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Fn(gener)-5 b(alize)g(d)42 b(Har)-5 b(dy)44 b(sp)-5 b(ac)g(e)49 b Fu(H)3006 1007 y Fy(p)2998 1081 y(E)3057 1054 y FB(\()p Fu(G)p FB(\))42 b(is)g(the)257 1175 y(closed)33 b(subspace)i(of)d Fu(L)1130 1139 y Fy(p)1170 1175 y FB(\()p Fu(G)p FB(\))g(de\014ned)i(as)f(follo)m(ws)960 1388 y Fu(H)1049 1340 y Fy(p)1041 1415 y(E)1101 1388 y FB(\()p Fu(G)p FB(\))27 b(=)h Fp(f)p Fu(f)38 b Fp(2)28 b Fu(L)1681 1346 y Fy(p)1721 1388 y FB(\()p Fu(G)p FB(\))f(:)1974 1361 y Fm(b)1956 1388 y Fu(f)11 b FB(\()p Fu(\037)p FB(\))28 b(=)f(0)32 b Fp(8)p Fu(\037)d Fp(2)f Fu(E)2681 1346 y Fy(c)2715 1388 y Fp(g)p Fu(:)404 1589 y FB(The)36 b(case)h(where)g Fu(G)e FB(is)g(connected)i(and)f Fu(E)j FB(=)33 b Fu(X)2267 1604 y Fx(+)2361 1589 y FB(is)i(due)i(to)e(Helson)h(and)f(Lo)m(w-)257 1710 y(denslager)i(\(see,)j(e.g.,)e([)1157 1710 y SDict begin H.S end 1157 1710 a FB(27)1255 1645 y SDict begin H.R end 1255 1645 a 1255 1710 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype 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y FB([)972 3728 y SDict begin H.S end 972 3728 a FB(24)1069 3664 y SDict begin H.R end 1069 3664 a 1069 3728 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.ijpam) cvn H.B /ANN pdfmark end 1069 3728 a FB(].)404 3848 y(Note)33 b(also)f(that)h(the) g(Hilb)s(ert)e(transform)h(is)h(a)f(con)m(tin)m(uous)i(map)e(from)g Fu(L)3183 3812 y Fx(1)3223 3848 y FB(\()p Fu(G)p FB(\))h(to)257 3969 y Fu(L)323 3932 y Fy(p)363 3969 y FB(\()p Fu(G)p FB(\))g(for)f(0)27 b Fu(<)h(p)f(<)h FB(1)k(\(see,)i(e.)f(g.,)f([)1627 3969 y SDict begin H.S end 1627 3969 a FB(27)1725 3904 y SDict begin H.R end 1725 3904 a 1725 3969 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Rud) cvn H.B /ANN pdfmark end 1725 3969 a FB(,)h(Theorem)g(8.7.6]\).)404 4089 y FC(De\014nition)k(2)c FB([)1042 4089 y SDict begin H.S end 1042 4089 a FB(9)1090 4025 y SDict begin H.R end 1090 4025 a 1090 4089 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Trudy) cvn H.B /ANN pdfmark end 1090 4089 a FB(].)44 b(W)-8 b(e)32 b(de\014ne)i(the)f(space)g Fu(B)5 b(M)10 b(O)s FB(\()p Fu(G)p FB(\))32 b(of)g Fn(functions)i(of)g(b)-5 b(ounde)g(d)257 4209 y(me)g(an)34 b(oscil)5 b(lation)34 b(on)h Fu(G)d FB(and)h(its)f(subspace)i Fu(B)5 b(M)10 b(O)s(A)p FB(\()p Fu(G)p FB(\),)33 b(as)g(follo)m(ws)1089 4411 y Fu(B)5 b(M)10 b(O)s FB(\()p Fu(G)p