110 3288 y Fu(+)2193 3209 y Fp(X)2194 3388 y Fe(p)p FD(\024)p Fq(j)2327 3288 y Fs(k)p FF(F)2429 3300 y Fe(p)2467 3288 y FF(P)2532 3300 y Fq(k)2573 3288 y FE(f)2614 3300 y Fr(2)2651 3288 y Fs(k)2693 3303 y Fq(L)2739 3286 y FC(2)2775 3288 y Fu(2)2817 3254 y FD(\000)2878 3227 y Fd(p)p 2878 3241 33 3 v 2880 3274 a FC(2)2925 3288 y Fs(k)p FF(F)3027 3300 y Fq(j)3062 3288 y FF(P)3127 3300 y Fq(k)3168 3288 y FE(f)3209 3300 y Fr(1)3246 3288 y Fs(k)3288 3303 y Fq(L)3334 3286 y FC(2)p 0 TeXcolorgray 3703 3288 a Fu(\(4.9\))p 0 TeXcolorgray 620 3528 a Fs(\002k)p FF(F)787 3540 y Fq(j)821 3528 y FF(P)886 3540 y Fq(k)927 3528 y FE(f)968 3540 y Fr(2)1005 3528 y Fs(k)1047 3543 y Fq(L)1093 3526 y FC(2)1129 3528 y Fu(2)1171 3493 y Fr(2)p Fq(j)1257 3528 y Fu(+)1340 3449 y Fp(X)1341 3627 y Fe(p)p FD(\025)p Fq(j)1473 3528 y Fs(k)p FF(F)1575 3540 y Fe(p)1614 3528 y FF(P)1679 3540 y Fq(k)1720 3528 y FE(f)1761 3540 y Fr(2)1798 3528 y Fs(k)1840 3543 y Fq(L)1886 3526 y FC(2)1922 3528 y Fs(k)p FF(F)2024 3540 y Fe(p)2062 3528 y FF(P)2127 3540 y Fq(k)2168 3528 y FE(f)2209 3540 y Fr(1)2246 3528 y Fs(k)2288 3543 y Fq(L)2334 3526 y FC(2)2370 3528 y Fs(k)p FF(F)2472 3540 y Fq(j)2506 3528 y FF(P)2571 3540 y Fq(k)2612 3528 y FE(f)2653 3540 y Fr(2)2690 3528 y Fs(k)2732 3543 y Fq(L)2778 3526 y FC(2)2814 3528 y Fu(2)2866 3471 y FC(3)p 2865 3480 29 3 v 2865 3513 a(2)2904 3493 y Fq(j)2957 3528 y Fu(+)18 b(2)3082 3493 y FD(\000)p Fq(j)s(N)3227 3528 y Fu(2)3269 3493 y FD(\000)p Fq(k)q(N)620 3754 y Fs(\002k)p FE(f)768 3766 y Fr(1)804 3754 y Fs(k)846 3781 y Fq(H)904 3754 y Fm(\000)p Fl(N)900 3801 y Fm(\000)p Fl(N)1007 3754 y Fs(k)p FE(f)1090 3766 y Fr(2)1126 3754 y Fs(k)1168 3720 y Fr(2)1168 3789 y Fq(H)1226 3762 y Fm(\000)p Fl(N)1222 3808 y Fm(\000)p Fl(N)1347 3754 y Fu(+)g(2)1472 3720 y FD(\000)p Fq(j)s(N)1617 3754 y Fu(2)1659 3720 y FD(\000)p Fq(k)q(N)1810 3754 y Fs(k)p FF(P)1917 3766 y Fq(k)1958 3754 y FE(f)1999 3766 y Fr(2)2036 3754 y Fs(k)2078 3773 y Fq(L)2124 3752 y FC(2)2124 3792 y Fm(\000)p Fl(N)2225 3754 y Fs(k)p FF(P)2332 3766 y Fq(k)2373 3754 y FE(f)2414 3766 y Fr(1)2451 3754 y Fs(k)2493 3773 y Fq(L)2539 3752 y FC(2)2539 3792 y Fm(\000)p Fl(N)2640 3754 y Fs(k)p FE(f)2723 3766 y Fr(2)2760 3754 y Fs(k)2802 3781 y Fq(H)2860 3754 y Fm(\000)p Fl(N)2856 3801 y Fm(\000)p Fl(N)2963 3754 y FE(:)0 3944 y Fu(Com)n(bining)33 b(the)h(previous)f (estimate)g(\()p 1 0 0 TeXcolorrgb 1284 3944 a SDict begin H.S end 1284 3944 a 1 0 0 TeXcolorrgb Fu(4.2)p 1 0 0 TeXcolorrgb 1391 3890 a SDict begin H.R end 1391 3890 a 1391 3944 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.2) cvn H.B /ANN pdfmark end 1391 3944 a 0 TeXcolorgray Fu(\)\()p 1 0 0 TeXcolorrgb 1455 3944 a SDict begin H.S end 1455 3944 a 1 0 0 TeXcolorrgb Fu(4.3)p 1 0 0 TeXcolorrgb 1561 3890 a SDict begin H.R end 1561 3890 a 1561 3944 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.3) cvn H.B /ANN pdfmark end 1561 3944 a 0 TeXcolorgray Fu(\))q(\()p 1 0 0 TeXcolorrgb 1626 3944 a SDict begin H.S end 1626 3944 a 1 0 0 TeXcolorrgb Fu(4.4)p 1 0 0 TeXcolorrgb 1732 3890 a SDict begin H.R end 1732 3890 a 1732 3944 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.4) cvn H.B /ANN pdfmark end 1732 3944 a 0 TeXcolorgray Fu(\)\()p 1 0 0 TeXcolorrgb 1796 3944 a SDict begin H.S end 1796 3944 a 1 0 0 TeXcolorrgb Fu(4.7)p 1 0 0 TeXcolorrgb 1903 3890 a SDict begin H.R end 1903 3890 a 1903 3944 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.7) cvn H.B /ANN pdfmark end 1903 3944 a 0 TeXcolorgray Fu(\))h(and)f(\()p 1 0 0 TeXcolorrgb 2168 3944 a SDict begin H.S end 2168 3944 a 1 0 0 TeXcolorrgb Fu(4.9)p 1 0 0 TeXcolorrgb 2275 3890 a SDict begin H.R end 2275 3890 a 2275 3944 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.9) cvn H.B /ANN pdfmark end 2275 3944 a 0 TeXcolorgray Fu(\))h(together)f(with)h(Prop)r(osition)p 1 0 0 TeXcolorrgb 3323 3944 a SDict begin H.S end 3323 3944 a 1 0 0 TeXcolorrgb Fu(2.1)p 1 0 0 TeXcolorrgb 3429 3890 a SDict begin H.R end 3429 3890 a 3429 3944 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (prop.2.1) cvn H.B /ANN pdfmark end 3429 3944 a 0 TeXcolorgray Fu(,)h(the)f(energy)0 4044 y(equation)27 b(comes)g(to)372 4240 y SDict begin H.S end 372 4240 a 372 4240 a SDict begin 12 H.A end 372 4240 a 372 4240 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.10) cvn /DEST pdfmark end 372 4240 a 382 4184 a Fu(1)p 382 4221 42 4 v 382 4297 a(2)459 4184 y FE(d)p 444 4221 74 4 v 444 4297 a(dt)527 4240 y Fs(k)p Ff(F)620 4252 y Fq(j)654 4240 y Fs(P)712 4252 y Fq(k)753 4240 y FE(f)794 4252 y Fr(2)830 4240 y 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FE(f)2886 1746 y Fr(2)2923 1734 y Fs(k)2965 1749 y Fq(L)3011 1732 y FC(2)3046 1734 y Fu(2)3088 1699 y FD(\000)3149 1673 y Fd(p)p 3149 1687 33 3 v 3151 1719 a FC(2)3197 1734 y Fs(k)p FF(F)3299 1746 y Fq(j)3333 1734 y FF(P)3398 1746 y Fq(k)3439 1734 y FE(f)3480 1746 y Fr(1)3517 1734 y Fs(k)3559 1749 y Fq(L)3605 1732 y FC(2)252 1974 y Fs(\002)18 b(k)p FF(F)437 1986 y Fq(j)471 1974 y FF(P)536 1986 y Fq(k)577 1974 y FE(f)618 1986 y Fr(2)655 1974 y Fs(k)697 1989 y Fq(L)743 1972 y FC(2)779 1974 y Fu(2)821 1939 y Fq(j)855 1974 y Fu(2)897 1939 y Fr(6)p Fq(k)994 1974 y Fk(.)1082 1895 y Fp(X)1100 2073 y Fq(j;k)1215 1895 y Fp(X)1217 2073 y Fe(p)p FD(\025)p Fq(j)1349 1974 y Fs(k)p FF(F)1451 1986 y Fe(p)1490 1974 y FF(P)1555 1986 y Fq(k)1596 1974 y FE(f)1637 1986 y Fr(1)1674 1974 y Fs(k)1716 1989 y Fq(L)1762 1972 y FC(2)1797 1974 y Fu(2)1849 1917 y FC(3)p 1849 1926 29 3 v 1849 1959 a(2)1887 1939 y Fe(p)1926 1974 y Fs(k)p FF(F)2028 1986 y Fe(p)2067 1974 y FF(P)2132 1986 y Fq(k)2173 1974 y FE(f)2214 1986 y Fr(2)2251 1974 y Fs(k)2293 1989 y Fq(L)2339 1972 y FC(2)2374 1974 y Fu(2)2416 1939 y FD(\000)2478 1917 y FC(1)p 2478 1926 V 2478 1959 a(2)2516 1939 y Fe(p)2555 1974 y Fs(k)p FF(F)2657 1986 y Fq(j)2691 1974 y FF(P)2756 1986 y Fq(k)2797 1974 y FE(f)2838 1986 y Fr(2)2875 1974 y Fs(k)2917 1989 y Fq(L)2963 1972 y FC(2)2999 1974 y Fu(2)3041 1939 y FD(\000)3102 1917 y FC(1)p 3102 1926 V 3102 1959 a(2)3141 1939 y Fq(j)3176 1974 y Fu(2)3218 1939 y Fq(j)s FD(\000)p Fe(p)3339 1974 y Fu(2)3381 1939 y Fr(6)p Fq(k)252 2215 y Fu(+)335 2136 y Fp(X)353 2315 y Fq(j;k)535 2136 y Fp(X)469 2314 y Fe(p)p FD(\024)p Fq(j)s Fr(+)p Fq(N)690 2322 y FC(0)736 2215 y Fs(k)p FF(F)838 2227 y Fe(p)876 2215 y FF(P)941 2227 y Fq(k)983 2215 y FE(f)1024 2227 y Fr(2)1060 2215 y Fs(k)1102 2230 y Fq(L)1148 2213 y FC(2)1184 2215 y Fu(2)1226 2181 y FD(\000)p Fe(p)1316 2215 y Fs(k)p FF(F)1418 2227 y Fq(j)1453 2215 y FF(P)1518 2227 y Fq(k)1559 2215 y FE(f)1600 2227 y Fr(1)1637 2215 y Fs(k)1679 2230 y Fq(L)1725 2213 y FC(2)1761 2215 y Fu(2)1803 2181 y Fr(2)p Fq(j)1870 2215 y Fs(k)p FF(F)1972 2227 y Fq(j)2007 2215 y FF(P)2072 2227 y Fq(k)2113 2215 y FE(f)2154 2227 y Fr(2)2191 2215 y Fs(k)2233 2230 y Fq(L)2279 2213 y FC(2)2314 2215 y Fu(2)2356 2181 y FD(\000)2418 2154 y Fl(j)p 2418 2168 V 2418 2201 a FC(2)2460 2215 y Fu(2)2502 2181 y Fr(6)p Fq(k)2599 2215 y Fk(.)k Fs(k)p FE(f)2769 2227 y Fr(1)2806 2215 y Fs(k)2848 2233 y Fq(H)2906 2213 y FC(3)2902 2251 y(3)2943 2215 y Fs(k)p FE(f)3026 2227 y Fr(2)3062 2215 y Fs(k)3104 2181 y Fr(2)3104 2279 y Fq(H)3162 2237 y Fm(\000)3217 2221 y FC(1)p 3217 2228 V 3217 2259 a(2)3158 2297 y(3)3264 2215 y FE(:)0 2452 y Fu(F)-7 b(or)27 b(the)h(fourth)g (term,)f(use)h(Cauc)n(h)n(y)e(inequalit)n(y)-7 b(,)27 b(w)n(e)h(ha)n(v)n(e)294 2537 y Fp(X)312 2716 y Fq(j;k)494 2537 y Fp(X)427 2715 y Fe(p)p FD(\024)p Fq(j)s Fr(+)p Fq(N)648 2723 y FC(0)695 2616 y Fs(k)p FF(F)797 2628 y Fe(p)835 2616 y FF(P)900 2628 y Fq(k)941 2616 y FE(f)982 2628 y Fr(1)1019 2616 y Fs(k)1061 2631 y Fq(L)1107 2614 y FC(2)1143 2616 y Fu(2)1195 2555 y Fd(p)p 1194 2569 33 3 v 1196 2602 a FC(2)1241 2616 y Fs(k)p FF(F)1343 2628 y Fq(j)1378 2616 y FF(P)1443 2628 y Fq(k)1484 2616 y FE(f)1525 2628 y Fr(2)1562 2616 y Fs(k)1604 2582 y Fr(2)1604 2638 y Fq(L)1650 2622 y FC(2)1686 2616 y Fu(2)1728 2582 y Fr(6)p Fq(k)1824 2616 y Fu(=)1912 2537 y Fp(X)1930 2716 y Fq(j;k)2112 2537 y Fp(X)2046 2715 y Fe(p)p FD(\024)p Fq(j)s Fr(+)p Fq(N)2267 2723 y FC(0)2313 2616 y Fs(k)p FF(F)2415 2628 y Fe(p)2453 2616 y FF(P)2518 2628 y Fq(k)2560 2616 y FE(f)2601 2628 y Fr(1)2637 2616 y Fs(k)2679 2631 y Fq(L)2725 2614 y FC(2)2761 2616 y Fu(2)2813 2555 y Fd(p)p 2813 2569 V 2815 2602 a FC(2)2860 2616 y Fu(2)2902 2582 y FD(\000)2964 2555 y Fl(j)p 2963 2569 29 3 v 2963 2602 a FC(2)3005 2616 y Fs(k)p FF(F)3107 2628 y Fq(j)3142 2616 y FF(P)3207 2628 y Fq(k)3248 2616 y FE(f)3289 2628 y Fr(2)3326 2616 y Fs(k)3368 2582 y Fr(2)3368 2638 y Fq(L)3414 2622 y FC(2)3450 2616 y Fu(2)3503 2555 y Fl(j)p 3501 2569 V 3501 2602 a FC(2)3543 2616 y Fu(2)3585 2582 y Fr(6)p Fq(k)215 2840 y Fk(.)p Fs(k)p FE(f)363 2852 y Fr(1)399 2840 y Fs(k)441 2859 y Fq(H)499 2839 y FC(3)495 2877 y(3)536 2840 y Fs(k)p FE(f)619 2852 y Fr(2)655 2840 y Fs(k)697 2897 y Fq(H)766 2839 y FC(1)p 766 2846 V 766 2877 a(2)761 2915 y(3)p 761 2922 V 761 2953 a(2)812 2840 y Fs(k)p FE(f)895 2852 y Fr(2)932 2840 y Fs(k)974 2897 y Fq(H)1032 2855 y Fm(\000)1087 2839 y FC(1)p 1087 2846 V 1087 2877 a(2)1028 2915 y(3)1156 2840 y Fk(.)23 b FE(")p Fs(k)p FE(f)1366 2852 y Fr(2)1402 2840 y Fs(k)1444 2806 y Fr(2)1444 2904 y Fq(H)1513 2846 y FC(1)p 1513 2853 V 1513 2884 a(2)1508 2922 y(3)p 1508 2929 V 1508 2961 a(2)1578 2840 y Fu(+)18 b FE(C)1720 2852 y Fq(")1755 2840 y Fs(k)p FE(f)1838 2852 y Fr(1)1875 2840 y Fs(k)1917 2806 y Fr(2)1917 2866 y Fq(H)1975 2846 y FC(3)1971 2884 y(3)2012 2840 y Fs(k)p FE(f)2095 2852 y Fr(2)2131 2840 y Fs(k)2173 2806 y Fr(2)2173 2904 y Fq(H)2231 2862 y Fm(\000)2286 2846 y FC(1)p 2286 2853 V 2286 2884 a(2)2227 2922 y(3)2333 2840 y 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2567 3273 a(2)2507 3311 y(3)2632 3229 y Fu(+)g FE(C)2774 3241 y Fq(")2809 3229 y Fs(k)p FE(f)2892 3241 y Fr(1)2929 3229 y Fs(k)2971 3194 y Fr(2)2971 3254 y Fq(H)3029 3234 y FC(3)3025 3272 y(3)3066 3229 y Fs(k)p FE(f)3149 3241 y Fr(2)3185 3229 y Fs(k)3227 3194 y Fr(2)3227 3293 y Fq(H)3285 3250 y Fm(\000)3340 3234 y FC(1)p 3340 3241 V 3340 3273 a(2)3281 3311 y(3)3387 3229 y FE(:)100 3470 y Fu(W)-7 b(e)24 b(remark)f(that)i Ff(I)743 3482 y Fr(2)804 3470 y Fu(can)f(b)r(e)h(handled)f(b)n(y)g(the)h(similar)f(manner.)35 b(Noticing)24 b(that)h Fs(k)p FE(f)2817 3482 y Fr(1)2853 3470 y Fs(k)2895 3497 y Fq(H)2953 3470 y Fm(\000)p Fl(N)2949 3517 y Fm(\000)p Fl(N)3068 3470 y Fu(+)12 b Fs(k)p FE(f)3228 3482 y Fr(2)3263 3470 y Fs(k)3305 3497 y Fq(H)3363 3470 y Fm(\000)p Fl(N)3359 3517 y Fm(\000)p Fl(N)3478 3470 y Fu(+)g Fs(k)p FE(f)3638 3482 y Fr(2)3674 3470 y Fs(k)3716 3489 y Fq(L)3762 3469 y FC(1)3762 3507 y(1)3809 3470 y Fu(+)0 3594 y Fs(k)p FE(f)83 3606 y Fr(1)119 3594 y Fs(k)161 3612 y Fq(L)207 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3897 a(2)2445 3777 y Fu(+)18 b FE(C)2587 3789 y Fq(")2623 3777 y Fs(k)p FE(f)2706 3789 y Fr(2)2742 3777 y Fs(k)2784 3743 y Fr(2)2784 3841 y Fq(H)2842 3799 y Fm(\000)2897 3783 y FC(1)p 2897 3790 V 2897 3821 a(2)2838 3859 y(3)2962 3777 y Fu(+)g Fs(k)p FE(f)3128 3789 y Fr(1)3165 3777 y Fs(k)3207 3795 y Fq(H)3265 3775 y FC(3)3261 3813 y(3)3302 3777 y Fs(k)p FE(f)3385 3789 y Fr(2)3421 3777 y Fs(k)3463 3834 y Fq(H)3521 3791 y Fm(\000)3576 3775 y FC(1)p 3576 3782 V 3576 3813 a(2)3517 3852 y(3)3623 3777 y FE(:)100 4027 y Fu(Let)26 b FE(X)7 b Fu(\()p FE(t)p Fu(\))23 b(:=)551 3965 y Fp(P)638 4052 y Fq(j;k)740 4027 y Fs(k)p Ff(F)833 4039 y Fq(j)867 4027 y Fs(P)925 4039 y Fq(k)966 4027 y FE(f)1007 4039 y Fr(2)1044 4027 y Fs(k)1086 3997 y Fr(2)1086 4054 y Fq(L)1132 4037 y FC(2)1167 4027 y Fu(2)1209 3997 y FD(\000)p Fq(j)1296 4027 y Fu(2)1338 3997 y Fr(6)p Fq(k)1411 4027 y FE(;)14 b(Y)19 b Fu(\()p FE(t)p Fu(\))k(:=)1743 3965 y Fp(P)1831 4052 y Fq(j;k)1932 4027 y Fs(k)p Ff(F)2025 4039 y Fq(j)2060 4027 y Fs(P)2118 4039 y Fq(k)2158 4027 y FE(f)2199 4039 y Fr(2)2236 4027 y Fs(k)2278 3997 y Fr(2)2278 4054 y Fq(L)2324 4037 y FC(2)2360 4027 y Fu(2)2402 3997 y Fq(j)2436 4027 y Fu(2)2478 3997 y Fr(3)p Fq(k)2552 4027 y Fu(.)36 b(Com)n(bining)26 b(the)g(estimates)g(of)h Ff(I)3677 4039 y Fr(1)3740 4027 y Fu(and)0 4129 y Ff(I)46 4141 y Fr(2)83 4129 y Fu(,)h(w)n(e)f(can)g (get)h(that)716 4301 y SDict begin H.S end 716 4301 a 716 4301 a SDict begin 12 H.A end 716 4301 a 716 4301 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.13) cvn /DEST pdfmark end 716 4301 a FE(X)792 4266 y FD(0)814 4301 y Fu(\()p FE(t)p Fu(\))19 b(+)1010 4208 y Fp(\020)1060 4301 y FE(\024)f Fs(\000)g FE(")g Fs(\000)g FE(C)1408 4313 y Fq(")1444 4301 y FE(X)1529 4244 y FC(1)p 1529 4253 29 3 v 1529 4286 a(2)1572 4208 y Fp(\021)1635 4301 y FE(Y)h Fu(\()p FE(t)p Fu(\))k Fs(\024)g FE(C)1972 4233 y Fp(\000)2078 4301 y Fu(sup)2024 4374 y Fq(t)p FD(2)p Fr([0)p Fq(;T)2205 4382 y FC(1)2237 4374 y Fr(])2270 4301 y Fs(k)p FE(f)2353 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Fr(3\))p Fq(k)3955 2345 y Fp(\001)3993 2412 y FE(:)0 2649 y Fu(Then)28 b(b)n(y)g(\()p 1 0 0 TeXcolorrgb 365 2649 a SDict begin H.S end 365 2649 a 1 0 0 TeXcolorrgb Fu(3.10)p 1 0 0 TeXcolorrgb 513 2595 a SDict begin H.R end 513 2595 a 513 2649 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.10) cvn H.B /ANN pdfmark end 513 2649 a 0 TeXcolorgray Fu(\))g(and)g(in)n (terp)r(olation)e(inequalit)n(y)h Fs(k)p FE(f)1696 2661 y Fr(2)1733 2649 y Fs(k)1775 2667 y Fq(L)1821 2647 y FC(2)1821 2686 y Fm(\000)p Fl(N)1945 2649 y Fs(\024)c FE(")p Fs(k)p FE(f)2155 2661 y Fr(2)2191 2649 y Fs(k)2233 2675 y Fq(H)2291 2648 y Fl(n)p FC(+1)2287 2709 y Fl(`)p Fm(\000)2367 2693 y FC(3)p 2367 2700 29 3 v 2367 2731 a(2)2432 2649 y Fu(+)18 b FE(C)2574 2661 y Fq(")2610 2649 y Fs(k)p FE(f)2693 2661 y Fr(2)2729 2649 y Fs(k)2771 2705 y Fq(H)2829 2663 y Fm(\000)2884 2647 y FC(1)p 2884 2654 V 2884 2685 a(2)2825 2723 y(3)2931 2649 y Fu(,)27 b(w)n(e)h(can)f(deduce)h(that)466 2857 y SDict begin H.S end 466 2857 a 466 2857 a SDict begin 12 H.A end 466 2857 a 466 2857 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.19) cvn /DEST pdfmark end 466 2857 a FE(X)542 2823 y FD(0)535 2878 y Fr(n)p Fq(;`)623 2857 y Fu(\()p FE(t)p Fu(\))19 b(+)g(\()p FE(\024)f Fs(\000)g FE(C)1060 2869 y Fr(n)p Fq(;`)1149 2857 y Fs(k)p FE(f)1232 2869 y Fr(2)1268 2857 y Fs(k)1310 2914 y Fq(H)1368 2872 y Fm(\000)1423 2856 y FC(1)p 1423 2863 29 3 v 1423 2894 a(2)1364 2932 y(3)1488 2857 y Fs(\000)g FE(")p Fu(\))p FE(Y)1690 2869 y Fr(n)p Fq(;`)1779 2857 y Fu(\()p FE(t)p Fu(\))24 b Fk(.)e FE(C)6 b Fu(\()54 b(sup)2081 2931 y Fq(t)p FD(2)p Fr([0)p Fq(;T)2262 2939 y FC(0)2295 2931 