FB(\))28 b(:=)g Fp(f)p Fu(f)k FB(+)23 b Fm(e)-56 b Fu(g)31 b FB(:)d Fu(f)5 b(;)17 b(g)31 b Fp(2)d Fu(L)2358 4370 y Fs(1)2433 4411 y FB(\()p Fu(G)p FB(\))p Fp(g)p Fu(;)1137 4613 y(B)5 b(M)10 b(O)s(A)p FB(\()p Fu(G)p FB(\))28 b(:=)g Fu(B)5 b(M)10 b(O)s FB(\()p Fu(G)p FB(\))22 b Fp(\\)g Fu(H)2396 4572 y Fx(1)2435 4613 y FB(\()p Fu(G)p FB(\))p Fu(;)665 4779 y Fp(k)p Fu(')p Fp(k)829 4794 y Fy(B)s(M)7 b(O)1048 4779 y FB(:=)27 b(inf)6 b Fp(fk)p Fu(f)11 b Fp(k)1505 4794 y Fs(1)1601 4779 y FB(+)22 b Fp(k)p Fu(g)t Fp(k)1850 4794 y 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b(O)1666 1237 y Fp(\024)28 b(k)p Fu(')p Fp(k)1935 1252 y Fs(\003)2002 1237 y Fp(\024)g FB(2)p Fp(k)p Fu(')p Fp(k)2320 1252 y Fy(B)s(M)7 b(O)2510 1237 y Fu(:)404 1480 y FC(De\014nition)50 b(3)p FB(.)75 b([)1124 1480 y SDict begin H.S end 1124 1480 a FB(21)1222 1416 y SDict begin H.R end 1222 1416 a 1222 1480 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Indag) cvn H.B /ANN pdfmark end 1222 1480 a FB(])43 b(W)-8 b(e)44 b(de\014ne)g(the)g (space)g Fu(H)2302 1444 y Fx(1)2294 1506 y Fw(R)2346 1480 y FB(\()p Fu(G)p FB(\))f(\(the)h Fn(r)-5 b(e)g(al)44 b Fu(H)3043 1444 y Fx(1)3127 1480 y Fn(sp)-5 b(ac)g(e)44 b(on)257 1601 y Fu(G)p FB(\))j(as)g(the)g(completion)e(of)h(the)i (space)g(P)m(ol)o(\()p Fu(G;)17 b Ft(R)5 b FB(\))52 b(of)47 b(real-v)-5 b(alued)45 b(trigonometric)257 1721 y(p)s(olynomials)30 b(on)i Fu(G)h FB(with)f(resp)s(ect)i(to)e(the)h(norm)1305 1929 y Fp(k)p Fu(q)t Fp(k)1452 1944 y Fx(1)p Fs(\003)1554 1929 y FB(:=)28 b 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Fn(Pr)-5 b(oje)g(ctors)38 b Fu(P)2439 2514 y Fs(\006)2498 2499 y Fn(,)h(and)e(the)h(Hilb)-5 b(ert)38 b(tr)-5 b(ans-)257 2619 y(form)35 b(ar)-5 b(e)34 b(b)-5 b(ounde)g(d)34 b(op)-5 b(er)g(ators)35 b(on)74 b Fu(H)1721 2583 y Fx(1)1713 2645 y Fw(R)1765 2619 y FB(\()p Fu(G)p FB(\);)404 2740 y(\(ii\))22 b Fn(r)-5 b(estrictions)28 b Fu(P)1125 2755 y Fs(\006)1184 2740 y Fp(j)p FB(P)m(ol)o(\()p Fu(G;)17 b Ft(R)5 b FB(\))33 b Fn(extend)27 b(to)h(b)-5 b(ounde)g(d)27 b(op)-5 b(er)g(ators)28 b Fu(P)2914 2704 y Fx(1)2900 2764 y Fs(\006)2986 2740 y Fn(fr)-5 b(om)27 b Fu(H)3298 2704 y Fx(1)3290 2765 y Fw(R)3342 2740 y FB(\()p Fu(G)p FB(\))257 2860 y Fn(to)35 b Fu(L)440 2824 y Fx(1)480 2860 y FB(\()p Fu(G)p FB(\))g Fn(and)532 3068 y Fp(k)p Fu(f)11 b Fp(k)691 3083 y Fx(1)p Fs(\003)792 3068 y FB(=)28 b Fp(k)p