y Fr(])2327 2857 y Fs(k)p FE(f)2410 2869 y Fr(1)2447 2857 y Fs(k)2489 2883 y Fq(H)2547 2856 y FC(n+2)2543 2903 y Fl(`)2658 2857 y Fu(\)\()p Fs(k)p FE(f)2805 2869 y Fr(2)2841 2857 y Fs(k)2883 2823 y Fr(2)2883 2921 y Fq(H)2941 2879 y Fm(\000)2997 2863 y FC(1)p 2997 2870 V 2997 2901 a(2)2937 2939 y(3)3062 2857 y Fu(+)18 b(1\))p FE(:)p 0 TeXcolorgray 420 w Fu(\(4.19\))p 0 TeXcolorgray 0 3071 a(F)-7 b(rom)32 b(\()p 1 0 0 TeXcolorrgb 255 3071 a SDict begin H.S end 255 3071 a 1 0 0 TeXcolorrgb Fu(4.17)p 1 0 0 TeXcolorrgb 402 3017 a SDict begin H.R end 402 3017 a 402 3071 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.17) cvn H.B /ANN pdfmark end 402 3071 a 0 TeXcolorgray Fu(\))q(,)h(w)n(e)f(ha)n(v) n(e)f(that)h(sup)1123 3091 y Fq(t)p FD(2)p Fr([0)p Fq(;T)1304 3099 y FC(0)1336 3091 y Fr(])1373 3071 y Fs(k)p FE(f)1456 3083 y Fr(2)1492 3071 y Fs(k)1534 3127 y Fq(H)1592 3085 y Fm(\000)1647 3069 y FC(1)p 1647 3076 29 3 v 1647 3107 a(2)1588 3145 y(3)1724 3071 y Fk(.)e Fu(2)p FE(")1900 3083 y Fr(1)1969 3071 y Fu(with)i FE(")2201 3083 y Fr(1)2271 3071 y Fu(su\016cien)n(tly)g(small,)g(and)g(\()p 1 0 0 TeXcolorrgb 3134 3071 a SDict begin H.S end 3134 3071 a 1 0 0 TeXcolorrgb Fu(4.16)p 1 0 0 TeXcolorrgb 3282 3017 a SDict begin H.R end 3282 3017 a 3282 3071 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.15) cvn H.B /ANN pdfmark end 3282 3071 a 0 TeXcolorgray Fu(\))h(holds,)f(th)n(us)g(w)n(e)0 3211 y(can)27 b(c)n(ho)r(ose)g(small)g FE(")g Fu(to)h(get)f(that)1070 3395 y FE(X)1146 3360 y FD(0)1139 3415 y Fr(n)p Fq(;`)1227 3395 y Fu(\()p FE(t)p Fu(\))19 b(+)1433 3338 y FE(\024)p 1433 3375 48 4 v 1436 3452 a Fu(4)1491 3395 y FE(X)1560 3407 y Fr(n)p Fq(;`)1648 3395 y Fu(\()p FE(t)p Fu(\))1742 3331 y Fr(1+)1883 3309 y FC(1)p 1837 3318 122 3 v 1837 3364 a(n+)1920 3347 y(1)p 1920 3354 29 3 v 1920 3386 a(2)1996 3395 y Fk(.)j FE(C)q(;)98 b(t)23 b Fs(2)g Fu(\(0)p FE(;)14 b(T)2555 3407 y Fr(0)2592 3395 y Fu(])p FE(;)0 3576 y Fu(Therefore,)26 b(\()p 1 0 0 TeXcolorrgb 431 3576 a SDict begin H.S end 431 3576 a 1 0 0 TeXcolorrgb Fu(4.18)p 1 0 0 TeXcolorrgb 579 3523 a SDict begin H.R end 579 3523 a 579 3576 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.18) cvn H.B /ANN pdfmark end 579 3576 a 0 TeXcolorgray Fu(\))i(holds)g(ture)f(if)h(w)n(e)f(let)h FE(`)23 b Fu(=)g FE(l)c Fs(\000)1633 3544 y Fr(3)p 1633 3558 34 4 v 1633 3605 a(2)1676 3576 y Fu(n)g Fs(\000)1834 3544 y Fr(3)p 1834 3558 V 1834 3605 a(4)1877 3576 y Fu(,)28 b(together)f(with)h(\()p 1 0 0 TeXcolorrgb 2477 3576 a SDict begin H.S end 2477 3576 a 1 0 0 TeXcolorrgb Fu(4.16)p 1 0 0 TeXcolorrgb 2624 3523 a SDict begin H.R end 2624 3523 a 2624 3576 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.15) cvn H.B /ANN pdfmark end 2624 3576 a 0 TeXcolorgray Fu(\))q(,)g(w)n(e)f(conclude)g(that) 436 3743 y SDict begin H.S end 436 3743 a 436 3743 a SDict begin 12 H.A end 436 3743 a 436 3743 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.20) cvn /DEST pdfmark end 436 3743 a Fs(k)p FE(f)9 b Fu(\()p FE(t)p Fu(\))p Fs(k)664 3755 y Fq(H)722 3736 y FC(n)718 3790 y Fl(l)p Fm(\000)793 3774 y FC(3)p 793 3781 29 3 v 793 3812 a(2)831 3790 y(n)p Fm(\000)917 3774 y FC(3)p 917 3781 V 917 3812 a(4)987 3743 y Fs(\024)22 b(k)p FE(f)1157 3755 y Fr(1)1194 3743 y Fu(\()p FE(t)p Fu(\))p Fs(k)1330 3755 y Fq(H)1388 3736 y FC(n)1384 3790 y Fl(l)p Fm(\000)1459 3774 y FC(3)p 1459 3781 V 1459 3812 a(2)1497 3790 y(n)p Fm(\000)1583 3774 y FC(3)p 1583 3781 V 1583 3812 a(4)1648 3743 y Fu(+)c Fs(k)p FE(f)1814 3755 y Fr(2)1851 3743 y Fu(\()p FE(t)p Fu(\))p Fs(k)1987 3755 y Fq(H)2045 3736 y FC(n)2041 3790 y Fl(l)p Fm(\000)2116 3774 y FC(3)p 2116 3781 V 2116 3812 a(2)2154 3790 y(n)p Fm(\000)2240 3774 y FC(3)p 2240 3781 V 2240 3812 a(4)2310 3743 y Fk(.)k FE(t)2427 3709 y FD(\000)2489 3686 y FC(n)p 2489 3695 32 3 v 2490 3729 a(2)2530 3709 y FD(\000)2592 3686 y FC(1)p 2592 3695 29 3 v 2592 3729 a(4)2653 3743 y Fu(+)c(1)p FE(;)97 b(t)23 b Fs(2)g Fu(\(0)p FE(;)14 b(T)3189 3755 y Fr(0)3225 3743 y Fu(])p FE(:)p 0 TeXcolorgray 391 w Fu(\(4.20\))p 0 TeXcolorgray 0 3929 a(In)28 b(particular,)e(it)i(holds)f(that)135 b(sup)990 4002 y Fq(t)p FD(2)p Fr([)p Fq(T)1118 4010 y FC(0)1151 4002 y Fq(=)p Fr(2)p Fq(;T)1277 4010 y FC(0)1309 4002 y Fr(])1342 3929 y Fs(k)p FE(f)9 b Fu(\()p FE(t)p Fu(\))p Fs(k)1570 3941 y Fq(H)1628 3921 y FC(n)1624 3975 y Fl(l)p Fm(\000)1698 3959 y FC(3)p 1698 3966 V 1698 3997 a(2)1736 3975 y(n)p Fm(\000)1822 3959 y FC(3)p 1822 3966 V 1822 3997 a(4)1892 3929 y Fk(.)22 b Fu(1.)100 4131 y(Step)28 b(2:)36 b(V)-7 b(alidit)n(y)28 b(of)34 b(\()p 1 0 0 TeXcolorrgb 837 4131 a SDict begin H.S end 837 4131 a 1 0 0 TeXcolorrgb Fu(4.14)p 1 0 0 TeXcolorrgb 985 4078 a SDict begin H.R end 985 4078 a 985 4131 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.14) cvn H.B /ANN pdfmark end 985 4131 a 0 TeXcolorgray Fu(\))28 b(o)n(v)n(er)e([0)p FE(;)14 b(T)e Fu(].)p 100 4165 1331 4 v 31 w(Note)33 b(that)h FE(T)43 b(>)32 b Fu(0)g(is)h(\014xed.)53 b(Since)33 b FE(f)41 b Fs(2)33 b FE(C)6 b Fu(\([0)p FE(;)14 b(T)e Fu(])p FE(;)i(H)3244 4082 y FD(\000)3304 4060 y FC(1)p 3304 4069 29 3 v 3304 4102 a(2)3237 4156 y 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2751 y Fs(P)3274 2763 y Fq(k)3315 2751 y FF(h)p Fs(i)3400 2763 y Fq(v)3439 2751 y FE(;)412 2881 y Ff(J)458 2893 y Fr(2)518 2881 y Fu(:=)23 b Fs(h)p Ff(F)712 2893 y Fq(j)747 2881 y Fs(P)805 2893 y Fq(k)845 2881 y FE(Q)p Fu(\()p FF(h)p FE(;)14 b FF(f)1062 2893 y Fr(1)1099 2881 y Fu(\))p FE(;)g Ff(F)1219 2893 y Fq(j)1254 2881 y Fs(P)1312 2893 y Fq(k)1353 2881 y FF(h)p Fs(i)1438 2893 y Fq(v)1496 2881 y Fu(+)k Fs(h)p Ff(F)1662 2893 y Fq(j)1697 2881 y Fs(P)1755 2893 y Fq(k)1796 2881 y FE(Q)p Fu(\()p FF(h)p FE(;)c FF(g)2032 2893 y Fr(2)2069 2881 y Fu(\))p FE(;)g Ff(F)2189 2893 y Fq(j)2224 2881 y Fs(P)2282 2893 y Fq(k)2322 2881 y FF(h)p Fs(i)2407 2893 y Fq(v)2447 2881 y FE(:)0 3042 y Fu(F)-7 b(or)27 b Ff(J)195 3054 y Fr(1)232 3042 y Fu(,)h(w)n(e)f(use)g(the)h(estimate)g(\()p 1 0 0 TeXcolorrgb 1056 3042 a SDict begin H.S end 1056 3042 a 1 0 0 TeXcolorrgb Fu(4.2)p 1 0 0 TeXcolorrgb 1162 2988 a SDict begin H.R end 1162 2988 a 1162 3042 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 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443 y FF(f)2982 455 y Fr(1)3020 443 y Fs(k)3062 458 y Fq(L)3108 441 y FC(2)3162 443 y Fu(+)g Fs(k)p FF(F)3347 455 y Fe(p)3385 443 y FF(P)3450 455 y Fq(k)3492 443 y FF(g)3540 455 y Fr(2)3577 443 y Fs(k)3619 458 y Fq(L)3665 441 y FC(2)3700 443 y Fu(\))372 678 y Fs(\002k)p FF(F)539 690 y Fq(j)573 678 y FF(P)638 690 y Fq(k)679 678 y FF(h)p Fs(k)774 693 y Fq(L)820 677 y FC(2)856 678 y Fu(2)908 622 y FC(3)p 908 631 29 3 v 908 664 a(2)946 644 y Fq(j)999 678 y Fu(+)g(2)1124 644 y FD(\000)p Fq(j)s(N)1269 678 y Fu(2)1311 644 y FD(\000)p Fq(k)q(N)1463 678 y Fu(\()p Fs(k)p FF(f)1566 690 y Fr(1)1603 678 y Fs(k)1645 705 y Fq(H)1703 678 y Fm(\000)p Fl(N)1699 725 y Fm(\000)p Fl(N)1824 678 y Fu(+)g Fs(k)p FF(g)1997 690 y Fr(2)2034 678 y Fs(k)2076 705 y Fq(H)2134 678 y Fm(\000)p Fl(N)2130 725 y Fm(\000)p Fl(N)2236 678 y Fu(\))p Fs(k)p FF(h)p Fs(k)2405 644 y Fr(2)2405 713 y Fq(H)2463 686 y Fm(\000)p Fl(N)2459 732 y Fm(\000)p Fl(N)225 847 y Fu(+)82 b(2)414 813 y FD(\000)p Fq(j)s(N)559 847 y Fu(2)601 813 y FD(\000)p Fq(k)q(N)753 847 y Fs(k)p FF(P)860 859 y Fq(k)900 847 y FF(h)p Fs(k)995 865 y Fq(L)1041 845 y FC(2)1041 885 y Fm(\000)p Fl(N)1143 847 y Fu(\()p Fs(k)p FF(P)1282 859 y Fq(k)1323 847 y FF(f)1352 859 y Fr(1)1389 847 y Fs(k)1431 865 y Fq(L)1477 845 y FC(2)1477 885 y Fm(\000)p Fl(N)1597 847 y Fu(+)18 b Fs(k)p FF(P)1787 859 y Fq(k)1828 847 y FF(g)1876 859 y Fr(2)1913 847 y Fs(k)1955 865 y Fq(L)2001 845 y FC(2)2001 885 y Fm(\000)p Fl(N)2102 847 y Fu(\))p Fs(k)p FF(h)p Fs(k)2271 874 y Fq(H)2329 847 y Fm(\000)p Fl(N)2325 894 y Fm(\000)p Fl(N)2432 847 y FE(:)p 0 TeXcolorgray 1248 w Fu(\(5.8\))p 0 TeXcolorgray 100 1034 a(While)31 b(for)f Ff(J)521 1004 y Fr(2)521 1055 y(2)590 1034 y Fu(and)g Ff(J)800 1004 y Fr(4)800 1055 y(2)837 1034 y Fu(,)i(w)n(e)f(\014rst)g(p)r(erform)f(the)h(dy)n (adic)f(decomp)r(osition)h(in)g(frequency)f(space)h(as)f(in)h(\()p 1 0 0 TeXcolorrgb 3415 1034 a SDict begin H.S end 3415 1034 a 1 0 0 TeXcolorrgb Fu(4.5)p 1 0 0 TeXcolorrgb 3521 981 a SDict begin H.R end 3521 981 a 3521 1034 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.5) cvn H.B /ANN pdfmark end 3521 1034 a 0 TeXcolorgray Fu(\))q(.)47 b(Rather)0 1134 y(than)32 b(directly)g(applying)h(\()p 1 0 0 TeXcolorrgb 882 1134 a SDict begin H.S end 882 1134 a 1 0 0 TeXcolorrgb Fu(2.15)p 1 0 0 TeXcolorrgb 1030 1080 a SDict begin H.R end 1030 1080 a 1030 1134 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.15) cvn H.B /ANN pdfmark end 1030 1134 a 0 TeXcolorgray Fu(\)\()p 1 0 0 TeXcolorrgb 1094 1134 a SDict begin H.S end 1094 1134 a 1 0 0 TeXcolorrgb Fu(2.17)p 1 0 0 TeXcolorrgb 1242 1080 a SDict begin H.R end 1242 1080 a 1242 1134 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.17) cvn H.B /ANN pdfmark end 1242 1134 a 0 TeXcolorgray Fu(\)\()p 1 0 0 TeXcolorrgb 1306 1134 a SDict begin H.S end 1306 1134 a 1 0 0 TeXcolorrgb Fu(2.19)p 1 0 0 TeXcolorrgb 1454 1080 a SDict begin H.R end 1454 1080 a 1454 1134 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.19) cvn H.B /ANN pdfmark end 1454 1134 a 0 TeXcolorgray Fu(\))f(and)g(\()p 1 0 0 TeXcolorrgb 1716 1134 a SDict begin H.S end 1716 1134 a 1 0 0 TeXcolorrgb Fu(2.20)p 1 0 0 TeXcolorrgb 1864 1080 a SDict begin H.R end 1864 1080 a 1864 1134 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.20) cvn H.B /ANN pdfmark end 1864 1134 a 0 TeXcolorgray Fu(\),)i(w)n(e)d(require)g(additional)h (estimates)g(for)f Ff(M)3348 1104 y Fr(1)3386 1134 y FE(;)14 b Ff(M)3510 1104 y Fr(2)3547 1134 y FE(;)g Ff(M)3671 1104 y Fr(3)3740 1134 y Fu(and)0 1234 y Ff(M)87 1203 y Fr(4)124 1234 y Fu(.)37 b(Noticing)28 b(that)g(for)f Ff(p)22 b Fs(\025)h Fu(0,)k(it)h(holds)f(that)147 1429 y Fs(k)p Ff(F)240 1441 y Fe(p)278 1429 y FE(a)322 1441 y Fe(l)365 1429 y Fs(\003)18 b Ff(F)476 1441 y Fe(p)515 1429 y FE(g)s Fs(k)600 1441 y Fq(L)646 1425 y Fm(1)733 1429 y Fs(\024)820 1316 y Fp(Z)917 1429 y FE(')p Fu(\(2)1045 1395 y FD(\000)p Fe(p)1136 1429 y FE(\030)t Fu(\)2)1250 1395 y Fr(2)p Fe(l)1322 1358 y Fp(\014)1322 1408 y(\014)1367 1429 y(b)-63 b FE(a)1394 1441 y Fr(0)1431 1429 y Fu(\(2)1505 1395 y Fe(l)1530 1429 y FE(\030)t Fu(\))1602 1358 y Fp(\014)1602 1408 y(\014)1644 1334 y(\014)1644 1383 y(\014)1644 1433 y(\014)1678 1407 y(d)1671 1429 y Ff(F)1722 1441 y Fe(p)1761 1429 y FE(g)s Fu(\()p FE(\030)t Fu(\))1908 1334 y Fp(\014)1908 1383 y(\014)1908 1433 y(\014)1950 1429 y FE(d\030)27 b Fk(.)2121 1441 y Fq(N)2207 1429 y Fu(2)2249 1395 y FD(\000)p Fr(2)p Fe(l)p Fq(N)2417 1429 y Fu(2)2459 1395 y FD(\000)p Fr(2)p Fe(p)p Fq(N)2642 1429 y Fs(k)p Ff(F)2735 1441 y Fe(p)2773 1429 y FE(g)s Fs(k)2858 1444 y Fq(L)2904 1427 y FC(2)2962 1429 y Fk(.)c Fs(k)p FE(g)s Fs(k)3177 1472 y Fq(H)3235 1443 y Fm(\000)3289 1427 y FC(1)p 3289 1434 29 3 v 3289 1465 a(2)3336 1429 y Fu(2)3378 1395 y FD(\000)p Fe(l)p Fq(N)3513 1429 y Fu(2)3555 1395 y FD(\000)p Fe(p)p Fq(N)3704 1429 y FE(;)0 1637 y Fu(where)k(w)n(e)g(use) h Fs(j)17 b Fp(b)-63 b FE(a)573 1649 y Fr(0)610 1637 y Fu(\()p FE(\021)s Fu(\))p Fs(j)24 b Fk(.)830 1649 y Fq(N)915 1637 y Fs(h)p FE(\021)s Fs(i)1023 1607 y FD(\000)p Fr(4)p Fq(N)1172 1637 y Fu(.)37 b(Th)n(us)763 1801 y SDict begin H.S end 763 1801 a 763 1801 a SDict begin 12 H.A end 763 1801 a 763 1801 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.9) cvn /DEST pdfmark end 763 1801 a -71 x Fp(\014)763 1780 y(\014)791 1801 y Ff(M)878 1767 y Fr(1)878 1821 y Fe(l)p Fq(;)p Fe(p)p Fq(;)p Fe(a)1012 1801 y Fu(\()p FE(g)s(;)14 b(h;)g(f)9 b Fu(\))1291 1730 y Fp(\014)1291 1780 y(\014)1341 1801 y Fk(.)1406 1813 y Fq(N)1492 1801 y Fu(2)1534 1767 y FD(\000)p Fe(l)p Fq(N)1669 1801 y Fu(2)1711 1767 y FD(\000)p Fe(p)p Fq(N)1860 1801 y Fu(2)1902 1767 y FD(\000)p Fe(a)p Fq(N)2051 1801 y Fs(k)p FE(g)s Fs(k)2178 1843 y Fq(H)2236 1815 y Fm(\000)2291 1799 y FC(1)p 2291 1806 29 3 v 2291 1837 a(2)2337 1801 y Fs(k)p Ff(F)2430 1813 y Fe(a)2469 1801 y FE(h)p Fs(k)2559 1816 y Fq(L)2605 1799 y FC(2)2640 1801 y Fs(k)2687 1779 y Fu(~)2682 1801 y Ff(F)2733 1813 y Fe(p)2771 1801 y FE(f)g Fs(k)2863 1816 y Fq(L)2909 1799 y FC(2)2945 1801 y FE(;)p 0 TeXcolorgray 735 w Fu(\(5.9\))p 0 TeXcolorgray 727 1883 a Fp(\014)727 1933 y(\014)754 1954 y Ff(M)841 1919 y Fr(4)841 1974 y Fe(l)p Fq(;)p Fe(p)p Fq(;)p Fe(m)992 1954 y Fu(\()p FE(g)s(;)14 b(h;)g(f)9 b Fu(\))1271 1883 y Fp(\014)1271 1933 y(\014)1321 1954 y Fk(.)1386 1966 y Fq(N)1472 1954 y Fu(2)1514 1919 y FD(\000)p Fe(l)p Fq(N)1649 1954 y Fu(2)1691 1919 y FD(\000)p Fe(p)p Fq(N)1841 1954 y Fu(2)1883 1919 y FD(\000)p Fe(m)p Fq(N)2049 1954 y Fs(k)p FE(g)s Fs(k)2176 1996 y Fq(H)2234 1968 y Fm(\000)2288 1952 y FC(1)p 2288 1959 V 2288 1990 a(2)2334 1954 y Fs(k)2381 1932 y Fu(~)2376 1954 y Ff(F)2427 1966 y Fe(p)2466 1954 y FE(h)p Fs(k)2556 1969 y Fq(L)2602 1952 y FC(2)2637 1954 y Fs(k)p Ff(F)2730 1966 y Fe(m)2785 1954 y FE(f)g Fs(k)2877 1969 y Fq(L)2923 1952 y FC(2)2958 1954 y FE(:)p 0 TeXcolorgray 681 w Fu(\(5.10\))p 0 TeXcolorgray 0 2130 a(F)-7 b(or)27 b Ff(M)236 2100 y Fr(2)273 2130 y Fu(,)h(w)n(e)f (\014rst)h(ha)n(v)n(e)e(that)333 2326 y Ff(M)420 2291 y Fr(2)420 2346 y Fe(l)p Fq(;)p Fe(a)499 2326 y Fu(\()p FE(g)s(;)14 b(h;)g(f)9 b Fu(\))23 b(=)f Fs(\000)967 2213 y Fp(Z)1050 2326 y Fu(\()p FE(a)1126 2338 y Fe(l)1170 2326 y Fs(\003)c(S)1280 2338 y Fe(a)p FD(\000)p Fq(N)1420 2346 y FC(0)1456 2326 y FE(g)s Fu(\))23 b(:)g Fs(r)p Ff(F)1720 2338 y Fe(a)1759 2326 y FE(h)18 b Fs(\012)g(r)1982 2304 y Fu(~)1977 2326 y Ff(F)2028 2338 y Fe(a)2067 2326 y FE(f)9 b(dv)21 b Fu(+)2304 2213 y Fp(Z)2387 2326 y Fu(\()p FE(b)2455 2338 y Fe(l)2499 2326 y Fs(\003)d(S)2609 2338 y Fe(a)p FD(\000)p Fq(N)2749 2346 y FC(0)2785 2326 y FE(g)s Fu(\))g Fs(\001)h Ff(F)2971 2338 y Fe(a)3010 2326 y FE(h)p Fs(r)3132 2304 y Fu(~)3127 2326 y Ff(F)3178 2338 y Fe(a)3216 2326 y FE(f)9 b(dv)s(:)0 2521 y Fu(F)-7 b(or)27 b(the)h(\014rst)f(term,)h(note)f(that)510 2604 y Fp(Z)593 2717 y Fu(\()p FE(a)669 2729 y Fe(l)712 2717 y Fs(\003)18 b(S)822 2729 y Fe(a)p FD(\000)p Fq(N)962 2737 y FC(0)998 2717 y FE(g)s Fu(\))23 b(:)h Fs(r)p Ff(F)1263 2729 y Fe(a)1301 2717 y FE(h)19 b Fs(\012)f(r)1525 2695 y Fu(~)1520 2717 y Ff(F)1571 2729 y Fe(a)1610 2717 y FE(f)9 b(dv)362 2934 y Fu(=)510 2821 y Fp(Z)g(Z)662 2934 y Fs(j)p FE(v)22 b Fs(\000)c FE(v)870 2946 y FD(\003)908 2934 y Fs(j)931 2900 y FD(\000)p