Fu(P)1009 3083 y Fs(\000)1068 3068 y Fu(f)11 b Fp(k)1177 3083 y Fx(1)p Fs(\003)1273 3068 y FB(+)22 b Fp(k)p Fu(P)1484 3083 y Fx(+)1543 3068 y Fu(f)11 b Fp(k)1652 3083 y Fx(1)p Fs(\003)1754 3068 y FB(=)27 b Fp(k)p Fu(P)1984 3027 y Fx(1)1970 3093 y Fs(\000)2029 3068 y Fu(f)11 b Fp(k)2138 3083 y Fx(1)2199 3068 y FB(+)22 b Fp(k)p Fu(P)2424 3027 y Fx(1)2410 3093 y(+)2469 3068 y Fu(f)11 b Fp(k)2578 3083 y Fx(1)2652 3068 y FB(\()p Fu(f)38 b Fp(2)28 b Fu(H)2959 3027 y Fx(1)2951 3093 y Fw(R)3003 3068 y FB(\()p Fu(G)p FB(\)\);)404 3397 y(\(iii\))h Fu(H)682 3361 y Fx(1)674 3423 y Fw(R)726 3397 y FB(\()p Fu(G)p FB(\))f(=)f(Im)p Fu(P)1190 3412 y Fs(\000)1271 3397 y Fk(u)c FB(Im)o Fu(P)1550 3412 y Fx(+)1641 3397 y FB(\()p Fn(the)35 b(dir)-5 b(e)g(ct)35 b(sum)g(of)f(close)-5 b(d)34 b(subsp)-5 b(ac)g(es)8 b FB(\);)404 3518 y(\(iv\))32 b Fp([)657 3533 y Fy(p>)p Fx(1)787 3518 y Fu(L)853 3481 y Fy(p)893 3518 y FB(\()p Fu(G;)17 b Ft(R)5 b FB(\))33 b Fp(\032)28 b Fu(H)1383 3481 y Fx(1)1375 3543 y Fw(R)1427 3518 y FB(\()p Fu(G)p FB(\))g Fp(\032)g Fu(L)1779 3481 y Fx(1)1819 3518 y FB(\()p Fu(G;)17 b Ft(R)t FB(\);)404 3652 y(\(v\))33 b Fp(k)p Fu(f)11 b Fp(k)723 3616 y Fs(\030)809 3652 y FB(:=)27 b Fp(k)p Fu(f)11 b Fp(k)1098 3667 y Fx(1)1159 3652 y FB(+)22 b Fp(k)1325 3626 y Fm(e)1307 3652 y Fu(f)11 b Fp(k)1416 3667 y Fx(1)1488 3652 y Fn(is)34 b(an)h(e)-5 b(quivalent)34 b(norm)g(in)40 b Fu(H)2659 3616 y Fx(1)2651 3677 y Fw(R)2703 3652 y FB(\()p Fu(G)p FB(\);)404 3772 y(\(vi\))32 b Fu(H)680 3736 y Fx(1)672 3798 y Fw(R)724 3772 y FB(\()p Fu(G)p FB(\))27 b(=)h(Re)p Fu(H)1212 3736 y Fx(1)1251 3772 y FB(\()p Fu(G)p FB(\).)404 3893 y(In)33 b([)553 3893 y SDict begin H.S end 553 3893 a FB(19)650 3829 y SDict begin H.R end 650 3829 a 650 3893 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.JMAA) cvn H.B /ANN pdfmark end 650 3893 a FB(])g(the)g(next)g(de\014nition)f(w)m(as)h(prop) s(osed.)404 4013 y FC(De\014nition)h(4)29 b FB([)1035 4013 y SDict begin H.S end 1035 4013 a FB(19)1133 3949 y SDict begin H.R end 1133 3949 a 1133 4013 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.JMAA) cvn H.B /ANN pdfmark end 1133 4013 a FB(].)42 b(Let)30 b(\(\012)p Fu(;)17 b(\026)p FB(\))29 b(b)s(e)g(a)g(measure)h(space,)h Fu(G)e FB(a)g(top)s(ological)c(group,)257 4133 y Fu(A)38 b FB(:)g(\012)g Fp(!)g FB(Aut\()p Fu(G)p FB(\))g(a)g(measurable)g(map,)i(and)e(\010)h (a)f(lo)s(cally)e