Fr(3)1020 2934 y FE(')p Fu(\(2)1148 2900 y FD(\000)p Fe(l)1225 2934 y Fu(\()p FE(v)k Fs(\000)c FE(v)1442 2946 y FD(\003)1481 2934 y Fu(\)\))c(\()p Fs(S)1641 2946 y Fe(a)p FD(\000)p Fq(N)1781 2954 y FC(0)1817 2934 y FE(g)s Fu(\))1892 2959 y FD(\003)1944 2867 y Fp(\000)1996 2934 y Fu(\()q FE(v)21 b Fs(\000)d FE(v)2213 2946 y FD(\003)2252 2934 y Fu(\))g Fs(\002)g(r)p Ff(F)2505 2946 y Fe(a)2544 2934 y FE(h)2592 2867 y Fp(\001)2649 2934 y Fs(\001)2690 2867 y Fp(\000)2728 2934 y Fu(\()q FE(v)j Fs(\000)d FE(v)2945 2946 y FD(\003)2984 2934 y Fu(\))g Fs(\002)g(r)3191 2913 y Fu(~)3186 2934 y Ff(F)3237 2946 y Fe(a)3276 2934 y FE(f)3326 2867 y Fp(\001)3364 2934 y FE(dv)3447 2946 y FD(\003)3486 2934 y FE(dv)s(:)0 3129 y Fu(Since)692 3160 y Fp(Z)789 3273 y Fs(j)p FE(v)k Fs(\000)c FE(v)997 3285 y FD(\003)1035 3273 y Fs(j)1058 3239 y FD(\000)p Fr(3)1148 3273 y FE(')p Fu(\(2)1276 3239 y FD(\000)p Fe(l)1353 3273 y Fu(\()p FE(v)k Fs(\000)c FE(v)1570 3285 y FD(\003)1608 3273 y Fu(\)\))c(\()q Fs(S)1769 3285 y Fe(a)p FD(\000)p Fq(N)1909 3293 y FC(0)1945 3273 y FE(g)s Fu(\))2020 3298 y FD(\003)2072 3273 y FE(dv)2155 3285 y FD(\003)2216 3273 y Fs(\024)2304 3160 y Fp(Z)2401 3178 y(\014)2401 3228 y(\014)2401 3277 y(\014)2457 3249 y Fo(\\)2429 3273 y Fs(j)k(\001)h(j)2535 3249 y FD(\000)p Fr(3)2624 3273 y FE(')p Fu(\(2)2752 3239 y Fe(l)2777 3273 y FE(\030)t Fu(\))2849 3178 y Fp(\014)2849 3228 y(\014)2849 3277 y(\014)2891 3273 y Fs(j)q Fp(b)-47 b FE(g)s Fu(\()p FE(\030)t Fu(\))p Fs(j)14 b FE(d\030)712 3470 y Fk(.)799 3391 y Fp(X)801 3570 y Fq(j)s FD(\025)p Fr(0)933 3470 y Fu(2)975 3436 y FD(\000)p Fr(3)p Fq(l)1085 3470 y Fu(2)1127 3436 y FD(\000)p Fq(j)1213 3470 y Fs(k)p Ff(F)1306 3482 y Fq(j)1341 3470 y FE(g)s Fs(k)1426 3485 y Fq(L)1472 3468 y FC(2)1525 3470 y Fu(+)k Fs(k)1721 3446 y Fo(\\)1650 3470 y Fs(j)g(\001)h(j)1756 3446 y FD(\000)p Fr(3)1845 3470 y FE(')p Fu(\()p Fs(\001)p Fu(\))q Fs(k)2029 3485 y Fq(L)2075 3468 y FC(1)2110 3470 y Fu(2)2152 3436 y FD(\000)p Fr(3)p Fq(l)2262 3470 y Fs(k)p Ff(F)2355 3482 y FD(\000)p Fr(1)2444 3470 y FE(g)s Fs(k)2529 3488 y Fq(L)2575 3468 y FC(2)2575 3506 y(2)2633 3470 y Fk(.)k Fu(2)2763 3436 y FD(\000)p Fr(3)p Fq(l)2873 3470 y Fs(k)p FE(g)s Fs(k)3000 3527 y Fq(H)3058 3484 y Fm(\000)3112 3468 y FC(1)p 3112 3475 V 3112 3507 a(2)3054 3545 y(2)3159 3470 y FE(;)0 3730 y Fu(where)k(w)n(e)g(use)h Fs(j)601 3707 y Fo(\\)529 3730 y Fs(j)18 b(\001)h(j)635 3706 y FD(\000)p Fr(3)724 3730 y FE(')p Fu(\()p Fs(\001)p Fu(\))q(\()p FE(\021)s Fu(\))p Fs(j)24 b Fk(.)e Fs(h)p FE(\021)s Fs(i)1216 3700 y FD(\000)p Fr(4)1334 3730 y Fu(and)28 b Fs(k)s Fu(^)-45 b FE(g)r Fu(\()p FE(\030)t Fu(\))p Fs(k)1726 3742 y Fq(L)1772 3726 y Fm(1)1859 3730 y Fk(.)23 b Fs(kh)p FE(v)s Fs(i)2096 3700 y Fr(2)2133 3730 y FE(g)s Fs(k)2218 3745 y Fq(L)2264 3728 y FC(2)2300 3730 y Fu(.)37 b(Th)n(us)27 b(w)n(e)g(get)587 3835 y Fp(\014)587 3885 y(\014)587 3935 y(\014)628 3817 y(Z)9 b(Z)780 3930 y Fs(j)p FE(v)22 b Fs(\000)c FE(v)988 3942 y FD(\003)1026 3930 y Fs(j)1049 3896 y FD(\000)p Fr(3)1139 3930 y FE(')p Fu(\(2)1267 3896 y FD(\000)p Fe(l)1344 3930 y Fu(\()p FE(v)k Fs(\000)c FE(v)1561 3942 y FD(\003)1599 3930 y Fu(\)\))c(\()q Fs(S)1760 3942 y Fe(a)p FD(\000)p Fq(N)1900 3950 y FC(0)1936 3930 y FE(g)s Fu(\))2011 3955 y FD(\003)2063 3930 y Fu(\()p FE(v)22 b Fs(\002)c(r)p Ff(F)2360 3942 y Fe(a)2399 3930 y FE(h)p Fu(\))g Fs(\001)2539 3838 y Fp(\020)2588 3930 y FE(v)k Fs(\002)c(r)2807 3909 y Fu(~)2802 3930 y Ff(F)2853 3942 y Fe(a)2892 3930 y FE(f)2942 3838 y Fp(\021)3005 3930 y FE(dv)3088 3942 y FD(\003)3127 3930 y FE(dv)3213 3835 y Fp(\014)3213 3885 y(\014)3213 3935 y(\014)439 4131 y Fk(.)83 b Fu(2)629 4096 y FD(\000)p Fr(3)p Fe(l)738 4131 y Fs(k)p FE(g)s Fs(k)865 4187 y Fq(H)923 4145 y Fm(\000)978 4129 y FC(1)p 978 4136 V 978 4167 a(2)919 4205 y(2)1024 4038 y Fp(\020)1074 4131 y Fs(k)14 b Fu(\()o Fs(\000)p Fu(\001)1295 4146 y Fn(S)1333 4129 y FC(2)1364 4131 y Fu(\))1406 4067 y FC(1)p 1406 4076 V 1406 4109 a(2)1462 4131 y Ff(F)1513 4143 y Fe(a)1552 4131 y FE(h)p Fs(k)1642 4146 y Fq(L)1688 4129 y FC(2)1742 4131 y Fu(+)k Fs(k)p Ff(F)1918 4143 y Fe(a)1956 4131 y FE(h)p Fs(k)2046 4146 y Fq(L)2092 4129 y FC(2)2128 4131 y Fu(2)2170 4096 y Fe(a)2208 4038 y Fp(\021\020)2307 4131 y Fs(k)c Fu(\()p Fs(\000)p Fu(\001)2529 4146 y Fn(S)2567 4129 y FC(2)2598 4131 y Fu(\))2640 4067 y FC(1)p 2640 4076 V 2640 4109 a(2)2696 4131 y Ff(F)2747 4143 y Fe(a)2786 4131 y FE(f)9 b Fs(k)2878 4146 y Fq(L)2924 4129 y FC(2)2977 4131 y Fu(+)18 b Fs(k)3107 4109 y Fu(~)3102 4131 y Ff(F)3153 4143 y Fe(a)3192 4131 y FE(f)9 b Fs(k)3284 4146 y Fq(L)3330 4129 y FC(2)3365 4131 y Fu(2)3407 4096 y Fe(a)3445 4038 y Fp(\021)3495 4131 y FE(:)0 4329 y Fu(By)27 b(the)h(similar)f(argumen) n(t,)g(w)n(e)g(can)g(deriv)n(e)g(that)587 4429 y Fp(\014)587 4479 y(\014)587 4528 y(\014)628 4411 y(Z)711 4524 y Fu(\()p FE(a)787 4536 y Fe(l)831 4524 y Fs(\003)18 b(S)941 4536 y Fe(a)p FD(\000)p Fq(N)1081 4544 y FC(0)1117 4524 y FE(g)s Fu(\))23 b(:)g Fs(r)p Ff(F)1381 4536 y Fe(a)1420 4524 y FE(h)18 b Fs(\012)g(r)1643 4503 y Fu(~)1638 4524 y Ff(F)1689 4536 y Fe(a)1728 4524 y FE(f)9 b(dv)1864 4429 y Fp(\014)1864 4479 y(\014)1864 4528 y(\014)439 4725 y Fk(.)83 b Fu(2)629 4690 y FD(\000)p Fr(3)p Fe(l)738 4725 y Fs(k)p FE(g)s Fs(k)865 4781 y Fq(H)923 4739 y Fm(\000)978 4723 y FC(1)p 978 4730 V 978 4761 a(2)919 4799 y(4)1024 4632 y Fp(\020)1074 4725 y Fs(k)14 b Fu(\()o Fs(\000)p Fu(\001)1295 4740 y Fn(S)1333 4723 y FC(2)1364 4725 y Fu(\))1406 4660 y FC(1)p 1406 4669 V 1406 4703 a(2)1462 4725 y Ff(F)1513 4737 y Fe(a)1552 4725 y FE(h)p Fs(k)1642 4740 y Fq(L)1688 4723 y FC(2)1742 4725 y Fu(+)k Fs(k)p Ff(F)1918 4737 y Fe(a)1956 4725 y FE(h)p Fs(k)2046 4740 y Fq(L)2092 4723 y FC(2)2128 4725 y Fu(2)2170 4690 y Fe(a)2208 4632 y Fp(\021\020)2307 4725 y Fs(k)c Fu(\()p Fs(\000)p Fu(\001)2529 4740 y Fn(S)2567 4723 y FC(2)2598 4725 y Fu(\))2640 4660 y FC(1)p 2640 4669 V 2640 4703 a(2)2696 4725 y Ff(F)2747 4737 y Fe(a)2786 4725 y FE(f)9 b Fs(k)2878 4740 y Fq(L)2924 4723 y FC(2)2977 4725 y Fu(+)18 b Fs(k)3107 4703 y Fu(~)3102 4725 y Ff(F)3153 4737 y Fe(a)3192 4725 y FE(f)9 b Fs(k)3284 4740 y Fq(L)3330 4723 y FC(2)3365 4725 y Fu(2)3407 4690 y Fe(a)3445 4632 y Fp(\021)3495 4725 y FE(:)0 4923 y Fu(and)666 5007 y Fp(\014)666 5056 y(\014)666 5106 y(\014)707 4989 y(Z)790 5102 y Fu(\()p FE(b)858 5114 y Fe(l)902 5102 y Fs(\003)18 b(S)1012 5114 y Fe(a)p FD(\000)p Fq(N)1152 5122 y FC(0)1188 5102 y FE(g)s Fu(\))g Fs(\001)h Ff(F)1374 5114 y Fe(a)1413 5102 y FE(h)p Fs(r)1535 5080 y Fu(~)1530 5102 y Ff(F)1581 5114 y Fe(a)1619 5102 y FE(f)9 b(dv)1755 5007 y Fp(\014)1755 5056 y(\014)1755 5106 y(\014)1806 5102 y Fk(.)23 b Fu(2)1936 5068 y FD(\000)p Fr(2)p Fe(l)2045 5102 y Fs(k)p FE(g)s Fs(k)2172 5158 y Fq(H)2230 5116 y Fm(\000)2285 5100 y FC(1)p 2285 5107 V 2285 5138 a(2)2226 5176 y(2)2331 5102 y Fs(k)p Ff(F)2424 5114 y Fe(a)2463 5102 y FE(h)p Fs(k)2553 5117 y Fq(L)2599 5100 y FC(2)2634 5102 y Fs(k)2681 5080 y Fu(~)2676 5102 y Ff(F)2727 5114 y Fe(a)2765 5102 y FE(f)9 b Fs(k)2857 5117 y Fq(L)2903 5100 y FC(2)2938 5102 y Fu(2)2980 5068 y Fe(a)3019 5102 y FE(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 36 36 TeXDict begin HPSdict begin 36 35 bop 0 TeXcolorgray 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 0 TeXcolorgray -8 -245 a SDict begin H.S end -8 -245 a -8 -245 a SDict begin H.R end -8 -245 a -8 -245 a SDict begin [/View [/XYZ H.V]/Dest (page.36) cvn /DEST pdfmark end -8 -245 a 0 TeXcolorgray 0 TeXcolorgray 0 -170 a Fr(36)1263 b(LING-BING)24 b(HE,)e(JIE)g(JI,)g(AND)g(YUE)g(LUO) p 0 TeXcolorgray 0 29 a Fu(Then)28 b(w)n(e)f(conclude)g(that)571 280 y SDict begin H.S end 571 280 a 571 280 a SDict begin 12 H.A end 571 280 a 571 280 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.11) cvn /DEST pdfmark end 571 280 a 166 w Fs(j)p Ff(M)847 245 y Fr(2)847 300 y Fe(l)p Fq(;)p Fe(a)927 280 y Fu(\()p FE(g)s(;)14 b(h;)g(f)9 b Fu(\))p Fs(j)23 b Fk(.)f Fu(2)1381 245 y FD(\000)p Fr(3)p Fe(l)1491 280 y Fs(k)p FE(g)s Fs(k)1618 336 y Fq(H)1676 294 y Fm(\000)1730 278 y FC(1)p 1730 285 29 3 v 1730 316 a(2)1672 354 y(4)1791 187 y Fp(\020)1840 280 y Fs(k)14 b Fu(\()p Fs(\000)p Fu(\001)2062 295 y Fn(S)2100 278 y FC(2)2131 280 y Fu(\))2173 216 y FC(1)p 2173 225 V 2173 258 a(2)2229 280 y Ff(F)2280 292 y Fe(a)2319 280 y FE(h)p Fs(k)2409 295 y Fq(L)2455 278 y FC(2)2509 280 y Fu(+)k Fs(k)p Ff(F)2685 292 y Fe(a)2723 280 y FE(h)p Fs(k)2813 295 y Fq(L)2859 278 y FC(2)2894 280 y Fu(2)2936 245 y Fe(a)2975 187 y Fp(\021)p 0 TeXcolorgray 3662 280 a Fu(\(5.11\))p 0 TeXcolorgray 737 483 a Fs(\002)816 390 y Fp(\020)865 483 y Fs(k)c Fu(\()p Fs(\000)p Fu(\001)1087 498 y Fn(S)1125 481 y FC(2)1156 483 y Fu(\))1198 419 y FC(1)p 1198 428 V 1198 461 a(2)33 b Fu(~)1254 483 y Ff(F)1305 495 y Fe(a)1344 483 y FE(f)9 b Fs(k)1436 498 y Fq(L)1482 481 y FC(2)1535 483 y Fu(+)18 b Fs(k)1665 461 y Fu(~)1660 483 y Ff(F)1711 495 y Fe(a)1750 483 y FE(f)9 b Fs(k)1842 498 y Fq(L)1888 481 y FC(2)1923 483 y Fu(2)1965 448 y Fe(a)2003 390 y Fp(\021)2071 483 y Fu(+)18 b(2)2196 448 y FD(\000)p Fr(2)p Fe(l)2306 483 y Fs(k)p FE(g)s Fs(k)2433 539 y Fq(H)2491 497 y Fm(\000)2545 481 y FC(1)p 2545 488 V 2545 519 a(2)2487 557 y(2)2592 483 y Fs(k)p Ff(F)2685 495 y Fe(a)2723 483 y FE(h)p Fs(k)2813 498 y Fq(L)2859 481 y FC(2)2895 483 y Fs(k)2942 461 y Fu(~)2937 483 y Ff(F)2988 495 y Fe(a)3026 483 y FE(f)9 b Fs(k)3118 498 y Fq(L)3164 481 y FC(2)3199 483 y Fu(2)3241 448 y Fe(a)3280 483 y FE(:)0 769 y Fu(As)28 b(a)f(matter)g(of)h(fact,)g (w)n(e)f(can)g(rep)r(eat)g(the)h(ab)r(o)n(v)n(e)e(ampli\014cation)i(to) f(obtain)0 1119 y SDict begin H.S end 0 1119 a 0 1119 a SDict begin 12 H.A end 0 1119 a 0 1119 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.12) cvn /DEST pdfmark end 0 1119 a 166 w Fs(j)p Ff(M)276 1085 y Fr(2)276 1140 y Fe(l)p Fq(;)p Fe(a)356 1119 y Fu(\()p FE(g)s(;)14 b(h;)g(f)9 b Fu(\))p Fs(j)22 b Fk(.)h Fu(min)920 1027 y Fp(n)976 1119 y Fu(2)1018 1085 y FD(\000)p Fr(3)p Fe(l)1127 1119 y Fs(k)p FE(g)s Fs(k)1254 1175 y Fq(H)1312 1133 y Fm(\000)1367 1117 y FC(1)p 1367 1124 29 3 v 1367 1155 a(2)1308 1193 y(4)1427 1027 y Fp(\020)1476 1119 y Fs(k)14 b Fu(\()p Fs(\000)p Fu(\001)1698 1134 y Fn(S)1736 1117 y FC(2)1767 1119 y Fu(\))1809 1055 y FC(1)p 1809 1064 V 1809 1097 a(2)1865 1119 y Ff(F)1916 1131 y Fe(a)1955 1119 y FE(h)p Fs(k)2045 1134 y Fq(L)2091 1117 y FC(2)2145 1119 y Fu(+)k Fs(k)p Ff(F)2321 1131 y Fe(a)2359 1119 y FE(h)p Fs(k)2449 1134 y Fq(L)2495 1117 y FC(2)2531 1119 y Fu(2)2573 1085 y Fe(a)2611 1027 y Fp(\021)p 0 TeXcolorgray 3737 1119 a Fu(\(5.12\))p 0 TeXcolorgray 166 1322 a Fs(\002)245 1230 y Fp(\020)294 1322 y Fs(k)c Fu(\()p Fs(\000)p Fu(\001)516 1337 y Fn(S)554 1320 y FC(2)584 1322 y Fu(\))627 1258 y FC(1)p 627 1267 V 627 1300 a(2)32 b Fu(~)683 1322 y Ff(F)734 1334 y Fe(a)772 1322 y FE(f)9 b Fs(k)864 1337 y Fq(L)910 1320 y FC(2)964 1322 y Fu(+)18 b Fs(k)1094 1300 y Fu(~)1089 1322 y Ff(F)1140 1334 y Fe(a)1178 1322 y FE(f)9 b Fs(k)1270 1337 y Fq(L)1316 1320 y FC(2)1352 1322 y Fu(2)1394 1288 y Fe(a)1432 1230 y Fp(\021)1500 1322 y Fu(+)18 b(2)1625 1288 y FD(\000)p Fr(2)p Fe(l)1735 1322 y Fs(k)p FE(g)s Fs(k)1862 1378 y Fq(H)1920 1336 y Fm(\000)1974 1320 y FC(1)p 1974 1327 V 1974 1358 a(2)1916 1396 y(4)2021 1322 y Fs(k)p Ff(F)2114 1334 y Fe(a)2152 1322 y FE(h)p Fs(k)2242 1337 y Fq(L)2288 1320 y FC(2)2323 1322 y Fs(k)2370 1300 y Fu(~)2365 1322 y Ff(F)2416 1334 y Fe(a)2455 1322 y FE(f)9 b Fs(k)2547 1337 y Fq(L)2593 1320 y FC(2)2628 1322 y Fu(2)2670 1288 y Fe(a)2708 1322 y FE(;)14 b Fu(2)2787 1288 y FD(\000)p Fe(l)2864 1322 y Fs(k)p FE(g)s Fs(k)2991 1378 y Fq(H)3049 1336 y Fm(\000)3103 1320 y FC(1)p 3103 1327 V 3103 1358 a(2)3045 1396 y(4)3150 1322 y Fs(k)p Ff(F)3243 1334 y Fe(a)3281 1322 y FE(h)p Fs(k)3371 1337 y Fq(L)3417 1320 y FC(2)3452 1322 y Fs(k)3499 1300 y Fu(~)3494 1322 y Ff(F)3545 1334 y Fe(a)3584 1322 y FE(f)9 b Fs(k)3676 1337 y Fq(L)3722 1320 y FC(2)3757 1322 y Fu(2)3799 1288 y Fr(2)p Fe(a)3870 1230 y Fp(o)3926 1322 y FE(:)100 1619 y Fu(The)39 b(term)g Ff(M)579 1589 y Fr(3)656 1619 y Fu(can)g(b)r(e)h(estimated)g(in)f(a)g(similar)g(w)n (a)n(y)f(and)h(has)g(the)h(same)f(b)r(ound)h(as)e(ab)r(o)n(v)n(e.)71 b(No)n(w)39 b(w)n(e)g(plug)0 1718 y(\()p 1 0 0 TeXcolorrgb 32 1718 a SDict begin H.S end 32 1718 a 1 0 0 TeXcolorrgb Fu(5.9)p 1 0 0 TeXcolorrgb 138 1665 a SDict begin H.R end 138 1665 a 138 1718 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.9) cvn H.B /ANN pdfmark end 138 1718 a 0 TeXcolorgray Fu(\))q(\()p 1 0 0 TeXcolorrgb 203 1718 a SDict begin H.S end 203 1718 a 1 0 0 TeXcolorrgb Fu(5.10)p 1 0 0 TeXcolorrgb 351 1665 a SDict begin H.R end 351 1665 a 351 1718 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.9) cvn H.B /ANN pdfmark end 351 1718 a 0 TeXcolorgray Fu(\))27 b(and)h(\()p 1 0 0 TeXcolorrgb 604 1718 a SDict begin H.S end 604 1718 a 1 0 0 TeXcolorrgb Fu(5.12)p 1 0 0 TeXcolorrgb 752 1665 a SDict begin H.R end 752 1665 a 752 1718 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.12) cvn H.B /ANN pdfmark end 752 1718 a 0 TeXcolorgray Fu(\))g(in)n(to)f(\()p 1 0 0 TeXcolorrgb 1012 1718 a SDict begin H.S end 1012 1718 a 1 0 0 TeXcolorrgb Fu(4.5)p 1 0 0 TeXcolorrgb 1118 1665 a SDict begin H.R end 1118 1665 a 1118 1718 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.5) cvn H.B /ANN pdfmark end 1118 1718 a 0 TeXcolorgray Fu(\))h(to)g(obtain)f(that)126 1989 y Fs(h)p FE(Q)224 2001 y Fe(l)249 1989 y Fu(\()p FF(h)p FE(;)14 b FF(g)419 2001 y Fr(2)456 1989 y Fu(\))p FE(;)g Fs(P)583 2001 y Fq(k)624 1989 y Ff(F)675 1955 y Fr(2)675 2010 y Fq(j)712 1989 y Fs(P)770 2001 y Fq(k)810 1989 y FF(h)p Fs(i)895 2001 y Fq(v)958 1989 y Fk(.)1023 2001 y Fq(N)1109 1989 y Fu(min)1261 1897 y Fp(n)1356 1910 y(X)1330 2089 y Fe(p)p FD(\025\000)p Fr(1)1516 1989 y Fu(2)1558 1955 y FD(\000)p Fr(3)p Fe(l)1667 1989 y Fs(k)p FF(h)p Fs(k)1804 2046 y Fq(H)1862 2003 y Fm(\000)1917 1987 y FC(1)p 1917 1994 29 3 v 1917 2026 a(2)1858 2064 y(4)1977 1897 y Fp(\020)2027 1989 y Fs(k)g Fu(\()p Fs(\000)p Fu(\001)2249 2004 y Fn(S)2287 1987 y FC(2)2317 1989 y Fu(\))2359 1925 y FC(1)p 2359 1934 V 2359 1967 a(2)2416 1989 y Ff(F)2467 2001 y Fe(p)2505 1989 y FF(g)2553 2001 y Fr(2)2590 1989 y Fs(k)2632 2004 y Fq(L)2678 1987 y FC(2)2732 1989 y Fu(+)k Fs(k)p Ff(F)2908 2001 y Fe(p)2947 1989 y FF(g)2995 2001 y Fr(2)3032 1989 y Fs(k)3074 2004 y Fq(L)3120 1987 y FC(2)3155 1989 y Fu(2)3197 1955 y Fe(p)3236 1897 y Fp(\021)144 2237 y Fs(\002)227 2144 y Fp(\020)277 2237 y Fs(k)c Fu(\()o Fs(\000)p Fu(\001)498 2252 y Fn(S)536 2235 y FC(2)567 2237 y Fu(\))609 2172 y FC(1)p 609 2181 V 609 2215 a(2)33 b Fu(~)665 2237 y Ff(F)716 2249 y Fe(p)755 2237 y Fs(P)813 2249 y Fq(k)854 2237 y Ff(F)905 2202 y Fr(2)905 2257 y Fq(j)942 2237 y Fs(P)1000 2249 y Fq(k)1040 2237 y FF(h)p Fs(k)1135 2252 y Fq(L)1181 2235 y FC(2)1235 2237 y Fu(+)18 b Fs(k)1365 2215 y Fu(~)1360 2237 y Ff(F)1411 2249 y Fe(p)1450 2237 y Fs(P)1508 2249 y Fq(k)1548 2237 y Ff(F)1599 2202 y Fr(2)1599 2257 y Fq(j)1636 2237 y Fs(P)1694 2249 y Fq(k)1735 2237 y FF(h)p Fs(k)1830 2252 y Fq(L)1876 2235 y FC(2)1911 2237 y Fu(2)1953 2202 y Fe(p)1992 2144 y Fp(\021)2060 2237 y Fu(+)2169 2158 y Fp(X)2143 2336 y Fe(p)p FD(\025\000)p Fr(1)2329 2237 y Fu(2)2371 2202 y FD(\000)p Fr(2)p Fe(l)2480 2237 y Fs(k)p FF(h)p Fs(k)2617 2293 y Fq(H)2675 2251 y Fm(\000)2730 2235 y FC(1)p 2730 2242 V 2730 2273 a(2)2671 2311 y(4)2777 2237 y Fs(k)p Ff(F)2870 2249 y Fe(p)2908 2237 y FF(g)2956 2249 y Fr(2)2993 2237 y Fs(k)3035 2252 y Fq(L)3081 2235 y FC(2)3116 2237 y Fs(k)3163 2215 y Fu(~)3158 2237 y Ff(F)3209 2249 y Fe(p)3248 2237 y Fs(P)3306 2249 y Fq(k)3346 2237 y Ff(F)3397 2202 y Fr(2)3397 2257 y Fq(j)3434 2237 y Fs(P)3492 2249 y Fq(k)3533 2237 y FF(h)p Fs(k)3628 2252 y Fq(L)3674 2235 y FC(2)3709 2237 y Fu(2)3751 2202 y Fe(p)3790 2237 y FE(;)165 2405 y Fp(X)139 2583 y Fe(p)p FD(\025\000)p Fr(1)325 2484 y Fu(2)367 2450 y FD(\000)p Fe(l)443 2484 y Fs(k)p FF(h)p Fs(k)580 2540 y Fq(H)638 2498 y Fm(\000)693 2482 y FC(1)p 693 2489 V 693 2520 a(2)634 2558 y(4)740 2484 y Fs(k)787 2462 y Fu(~)782 2484 y Ff(F)833 2496 y Fe(p)871 2484 y FF(g)919 2496 y Fr(2)956 2484 y Fs(k)998 2499 y Fq(L)1044 2482 y FC(2)1080 2484 y Fs(k)1127 2462 y Fu(~)1122 2484 y Ff(F)1173 2496 y Fe(p)1211 2484 y Fs(P)1269 2496 y Fq(k)1310 2484 y Ff(F)1361 2450 y Fr(2)1361 2504 y Fq(j)1398 2484 y Fs(P)1456 2496 y Fq(k)1496 2484 y FF(h)p Fs(k)1591 2499 y Fq(L)1637 2482 y FC(2)1673 2484 y Fu(2)1715 2450 y Fr(2)p Fe(p)1786 2392 y Fp(o)61 2731 y Fu(+)217 2652 y Fp(X)139 2830 y Fe(m)p FD(\024)p Fe(p)p Fr(+)p Fq(N)382 2838 y FC(0)428 2731 y Fu(2)470 2697 y FD(\000)p Fe(l)p Fq(N)605 2731 y Fu(2)647 2697 y FD(\000)p Fr(2)p Fe(p)p Fq(N)829 2731 y Fu(2)871 2697 y FD(\000)p Fr(2)p Fe(m)p Fq(N)1070 2731 y Fs(k)p FF(h)p Fs(k)1207 2787 y Fq(H)1265 2745 y Fm(\000)1320 2729 y FC(1)p 1320 2736 V 1320 2767 a(2)1261 2806 y(2)1380 2639 y Fp(\020)1430 2731 y Fs(k)p Ff(F)1523 2743 y Fe(m)1578 2731 y FF(g)1626 2743 y Fr(2)1663 2731 y Fs(k)1705 2746 y Fq(L)1751 2729 y FC(2)1787 2731 y Fs(k)1834 2709 y Fu(~)1829 2731 y Ff(F)1880 2743 y Fe(p)1918 2731 y Fs(P)1976 2743 y Fq(k)2016 2731 y Ff(F)2067 2697 y Fr(2)2067 2752 y Fq(j)2104 2731 y Fs(P)2162 2743 y Fq(k)2203 2731 y FF(h)p Fs(k)2298 2746 y Fq(L)2344 2729 y FC(2)2398 2731 y Fu(+)g Fs(k)2528 2709 y Fu(~)2523 2731 y Ff(F)2574 2743 y Fe(p)2612 2731 y FF(g)2660 2743 y Fr(2)2697 2731 y Fs(k)2739 2746 y Fq(L)2785 2729 y FC(2)2821 2731 y Fs(k)p Ff(F)2914 2743 y Fe(m)2969 2731 y Fs(P)3027 2743 y Fq(k)3068 2731 y Ff(F)3119 2697 y Fr(2)3119 2752 y Fq(j)3156 2731 y Fs(P)3214 2743 y Fq(k)3254 2731 y FF(h)p Fs(k)3349 2746 y Fq(L)3395 2729 y FC(2)3431 2639 y Fp(\021)3494 2731 y FE(:)0 3067 y Fu(Next,)28 b(w)n(e)f(p)r(erform)g(the)h(dy)n (adic)f(decomp)r(osition)g(in)h(phase)f(space)g(as)g(in)h(\()p 1 0 0 TeXcolorrgb 2396 3067 a SDict begin H.S end 2396 3067 a 1 0 0 TeXcolorrgb Fu(4.6)p 1 0 0 TeXcolorrgb 2502 3013 a SDict begin H.R end 2502 3013 a 2502 3067 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.6) cvn H.B /ANN pdfmark end 2502 3067 a 0 TeXcolorgray Fu(\))g(and)g(use)f(ab)r(o)n(v)n(e)f (estimate)i(to)f(get)302 3338 y SDict begin H.S end 302 3338 a 302 3338 a SDict begin 12 H.A end 302 3338 a 302 3338 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.13) cvn /DEST pdfmark end 302 3338 a 230 w Ff(J)578 3303 y Fr(2)578 3358 y(2)639 3338 y Fk(.)704 3350 y Fq(N)789 3338 y Fu(2)831 3303 y FD(\000)p Fr(3)p Fq(k)957 3338 y Fs(k)p FF(h)p Fs(k)1094 3394 y Fq(H)1152 3352 y Fm(\000)1206 3336 y FC(1)p 1206 3343 29 3 v 1206 3374 a(2)1148 3412 y(5)1253 3245 y Fp(\020)1302 3338 y Fs(k)14 b Fu(\()p Fs(\000)p Fu(\001)1524 3353 y Fn(S)1562 3336 y FC(2)1593 3338 y Fu(\))1635 3274 y FC(1)p 1635 3283 V 1635 3316 a(2)1691 3338 y FF(F)1751 3350 y Fq(j)1786 3338 y FF(P)1851 3350 y Fq(k)1892 3338 y FF(g)1940 3350 y Fr(2)1977 3338 y Fs(k)2019 3353 y Fq(L)2065 3336 y FC(2)2119 3338 y Fu(+)k Fs(k)p FF(F)2304 3350 y Fq(j)2339 3338 y FF(P)2404 3350 y Fq(k)2445 3338 y FF(g)2493 3350 y Fr(2)2530 3338 y Fs(k)2572 3353 y Fq(L)2618 3336 y FC(2)2654 3338 y Fu(2)2696 3303 y Fq(j)2730 3245 y Fp(\021)p 0 TeXcolorgray 3662 3338 a Fu(\(5.13\))p 0 TeXcolorgray 532 3541 a Fs(\002)597 3448 y Fp(\020)647 3541 y Fs(k)c Fu(\()o Fs(\000)p Fu(\001)868 3556 y Fn(S)906 3539 y FC(2)937 3541 y Fu(\))979 3477 y FC(1)p 979 3486 V 979 3519 a(2)1035 3541 y FF(F)1095 3553 y Fq(j)1130 3541 y FF(P)1195 3553 y Fq(k)1236 3541 y FF(h)p Fs(k)1331 3556 y Fq(L)1377 3539 y FC(2)1432 3541 y Fu(+)k Fs(k)p FF(F)1617 3553 y Fq(j)1651 3541 y FF(P)1716 3553 y Fq(k)1757 3541 y FF(h)p Fs(k)1852 3556 y Fq(L)1898 3539 y FC(2)1934 3541 y Fu(2)1976 3506 y Fq(j)2010 3448 y Fp(\021)2078 3541 y Fu(+)g(2)2203 3506 y FD(\000)p Fr(2)p Fq(k)2329 3541 y Fs(k)p FF(h)p Fs(k)2466 3597 y Fq(H)2524 3555 y Fm(\000)2579 3539 y FC(1)p 2579 3546 V 2579 3577 a(2)2520 3615 y(5)2625 3541 y Fs(k)p FF(F)2727 3553 y Fq(j)2762 3541 y FF(P)2827 3553 y Fq(k)2868 3541 y FF(g)2916 3553 y Fr(2)2953 3541 y Fs(k)2995 3556 y Fq(L)3041 3539 y FC(2)3076 3541 y Fs(k)p FF(F)3178 3553 y Fq(j)3213 3541 y FF(P)3278 3553 y Fq(k)3319 3541 y FF(h)p Fs(k)3414 3556 y Fq(L)3460 3539 y FC(2)3496 3541 y Fu(2)3538 3506 y Fq(j)385 3723 y Fu(+)82 b(2)574 3689 y FD(\000)p Fq(k)667 3723 y Fu(2)709 3689 y FD(\000)p Fq(j)s(N)854 3723 y Fs(k)p FF(h)p Fs(k)991 3779 y Fq(H)1049 3737 y Fm(\000)1104 3721 y FC(1)p 1104 3728 V 1104 3759 a(2)1045 3797 y(5)1150 3723 y Fs(k)p FF(P)1257 3735 y Fq(k)1298 3723 y FF(g)1346 3735 y Fr(2)1383 3723 y Fs(k)1425 3739 y Fq(H)1483 3723 y Fm(\000)p Fl(N)13 b Fs(k)p FF(P)1693 3735 y Fq(k)1733 3723 y FF(h)p Fs(k)1828 3739 y Fq(H)1886 3723 y Fm(\000)p Fl(N)31 b Fu(+)19 b(2)2133 3689 y FD(\000)p Fq(j)s(N)2278 3723 y Fu(2)2320 3689 y FD(\000)p Fq(k)q(N)2471 3723 y Fs(k)p FF(h)p Fs(k)2608 3779 y Fq(H)2666 3737 y Fm(\000)2721 3721 y FC(1)p 2721 3728 V 2721 3759 a(2)2662 3797 y(5)2767 3723 y Fs(k)p FF(g)2857 3735 y Fr(2)2894 3723 y Fs(k)2936 3750 y Fq(H)2994 3723 y Fm(\000)p Fl(N)2990 3769 y Fm(\000)p Fl(N)3096 3723 y Fs(k)p FF(h)p Fs(k)3233 3750 y Fq(H)3291 3723 y Fm(\000)p Fl(N)3287 3769 y Fm(\000)p Fl(N)3394 3723 y FE(:)0 4010 y Fu(By)27 b(the)h(same)f(pro)r(cess,)g(w)n(e)g(can)g (deriv)n(e)g(that)259 4255 y SDict begin H.S end 259 4255 a 259 4255 a SDict begin 12 H.A end 259 4255 a 259 4255 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.14) cvn /DEST pdfmark end 259 4255 a 231 w Ff(J)536 4221 y Fr(4)536 4276 y(2)596 4255 y Fk(.)22 b Fu(2)725 4221 y FD(\000)p Fq(k)818 4255 y Fs(k)p FF(h)p Fs(k)955 4312 y Fq(H)1013 4270 y Fm(\000)1067 4254 y FC(1)p 1067 4261 29 3 v 1067 4292 a(2)1009 4330 y(5)1114 4255 y Fs(k)p FF(F)1216 4267 y Fq(j)1250 4255 y FF(P)1315 4267 y Fq(k)1357 4255 y FF(f)1386 4267 y Fr(1)1423 4255 y Fs(k)1465 4270 y Fq(L)1511 4254 y FC(2)1547 4255 y Fs(k)p FF(F)1649 4267 y Fq(j)1683 4255 y FF(P)1748 4267 y Fq(k)1789 4255 y FF(h)p Fs(k)1884 4270 y Fq(L)1930 4254 y FC(2)1966 4255 y Fu(2)2008 4221 y Fr(2)p Fq(j)2094 4255 y Fu(+)c(2)2219 4221 y FD(\000)p Fq(k)2311 4255 y Fu(2)2353 4221 y FD(\000)p Fq(j)s(N)2498 4255 y Fs(k)p FF(h)p Fs(k)2635 4312 y Fq(H)2693 4270 y Fm(\000)2748 4254 y FC(1)p 2748 4261 V 2748 4292 a(2)2689 4330 y(5)2795 4255 y Fs(k)p FF(P)2902 4267 y Fq(k)2942 4255 y FF(f)2971 4267 y Fr(1)3009 4255 y Fs(k)3051 4272 y Fq(H)3109 4255 y Fm(\000)p Fl(N)13 b Fs(k)p FF(P)3319 4267 y Fq(k)3359 4255 y FF(h)p Fs(k)3454 4272 y Fq(H)3512 4255 y Fm(\000)p Fl(N)342 4438 y Fu(+)83 b(2)532 4403 y FD(\000)p Fq(j)s(N)677 4438 y Fu(2)719 4403 y FD(\000)p Fq(k)q(N)870 4438 y Fs(k)p FF(h)p Fs(k)1007 4494 y Fq(H)1065 4452 y Fm(\000)1120 4436 y FC(1)p 1120 4443 V 1120 4474 a(2)1061 4512 y(5)1166 4438 y Fs(k)p FF(f)1237 4450 y Fr(1)1274 4438 y Fs(k)1316 4464 y Fq(H)1374 4437 y Fm(\000)p Fl(N)1370 4484 y Fm(\000)p Fl(N)1477 4438 y Fs(k)p FF(h)p Fs(k)1614 4464 y Fq(H)1672 4437 y Fm(\000)p Fl(N)1668 4484 y Fm(\000)p Fl(N)1774 4438 y FE(:)p 0 TeXcolorgray 1865 w Fu(\(5.14\))p 0 TeXcolorgray 0 4735 a(Plug)39 b(all)g(the)g(estimates)g(of)g Ff(J)1017 4705 y Fr(1)1017 4756 y(1)1055 4735 y FE(;)14 b Ff(J)1138 4705 y Fr(2)1138 4756 y(1)1214 4735 y Fu(and)39 b Ff(J)1433 4705 y Fq(i)1433 4756 y Fr(2)1470 4735 y FE(;)14 b(i)42 b Fu(=)g(1)p FE(;)14 b Fu(2)p FE(;)g Fu(3)p FE(;)g Fu(4,)40 b(\()p 1 0 0 TeXcolorrgb 2059 4735 a SDict begin H.S end 2059 4735 a 1 0 0 TeXcolorrgb Fu(5.6)p 1 0 0 TeXcolorrgb 2166 4682 a SDict begin H.R end 2166 4682 a 2166 4735 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.6) cvn H.B /ANN pdfmark end 2166 4735 a 0 TeXcolorgray Fu(\)\()p 1 0 0 TeXcolorrgb 2230 4735 a SDict begin H.S end 2230 4735 a 1 0 0 TeXcolorrgb Fu(5.7)p 1 0 0 TeXcolorrgb 2336 4682 a SDict begin H.R end 2336 4682 a 2336 4735 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.7) cvn H.B /ANN pdfmark end 2336 4735 a 0 TeXcolorgray Fu(\))q(\()p 1 0 0 TeXcolorrgb 2401 4735 a SDict begin H.S end 2401 4735 a 1 0 0 TeXcolorrgb Fu(5.8)p 1 0 0 TeXcolorrgb 2507 4682 a SDict begin H.R end 2507 4682 a 2507 4735 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.8) cvn H.B /ANN pdfmark end 2507 4735 a 0 TeXcolorgray Fu(\)\()p 1 0 0 TeXcolorrgb 2571 4735 a SDict begin H.S end 2571 4735 a 1 0 0 TeXcolorrgb Fu(5.13)p 1 0 0 TeXcolorrgb 2719 4682 a SDict begin H.R end 2719 4682 a 2719 4735 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.13) cvn H.B /ANN pdfmark end 2719 4735 a 0 TeXcolorgray Fu(\))g(and)f(\()p 1 0 0 TeXcolorrgb 2996 4735 a SDict begin H.S end 2996 4735 a 1 0 0 TeXcolorrgb Fu(5.14)p 1 0 0 TeXcolorrgb 3144 4682 a SDict begin H.R end 3144 4682 a 3144 4735 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.14) cvn H.B /ANN pdfmark end 3144 4735 a 0 TeXcolorgray Fu(\))h(in)n(to)f(\()p 1 0 0 TeXcolorrgb 3428 4735 a SDict begin H.S end 3428 4735 a 1 0 0 TeXcolorrgb Fu(5.5)p 1 0 0 TeXcolorrgb 3534 4682 a SDict begin H.R end 3534 4682 a 3534 4735 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.5) cvn H.B /ANN pdfmark end 3534 4735 a 0 TeXcolorgray Fu(\),)k(and)c(in)0 4835 y(conjunction)28 b(with)g(Prop)r(osition)p 1 0 0 TeXcolorrgb 1086 4835 a SDict begin H.S end 1086 4835 a 1 0 0 TeXcolorrgb Fu(2.1)p 1 0 0 TeXcolorrgb 1192 4781 a SDict begin H.R end 1192 4781 a 1192 4835 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (prop.2.1) cvn H.B /ANN pdfmark end 1192 4835 a 0 TeXcolorgray Fu(,)g(w)n(e)f(pro)r(ceed)g(with)h (some)f(simpli\014cations)g(to)h(deriv)n(e)e(the)i(follo)n(wing)f (result:)400 5120 y SDict begin H.S end 400 5120 a 400 5120 a SDict begin 12 H.A end 400 5120 a 400 5120 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.15) cvn /DEST pdfmark end 400 5120 a 410 5063 a Fu(1)p 410 5100 42 4 v 410 5176 a(2)486 5063 y FE(d)p 471 5100 74 4 v 471 5176 a(dt)555 5120 y Fs(k)p Ff(F)648 5132 y Fq(j)682 5120 y Fs(P)740 5132 y Fq(k)780 5120 y FF(h)p Fs(k)875 5085 y Fr(2)875 5142 y Fq(L)921 5125 y FC(2)975 5120 y Fu(+)18 b FE(\024)p Fs(k)p Ff(F)1199 5132 y Fq(j)1234 5120 y Fs(P)1292 5132 y Fq(k)1332 5120 y FF(h)p Fs(k)1427 5085 y Fr(2)1427 5142 y Fq(L)1473 5125 y FC(2)1509 5120 y Fu(2)1551 5085 y Fr(2)p Fq(j)1618 5120 y Fu(2)1660 5085 y FD(\000)p Fr(3)p Fq(k)1804 5120 y Fu(+)g FE(\024)p Fs(k)p Fu(\()p Fs(\000)p Fu(\001)2143 5137 y Fn(S)2175 5118 y FC(2)2211 5120 y Fu(\))2253 5063 y FC(1)p 2254 5072 29 3 v 2254 5105 a(2)2296 5120 y Ff(F)2347 5132 y Fq(j)2382 5120 y Fs(P)2440 5132 y Fq(k)2480 5120 y FF(h)p Fs(k)2575 5085 y Fr(2)2575 5142 y Fq(L)2621 5125 y FC(2)2657 5120 y Fu(2)2699 5085 y FD(\000)p Fr(3)p Fq(k)2847 5120 y Fk(.)2912 5132 y Fq(N)2998 5120 y Ff(K)3053 5132 y Fr(1)3109 5120 y Fu(+)g Ff(K)3247 5132 y Fr(2)3285 5120 y FE(;)p 0 TeXcolorgray 354 w Fu(\(5.15\))p 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 37 37 TeXDict begin HPSdict begin 37 36 bop 0 TeXcolorgray 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 0 TeXcolorgray -8 -245 a SDict begin H.S end -8 -245 a -8 -245 a SDict begin H.R end -8 -245 a -8 -245 a SDict begin [/View [/XYZ H.V]/Dest (page.37) cvn /DEST pdfmark end -8 -245 a 0 TeXcolorgray 0 TeXcolorgray 1335 -165 a Fr(LAND)n(A)n(U)30 b(EQUA)-5 b(TION)28 b(IN)h Fq(H)2196 -193 y Fm(\000)2252 -209 y FC(1)p 2252 -202 29 3 v 2252 -171 a(2)2324 -165 y Fr(SP)-5 b(A)n(CE)1268 b(37)p 0 TeXcolorgray 0 29 a Fu(where)397 212 y Ff(K)452 224 y Fr(1)513 212 y Fu(=)600 133 y Fp(X)602 311 y Fe(p)p FD(\025)p Fq(j)734 212 y Fs(k)p FF(F)836 224 y Fe(p)875 212 y FF(P)940 224 y Fq(k)981 212 y FF(f)1010 224 y Fr(2)1047 212 y Fs(k)1089 227 y Fq(L)1135 210 y FC(2)1171 212 y Fs(k)p FF(F)1273 224 y Fe(p)1311 212 y FF(P)1376 224 y Fq(k)1418 212 y FF(h)p Fs(k)1513 227 y Fq(L)1559 210 y FC(2)1594 212 y Fs(k)p FF(F)1696 224 y Fq(j)1731 212 y FF(P)1796 224 y Fq(k)1837 212 y FF(h)p Fs(k)1932 227 y Fq(L)1978 210 y FC(2)2014 212 y Fu(2)2066 155 y FC(3)p 2065 164 V 2065 198 a(2)2103 178 y Fq(j)2157 212 y Fu(+)2306 133 y Fp(X)2240 311 y Fe(p)p FD(\024)p Fq(j)s 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Fq(j)1714 691 y Fu(+)1864 612 y Fp(X)1797 790 y Fe(p)p FD(\024)p Fq(j)s Fr(+)p Fq(N)2018 798 y FC(0)2064 691 y Fs(k)p FF(F)2166 703 y Fe(p)2205 691 y FF(P)2270 703 y Fq(k)2311 691 y FF(h)p Fs(k)2406 706 y Fq(L)2452 689 y FC(2)2488 691 y Fu(2)2530 656 y FD(\000)2591 630 y Fd(p)p 2591 644 V 2593 676 a FC(2)2638 691 y Fs(k)p FF(F)2740 703 y Fq(j)2775 691 y FF(P)2840 703 y Fq(k)2881 691 y FF(f)2910 703 y Fr(1)2947 691 y Fs(k)2989 706 y Fq(L)3035 689 y FC(2)3071 691 y Fs(k)p FF(F)3173 703 y Fq(j)3207 691 y FF(P)3272 703 y Fq(k)3313 691 y FF(h)p Fs(k)3408 706 y Fq(L)3454 689 y FC(2)3490 691 y Fu(2)3532 656 y Fr(2)p Fq(j)249 930 y Fu(+)397 851 y Fp(X)398 1029 y Fe(p)p FD(\025)p Fq(j)531 930 y Fs(k)p FF(F)633 942 y Fe(p)671 930 y FF(P)736 942 y Fq(k)777 930 y FF(g)825 942 y Fr(2)862 930 y Fs(k)904 945 y Fq(L)950 928 y FC(2)986 930 y Fs(k)p FF(F)1088 942 y Fe(p)1127 930 y FF(P)1192 942 y Fq(k)1233 930 y FF(h)p Fs(k)1328 945 y Fq(L)1374 928 y FC(2)1409 930 y Fs(k)p FF(F)1511 942 y Fq(j)1546 930 y FF(P)1611 942 y Fq(k)1652 930 y FF(h)p Fs(k)1747 945 y Fq(L)1793 928 y FC(2)1829 930 