Fu(\026)p FB(-in)m(tegrable)h(function)257 4254 y(on)32 b(\012.)44 b(W)-8 b(e)32 b(de\014ne)h(the)g Fn(Hausdor\013)h(op)-5 b(er)g(ator)42 b FB(with)32 b(the)g(k)m(ernel)h (\010)f(o)m(v)m(er)h(the)g(group)e Fu(G)257 4374 y FB(b)m(y)j(the)f (form)m(ula)1038 4625 y(\()p Fp(H)1160 4640 y Fx(\010)p Fy(;A)1288 4625 y Fu(f)11 b FB(\)\()p Fu(x)p FB(\))28 b(=)1647 4490 y Fm(Z)1702 4715 y Fx(\012)1774 4625 y FB(\010\()p Fu(u)p FB(\))p Fu(f)11 b FB(\()p Fu(A)p FB(\()p Fu(u)p FB(\)\()p Fu(x)p FB(\)\))p Fu(d\026)p FB(\()p Fu(u)p FB(\))p Fu(:)404 4881 y FB(In)33 b(particular,)e(w)m(e)i(get)g (a)f(class)h(of)f Fn(discr)-5 b(ete)34 b(Hausdor\013)h(op)-5 b(er)g(ators)40 b FB(of)32 b(the)h(form)1372 5101 y Fu(f)38 b Fp(7!)1585 5006 y Fm(X)1588 5218 y Fy(u)p Fs(2)p Fx(\012)1746 5101 y FB(\010\()p Fu(u)p FB(\)\()p Fu(f)33 b Fp(\016)21 b Fu(A)p FB(\()p Fu(u)p FB(\)\))p 257 5297 1296 4 v 370 5358 a Fr(2)407 5388 y SDict begin H.S end 407 5388 a 407 5388 a SDict begin H.R end 407 5388 a 407 5388 a SDict begin [/View [/XYZ H.V]/Dest (Hfootnote.2) cvn /DEST pdfmark end 407 5388 a Fq(Here)27 b(w)n(e)g(correct)f(a)i(t)n(yp)r(o)f (made)f(in)h([)1604 5388 y SDict begin H.S end 1604 5388 a Fq(9)1646 5335 y SDict begin H.R end 1646 5335 a 1646 5388 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Trudy) cvn H.B /ANN pdfmark end 1646 5388 a Fq(,)h(p.)37 b(139].)1852 5637 y FB(5)p eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 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Fs(\003)3465 4449 y Fn(,)257 4569 y(and)d(a)f(top)-5 b(olo)g(gic)g(al)34 b(isomorphism)f(of)i FB(\()p Fu(B)5 b(M)10 b(O)s(A)p FB(\()p Fu(G)p FB(\))p Fu(;)17 b Fp(k)k(\001)h(k)2431 4584 y Fy(B)s(M)7 b(O)2622 4569 y FB(\))35 b Fn(and)f Fu(H)2973 4533 y Fx(1)3012 4569 y FB(\()p Fu(G)p FB(\))3165 4533 y Fs(\003)3204 4569 y Fn(.)404 4689 y FC(Theorem)46 b(3.)67 b Fn(L)-5 b(et)42 b Fu(A)p FB(\()p Fu(u)p FB(\))e Fp(2)h FB(Aut\()p Fu(G)p FB(\))1886 4653 y Fx(+)1987 4689 y Fn(for)h Fu(\026)p Fn(-a.)f(e.)65 b Fu(u)40 b Fp(2)h FB(\012)p Fn(.)66 b(The)41 b(Hausdor\013)257 4810 y(op)-5 b(er)g(ator)35 b Fp(H)725 4825 y Fx(\010)p Fy(;A)888 4810 y Fn(is)f(b)-5 b(ounde)g(d)35 b(on)f(the)h(sp)-5 b(ac)g(e)34 b Fu(B)5 b(M)10 b(O)s(A)p FB(\()p Fu(G)p FB(\))36 b Fn(if)e(and)h(only)g(if)f FB(\010)29 b Fp(2)f Fu(L)3291 4774 y Fx(1)3331 4810 y FB(\()p Fu(\026)p FB(\))p Fn(.)257 4930 y(Mor)-5 