y Fu(2)1881 873 y FC(3)p 1880 882 29 3 v 1880 916 a(2)1918 896 y Fq(j)1972 930 y Fu(+)2122 851 y Fp(X)2055 1029 y Fe(p)p FD(\024)p Fq(j)s Fr(+)p Fq(N)2276 1037 y FC(0)2322 930 y Fs(k)p FF(F)2424 942 y Fe(p)2463 930 y FF(P)2528 942 y Fq(k)2569 930 y FF(g)2617 942 y Fr(2)2654 930 y Fs(k)2696 945 y Fq(L)2742 928 y FC(2)2777 930 y Fu(2)2829 873 y FC(3)p 2829 882 V 2829 916 a(2)2867 896 y Fe(p)2906 930 y Fs(k)p FF(F)3008 942 y Fq(j)3043 930 y FF(P)3108 942 y Fq(k)3149 930 y FF(h)p Fs(k)3244 896 y Fr(2)3244 952 y Fq(L)3290 936 y FC(2)249 1177 y Fu(+)464 1099 y Fp(X)397 1277 y Fe(p)p FD(\024)p Fq(j)s Fr(+)p Fq(N)618 1285 y FC(0)664 1177 y Fs(k)p FF(F)766 1189 y Fe(p)805 1177 y FF(P)870 1189 y Fq(k)911 1177 y FF(h)p Fs(k)1006 1192 y Fq(L)1052 1176 y FC(2)1088 1177 y Fu(2)1130 1143 y FD(\000)1191 1117 y Fd(p)p 1191 1131 33 3 v 1193 1163 a FC(2)1238 1177 y Fs(k)p FF(F)1340 1189 y Fq(j)1375 1177 y FF(P)1440 1189 y Fq(k)1481 1177 y FF(g)1529 1189 y Fr(2)1566 1177 y Fs(k)1608 1192 y Fq(L)1654 1176 y FC(2)1689 1177 y Fs(k)p FF(F)1791 1189 y Fq(j)1826 1177 y FF(P)1891 1189 y Fq(k)1932 1177 y FF(h)p Fs(k)2027 1192 y Fq(L)2073 1176 y FC(2)2109 1177 y Fu(2)2151 1143 y Fr(2)p Fq(j)2237 1177 y Fu(+)18 b(2)2362 1143 y FD(\000)p Fr(3)p Fq(k)2487 1177 y Fs(k)p FF(h)p Fs(k)2624 1234 y Fq(H)2682 1192 y Fm(\000)2737 1176 y FC(1)p 2737 1183 29 3 v 2737 1214 a(2)2678 1252 y(5)2783 1085 y Fp(\020)2833 1177 y Fs(k)c Fu(\()p Fs(\000)p Fu(\001)3055 1192 y Fn(S)3093 1176 y FC(2)3123 1177 y Fu(\))3165 1113 y FC(1)p 3165 1122 V 3165 1156 a(2)3222 1177 y FF(F)3282 1189 y Fq(j)3317 1177 y FF(P)3382 1189 y Fq(k)3423 1177 y FF(g)3471 1189 y Fr(2)3508 1177 y Fs(k)3550 1192 y Fq(L)3596 1176 y FC(2)249 1425 y Fu(+)83 b Fs(k)p FF(F)499 1437 y Fq(j)534 1425 y FF(P)599 1437 y Fq(k)640 1425 y FF(g)688 1437 y Fr(2)725 1425 y Fs(k)767 1440 y Fq(L)813 1423 y FC(2)848 1425 y Fu(2)890 1391 y Fq(j)925 1333 y Fp(\021)13 b(\020)1038 1425 y Fs(k)h Fu(\()o Fs(\000)p Fu(\001)1259 1440 y Fn(S)1297 1423 y FC(2)1328 1425 y Fu(\))1370 1361 y FC(1)p 1370 1370 V 1370 1403 a(2)1426 1425 y FF(F)1486 1437 y Fq(j)1521 1425 y FF(P)1586 1437 y Fq(k)1628 1425 y FF(h)p Fs(k)1723 1440 y Fq(L)1769 1423 y FC(2)1823 1425 y Fu(+)k Fs(k)p FF(F)2008 1437 y Fq(j)2042 1425 y FF(P)2107 1437 y Fq(k)2148 1425 y FF(h)p Fs(k)2243 1440 y Fq(L)2289 1423 y FC(2)2325 1425 y Fu(2)2367 1391 y Fq(j)2402 1333 y Fp(\021)2474 1425 y Fu(:=)23 b Ff(K)2640 1391 y Fq(i)2640 1446 y Fr(1)2678 1425 y FE(;)14 b(i)22 b Fu(=)h(1)p FE(;)14 b Fs(\001)g(\001)g(\001)27 b FE(;)14 b Fu(10)p FE(;)0 1628 y Fu(and)379 1810 y Ff(K)434 1822 y Fr(2)495 1810 y Fu(=)23 b(2)625 1776 y FD(\000)p Fr(2)p Fq(k)750 1810 y Fs(k)p FF(h)p Fs(k)887 1867 y Fq(H)945 1825 y Fm(\000)1000 1808 y FC(1)p 1000 1815 V 1000 1847 a(2)941 1885 y(5)1046 1810 y Fs(k)p FF(F)1148 1822 y Fq(j)1183 1810 y FF(P)1248 1822 y Fq(k)1289 1810 y FF(g)1337 1822 y Fr(2)1374 1810 y Fs(k)1416 1825 y Fq(L)1462 1808 y FC(2)1498 1810 y Fs(k)p FF(F)1600 1822 y Fq(j)1634 1810 y FF(P)1699 1822 y Fq(k)1740 1810 y FF(h)p Fs(k)1835 1825 y Fq(L)1881 1808 y FC(2)1917 1810 y Fu(2)1959 1776 y Fq(j)2012 1810 y Fu(+)18 b(2)2137 1776 y FD(\000)p Fq(k)2229 1810 y Fs(k)p FF(h)p Fs(k)2366 1867 y Fq(H)2424 1825 y Fm(\000)2479 1808 y FC(1)p 2479 1815 V 2479 1847 a(2)2420 1885 y(5)2525 1810 y Fs(k)p FF(F)2627 1822 y Fq(j)2662 1810 y FF(P)2727 1822 y Fq(k)2768 1810 y FF(f)2797 1822 y Fr(1)2835 1810 y Fs(k)2877 1825 y Fq(L)2923 1808 y FC(2)2958 1810 y Fs(k)p FF(F)3060 1822 y Fq(j)3095 1810 y FF(P)3160 1822 y Fq(k)3201 1810 y FF(h)p Fs(k)3296 1825 y Fq(L)3342 1808 y FC(2)3377 1810 y Fu(2)3419 1776 y Fr(2)p Fq(j)232 1993 y Fu(+)82 b(2)421 1958 y FD(\000)p Fq(j)s(N)566 1993 y Fu(2)608 1958 y FD(\000)p Fq(k)q(N)759 1993 y Fs(k)p FF(f)830 2005 y Fr(1)867 1993 y Fs(k)909 2011 y Fq(L)955 1991 y FC(2)955 2030 y Fm(\000)p Fl(N)1057 1993 y Fs(k)p FF(h)p Fs(k)1194 2011 y Fq(L)1240 1991 y FC(2)1240 2030 y Fm(\000)p Fl(N)1341 1993 y Fs(k)p FF(h)p Fs(k)1478 2019 y Fq(H)1536 1992 y Fm(\000)p Fl(N)1532 2039 y Fm(\000)p Fl(N)1657 1993 y Fu(+)18 b(2)1782 1958 y FD(\000)p Fr(2)p Fq(k)1907 1993 y Fs(k)p FF(f)1978 2005 y Fr(1)2015 1993 y Fs(k)2057 2011 y Fq(L)2103 1991 y FC(1)2103 2029 y(2)2139 1993 y Fs(k)p FF(F)2241 2005 y Fq(j)2275 1993 y FF(P)2340 2005 y Fq(k)2381 1993 y FF(h)p Fs(k)2476 1958 y Fr(2)2476 2015 y Fq(L)2522 1998 y FC(2)2558 1993 y Fu(2)2600 1958 y Fq(j)2653 1993 y Fu(+)g(2)2778 1958 y FD(\000)p Fr(2)p Fq(k)2903 1993 y Fu(2)2945 1958 y FD(\000)p Fq(j)s(N)3090 1993 y Fs(k)p FF(f)3161 2005 y Fr(1)3198 1993 y Fs(k)3240 2011 y Fq(L)3286 1991 y FC(1)3286 2029 y(2)3322 1993 y Fs(k)p FF(P)3429 2005 y Fq(k)3470 1993 y FF(h)p Fs(k)3565 1958 y Fr(2)3565 2017 y Fq(H)3623 2000 y Fm(\000)p Fl(N)272 2382 y Fu(+)82 b(2)461 2348 y FD(\000)p Fq(j)s(N)607 2382 y Fu(2)649 2348 y FD(\000)p Fq(k)q(N)800 2382 y Fs(k)p FF(f)871 2394 y Fr(1)908 2382 y Fs(k)950 2401 y Fq(L)996 2381 y FC(1)996 2419 y(2)1031 2382 y Fs(k)p FF(h)p Fs(k)1168 2348 y Fr(2)1168 2417 y Fq(H)1226 2390 y Fm(\000)p Fl(N)1222 2436 y Fm(\000)p Fl(N)1347 2382 y Fu(+)18 b(2)1472 2348 y FD(\000)p Fq(j)s(N)1617 2382 y Fu(2)1659 2348 y FD(\000)p Fq(k)q(N)1810 2382 y Fs(k)p FF(f)1881 2394 y Fr(2)1918 2382 y Fs(k)1960 2401 y Fq(L)2006 2381 y FC(2)2006 2420 y Fm(\000)p Fl(N)2108 2382 y Fs(k)p FF(h)p Fs(k)2245 2401 y Fq(L)2291 2381 y FC(2)2291 2420 y Fm(\000)p Fl(N)2392 2382 y Fs(k)p FF(h)p Fs(k)2529 2409 y Fq(H)2587 2382 y Fm(\000)p Fl(N)2583 2429 y Fm(\000)p Fl(N)2708 2382 y Fu(+)g(2)2833 2348 y FD(\000)p Fr(2)p Fq(k)2958 2382 y Fs(k)p FF(f)3029 2394 y Fr(2)3066 2382 y Fs(k)3108 2401 y Fq(L)3154 2381 y FC(1)3154 2419 y(2)3189 2382 y Fs(k)p FF(F)3291 2394 y Fq(j)3326 2382 y FF(P)3391 2394 y Fq(k)3432 2382 y FF(h)p Fs(k)3527 2348 y Fr(2)3527 2405 y Fq(L)3573 2388 y FC(2)3609 2382 y Fu(2)3651 2348 y Fq(j)272 2551 y Fu(+)82 b(2)461 2517 y FD(\000)p Fr(2)p Fq(k)587 2551 y Fu(2)629 2517 y FD(\000)p Fq(j)s(N)774 2551 y Fs(k)p FF(f)845 2563 y Fr(2)882 2551 y Fs(k)924 2569 y Fq(L)970 2549 y FC(1)970 2587 y(2)1005 2551 y Fs(k)p FF(P)1112 2563 y Fq(k)1153 2551 y FF(h)p Fs(k)1248 2517 y Fr(2)1248 2575 y Fq(H)1306 2558 y Fm(\000)p Fl(N)1427 2551 y Fu(+)18 b(2)1552 2517 y FD(\000)p Fq(j)s(N)1698 2551 y Fu(2)1740 2517 y FD(\000)p Fq(k)q(N)1891 2551 y Fs(k)p FF(f)1962 2563 y Fr(2)1999 2551 y Fs(k)2041 2569 y Fq(L)2087 2549 y FC(1)2087 2587 y(2)2122 2551 y Fs(k)p FF(h)p Fs(k)2259 2517 y Fr(2)2259 2585 y Fq(H)2317 2558 y Fm(\000)p Fl(N)2313 2605 y Fm(\000)p Fl(N)2438 2551 y Fu(+)g(2)2563 2517 y FD(\000)p Fq(j)s(N)2708 2551 y Fu(2)2750 2517 y FD(\000)p Fq(k)q(N)2901 2551 y Fs(k)p FF(g)2991 2563 y Fr(2)3028 2551 y Fs(k)3070 2578 y Fq(H)3128 2551 y Fm(\000)p Fl(N)3124 2597 y Fm(\000)p Fl(N)3231 2551 y Fs(k)p FF(h)p Fs(k)3368 2517 y Fr(2)3368 2585 y Fq(H)3426 2558 y Fm(\000)p Fl(N)3422 2605 y Fm(\000)p Fl(N)272 2720 y Fu(+)82 b(2)461 2686 y FD(\000)p Fq(j)s(N)607 2720 y Fu(2)649 2686 y FD(\000)p Fq(k)q(N)800 2720 y Fs(k)p FF(h)p Fs(k)937 2746 y Fq(H)995 2720 y Fm(\000)p Fl(N)991 2766 y Fm(\000)p Fl(N)1097 2720 y Fs(k)p FF(P)1204 2732 y Fq(k)1245 2720 y FF(h)p Fs(k)1340 2738 y Fq(L)1386 2718 y FC(2)1386 2758 y Fm(\000)p Fl(N)1487 2720 y Fs(k)p FF(P)1594 2732 y Fq(k)1635 2720 y FF(g)1683 2732 y Fr(2)1720 2720 y Fs(k)1762 2738 y Fq(L)1808 2718 y FC(2)1808 2758 y Fm(\000)p Fl(N)1928 2720 y Fu(+)18 b(2)2053 2686 y FD(\000)p Fq(k)2145 2720 y Fu(2)2187 2686 y FD(\000)p Fq(j)s(N)2332 2720 y Fs(k)p FF(h)p Fs(k)2469 2776 y Fq(H)2527 2734 y Fm(\000)2582 2718 y FC(1)p 2582 2725 V 2582 2756 a(2)2523 2794 y(5)2629 2720 y Fs(k)p FF(P)2736 2732 y Fq(k)2776 2720 y FF(g)2824 2732 y Fr(2)2861 2720 y Fs(k)2903 2736 y Fq(H)2961 2720 y Fm(\000)p Fl(N)13 b Fs(k)p FF(P)3171 2732 y Fq(k)3212 2720 y FF(h)p Fs(k)3307 2736 y Fq(H)3365 2720 y Fm(\000)p Fl(N)272 2902 y Fu(+)82 b(2)461 2868 y FD(\000)p Fq(j)s(N)607 2902 y Fu(2)649 2868 y FD(\000)p Fq(k)q(N)800 2902 y Fs(k)p FF(h)p Fs(k)937 2958 y Fq(H)995 2916 y Fm(\000)1049 2900 y FC(1)p 1049 2907 V 1049 2938 a(2)991 2976 y(5)1096 2902 y Fs(k)p FF(g)1186 2914 y Fr(2)1222 2902 y Fs(k)1264 2929 y Fq(H)1322 2902 y Fm(\000)p Fl(N)1318 2948 y Fm(\000)p Fl(N)1425 2902 y Fs(k)p FF(h)p Fs(k)1562 2929 y Fq(H)1620 2902 y Fm(\000)p Fl(N)1616 2948 y Fm(\000)p Fl(N)1741 2902 y Fu(+)18 b(2)1866 2868 y FD(\000)p Fq(k)1958 2902 y Fu(2)2000 2868 y FD(\000)p Fq(j)s(N)2145 2902 y Fs(k)p FF(h)p Fs(k)2282 2958 y Fq(H)2340 2916 y Fm(\000)2395 2900 y FC(1)p 2395 2907 V 2395 2938 a(2)2336 2976 y(5)2442 2902 y Fs(k)p FF(P)2549 2914 y Fq(k)2589 2902 y FF(f)2618 2914 y Fr(1)2656 2902 y Fs(k)2698 2918 y Fq(H)2756 2902 y Fm(\000)p Fl(N)13 b Fs(k)p FF(P)2966 2914 y Fq(k)3006 2902 y FF(h)p Fs(k)3101 2918 y Fq(H)3159 2902 y Fm(\000)p Fl(N)272 3084 y Fu(+)82 b(2)461 3050 y FD(\000)p Fq(j)s(N)607 3084 y Fu(2)649 3050 y FD(\000)p Fq(k)q(N)800 3084 y Fs(k)p FF(h)p Fs(k)937 3141 y Fq(H)995 3098 y Fm(\000)1049 3082 y FC(1)p 1049 3089 V 1049 3121 a(2)991 3159 y(5)1096 3084 y Fs(k)p FF(f)1167 3096 y Fr(1)1204 3084 y Fs(k)1246 3111 y Fq(H)1304 3084 y Fm(\000)p Fl(N)1300 3131 y Fm(\000)p Fl(N)1407 3084 y Fs(k)p FF(h)p Fs(k)1544 3111 y Fq(H)1602 3084 y Fm(\000)p Fl(N)1598 3131 y Fm(\000)p Fl(N)1704 3084 y FE(:)100 3320 y Fu(W)-7 b(e)27 b(c)n(ho)r(ose)f FE(N)32 b Fu(=)23 b(10)j(and)h(m)n(ultiply)h(2) 1337 3290 y FD(\000)p Fq(j)1423 3320 y FE(j)1462 3290 y Fr(2)p Fh(s)1523 3320 y Fu(2)1565 3290 y Fr(10)p Fq(k)1699 3320 y Fu(on)f(b)r(oth)g(sides)g(of)34 b(\()p 1 0 0 TeXcolorrgb 2341 3320 a SDict begin H.S end 2341 3320 a 1 0 0 TeXcolorrgb Fu(5.15)p 1 0 0 TeXcolorrgb 2489 3267 a SDict begin H.R end 2489 3267 a 2489 3320 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.5.15) cvn H.B /ANN pdfmark end 2489 3320 a 0 TeXcolorgray Fu(\))27 b(and)g(sum)h(up)f(with)h(resp)r(ect)f (to)g FE(j)32 b Fu(and)27 b FE(k)s Fu(.)0 3420 y(F)-7 b(or)27 b(the)h(\014rst)f(three)h(terms)f(of)h Ff(K)1057 3432 y Fr(1)1094 3420 y Fu(,)g(w)n(e)f(ha)n(v)n(e)130 3614 y SDict begin H.S end 130 3614 a 130 3614 a SDict begin 12 H.A end 130 3614 a 130 3614 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.16) cvn /DEST pdfmark end 130 3614 a 361 3535 a Fp(X)379 3714 y Fq(j;k)481 3614 y Fu(\()p Ff(K)568 3580 y Fr(1)568 3635 y(1)624 3614 y Fu(+)18 b Ff(K)762 3580 y Fr(2)762 3635 y(1)818 3614 y Fu(+)g Ff(K)956 3580 y Fr(3)956 3635 y(1)994 3614 y Fu(\)2)1068 3580 y FD(\000)p Fq(j)1155 3614 y FE(j)1194 3580 y Fr(2)p Fh(s)1255 3614 y Fu(2)1297 3580 y Fr(10)p Fq(k)1426 3614 y Fk(.)1514 3535 y Fp(X)1532 3714 y Fq(j;k)1648 3535 y Fp(X)1649 3714 y Fe(p)p FD(\025)p Fq(j)1781 3614 y Fs(k)p FF(F)1883 3626 y Fe(p)1922 3614 y FF(P)1987 3626 y Fq(k)2028 3614 y FF(f)2057 3626 y Fr(2)2095 3614 y Fs(k)2137 3629 y Fq(L)2183 3612 y FC(2)2218 3614 y Fu(2)2270 3553 y Fd(p)p 2270 3567 33 3 v 2272 3600 a FC(2)2317 3614 y FE(p)2359 3580 y Fh(s)2386 3614 y Fs(k)p FF(F)2488 3626 y Fe(p)2527 3614 y FF(P)2592 3626 y Fq(k)2633 3614 y FF(h)p Fs(k)2728 3629 y Fq(L)2774 3612 y FC(2)2810 3614 y Fu(2)2862 3553 y Fd(p)p 2861 3567 V 2863 3600 a FC(2)2908 3614 y FE(p)2950 3580 y Fh(s)2978 3614 y Fs(k)p FF(F)3080 3626 y Fq(j)3114 3614 y FF(P)3179 3626 y Fq(k)3220 3614 y FF(h)p Fs(k)3315 3629 y Fq(L)3361 3612 y FC(2)361 3854 y Fs(\002)p Fu(2)468 3820 y FD(\000)529 3798 y FC(1)p 529 3807 29 3 v 529 3840 a(2)567 3820 y Fq(j)602 3854 y Fu(2)644 3820 y Fq(j)s FD(\000)p Fe(p)765 3854 y Ff(p)807 3820 y FD(\000)p Fr(2)p Fh(s)919 3854 y FE(j)958 3820 y Fr(2)p Fh(s)1019 3854 y Fu(2)1061 3820 y Fr(10)p Fq(k)1186 3854 y Fu(+)1269 3775 y Fp(X)1287 3954 y Fq(j;k)1470 3775 y Fp(X)1403 3953 y Fe(p)p FD(\024)p Fq(j)s Fr(+)p Fq(N)1624 3961 y FC(0)1670 3854 y Fs(k)p FF(F)1772 3866 y Fe(p)1811 3854 y FF(P)1876 3866 y Fq(k)1917 3854 y FF(f)1946 3866 y Fr(2)1984 3854 y Fs(k)2026 3869 y Fq(L)2072 3852 y FC(2)2107 3854 y Fu(2)2149 3820 y FD(\000)2211 3793 y Fd(p)p 2211 3807 33 3 v 2213 3840 a FC(2)2258 3854 y Fs(k)p FF(F)2360 3866 y Fq(j)2394 3854 y FF(P)2459 3866 y Fq(k)2500 3854 y FF(h)p Fs(k)2595 3820 y Fr(2)2595 3876 y Fq(L)2641 3860 y FC(2)2677 3854 y Fu(2)2719 3820 y Fq(j)2753 3854 y Fu(2)2795 3820 y Fe(p)p FD(\000)p Fq(j)2917 3854 y FE(j)2956 3820 y Fr(2)p Fh(s)3017 3854 y Fu(2)3059 3820 y Fr(10)p Fq(k)213 4096 y Fu(+)361 4017 y Fp(X)379 4195 y Fq(j;k)561 4017 y Fp(X)495 4195 y Fe(p)p FD(\024)p Fq(j)s Fr(+)p Fq(N)716 4203 y FC(0)762 4096 y Fs(k)p FF(F)864 4108 y Fe(p)902 4096 y FF(P)967 4108 y Fq(k)1009 4096 y FF(h)p Fs(k)1104 4111 y Fq(L)1150 4094 y FC(2)1185 4096 y Fu(2)1227 4061 y FD(\000)1289 4035 y Fd(p)p 1289 4049 V 1291 4081 a FC(2)1336 4096 y Fs(k)p FF(F)1438 4108 y Fq(j)1472 4096 y FF(P)1537 4108 y Fq(k)1578 4096 y FF(f)1607 4108 y Fr(2)1645 4096 y Fs(k)1687 4111 y Fq(L)1733 4094 y FC(2)1768 4096 y Fu(2)1821 4034 y Fl(j)p 1820 4048 29 3 v 1820 4081 a FC(2)1862 4096 y FE(j)1901 4061 y Fh(s)1929 4096 y Fs(k)p FF(F)2031 4108 y Fq(j)2066 4096 y FF(P)2131 4108 y Fq(k)2172 4096 y FF(h)p Fs(k)2267 4111 y Fq(L)2313 4094 y FC(2)2348 4096 y Fu(2)2401 4034 y Fl(j)p 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(hand)g(side,)h(w)n(e)f(ha)n(v)n(e)316 1488 y Fp(\014)316 1538 y(\014)316 1587 y(\014)316 1637 y(\014)344 1495 y(Z)9 b(Z)506 1552 y Fu(\()p FE(v)22 b Fs(\000)c FE(v)723 1564 y FD(\003)761 1552 y Fu(\))h Fs(\001)f FE(v)893 1564 y FD(\003)p 506 1589 427 4 v 566 1665 a Fs(j)p FE(v)j Fs(\000)d FE(v)773 1677 y FD(\003)812 1665 y Fs(j)835 1641 y Fr(3)942 1608 y FE(f)983 1620 y FD(\003)1021 1608 y FE(f)1071 1574 y Fr(2)1107 1608 y Fs(h)p FE(v)s Fs(i)1214 1574 y Fr(2)p Fq(`)p FD(\000)p Fr(2)1365 1608 y FE(dv)1448 1620 y FD(\003)1487 1608 y FE(dv)1573 1488 y Fp(\014)1573 1538 y(\014)1573 1587 y(\014)1573 1637 y(\014)1624 1608 y Fs(\024)1712 1495 y Fp(Z)9 b(Z)1864 1608 y Fs(j)p FE(v)22 b Fs(\000)c FE(v)2072 1620 y FD(\003)2110 1608 y Fs(j)2133 1574 y FD(\000)p Fr(2)2222 1608 y Fs(j)p FE(v)2285 1620 y FD(\003)2324 1608 y Fs(j)p FE(f)2388 1620 y FD(\003)2426 1608 y FE(f)2476 1574 y Fr(2)2512 1608 y Fs(h)p FE(v)s Fs(i)2619 1574 y Fr(2)p Fq(`)p FD(\000)p Fr(2)2770 1608 y FE(dv)2853 1620 y FD(\003)2892 1608 y FE(dv)238 1833 y Fu(=)316 1720 y Fp(Z)9 b(Z)468 1833 y Fs(j)p FE(v)22 b Fs(\000)c FE(v)676 1845 y FD(\003)714 1833 y Fs(j)737 1799 y FD(\000)p Fr(2)826 1833 y FF(1)874 1848 y FD(j)p Fq(v)r FD(\000)p Fq(v)1014 1856 y Fm(\003)1049 1848 y FD(j\024)p Fr(1)1158 1833 y Fs(j)p FE(v)1221 1845 y FD(\003)1259 1833 y Fs(j)p FE(f)1323 1845 y FD(\003)1361 1833 y FE(f)1411 1799 y Fr(2)1448 1833 y Fs(h)p FE(v)s Fs(i)1555 1799 y Fr(2)p Fq(`)p FD(\000)p Fr(2)1706 1833 y FE(dv)1789 