b(e)g(over,)1290 5051 y Fp(kH)1424 5066 y Fx(\010)p Fy(;A)1551 5051 y Fp(k)1601 5066 y Fs(L)p Fx(\()p Fy(B)s(M)7 b(O)r(A)p Fx(\))1976 5051 y Fp(\024)28 b(k)p FB(\010)p Fp(k)2251 5067 y Fy(L)2299 5048 y Fl(1)2334 5067 y Fx(\()p Fy(\026)p Fx(\))2436 5051 y Fu(:)404 5225 y FB(Pro)s(of.)78 b(Since)44 b FC(1)k Fp(2)g Fu(B)5 b(M)10 b(O)s(A)p FB(\()p Fu(G)p FB(\),)48 b(the)c("only)g(if")f(part)h(is)g (ob)m(vious.)79 b(No)m(w)45 b(let)257 5345 y(\010)51 b Fp(2)g Fu(L)561 5309 y Fx(1)600 5345 y FB(\()p Fu(\026)p FB(\).)83 b(In)46 b(view)g(of)f(Theorem)i(2)e(for)g(the)i(pro)s(of)d (it)h(su\016ces)j(to)d(sho)m(w)i(that)1852 5637 y(8)p eop end end %%Page: 9 9 TeXDict begin HPSdict begin 9 8 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 249 158 a SDict begin H.S end 249 158 a 249 158 a SDict begin H.R end 249 158 a 249 158 a SDict begin [/View [/XYZ H.V]/Dest (page.9) cvn /DEST pdfmark end 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Fs(\003)p 2991 3270 51 3 v 2991 3324 a Fx(\010)p Fy(;A)3115 3306 y Fi(\000)p Fl(1)3201 3284 y FB(\()p Fu(F)3302 3299 y Fy(')3352 3284 y FB(\))29 b(=)257 3420 y Fu(F)320 3435 y Fy( )373 3420 y FB(.)44 b(If)32 b(w)m(e)i(iden)m(tify)e(\(again)f(b)m(y)i(Theorem)g(A\))g Fu(F)2092 3435 y Fy(')2175 3420 y FB(with)f Fu(')g FB(and)h Fu(F)2746 3435 y Fy( )2831 3420 y FB(with)f Fu( )37 b FB(w)m(e)c(ha)m(v)m(e)1480 3638 y Fp(H)1564 3653 y Fx(\010)p Fy(;A)1692 3638 y Fu(')28 b FB(=)f Fp(H)1972 3597 y Fs(\003)p 1971 3619 V 1971 3673 a Fx(\010)p Fy(;A)2095 3655 y Fi(\000)p Fl(1)2181 3638 y Fu(';)257 3856 y FB(whic)m(h)33 b(completes)g(the)g (pro)s(of.)404 3977 y(F)-8 b(or)31 b(the)i(next)h(corollary)d(w)m(e)i (need)h(the)f(follo)m(wing)404 4097 y FC(De\014nition)41 b(5.)51 b FB([)1095 4097 y SDict begin H.S end 1095 4097 a FB(27)1193 4033 y SDict begin H.R end 1193 4033 a 1193 4097 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Rud) cvn H.B 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FB(\()p Fu(\026)p FB(\))p Fn(.)42 b(Then)28 b Fp(H)1904 4594 y Fx(\010)p Fy(;A)2060 4579 y Fn(is)g(a)g(b)-5 b(ounde)g(d)28 b(op)-5 b(er)g(ator)28 b(fr)-5 b(om)28 b Fu(H)3291 4542 y Fx(2)3283 4604 y Fy(E)3342 4579 y FB(\()p Fu(G)p FB(\))257 4699 y Fn(into)35 b FB(\()p Fu(B)5 b(M)10 b(O)s(A)p FB(\()p Fu(G)p FB(\))p Fu(;)17 b Fp(k)22 b(\001)g(k)1200 4714 y Fy(B)s(M)7 b(O)1390 4699 y FB(\))35 b Fn(and)1155 4929 y Fp(kH)1289 4944 y