1845 y FD(\003)1828 1833 y FE(dv)j Fu(+)2015 1720 y Fp(Z)10 b(Z)2168 1833 y Fs(j)p FE(v)21 b Fs(\000)d FE(v)2375 1845 y FD(\003)2414 1833 y Fs(j)2437 1799 y FD(\000)p Fr(2)2526 1833 y FF(1)2574 1848 y FD(j)p Fq(v)r FD(\000)p Fq(v)2714 1856 y Fm(\003)2749 1848 y FD(j\025)p Fr(1)2857 1833 y Fs(j)p FE(v)2920 1845 y FD(\003)2959 1833 y Fs(j)p FE(f)3023 1845 y FD(\003)3061 1833 y FE(f)3111 1799 y Fr(2)3148 1833 y Fs(h)p FE(v)s Fs(i)3255 1799 y Fr(2)p Fq(`)p FD(\000)p Fr(2)3405 1833 y FE(dv)3488 1845 y FD(\003)3527 1833 y FE(dv)s(:)0 2007 y Fu(Moreo)n(v)n(er,)477 2068 y Fp(Z)9 b(Z)629 2181 y Fs(j)p FE(v)22 b Fs(\000)c FE(v)837 2193 y FD(\003)875 2181 y Fs(j)898 2146 y FD(\000)p Fr(2)987 2181 y FF(1)1035 2196 y FD(j)p Fq(v)r FD(\000)p Fq(v)1175 2204 y Fm(\003)1210 2196 y FD(j\024)p Fr(1)1319 2181 y Fs(j)p FE(v)1382 2193 y FD(\003)1420 2181 y Fs(j)p FE(f)1484 2193 y FD(\003)1522 2181 y FE(f)1572 2146 y Fr(2)1609 2181 y Fs(h)p FE(v)s Fs(i)1716 2146 y Fr(2)p Fq(`)p FD(\000)p Fr(2)1867 2181 y FE(dv)1950 2193 y FD(\003)1988 2181 y FE(dv)27 b Fk(.)22 b Fs(k)14 b(j\001j)2310 2139 y FD(\000)p Fr(2)2413 2181 y FF(1)2461 2196 y FD(j)o(\001j\024)p Fr(1)2609 2181 y Fs(k)2651 2223 y Fq(L)2706 2179 y FC(3)p 2706 2186 29 3 v 2706 2217 a(2)2744 2195 y Fl(;)p Fm(1)2828 2181 y Fs(k)p FE(f)9 b Fs(k)2962 2207 y Fq(L)3008 2179 y FC(3)p Fl(;)p FC(1)3008 2225 y(3)3090 2181 y Fs(k)p FE(f)g Fs(k)3224 2146 y Fr(2)3224 2206 y Fq(L)3270 2186 y FC(2)3270 2227 y Fl(`)p Fm(\000)p FC(2)3374 2181 y Fu(;)291 2299 y Fp(Z)g(Z)443 2412 y Fs(j)p FE(v)22 b Fs(\000)c FE(v)651 2424 y FD(\003)690 2412 y Fs(j)713 2378 y FD(\000)p Fr(2)802 2412 y FF(1)849 2427 y FD(j)p Fq(v)r FD(\000)p Fq(v)989 2435 y Fm(\003)1024 2427 y FD(j\025)p Fr(1)1133 2412 y Fs(j)p FE(v)1196 2424 y FD(\003)1235 2412 y Fs(j)p FE(f)1299 2424 y FD(\003)1336 2412 y FE(f)1386 2378 y Fr(2)1423 2412 y Fs(h)p FE(v)s Fs(i)1530 2378 y Fr(2)p Fq(`)p FD(\000)p Fr(2)1681 2412 y FE(dv)1764 2424 y FD(\003)1803 2412 y FE(dv)26 b Fk(.)2000 2299 y Fp(Z)9 b(Z)2138 2412 y Fs(h)p FE(v)2210 2424 y FD(\003)2249 2412 y Fs(i)2281 2378 y Fr(3)2318 2412 y FE(f)2359 2424 y FD(\003)2397 2412 y FE(f)2447 2378 y Fr(2)2484 2412 y Fs(h)p FE(v)s Fs(i)2591 2378 y Fr(2)p Fq(`)p FD(\000)p Fr(4)2742 2412 y FE(dv)2825 2424 y FD(\003)2864 2412 y FE(dv)26 b Fs(\024)d(k)p FE(f)9 b Fs(k)3195 2430 y Fq(L)3241 2410 y FC(1)3241 2448 y(3)3275 2412 y Fs(k)p FE(f)g Fs(k)3409 2378 y Fr(2)3409 2438 y Fq(L)3455 2418 y FC(2)3455 2458 y Fl(`)p Fm(\000)p FC(2)3560 2412 y FE(;)0 2579 y Fu(where)19 b(w)n(e)h(use)f(the)h(fact)g(that)g Fs(h)p FE(v)1017 2591 y FD(\003)1056 2579 y Fs(i)j(\030)g(h)p FE(v)s Fs(i)d Fu(in)g(regime)f Fs(fj)p FE(v)6 b Fs(\000)s FE(v)1894 2591 y FD(\003)1931 2579 y Fs(j)23 b(\024)g Fu(1)p Fs(g)c Fu(and)g Fs(j)p FE(v)6 b Fs(\000)s FE(v)2498 2591 y FD(\003)2536 2579 y Fs(j)23 b(\025)g(h)p FE(v)s Fs(ih)p FE(v)2849 2591 y FD(\003)2888 2579 y Fs(i)2920 2548 y FD(\000)p Fr(1)3029 2579 y Fu(in)d(regime)f Fs(fj)p FE(v)6 b Fs(\000)s FE(v)3597 2591 y FD(\003)3634 2579 y Fs(j)23 b(\025)g Fu(1)p Fs(g)p Fu(.)0 2678 y(F)-7 b(or)27 b(the)h(third)g(term)f(on)h(the)g(righ)n(t)e(hand)i(side,)g(w)n(e)f(ha) n(v)n(e)290 2750 y Fp(Z)10 b(Z)452 2807 y Fs(j)p FE(v)22 b Fs(\002)c FE(v)660 2819 y FD(\003)699 2807 y Fs(j)722 2777 y Fr(2)p 452 2844 307 4 v 452 2920 a Fs(j)p FE(v)k Fs(\000)c FE(v)660 2932 y FD(\003)699 2920 y Fs(j)722 2896 y Fr(3)769 2863 y FE(f)810 2875 y FD(\003)848 2863 y FE(f)898 2829 y Fr(2)934 2863 y Fs(h)p FE(v)s Fs(i)1041 2829 y Fr(2)p Fq(`)p FD(\000)p Fr(4)1192 2863 y FE(dv)1275 2875 y FD(\003)1314 2863 y FE(dv)26 b Fs(\024)1511 2750 y Fp(Z)9 b(Z)1663 2863 y Fs(j)p FE(v)22 b Fs(\000)c FE(v)1871 2875 y FD(\003)1909 2863 y Fs(j)1932 2829 y FD(\000)p Fr(1)2022 2863 y Fs(j)p FE(v)2085 2875 y FD(\003)2123 2863 y Fs(j)2146 2829 y Fr(2)2183 2863 y FE(f)2224 2875 y FD(\003)2262 2863 y FE(f)2312 2829 y Fr(2)2349 2863 y Fs(h)p FE(v)s Fs(i)2456 2829 y Fr(2)p Fq(`)p FD(\000)p Fr(4)2607 2863 y FE(dv)2690 2875 y FD(\003)2728 2863 y FE(dv)212 3087 y Fu(=)290 2974 y Fp(Z)10 b(Z)442 3087 y Fs(j)p FE(v)22 b Fs(\000)c FE(v)650 3099 y FD(\003)689 3087 y Fs(j)712 3053 y FD(\000)p Fr(1)801 3087 y FF(1)849 3102 y FD(j)p Fq(v)r FD(\000)p Fq(v)989 3110 y Fm(\003)1023 3102 y FD(j\024)p Fr(1)1132 3087 y Fs(j)p FE(v)1195 3099 y FD(\003)1234 3087 y Fs(j)1257 3053 y Fr(2)1294 3087 y FE(f)1335 3099 y FD(\003)1373 3087 y FE(f)1423 3053 y Fr(2)1460 3087 y Fs(h)p FE(v)s Fs(i)1567 3053 y Fr(2)p Fq(`)p FD(\000)p Fr(4)1717 3087 y FE(dv)1800 3099 y FD(\003)1839 3087 y FE(dv)k Fu(+)2027 2974 y Fp(Z)9 b(Z)2179 3087 y Fs(j)p FE(v)22 b Fs(\000)c FE(v)2387 3099 y FD(\003)2425 3087 y Fs(j)2448 3053 y FD(\000)p Fr(1)2538 3087 y FF(1)2585 3102 y FD(j)p Fq(v)r FD(\000)p Fq(v)2725 3110 y Fm(\003)2760 3102 y FD(j\025)p Fr(1)2869 3087 y Fs(j)p FE(v)2932 3099 y FD(\003)2970 3087 y Fs(j)2993 3053 y Fr(2)3031 3087 y FE(f)3072 3099 y FD(\003)3110 3087 y FE(f)3160 3053 y Fr(2)3196 3087 y Fs(h)p FE(v)s Fs(i)3303 3053 y Fr(2)p Fq(`)p FD(\000)p Fr(4)3454 3087 y FE(dv)3537 3099 y FD(\003)3576 3087 y FE(dv)212 3271 y Fs(\024)o(k)c(j\001j)401 3229 y FD(\000)p Fr(1)504 3271 y FF(1)552 3286 y FD(j)o(\001j\024)p Fr(1)700 3271 y Fs(k)742 3286 y Fq(L)788 3269 y FC(3)p Fl(;)p Fm(1)899 3271 y Fs(k)p FE(f)9 b Fs(k)1033 3327 y Fq(L)1087 3269 y FC(3)p 1087 3276 29 3 v 1087 3307 a(2)1125 3285 y Fl(;)p FC(1)1079 3345 y(3)1181 3271 y Fs(k)p FE(f)g Fs(k)1315 3237 y Fr(2)1315 3297 y Fq(L)1361 3277 y FC(2)1361 3331 y Fl(`)p Fm(\000)1439 3315 y FC(5)p 1439 3322 V 1439 3353 a(2)1504 3271 y Fu(+)18 b Fs(k)p FE(f)9 b Fs(k)1721 3289 y Fq(L)1767 3269 y FC(1)1767 3307 y(3)1802 3271 y Fs(k)p FE(f)g Fs(k)1936 3237 y Fr(2)1936 3297 y Fq(L)1982 3277 y FC(2)1982 3331 y Fl(`)p Fm(\000)2059 3315 y FC(5)p 2059 3322 V 2059 3353 a(2)2106 3271 y FE(:)0 3455 y Fu(Com)n(bibing)27 b(all)g(ab)r(o)n(v)n(e)g(estimates,)g(w)n(e)g (conclude)g(that)16 3581 y(1)p 16 3618 42 4 v 16 3694 a(2)93 3581 y FE(d)p 78 3618 74 4 v 78 3694 a(dt)161 3637 y Fs(k)p FE(f)9 b Fs(k)295 3603 y Fr(2)295 3663 y Fq(L)341 3643 y FC(2)341 3683 y Fl(`)394 3637 y Fu(+)18 b FE(\024`)p Fs(k)p FE(f)9 b Fs(k)694 3603 y Fr(2)694 3663 y Fq(L)740 3643 y FC(2)740 3697 y Fl(`)p Fm(\000)817 3681 y FC(3)p 817 3688 29 3 v 817 3719 a(2)887 3637 y Fk(.)22 b FE(`)1023 3545 y Fp(\020)1072 3637 y Fs(k)p FE(f)9 b Fs(k)1206 3663 y Fq(L)1252 3635 y FC(3)p Fl(;)p FC(1)1252 3681 y(3)1353 3637 y Fu(+)18 b Fs(k)p FE(f)9 b Fs(k)1570 3655 y Fq(L)1616 3635 y FC(1)1616 3673 y(3)1651 3545 y Fp(\021)1714 3637 y Fs(k)p FE(f)g Fs(k)1848 3603 y Fr(2)1848 3663 y Fq(L)1894 3643 y FC(2)1894 3683 y Fl(`)p Fm(\000)p FC(2)2017 3637 y Fu(+)18 b FE(`)2135 3603 y Fr(2)2186 3520 y Fp(\022)2247 3637 y Fs(k)p FE(f)9 b Fs(k)2381 3693 y Fq(L)2435 3635 y FC(3)p 2435 3642 V 2435 3673 a(2)2473 3651 y Fl(;)p FC(1)2427 3711 y(3)2547 3637 y Fu(+)18 b Fs(k)p FE(f)9 b Fs(k)2764 3655 y Fq(L)2810 3635 y FC(1)2810 3673 y(3)2845 3520 y Fp(\023)2920 3637 y Fs(k)p FE(f)g Fs(k)3054 3603 y Fr(2)3054 3663 y Fq(L)3100 3643 y FC(2)3100 3697 y Fl(`)p Fm(\000)3178 3681 y FC(5)p 3178 3688 V 3178 3719 a(2)3243 3637 y Fu(+)18 b Fs(k)p FE(f)9 b Fs(k)3460 3603 y Fr(2)3460 3663 y Fq(L)3506 3643 y FC(3)3506 3681 y(3)3540 3637 y Fs(k)p FE(f)g Fs(k)3674 3603 y Fr(2)3674 3663 y Fq(L)3720 3643 y FC(2)3720 3697 y Fl(`)p Fm(\000)3798 3681 y FC(3)p 3798 3688 V 3798 3719 a(2)3845 3637 y FE(:)100 3832 y Fu(Since)27 b FE(f)357 3844 y Fr(0)422 3832 y Fu(p)r(ossesses)f FE(L)834 3801 y Fr(1)899 3832 y Fu(exp)r(onen)n(tial)h(momen)n(t,)g(b)n(y)h(virtue)f (of)g([)p 0 1 0 TeXcolorrgb 2162 3832 a SDict begin H.S end 2162 3832 a 0 1 0 TeXcolorrgb Fu(4)p 0 1 0 TeXcolorrgb 2204 3778 a SDict begin H.R end 2204 3778 a 2204 3832 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.CDH) cvn H.B /ANN pdfmark end 2204 3832 a 0 TeXcolorgray Fu(],)h FE(f)9 b Fu(\()p FE(t)p Fu(\))23 b Fs(2)g FE(L)2580 3801 y FD(1)2650 3832 y Fu(\([0)p FE(;)14 b Fs(1)p Fu(\))p FE(;)g(L)2993 3801 y Fr(1)2993 3852 y(40)3063 3832 y Fu(\).)37 b(Th)n(us)28 b(for)f FE(t)c Fs(2)g Fu([1)p FE(;)14 b Fs(1)p Fu(\),)957 3995 y Fs(k)p FE(f)9 b Fu(\()p FE(t)p Fu(\))p Fs(k)1185 4013 y Fq(L)1231 3993 y FC(2)1231 4031 y(10)1318 3995 y Fs(\024)23 b(k)p FE(f)9 b Fu(\()p FE(t)p Fu(\))p Fs(k)1644 3924 y FC(1)p 1643 3933 29 3 v 1643 3966 a(4)1634 4026 y Fq(L)1680 4006 y FC(1)1680 4044 y(40)1743 3995 y Fs(k)p FE(f)g Fu(\()p FE(t)p Fu(\))p Fs(k)1981 3924 y FC(3)p 1980 3933 V 1980 3966 a(4)1971 4023 y Fq(L)2017 4006 y FC(3)2076 3995 y Fk(.)22 b FE(I)7 b Fu(\()p FE(f)i Fu(\()p FE(t)p Fu(\)\))2424 3938 y FC(3)p 2425 3947 V 2425 3981 a(4)2490 3995 y Fs(\024)23 b FE(I)7 b Fu(\()p FE(f)i Fu(\(1\)\))2851 3938 y FC(3)p 2851 3947 V 2851 3981 a(4)2893 3995 y FE(;)0 4157 y Fu(namely)27 b FE(f)9 b Fu(\()p FE(t)p Fu(\))23 b Fs(2)h FE(L)591 4127 y FD(1)661 4157 y Fu(\([0)p FE(;)14 b Fs(1)p Fu(\))p FE(;)g(L)1004 4127 y Fr(2)1004 4178 y(10)1074 4157 y Fu(\).)37 b(Due)28 b(to)f(\()p 1 0 0 TeXcolorrgb 1473 4157 a SDict begin H.S end 1473 4157 a 1 0 0 TeXcolorrgb Fu(1.23)p 1 0 0 TeXcolorrgb 1621 4103 a SDict begin H.R end 1621 4103 a 1621 4157 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.1.23) cvn H.B /ANN pdfmark end 1621 4157 a 0 TeXcolorgray Fu(\))h(and)g(the)g(en)n(trop)n(y)e(metho)r(d)i (in)g([)p 0 1 0 TeXcolorrgb 2710 4157 a SDict begin H.S end 2710 4157 a 0 1 0 TeXcolorrgb Fu(4)p 0 1 0 TeXcolorrgb 2751 4103 a SDict begin H.R end 2751 4103 a 2751 4157 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.CDH) cvn H.B /ANN pdfmark end 2751 4157 a 0 TeXcolorgray Fu(])g(w)n(e)f(can)h(get) f(for)g FE(t)c Fs(2)h Fu(\(0)p FE(;)14 b Fs(1)p Fu(\),)1470 4306 y SDict begin H.S end 1470 4306 a 1470 4306 a SDict begin 12 H.A end 1470 4306 a 1470 4306 a SDict begin [/View [/XYZ H.V]/Dest (equation.7.1) cvn /DEST pdfmark end 1470 4306 a Fs(k)p FE(f)9 b Fu(\()p FE(t)p Fu(\))p Fs(k)1698 4332 y Fq(L)1744 4304 y FC(3)p Fl(;)p FC(1)1744 4350 y(3)1850 4306 y Fk(.)22 b FE(t)1967 4272 y FD(\000)2029 4249 y FC(1)p 2029 4258 29 3 v 2029 4292 a(4)2090 4306 y Fu(+)c(1)p FE(:)p 0 TeXcolorgray 1465 w Fu(\(7.1\))p 0 TeXcolorgray 0 4458 a(Then)28 b(w)n(e)f(obtain)g(that)h(for)f FE(t)c Fs(2)h Fu(\(0)p FE(;)14 b Fs(1)p Fu(\),)367 4637 y SDict begin H.S end 367 4637 a 367 4637 a SDict begin 12 H.A end 367 4637 a 367 4637 a SDict begin [/View [/XYZ H.V]/Dest (equation.7.2) cvn /DEST pdfmark end 367 4637 a 377 4581 a Fu(1)p 377 4618 42 4 v 377 4694 a(2)453 4581 y FE(d)p 438 4618 74 4 v 438 4694 a(dt)521 4637 y Fs(k)p FE(f)9 b Fs(k)655 4603 y Fr(2)655 4663 y Fq(L)701 4643 y FC(2)701 4683 y Fl(`)755 4637 y Fu(+)18 b FE(\024`)p Fs(k)p FE(f)9 b Fs(k)1055 4603 y Fr(2)1055 4663 y Fq(L)1101 4643 y FC(2)1101 4697 y Fl(`)p Fm(\000)1178 4681 y FC(3)p 1178 4688 29 3 v 1178 4719 a(2)1247 4637 y Fk(.)1335 4545 y Fp(\020)1385 4637 y FE(t)1415 4603 y FD(\000)1476 4581 y FC(1)p 1476 4590 V 1476 4623 a(4)1537 4637 y Fu(+)18 b(1)1662 4545 y Fp(\021)1725 4637 y FE(`)p Fs(k)p FE(f)9 b Fs(k)1894 4603 y Fr(2)1894 4663 y Fq(L)1940 4643 y FC(2)1940 4683 y Fl(`)p Fm(\000)p FC(2)2063 4637 y Fu(+)18 b FE(`)2181 4603 y Fr(2)2218 4637 y Fs(k)p FE(f)9 b Fs(k)2352 4603 y Fr(2)2352 4663 y Fq(L)2398 4643 y FC(2)2398 4697 y Fl(`)p Fm(\000)2476 4681 y FC(5)p 2476 4688 V 2476 4719 a(2)2540 4637 y Fu(+)2623 4545 y Fp(\020)2673 4637 y FE(t)2703 4603 y FD(\000)2765 4581 y FC(1)p 2765 4590 V 2765 4623 a(2)2826 4637 y Fu(+)18 b(1)2951 4545 y Fp(\021)3014 4637 y Fs(k)p FE(f)9 b Fs(k)3148 4603 y 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2420 a 1 0 0 TeXcolorrgb Fu(2.1)p 1 0 0 TeXcolorrgb 1190 2367 a SDict begin H.R end 1190 2367 a 1190 2420 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (prop.2.1) cvn H.B /ANN pdfmark end 1190 2420 a 0 TeXcolorgray 27 w Fu(and)g(\()p 1 0 0 TeXcolorrgb 1411 2420 a SDict begin H.S end 1411 2420 a 1 0 0 TeXcolorrgb Fu(8.4)p 1 0 0 TeXcolorrgb 1517 2367 a SDict begin H.R end 1517 2367 a 1517 2420 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.8.4) cvn H.B /ANN pdfmark end 1517 2420 a 0 TeXcolorgray Fu(\),)g(w)n(e)f(ha)n(v)n(e)g(that) 892 2519 y Fp(Z)975 2632 y Fu(\()p FE(a)19 b Fs(\003)f FE(f)9 b Fu(\))23 b(:)g Fs(r)p FE(F)1415 2598 y Fr(\()p Fq(\013)p Fr(\))1533 2632 y Fs(\012)18 b(r)p FE(F)1750 2598 y Fr(\()p Fq(\013)p Fr(\))1849 2632 y FE(dv)26 b Fs(\025)d FE(\024)p Fs(j)-9 b(j)g(j)p FE(F)2210 2598 y Fr(\()p Fq(\013)p Fr(\))2309 2632 y Fs(j)g(j)g(j)2361 2575 y Fr(2)2416 2632 y Fs(\000)18 b FE(C)6 b 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3616 a 679 3670 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.12) cvn H.B /ANN pdfmark end 679 3670 a 0 TeXcolorgray Fu(\))h(and)f(\()p 1 0 0 TeXcolorrgb 932 3670 a SDict begin H.S end 932 3670 a 1 0 0 TeXcolorrgb Fu(8.4)p 1 0 0 TeXcolorrgb 1039 3616 a SDict begin H.R end 1039 3616 a 1039 3670 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.8.4) cvn H.B /ANN pdfmark end 1039 3670 a 0 TeXcolorgray Fu(\),)h(w)n(e)f(ha)n(v)n(e)f(that) 131 3769 y Fp(\014)131 3819 y(\014)131 3868 y(\014)158 3864 y Fs(h)p FE(Q)p Fu(\()p FE(@)337 3830 y Fq(\013)380 3838 y FC(1)417 3864 y FE(f)t(;)14 b(F)564 3830 y Fr(\()p Fq(\013)633 3838 y FC(2)665 3830 y Fr(\))695 3864 y Fu(\))p FE(;)g(F)829 3830 y Fr(\()p Fq(\013)p Fr(\))929 3864 y Fs(i)961 3876 y Fq(v)1000 3769 y Fp(\014)1000 3819 y(\014)1000 3868 y(\014)1051 3864 y Fk(.)1139 3772 y Fp(\020)1188 3864 y Fs(k)p FE(@)1279 3830 y Fq(\013)1322 3838 y FC(1)1358 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y Fm(\000)2252 -209 y FC(1)p 2252 -202 29 3 v 2252 -171 a(2)2324 -165 y Fr(SP)-5 b(A)n(CE)1268 b(47)p 0 TeXcolorgray 100 29 a Fu(No)n(w)27 b(m)n(ultiply)h FE(F)686 -1 y Fr(\()p Fq(\013)p Fr(\))813 29 y Fu(on)f(b)r(oth)h(sides) f(of)34 b(\()p 1 0 0 TeXcolorrgb 1456 29 a SDict begin H.S end 1456 29 a 1 0 0 TeXcolorrgb Fu(8.5)p 1 0 0 TeXcolorrgb 1563 -24 a SDict begin H.R end 1563 -24 a 1563 29 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.8.5) cvn H.B /ANN pdfmark end 1563 29 a 0 TeXcolorgray Fu(\))28 b(and)f(in)n(tegrate)f(o)n(v)n(er)g Fo(R)2370 -7 y Fr(3)2370 50 y Fq(v)2410 29 y Fu(,)h(w)n(e)h(can)f(deriv)n(e)g(from)g(ab)r(o)n(v) n(e)f(that)0 257 y SDict begin H.S end 0 257 a 0 257 a SDict begin 12 H.A end 0 257 a 0 257 a SDict begin [/View [/XYZ H.V]/Dest (equation.8.6) cvn /DEST pdfmark end 0 257 a 241 200 a Fu(1)p 241 238 42 4 v 241 314 a(2)317 200 y FE(d)p 302 238 74 4 v 302 314 a(dt)385 257 y Fs(k)p FE(F)492 222 y Fr(\()p Fq(\013)p 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314 y Fs(j)p Fu(!)p Fs(j)p FE(\013)3034 326 y Fr(2)3071 314 y Fs(j)p Fu(!)3127 257 y Fs(k)p FE(F)3234 222 y Fr(\()p Fq(\013)3303 230 y FC(1)3335 222 y Fr(\))3365 257 y Fs(k)3407 272 y Fq(L)3453 255 y FC(2)3489 257 y Fs(j)-9 b(j)g(j)p FE(F)3605 222 y Fr(\()p Fq(\013)3674 230 y FC(2)3707 222 y Fr(\))3737 257 y Fs(j)g(j)g(jj)g(j)g(j)p FE(F)3904 222 y Fr(\()p Fq(\013)p Fr(\))4004 257 y Fs(j)g(j)g(j)83 587 y Fu(+)319 508 y Fp(X)231 682 y Fq(\013)p Fr(=)p Fq(\013)368 690 y FC(1)400 682 y Fr(+)p Fq(\013)494 690 y FC(2)649 531 y Fs(j)p FE(\013)p Fs(j)p Fu(!)p 551 568 V 551 644 a Fs(j)p FE(\013)627 656 y Fr(1)664 644 y Fs(j)p Fu(!)p Fs(j)p FE(\013)786 656 y Fr(2)824 644 y Fs(j)p Fu(!)880 587 y Fs(k)p FE(F)987 553 y Fr(\()p Fq(\013)1056 561 y FC(1)1088 553 y Fr(\))1118 587 y Fs(k)1160 602 y Fq(L)1206 585 y FC(2)1241 587 y Fs(k)p FE(F)1348 553 y Fr(\()p Fq(\013)1417 561 y FC(2)1449 553 y Fr(\))1479 587 y Fs(k)1521 602 y Fq(L)1567 585 y FC(2)1603 587 y Fs(j)g(j)g(j)p FE(F)1719 553 y Fr(\()p Fq(\013)p Fr(\))1818 587 y Fs(j)g(j)g(j)19 b Fu(+)2059 508 y Fp(X)1971 682 y Fq(\013)p Fr(=)p Fq(\013)2108 690 y FC(1)2141 682 y Fr(+)p Fq(\013)2235 690 y FC(2)2389 531 y Fs(j)p FE(\013)p Fs(j)p Fu(!)p 2291 568 V 2291 644 a Fs(j)p FE(\013)2367 656 y Fr(1)2405 644 y Fs(j)p Fu(!)p Fs(j)p FE(\013)2527 656 y Fr(2)2564 644 y Fs(j)p Fu(!)2620 587 y Fs(k)p FE(F)2727 553 y Fr(\()p Fq(\013)2796 561 y FC(1)2828 553 y Fr(\))2858 587 y Fs(k)2900 602 y Fq(L)2946 585 y FC(2)2982 587 y Fs(k)p FE(F)3089 553 y Fr(\()p Fq(\013)3158 561 y FC(2)3190 553 y Fr(\))3220 587 y Fs(k)3262 602 y Fq(L)3308 585 y FC(2)3344 587 y Fs(k)p FE(F)3451 553 y Fr(\()p Fq(\013)p Fr(\))3549 587 y Fs(k)3591 602 y Fq(L)3637 585 y FC(2)3673 587 y FE(:)p 0 TeXcolorgray 188 w Fu(\(8.6\))p 0 TeXcolorgray 100 871 a(Let)27 b(0)c FE(<)g(\032)g(<)f Fu(1)27 b(and)h(recall)e(that) i FE(\027)h Fu(=)1353 831 y Fr(1+3)p Fq(=\014)p 1353 852 192 4 v 1433 899 a Fr(2)1555 871 y Fu(,)f(w)n(e)f(de\014ne)701 1129 y Fs(j)p FE(f)9 b Fs(j)797 1141 y Fq(\032;\013)898 1129 y Fu(\()p FE(t)p Fu(\))24 b(=)1114 1073 y(\()p FE(\032)p Ff(r)p Fu(\()p FE(t)p Fu(\)\))1348 1031 y FD(j)p Fq(\013)p FD(j)p 1114 1110 322 4 v 1160 1187 a Fu(\()p Fs(j)p FE(\013)p Fs(j)p Fu(!\))1347 1145 y Fq(\027)1445 1129 y Fs(k)p FE(F)1552 1095 y Fr(\()p Fq(\013)p Fr(\))1650 1129 y Fu(\()p FE(t)p Fu(\))p Fs(k)1786 1144 y Fq(L)1832 1127 y FC(2)1869 1129 y FE(;)96 b Fs(j)-9 b(j)g(j)p FE(f)9 b Fs(j)-9 b(j)g(j)2141 1154 y Fq(\032;\013)2242 1129 y Fu(\()p FE(t)p Fu(\))24 b(=)2457 1073 y(\()q FE(\032)p Ff(r)p Fu(\()p FE(t)p Fu(\)\))2692 1031 y FD(j)p Fq(\013)p FD(j)p 2457 1110 V 2504 1187 a Fu(\()p Fs(j)p FE(\013)p Fs(j)p Fu(!\))2691 1145 y Fq(\027)2788 1129 y Fs(j)-9 b(j)g(j)p FE(F)2904 1095 y Fr(\()p Fq(\013)p Fr(\))3004 1129 y Fu(\()p FE(t)p Fu(\))p Fs(j)g(j)g(j)q FE(;)774 1382 y Fs(j)p FE(f)9 b Fs(j)870 1394 y Fq(\014)s(;\032;\013)1031 1382 y Fu(\()p FE(t)p Fu(\))24 b(=)1246 1326 y(\()q FE(\032)p Ff(r)p Fu(\()p FE(t)p Fu(\)\))1481 1284 y FD(j)p Fq(\013)p FD(j)p 1246 1363 V 1293 1440 a Fu(\()p Fs(j)p FE(\013)p Fs(j)p Fu(!\))1480 1398 y Fq(\027)1577 1382 y Fs(k)p FE(F)1684 1348 y Fr(\()p Fq(\013)p Fr(\))1783 1382 y Fu(\()p FE(t)p Fu(\))p Fs(k)1919 1400 y Fq(L)1965 1380 y FC(2)1975 1420 y Fl(\014)p 1975 1432 35 3 v 1978 1464 a FC(2)2028 1382 y FE(;)97 b Fs(j)p FE(f)9 b Fs(j)2244 1394 y Fq(\032;N)2361 1382 y Fu(\()p FE(t)p Fu(\))23 b(=)99 b(sup)2566 1456 y Fr(0)p Fq(<)p FD(j)p Fq(\013)p FD(j\024)p Fq(N)2858 1382 y Fs(j)p FE(f)9 b Fs(j)2954 1394 y Fq(\032;\013)3055 1382 y Fu(\()p FE(t)p Fu(\))p FE(:)0 1690 y Fu(Then)28 b(due)g(to)f(\()p 1 0 0 TeXcolorrgb 507 1690 a SDict begin H.S end 507 1690 a 1 0 0 TeXcolorrgb Fu(8.3)p 1 0 0 TeXcolorrgb 613 1637 a SDict begin H.R end 613 1637 a 613 1690 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.8.3) cvn H.B /ANN pdfmark end 613 1690 a 0 TeXcolorgray Fu(\))h(and)g(the)g(de\014nition)g(of)f Ff(r)p Fu(\()p FE(t)p Fu(\))d(=)1679 1598 y Fp(\020)1821 1658 y Fq(t)p 1738 1672 190 4 v 1738 1719 a Fr(log)13 b(1)p Fq(=t)1938 1598 y Fp(\021)1997 1593 y FC(1)p 1997 1602 29 3 v 1997 1636 a(2)2040 1690 y Fu(,)27 b(w)n(e)h(ha)n(v)n(e)e (that)679 1904 y SDict begin H.S end 679 1904 a 679 1904 a SDict begin 12 H.A end 679 1904 a 679 1904 a SDict begin [/View [/XYZ H.V]/Dest (equation.8.7) cvn /DEST pdfmark end 679 1904 a 47 w Fu(sup)679 1978 y Fq(\034)7 b FD(2)p Fr(\(0)p Fq(;)p Fr(1\))913 1904 y Fs(j)p FE(f)i Fs(j)1009 1870 y Fr(2)1009 1925 y Fq(\032;)p Fr(0)1100 1904 y Fu(\()p FE(\034)g Fu(\))24 b Fk(.)f Fu(1)p FE(;)96 b Fu(and)101 b(lim)1713 1955 y Fq(t)p FD(!)p Fr(0)1851 1904 y Fs(j)p FE(f)9 b Fs(j)1947 1916 y Fq(\032;\013)2048 1904 y Fu(\()p FE(t)p Fu(\))24 b(=)e(0)p FE(;)97 b Fu(for)27 b(an)n(y)82 b Fs(j)p FE(\013)p Fs(j)24 b FE(>)e Fu(0)p FE(:)p 0 TeXcolorgray 674 w Fu(\(8.7\))p 0 TeXcolorgray 100 2151 a(Fix)27 b FE(N)32 b(>)23 b Fu(0,)k(let)h(0)23 b Fs(\024)f(j)p FE(\013)p Fs(j)i(\024)e FE(N)9 b Fu(,)28 b(from)f(\()p 1 0 0 TeXcolorrgb 1364 2151 a SDict begin H.S end 1364 2151 a 1 0 0 TeXcolorrgb Fu(8.6)p 1 0 0 TeXcolorrgb 1471 2097 a SDict begin H.R end 1471 2097 a 1471 2151 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.8.6) cvn H.B /ANN pdfmark end 1471 2151 a 0 TeXcolorgray Fu(\))h(and)f(direct)g(computation,)h(w)n(e)f(can)g(deriv)n(e)g(that) 246 2349 y(1)p 246 2386 42 4 v 246 2462 a(2)322 2349 y FE(d)p 307 2386 74 4 v 307 2462 a(dt)390 2405 y Fs(j)p FE(f)9 b Fs(j)486 2371 y Fr(2)486 2425 y Fq(\032;\013)587 2405 y Fu(\()p FE(t)p Fu(\))19 b(+)f Ft(a)823 2371 y FD(0)846 2405 y Fu(\()p FE(t)p Fu(\))p Fs(j)p FE(f)9 b Fs(j)1036 2371 y Fr(2)1036 2425 y Fq(\014)s(;\032;\013)1199 2405 y Fu(\()p FE(t)p Fu(\))19 b(+)1405 2349 y FE(\024)p 1405 2386 48 4 v 1408 2462 a Fu(2)1462 2405 y Fs(j)-9 b(j)g(j)p FE(f)9 b Fs(j)-9 b(j)g(j)1615 2363 y Fr(2)1615 2430 y Fq(\032;\013)1716 2405 y Fu(\()p FE(t)p Fu(\))24 b Fk(.)e FE(t)1951 2371 y FD(\000)2013 2348 y FC(1)p 2013 2357 29 3 v 2013 2391 a(2)2055 2405 y Fs(j)p FE(f)9 b Fs(j)2151 2371 y Fr(2)2151 2425 y Fq(\032;\013)2253 2405 y Fu(\()p FE(t)p Fu(\))19 b(+)2474 2349 y FE(d)p 2459 2386 74 4 v 2459 2462 a(dt)2556 2263 y Fp( )2631 2349 y Fu(\()q FE(\032)p Ff(r)p Fu(\()p FE(t)p Fu(\)\))2866 2307 y Fr(2)p FD(j)p Fq(\013)p FD(j)p 2631 2386 355 4 v 2678 2475 a Fu(\()p Fs(j)p FE(\013)p Fs(j)p Fu(!\))2865 2433 y Fr(2)p Fq(\027)2996 2263 y Fp(!)3075 2405 y Fs(k)p FE(F)3182 2371 y Fr(\()p Fq(\013)p Fr(\))3281 2405 y Fs(k)3323 2371 y Fr(2)3323 2427 y Fq(L)3369 2411 y FC(2)254 2679 y Fu(+)425 2600 y Fp(X)337 2775 y Fq(\013)p Fr(=)p Fq(\013)474 2783 y FC(1)507 2775 y Fr(+)p Fq(\013)601 2783 y FC(2)385 2835 y FD(j)p Fq(\013)448 2843 y FC(1)480 2835 y FD(j)p Fq(>)p Fr(0)647 2562 y Fp(\022)816 2623 y Fs(j)p FE(\013)p Fs(j)p Fu(!)p 718 2660 320 4 v 718 2736 a Fs(j)p FE(\013)794 2748 y Fr(1)832 2736 y Fs(j)p Fu(!)p Fs(j)p FE(\013)954 2748 y Fr(2)991 2736 y Fs(j)p Fu(!)1047 2562 y Fp(\023)1108 2579 y Fr(1)p FD(\000)p Fq(\027)1248 2679 y Fs(j)p FE(f)9 b Fs(j)1344 2691 y Fq(\032;\013)1441 2699 y FC(1)1478 2679 y Fu(\()p FE(t)p Fu(\))p Fs(j)-9 b(j)g(j)p FE(f)9 b Fs(j)-9 b(j)g(j)1725 2704 y Fq(\032;\013)1822 2712 y FC(2)1859 2679 y Fu(\()p FE(t)p Fu(\))p Fs(j)g(j)g(j)p FE(f)9 b Fs(j)-9 b(j)g(j)2105 2704 y Fq(\032;\013)2207 2679 y Fu(\()p FE(t)p Fu(\))19 b(+)2491 2600 y Fp(X)2403 2775 y Fq(\013)p Fr(=)p Fq(\013)2540 2783 y FC(1)2573 2775 y Fr(+)p Fq(\013)2667 2783 y FC(2)2713 2562 y Fp(\022)2882 2623 y Fs(j)p FE(\013)p Fs(j)p Fu(!)p 2784 2660 V 2784 2736 a Fs(j)p FE(\013)2860 2748 y Fr(1)2897 2736 y Fs(j)p Fu(!)p Fs(j)p FE(\013)3019 2748 y Fr(2)3057 2736 y Fs(j)p Fu(!)3113 2562 y Fp(\023)3174 2579 y Fr(1)p FD(\000)p Fq(\027)3314 2679 y Fs(j)p FE(f)9 b Fs(j)3410 2691 y Fq(\032;\013)3507 2699 y FC(1)3544 2679 y Fu(\()p FE(t)p Fu(\))254 3028 y Fs(\002)18 b(j)p FE(f)9 b Fs(j)433 3040 y Fq(\032;\013)530 3048 y FC(2)567 3028 y Fu(\()p FE(t)p Fu(\))p Fs(j)-9 b(j)g(j)p FE(f)9 b Fs(j)-9 b(j)g(j)813 3053 y Fq(\032;\013)915 3028 y Fu(\()p FE(t)p Fu(\))19 b(+)1199 2949 y Fp(X)1111 3124 y Fq(\013)p Fr(=)p Fq(\013)1248 3132 y FC(1)1281 3124 y Fr(+)p Fq(\013)1375 3132 y FC(2)1421 2911 y Fp(\022)1590 2972 y Fs(j)p FE(\013)p Fs(j)p Fu(!)p 1492 3009 V 1492 3085 a Fs(j)p FE(\013)1568 3097 y Fr(1)1605 3085 y Fs(j)p Fu(!)p Fs(j)p FE(\013)1727 3097 y Fr(2)1765 3085 y Fs(j)p Fu(!)1821 2911 y Fp(\023)1882 2928 y Fr(1)p FD(\000)p Fq(\027)2022 3028 y Fs(j)p FE(f)9 b Fs(j)2118 3040 y Fq(\032;\013)2215 3048 y FC(1)2252 3028 y Fu(\()p FE(t)p Fu(\))p Fs(j)p FE(f)g Fs(j)2442 3040 y Fq(\032;\013)2539 3048 y FC(2)2576 3028 y Fu(\()p FE(t)p Fu(\))p Fs(j)p FE(f)g Fs(j)2766 3040 y Fq(\032;\013)2868 3028 y Fu(\()p FE(t)p Fu(\))p FE(:)100 3286 y Fs(\017)27 b Fu(F)-7 b(or)27 b FE(\013)c Fu(=)g(0,)k(w)n(e)g(can)h(deriv)n(e)e(from)i(ab)r(o)n(v)n (e)e(that)299 3448 y(1)p 299 3485 42 4 v 299 3561 a(2)376 3448 y FE(d)p 361 3485 74 4 v 361 3561 a(dt)444 3504 y Fs(j)p FE(f)9 b Fs(j)540 3470 y Fr(2)540 3525 y Fq(\032;)p Fr(0)631 3504 y Fu(\()p FE(t)p Fu(\))19 b(+)f Ft(a)867 3470 y FD(0)890 3504 y Fu(\()p FE(t)p Fu(\))p Fs(j)p FE(f)9 b Fs(j)1080 3470 y Fr(2)1080 3525 y Fq(\014)s(;\032;)p Fr(0)1232 3504 y Fu(\()p FE(t)p Fu(\))19 b(+)1438 3448 y FE(\024)p 1438 3485 48 4 v 1441 3561 a Fu(2)1496 3504 y Fs(j)-9 b(j)g(j)p FE(f)9 b Fs(j)-9 b(j)g(j)1648 3462 y Fr(2)1648 3529 y Fq(\032;)p Fr(0)1739 3504 y Fu(\()p FE(t)p Fu(\))84 b Fk(.)f Fs(j)p FE(f)9 b Fs(j)2161 3470 y Fr(2)2161 3525 y Fq(\032;)p Fr(0)2252 3504 y Fu(\()p FE(t)p Fu(\))p Fs(j)-9 b(j)g(j)p FE(f)9 b Fs(j)-9 b(j)g(j)2498 3529 y Fq(\032;)p Fr(0)2590 3504 y Fu(\()p FE(t)p Fu(\))19 b(+)f FE(t)2816 3470 y FD(\000)2878 3447 y FC(1)p 2878 3456 29 3 v 2878 3490 a(2)2920 3504 y Fs(j)p FE(f)9 b Fs(j)3016 3470 y Fr(2)3016 3525 y Fq(\032;)p Fr(0)3107 3504 y Fu(\()p FE(t)p Fu(\))19 b(+)f Fs(j)p FE(f)9 b Fs(j)3399 3470 y Fr(3)3399 3525 y Fq(\032;)p Fr(0)3490 3504 y Fu(\()p FE(t)p Fu(\))1917 3678 y Fk(.)83 b FE(")p Fs(j)-9 b(j)g(j)p FE(f)9 b Fs(j)-9 b(j)g(j)2256 3636 y Fr(2)2256 3702 y Fq(\032;)p Fr(0)2347 3678 y Fu(\()p FE(t)p Fu(\))19 b(+)f FE(C)2602 3690 y Fq(")2638 3678 y Fs(j)p FE(f)9 b Fs(j)2734 3643 y Fr(4)2734 3698 y Fq(\032;)p Fr(0)2825 3678 y Fu(\()p FE(t)p Fu(\))19 b(+)f FE(t)3051 3643 y FD(\000)3113 3621 y FC(1)p 3113 3630 V 3113 3663 a(2)3155 3678 y Fs(j)p FE(f)9 b Fs(j)3251 3643 y Fr(2)3251 3698 y Fq(\032;)p Fr(0)3342 3678 y Fu(\()p FE(t)p Fu(\))p FE(:)0 3869 y Fu(By)27 b(c)n(ho)r(osing)f(a)i(small)f FE(")g Fu(and)h(the)g(fact)f(that)h(sup)1593 3890 y Fq(\034)7 b FD(2)p Fr(\(0)p Fq(;)p Fr(1\))1831 3869 y Fs(j)p FE(f)i Fs(j)1927 3839 y Fr(2)1927 3890 y Fq(\032;)p Fr(0)2018 3869 y Fu(\()p FE(\034)g Fu(\))25 b Fk(.)d Fu(1)27 b(as)g(giv)n(en)g (in)h(\()p 1 0 0 TeXcolorrgb 2756 3869 a SDict begin H.S end 2756 3869 a 1 0 0 TeXcolorrgb Fu(8.7)p 1 0 0 TeXcolorrgb 2862 3816 a SDict begin H.R end 2862 3816 a 2862 3869 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.8.7) cvn H.B /ANN pdfmark end 2862 3869 a 0 TeXcolorgray Fu(\))q(,)f(w)n(e)h(obtain)f(that)869 4050 y(1)p 869 4087 42 4 v 869 4163 a(2)945 4050 y FE(d)p 930 4087 74 4 v 930 4163 a(dt)1014 4106 y Fs(j)p FE(f)9 b Fs(j)1110 4072 y Fr(2)1110 4126 y Fq(\032;)p Fr(0)1201 4106 y Fu(\()p FE(t)p Fu(\))19 b(+)f Ft(a)1437 4072 y FD(0)1460 4106 y Fu(\()p FE(t)p Fu(\))p Fs(j)p FE(f)9 b Fs(j)1650 4072 y Fr(2)1650 4126 y Fq(\014)s(;\032;)p Fr(0)1802 4106 y Fu(\()p FE(t)p Fu(\))19 b(+)2008 4050 y FE(\024)p 2008 4087 48 4 v 2011 4163 a Fu(4)2066 4106 y Fs(j)-9 b(j)g(j)p FE(f)9 b Fs(j)-9 b(j)g(j)2218 4064 y Fr(2)2218 4131 y Fq(\032;)p Fr(0)2309 4106 y Fu(\()p FE(t)p Fu(\))24 b Fk(.)f Fu(1)p FE(;)96 b(t)23 b Fs(2)g Fu(\(0)p FE(;)14 b Fu(1\))p FE(:)0 4309 y Fu(It)28 b(leads)f(to)1100 4526 y SDict begin H.S end 1100 4526 a 1100 4526 a SDict begin 12 H.A end 1100 4526 a 1100 4526 a SDict begin [/View [/XYZ H.V]/Dest (equation.8.8) cvn /DEST pdfmark end 1100 4526 a -113 x Fp(Z)1183 4434 y Fr(1)1146 4602 y(0)1234 4459 y Fp(\000)1272 4526 y FE(\024)p Fs(j)-9 b(j)g(j)p FE(f)9 b Fs(j)-9 b(j)g(j)1472 4485 y Fr(2)1472 4551 y Fq(\032;)p Fr(0)1563 4526 y Fu(\()p FE(\034)9 b Fu(\))20 b(+)e Ft(a)1815 4492 y FD(0)1838 4526 y Fu(\()p FE(\034)9 b Fu(\))p Fs(j)p FE(f)g Fs(j)2043 4492 y Fr(2)2043 4547 y Fq(\014)s(;\032;)p Fr(0)2196 4526 y Fu(\()p FE(\034)g Fu(\))2305 4459 y Fp(\001)2344 4526 y FE(d\034)33 b Fk(.)23 b Fu(1)p FE(:)p 0 TeXcolorgray 1094 w Fu(\(8.8\))p 0 TeXcolorgray 100 4760 a Fs(\017)28 b Fu(F)-7 b(or)28 b(0)d FE(<)g Fs(j)p FE(\013)p Fs(j)h(\024)e FE(N)9 b Fu(,)29 b(in)n(tegrating)f(o)n(v)n(er)f(\(0)p FE(;)14 b(t)p Fu(\))29 b(with)g FE(t)d Fs(\024)e FE(T)37 b Fs(\024)2144 4728 y Fr(1)p 2144 4742 34 4 v 2144 4789 a(4)2187 4760 y Fu(,)30 b(b)n(y)e(the)i(fact)f(that)g(lim)2962 4772 y Fq(t)p FD(!)p Fr(0)3105 4760 y Fs(j)p FE(f)9 b Fs(j)3201 4772 y Fq(\032;\013)3302 4760 y Fu(\()p FE(t)p Fu(\))25 b(=)g(0)k(as)f(giv)n(en)0 4860 y(in)g(\()p 1 0 0 TeXcolorrgb 129 4860 a SDict begin H.S end 129 4860 a 1 0 0 TeXcolorrgb Fu(8.7)p 1 0 0 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b(j)g(j)997 3400 y Fr(2)997 3467 y Fq(\032;)p Fr(0)1089 3442 y Fu(\()p FE(\034)9 b Fu(\))p FE(d\034)29 b Fu(+)1389 3329 y Fp(Z)1472 3349 y Fq(T)1435 3518 y Fr(0)1538 3375 y Fp(\000)1576 3442 y Fs(j)p FE(f)9 b Fs(j)1672 3408 y Fr(4)1672 3462 y Fq(\032;N)1789 3442 y Fu(\()p FE(\034)g Fu(\))20 b(+)e FE(\034)2046 3408 y FD(\000)2108 3385 y FC(1)p 2108 3394 29 3 v 2108 3428 a(2)2151 3442 y Fs(j)p FE(f)9 b Fs(j)2247 3408 y Fr(2)2247 3462 y Fq(\032;N)2364 3442 y Fu(\()p FE(\034)g Fu(\))2473 3375 y Fp(\001)2512 3442 y FE(d\034)c(:)0 3655 y Fu(By)32 b(taking)f(the)h(supreme)g(o)n(v)n(er)e(0)g FE(<)g Fs(j)p FE(\013)p Fs(j)h(\024)f FE(N)41 b Fu(and)32 b FE(t)e Fs(2)h Fu(\(0)p FE(;)14 b(T)e Fu(\))31 b(on)h(the)g(left)h (hand)f(side)g(and)g(c)n(ho)r(osing)e FE(\032)i Fu(su\016cien)n(tly)0 3755 y(small,)27 b(w)n(e)h(ha)n(v)n(e)73 3972 y SDict begin H.S end 73 3972 a 73 3972 a SDict begin 12 H.A end 73 3972 a 73 3972 a SDict begin [/View [/XYZ H.V]/Dest (equation.8.9) cvn /DEST pdfmark end 73 3972 a 209 w Fu(sup)239 4041 y Fr(0)p Fq(<t<T)463 3972 y Fs(j)p FE(f)9 b Fs(j)559 3937 y Fr(2)559 3992 y Fq(\032;N)676 3972 y Fu(\()p FE(t)p Fu(\))19 b(+)872 3855 y Fp(\022)943 3915 y FE(\024)p 943 3953 48 4 v 946 4029 a Fu(2)1019 3972 y Fs(\000)f FE(C)1161 3984 y Fr(1)1262 3972 y Fu(sup)1213 4041 y Fr(0)p Fq(<\034)7 b(<T)1449 3972 y Fs(j)p FE(f)i Fs(j)1545 3984 y Fq(\032;N)1662 3972 y Fu(\()p FE(\034)g Fu(\))1771 3855 y Fp(\023)1924 3972 y Fu(sup)1847 4045 y Fr(0)p Fq(<)p FD(j)p Fq(\013)p FD(j\024)p Fq(N)2139 3859 y Fp(Z)2222 3879 y Fq(T)2185 4047 y Fr(0)2288 3972 y Fs(j)-9 b(j)g(j)p FE(f)9 b Fs(j)-9 b(j)g(j)2441 3930 y Fr(2)2441 3997 y Fq(\032;\013)2542 3972 y Fu(\()p FE(\034)9 b Fu(\))p FE(d\034)34 b Fs(\024)23 b FE(C)2911 3984 y Fr(2)2962 3855 y Fp(\022)3024 3972 y Fu(1)17 b(+)67 b(sup)3166 4041 y Fr(0)p Fq(<\034)7 b(<T)3403 3972 y Fs(j)p FE(f)i Fs(j)3499 3984 y Fq(\032;N)3616 3972 y Fu(\()p FE(\034)g Fu(\))3725 3855 y Fp(\023)239 4233 y Fs(\002)318 4120 y Fp(Z)401 4141 y Fq(T)364 4309 y 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(section.9) cvn /DEST pdfmark end 0 1241 a 1691 1341 a Fu(9.)41 b Fv(Appendix)100 1490 y Fu(In)27 b(the)h(App)r(endix,)h(w)n(e)e(giv)n (e)g(some)g(useful)h(lemmas)f(on)g(pseudo-Di\013eren)n(tial)g(op)r (erator)f(and)i(basic)f(comm)n(utators.)0 1525 y SDict begin H.S end 0 1525 a 0 1525 a SDict begin 12 H.A end 0 1525 a 0 1525 a SDict begin [/View [/XYZ H.V]/Dest (lem.9.1) cvn /DEST pdfmark end 0 1525 a 0 TeXcolorgray 100 x FF(Lemma)j(9.1)g Fu(\(see)e([)p 0 1 0 TeXcolorrgb 682 1625 a SDict begin H.S end 682 1625 a 0 1 0 TeXcolorrgb Fu(23)p 0 1 0 TeXcolorrgb 765 1571 a SDict begin H.R end 765 1571 a 765 1625 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.HE) cvn H.B /ANN pdfmark end 765 1625 a 0 TeXcolorgray Fu(]\))p FF(.)p 0 TeXcolorgray 41 w Fj(L)l(et)i FE(s;)14 b(r)25 b Fs(2)f Fo(R)35 b Fj(and)c FE(a)p Fu(\()p FE(v)s Fu(\))p FE(;)14 b(b)p Fu(\()p FE(\030)t Fu(\))24 b Fs(2)f FE(C)1995 1595 y FD(1)2095 1625 y Fj(satisfy)31 b(for)g(any)f FE(\013)23 b Fs(2)h Fo(Z)2862 1595 y Fr(3)2862 1645 y(+)2911 1625 y Fj(,)963 1773 y SDict begin H.S end 963 1773 a 963 1773 a SDict begin 12 H.A end 963 1773 a 963 1773 a SDict begin [/View [/XYZ H.V]/Dest (equation.9.1) cvn /DEST pdfmark end 963 1773 a Fs(j)p FE(D)1057 1739 y Fq(\013)1055 1794 y(v)1105 1773 y FE(a)p Fu(\()p FE(v)s Fu(\))p Fs(j)f(\024)g FE(C)1449 1785 y Fr(1)p Fq(;\013)1549 1773 y Fs(h)p FE(v)s Fs(i)1656 1739 y Fq(r)r FD(\000j)p Fq(\013)p FD(j)1829 1773 y FE(;)43 b Fs(j)p FE(D)1989 1739 y Fq(\013)1987 1794 y(\030)2037 1773 y FE(b)p Fu(\()p FE(\030)t Fu(\))p Fs(j)23 b(\024)g FE(C)2370 1785 y Fr(2)p Fq(;\013)2470 1773 y Fs(h)p FE(\030)t Fs(i)2574 1739 y Fq(s)p FD(\000j)p Fq(\013)p FD(j)p 0 TeXcolorgray 3703 1773 a Fu(\(9.1\))p 0 TeXcolorgray 0 1908 a Fj(for)42 b(c)l(onstants)e FE(C)579 1920 y Fr(1)p Fq(;\013)680 1908 y FE(;)14 b(C)776 1920 y Fr(2)p Fq(;\013)876 1908 y Fj(.)73 b(Then)42 b(ther)l(e)f(exists)f(a)i(c)l(onstant)e FE(C)48 b Fj(dep)l(ending)42 b(only)g(on)f FE(s;)14 b(r)44 b Fj(and)d(\014nite)g(numb)l(ers)f(of)0 2014 y FE(C)59 2026 y Fr(1)p Fq(;\013)159 2014 y FE(;)14 b(C)255 2026 y Fr(2)p Fq(;\013)386 2014 y Fj(such)29 b(that)h(for)h(any)f FE(f)h Fs(2)24 b Fi(S)16 b Fu(\()p Fo(R)1367 1977 y Fr(3)1404 2014 y Fu(\))p Fj(,)565 2149 y Fs(k)p FE(a)p Fu(\()p FE(v)s Fu(\))p FE(b)p Fu(\()p FE(D)r Fu(\))p FE(f)9 b Fs(k)1021 2164 y Fq(L)1067 2147 y FC(2)1126 2149 y Fs(\024)23 b FE(C)6 b Fs(kh)p FE(D)r Fs(i)1456 2114 y Fq(s)1491 2149 y Fs(h)p FE(v)s Fs(i)1598 2114 y Fq(r)1636 2149 y FE(f)j Fs(k)1728 2164 y Fq(L)1774 2147 y FC(2)1809 2149 y FE(;)44 b Fs(k)p FE(b)p Fu(\()p FE(D)r Fu(\))p FE(a)p Fu(\()p FE(v)s Fu(\))p FE(f)9 b Fs(k)2332 2164 y Fq(L)2378 2147 y FC(2)2436 2149 y Fs(\024)23 b FE(C)6 b Fs(kh)p FE(v)s Fs(i)2738 2114 y Fq(r)2775 2149 y Fs(h)p FE(D)r Fs(i)2910 2114 y Fq(s)2947 2149 y FE(f)j Fs(k)3039 2164 y Fq(L)3085 2147 y FC(2)3120 2149 y FE(:)0 2283 y Fj(As)29 b(a)h(dir)l(e)l(ct)h(c)l(onse)l(quenc)l(e,)e(we)h(get)g (that)g Fs(kh)p FE(D)r Fs(i)1515 2253 y Fq(m)1578 2283 y Fs(h)p FE(v)s Fs(i)1685 2253 y Fq(l)1711 2283 y FE(f)9 b Fs(k)1803 2298 y Fq(L)1849 2281 y FC(2)1908 2283 y Fs(\030)22 b(kh)p FE(v)s Fs(i)2144 2253 y Fq(l)2170 2283 y Fs(h)p FE(D)r Fs(i)2305 2253 y Fq(m)2369 2283 y FE(f)9 b Fs(k)2461 2298 y Fq(L)2507 2281 y FC(2)2565 2283 y Fs(\030)23 b(k)p FE(f)9 b Fs(k)2787 2295 y Fq(H)2845 2275 y Fl(m)2841 2316 y(l)2903 2283 y Fj(.)0 2330 y SDict begin H.S end 0 2330 a 0 2330 a SDict begin 12 H.A end 0 2330 a 0 2330 a SDict begin [/View [/XYZ H.V]/Dest (lem.9.2) cvn /DEST pdfmark end 0 2330 a 0 TeXcolorgray 99 x FF(Lemma)34 b(9.2)h Fu(\(see)d([)p 0 1 0 TeXcolorrgb 695 2429 a SDict begin H.S end 695 2429 a 0 1 0 TeXcolorrgb Fu(5)p 0 1 0 TeXcolorrgb 736 2376 a SDict begin H.R end 736 2376 a 736 2429 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.HJ1) cvn H.B /ANN pdfmark end 736 2429 a 0 TeXcolorgray Fu(]\))p FF(.)p 0 TeXcolorgray 44 w Fj(L)l(et)h FE(l)r(;)14 b(s;)g(r)31 b Fs(2)f Fo(R)6 b FE(;)14 b(M)9 b Fu(\()p FE(\030)t Fu(\))30 b Fs(2)g FE(S)1764 2399 y Fq(r)1759 2450 y Fr(1)p Fq(;)p Fr(0)1882 2429 y Fj(and)k Fu(\010\()p FE(v)s Fu(\))c Fs(2)g FE(S)2385 2399 y Fq(l)2380 2450 y Fr(1)p Fq(;)p Fr(0)2470 2429 y Fj(.)50 b(Then)34 b(ther)l(e)f(exists)g(a)h(c)l(onstant)e FE(C)40 b Fj(such)0 2532 y(that)30 b Fs(k)p Fu([)p FE(M)9 b Fu(\()p FE(D)426 2544 y Fq(v)464 2532 y Fu(\))p FE(;)14 b Fu(\010\()p FE(v)s Fu(\)])p FE(f)9 b Fs(k)815 2544 y Fq(H)873 2528 y Fl(s)933 2532 y Fs(\024)23 b FE(C)6 b Fs(k)p FE(f)j Fs(k)1220 2558 y Fq(H)1278 2531 y Fl(r)q FC(+)p Fl(s)p Fm(\000)p FC(1)1274 2578 y Fl(l)p Fm(\000)p FC(1)1459 2532 y Fj(.)39 b(Mor)l(e)l(over,)31 b(for)g(any)f FE(N)i Fs(2)23 b Fo(N)10 b FE(;)784 2738 y SDict begin H.S end 784 2738 a 784 2738 a SDict begin 12 H.A end 784 2738 a 784 2738 a SDict begin [/View [/XYZ H.V]/Dest (equation.9.2) cvn /DEST pdfmark end 784 2738 a FE(M)f Fu(\()p FE(D)975 2750 y Fq(v)1014 2738 y Fu(\)\010)23 b(=)g(\010)p FE(M)9 b Fu(\()p FE(D)1468 2750 y Fq(v)1507 2738 y Fu(\))19 b(+)1720 2659 y Fp(X)1641 2841 y Fr(1)p FD(\024j)p Fq(\013)p FD(j\024)p Fq(N)1960 2681 y Fu(1)p 1943 2718 77 4 v 1943 2795 a FE(\013)p Fu(!)2029 2738 y(\010)2089 2750 y Fq(\013)2137 2738 y FE(M)2227 2703 y Fq(\013)2273 2738 y Fu(\()p FE(D)2374 2750 y Fq(v)2414 2738 y Fu(\))g(+)f FE(r)2585 2750 y Fq(N)2648 2738 y Fu(\()p FE(v)s(;)c(D)2829 2750 y Fq(v)2869 2738 y Fu(\))p FE(;)p 0 TeXcolorgray 779 w Fu(\(9.2\))p 0 TeXcolorgray 0 2982 a Fj(wher)l(e)26 b Fu(\010)290 2994 y Fq(\013)338 2982 y Fu(\()p FE(v)s Fu(\))d(=)g FE(@)605 2952 y Fq(\013)600 3002 y(v)652 2982 y Fu(\010)p FE(;)40 b(M)865 2952 y Fq(\013)912 2982 y Fu(\()p FE(\030)t Fu(\))23 b(=)g FE(@)1176 2952 y Fq(\013)1171 3005 y(\030)1223 2982 y FE(M)9 b Fu(\()p FE(\030)t Fu(\))26 b Fj(and)g Fs(h)p FE(v)s Fs(i)1707 2952 y Fq(N)6 b FD(\000)p Fq(l)1844 2982 y FE(r)1881 2994 y Fq(N)1944 2982 y Fu(\()p FE(v)s(;)14 b(\030)t Fu(\))24 b Fs(2)g FE(S)2287 2946 y Fq(r)r FD(\000)p Fq(N)2282 3004 y Fr(1)p Fq(;)p Fr(0)2434 2982 y Fj(.)37 b(Mor)l(e)l(over,)29 b(for)d(any)g FE(\014)t(;)14 b(\014)3306 2952 y FD(0)3353 2982 y Fs(2)23 b Fo(Z)3493 2952 y Fr(3)3493 3002 y(+)3542 2982 y Fj(,)k(we)f(have)249 3148 y SDict begin H.S end 249 3148 a 249 3148 a SDict begin 12 H.A end 249 3148 a 249 3148 a SDict begin [/View [/XYZ H.V]/Dest (equation.9.3) cvn /DEST pdfmark end 249 3148 a Fs(j)p FE(@)321 3114 y Fq(\014)316 3168 y(v)365 3148 y FE(@)414 3108 y Fq(\014)455 3083 y Fm(0)409 3173 y Fq(\030)481 3148 y FE(r)518 3160 y Fq(N)581 3148 y Fu(\()p FE(v)s(;)14 b(\030)t Fu(\))p Fs(j)24 b(\024)f FE(C)959 3160 y Fq(\014)s(;\014)1061 3144 y Fm(0)1087 3148 y Fs(h)p FE(\030)t Fs(i)1191 3114 y Fq(r)r FD(\000)p Fq(N)6 b FD(\000j)p Fq(\014)s FD(j)1471 3148 y Fs(h)p FE(v)s Fs(i)1578 3114 y Fq(l)p FD(\000)p Fq(N)g FD(\000j)p Fq(\014)1823 3089 y Fm(0)1846 3114 y FD(j)1869 3148 y FE(;)99 b Fs(k)p FE(r)2070 3160 y Fr(2)p Fq(N)6 b Fr(+1)2250 3148 y Fu(\()p FE(v)s(;)14 b(D)2431 3160 y Fq(v)2471 3148 y Fu(\))p Fs(h)p FE(D)r Fs(i)2638 3114 y Fq(N)2702 3148 y Fs(h)p FE(v)s Fs(i)2809 3114 y Fq(N)2873 3148 y FE(f)9 b Fs(k)2965 3163 y Fq(L)3011 3146 y FC(2)3069 3148 y Fs(\024)22 b FE(C)6 b Fs(k)p FE(f)j Fs(k)3355 3163 y Fq(L)3401 3146 y FC(2)3436 3148 y FE(:)p 0 TeXcolorgray 244 w Fu(\(9.3\))p 0 TeXcolorgray 0 3290 a Fj(F)-6 b(urthermor)l(e,)30 b(using)g(\()p 1 0 0 TeXcolorrgb 758 3290 a SDict begin H.S end 758 3290 a 1 0 0 TeXcolorrgb Fj(9.2)p 1 0 0 TeXcolorrgb 868 3236 a SDict begin H.R end 868 3236 a 868 3290 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.9.2) cvn H.B /ANN pdfmark end 868 3290 a 0 TeXcolorgray Fj(\))g(r)l(ep)l(e)l(ate)l(d)t(ly,)h(we)f(c)l(an) g(also)g(obtain)h(that)718 3441 y SDict begin H.S end 718 3441 a 718 3441 a SDict begin 12 H.A end 718 3441 a 718 3441 a SDict begin [/View [/XYZ H.V]/Dest (equation.9.4) cvn /DEST pdfmark end 718 3441 a FE(M)9 b Fu(\()p FE(D)909 3453 y Fq(v)948 3441 y Fu(\)\010)23 b(=)g(\010)p FE(M)9 b Fu(\()p FE(D)1402 3453 y Fq(v)1441 3441 y Fu(\))18 b(+)1654 3362 y Fp(X)1574 3544 y Fr(1)p FD(\024j)p Fq(\013)p FD(j\024)p Fq(N)1867 3441 y FE(C)1926 3453 y Fq(\013)1973 3441 y FE(M)2063 3407 y Fq(\013)2110 3441 y Fu(\()p FE(D)2211 3453 y Fq(v)2251 3441 y Fu(\)\010)2343 3453 y Fq(\013)2409 3441 y Fu(+)g FE(C)2551 3453 y Fq(N)2614 3441 y FE(r)2651 3453 y Fq(N)2714 3441 y Fu(\()p FE(v)s(;)c(D)2895 3453 y Fq(v)2935 3441 y Fu(\))p FE(:)p 0 TeXcolorgray 713 w Fu(\(9.4\))p 0 TeXcolorgray 0 3568 a SDict begin H.S end 0 3568 a 0 3568 a SDict begin 12 H.A end 0 3568 a 0 3568 a SDict begin [/View [/XYZ H.V]/Dest (rmk.9.1) cvn /DEST pdfmark end 0 3568 a 0 TeXcolorgray 100 x FF(Remark)24 b(9.1.)p 0 TeXcolorgray 35 w Fj(We)g(emphasize)i(that)e(in)g(the)h (statement)e(of)i(L)l(emma)p 1 0 0 TeXcolorrgb 2222 3668 a SDict begin H.S end 2222 3668 a 1 0 0 TeXcolorrgb Fj(9.2)p 1 0 0 TeXcolorrgb 2333 3614 a SDict begin H.R end 2333 3614 a 2333 3668 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (lem.9.2) cvn H.B /ANN pdfmark end 2333 3668 a 0 TeXcolorgray Fj(,)g(the)g(c)l(onstant)e FE(C)31 b Fj(app)l(e)l(aring)26 b(in)e(the)g(ine)l(quality)0 3767 y(dep)l(ends)32 b(only)f(on)g FE(C)671 3779 y Fr(1)p Fq(;\013)771 3767 y FE(;)14 b(C)867 3779 y Fr(2)p Fq(;\013)998 3767 y Fj(in)38 b Fu(\()p 1 0 0 TeXcolorrgb 1140 3767 a SDict begin H.S end 1140 3767 a 1 0 0 TeXcolorrgb Fu(9.1)p 1 0 0 TeXcolorrgb 1246 3714 a SDict begin H.R end 1246 3714 a 1246 3767 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.9.1) cvn H.B /ANN pdfmark end 1246 3767 a 0 TeXcolorgray Fu(\))31 b Fj(with)g FE(a)25 b Fu(=)g(\010)30 b Fj(and)i FE(b)24 b Fu(=)g FE(M)40 b Fj(and)31 b(also)h(the)f(c)l(onstants)f FE(C)3069 3779 y Fq(\014)s(;\014)3171 3763 y Fm(0)3227 3767 y Fj(for)i FE(r)3398 3779 y Fq(N)3461 3767 y Fu(\()p FE(v)s(;)14 b(\030)t Fu(\))p Fj(.)43 b(This)0 3867 y(fact)33 b(is)g(crucial)h(for)f (the)g(estimates)g(of)g(c)l(ommutators)f(and)h(the)g(pr)l(o\014les)g (of)h(weighte)l(d)f(Sob)l(olev)h(sp)l(ac)l(es.)48 b(F)-6 b(or)33 b(instanc)l(e,)0 3966 y(if)f FE(M)9 b Fu(\()p FE(D)273 3978 y Fq(v)312 3966 y Fu(\))31 b Fj(and)h Fu(\010\()p FE(v)s Fu(\))f Fj(ar)l(e)h(chosen)f(to)g(b)l(e)g(the)g(lo)l(c)l(alize)l (d)i(op)l(er)l(ators)f Ff(F)2231 3978 y Fq(j)2297 3966 y Fj(and)f Fs(P)2517 3978 y Fq(k)2558 3966 y Fj(,)g(the)g(c)l(onstant)g FE(C)37 b Fj(in)31 b(L)l(emma)p 1 0 0 TeXcolorrgb 3580 3966 a SDict begin H.S end 3580 3966 a 1 0 0 TeXcolorrgb Fj(9.2)p 1 0 0 TeXcolorrgb 3690 3913 a SDict begin H.R end 3690 3913 a 3690 3966 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (lem.9.2) cvn H.B /ANN pdfmark end 3690 3966 a 0 TeXcolorgray 31 w Fj(do)l(es)0 4069 y(not)e(dep)l(end)i(on)f FE(j)35 b Fj(and)30 b FE(k)s Fj(.)38 b(Inde)l(e)l(d,)31 b(for)g(any)f FE(k)c Fs(\025)c Fu(0)p FE(;)14 b(N)31 b Fs(2)24 b Fo(N)10 b Fj(,)30 b Fu(2)2028 4039 y Fq(N)6 b(k)2127 4069 y FE(')p Fu(\(2)2255 4039 y FD(\000)p Fq(k)2348 4069 y FE(v)s Fu(\))30 b Fj(satis\014es)g(that)g(for)g(any)h FE(\013)23 b Fs(2)g Fo(Z)3443 4039 y Fr(3)3443 4089 y(+)3492 4069 y Fj(,)749 4217 y SDict begin H.S end 749 4217 a 749 4217 a SDict begin 12 H.A end 749 4217 a 749 4217 a SDict begin [/View [/XYZ H.V]/Dest (equation.9.5) cvn /DEST pdfmark end 749 4217 a Fs(j)p FE(D)843 4183 y Fq(\013)841 4238 y(v)890 4217 y Fu(2)932 4183 y Fq(N)6 b(k)1032 4217 y FE(')p Fu(\(2)1160 4183 y FD(\000)p Fq(k)1252 4217 y FE(v)s Fu(\))p Fs(j)24 b(\024)f FE(C)1521 4229 y Fq(N)s(;\013)1643 4217 y Fs(h)p FE(v)s Fs(i)1750 4183 y Fq(N)6 b FD(\000j)p Fq(\013)p FD(j)1949 4217 y Fs(j)p FE(')2026 4229 y Fq(\013)2073 4217 y Fu(\(2)2147 4183 y FD(\000)p Fq(k)2240 4217 y FE(v)s Fu(\))p Fs(j)23 b(\024)g FE(C)2508 4229 y Fq(N)s(;\013)2631 4217 y Fs(h)p FE(v)s Fs(i)2738 4183 y Fq(N)6 b FD(\000j)p Fq(\013)p FD(j)2936 4217 y FE(:)p 0 TeXcolorgray 744 w Fu(\(9.5\))p 0 TeXcolorgray 0 4252 a SDict begin H.S end 0 4252 a 0 4252 a SDict begin 12 H.A end 0 4252 a 0 4252 a SDict begin [/View [/XYZ H.V]/Dest (lem.9.3) cvn /DEST pdfmark end 0 4252 a 0 TeXcolorgray 100 x FF(Lemma)30 b(9.3.)p 0 TeXcolorgray 40 w Fj(\(Bernstein)f(ine)l(quality\).)40 b(Ther)l(e)30 b(exists)g(a)g(c)l(onstant)f FE(C)36 b Fj(indep)l(endent)30 b(of)h FE(j)k Fj(and)30 b FE(f)38 b Fj(such)30 b(that)100 4452 y(\(1\))g(F)-6 b(or)30 b(any)g FE(s)23 b Fs(2)g Fo(R)36 b Fj(and)30 b FE(j)e Fs(\025)23 b Fu(0)p Fj(,)868 4588 y FE(C)933 4554 y FD(\000)p Fr(1)1022 4588 y Fu(2)1064 4554 y Fq(j)s(s)1130 4588 y Fs(k)p Ff(F)1223 4600 y Fq(j)1257 4588 y FE(f)9 b Fs(k)1349 4606 y Fq(L)1395 4589 y FC(2)1426 4606 y Fr(\()p Fn(R)1494 4586 y FC(3)1526 4606 y Fr(\))1580 4588 y Fs(\024)22 b(k)p Ff(F)1760 4600 y Fq(j)1794 4588 y FE(f)9 b Fs(k)1886 4606 y Fq(H)1944 4589 y Fl(s)1976 4606 y Fr(\()p Fn(R)2044 4586 y FC(3)2077 4606 y Fr(\))2130 4588 y Fs(\024)22 b FE(C)6 b Fu(2)2324 4554 y Fq(j)s(s)2391 4588 y Fs(k)p Ff(F)2484 4600 y Fq(j)2518 4588 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begin [/View [/XYZ H.V]/Dest (equation.9.7) cvn /DEST pdfmark end 163 2668 a 216 2564 a FD(1)189 2589 y Fp(X)163 2768 y Fq(k)q Fr(=)p FD(\000)p Fr(1)350 2668 y Fu(2)392 2634 y Fr(2)p Fq(k)q(l)486 2668 y Fs(k)p FF(P)593 2680 y Fq(k)634 2668 y FE(f)9 b Fs(k)726 2634 y Fr(2)726 2689 y Fq(H)784 2672 y Fl(m)866 2668 y Fs(\024)23 b FE(C)1013 2680 y Fq(m;l)1184 2564 y FD(1)1157 2589 y Fp(X)1131 2768 y Fq(k)q Fr(=)p FD(\000)p Fr(1)1318 2668 y Fu(2)1360 2634 y Fr(2)p Fq(k)q(l)1454 2668 y Fs(kP)1554 2680 y Fq(k)1594 2668 y FE(f)9 b Fs(k)1686 2634 y Fr(2)1686 2689 y Fq(H)1744 2672 y Fl(m)1804 2668 y FE(;)1976 2564 y FD(1)1949 2589 y Fp(X)1925 2766 y Fq(j)s Fr(=)p FD(\000)p Fr(1)2106 2668 y Fu(2)2148 2634 y Fr(2)p Fq(j)s(m)2275 2668 y Fs(k)p FF(F)2377 2680 y Fq(j)2411 2668 y FE(f)g Fs(k)2503 2634 y Fr(2)2503 2694 y Fq(L)2549 2674 y FC(2)2549 2714 y Fl(l)2607 2668 y Fs(\024)23 b FE(C)2754 2680 y Fq(m;l)2923 2564 y FD(1)2896 2589 y Fp(X)2872 2766 y Fq(j)s Fr(=)p FD(\000)p Fr(1)3053 2668 y Fu(2)3095 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