Fx(\010)p Fy(;A)1416 4929 y Fp(k)1466 4950 y Fy(H)1529 4927 y Fl(2)1524 4973 y Fj(E)1576 4950 y Fs(!)p Fy(B)s(M)7 b(O)r(A)1918 4929 y Fp(\024)28 b FB(3)2072 4838 y Fm(p)p 2171 4838 143 4 v 2171 4929 a Fu(K)2254 4944 y Fy(E)2314 4929 y Fp(k)p FB(\010)p Fp(k)2484 4946 y Fy(L)2532 4927 y Fl(1)2571 4929 y Fu(:)404 5147 y FB(Pro)s(of.)64 b(Let)41 b Fu(P)14 b(ol)1066 5162 y Fy(E)1124 5147 y FB(\()p Fu(G)p FB(\))40 b(:=)g(span)1656 5171 y Fw(C)1708 5147 y FB(\()p Fu(E)6 b FB(\))40 b(b)s(e)g(the)g(space)h(of)f Fu(E)6 b FB(-p)s(olynomials.)62 b(It)40 b(is)257 5268 y(kno)m(wn)24 b([)583 5268 y SDict begin H.S end 583 5268 a FB(6)632 5204 y SDict begin H.R end 632 5204 a 632 5268 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.DS2019) cvn H.B /ANN pdfmark end 632 5268 a FB(,)g(Prop)s(ositiuon)d(3.14],)j([)1531 5268 y SDict begin H.S end 1531 5268 a FB(23)1629 5204 y SDict begin H.R end 1629 5204 a 1629 5268 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.MatSb) cvn H.B /ANN pdfmark end 1629 5268 a FB(,)h(Lemma)c(1])h(that)g Fu(P)14 b(ol)2471 5283 y Fy(E)2530 5268 y FB(\()p Fu(G)p FB(\))22 b(is)g(a)g(dense)i(subspace)257 5388 y(of)35 b Fu(H)460 5341 y Fy(p)452 5415 y(E)512 5388 y FB(\()p Fu(G)p FB(\))g(for)f(all)f Fu(p)f Fp(2)h FB([1)p Fu(;)17 b Fp(1)p FB(\).)51 b(Since)35 b Fu(E)j Fp(\032)33 b Fu(X)2063 5403 y Fx(+)2122 5388 y FB(,)j(w)m(e)g(ha)m(v)m(e)h Fu(H)2648 5341 y Fy(p)2640 5415 y(E)2699 5388 y FB(\()p Fu(G)p FB(\))32 b Fp(\032)h Fu(H)3083 5352 y Fy(p)3122 5388 y FB(\()p Fu(G)p FB(\).)51 b(Let)1852 5637 y(9)p eop end end %%Page: 10 10 TeXDict begin HPSdict begin 10 9 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 249 158 a SDict begin H.S end 249 158 a 249 158 a SDict begin H.R end 249 158 a 249 158 a SDict begin [/View [/XYZ H.V]/Dest (page.10) cvn /DEST pdfmark end 249 158 a 257 573 a Fu(')38 b Fp(2)f Fu(P)14 b(ol)615 588 y Fy(E)674 573 y FB(\()p Fu(G)p FB(\).)60 b(Then)40 b Fu(')d Fp(2)h Fu(H)1469 537 y Fx(1)1461 598 y Fy(E)1520 573 y FB(\()p Fu(G)p FB(\))26 b Fp(\\)g Fu(H)1880 537 y Fx(2)1919 573 y FB(\()p Fu(G)p FB(\))38 b(and)g(b)m(y)h([)2473 573 y SDict begin H.S end 2473 573 a FB(21)2571 509 y SDict begin H.R end 2571 509 a 2571 573 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Indag) cvn H.B /ANN pdfmark end 2571 573 a FB(,)h(Theorem)e(3])g(one)h(has)257 693 y Fu(')28 b Fp(2)g Fu(B)5 b(M)10 b(O)s(A)p FB(\()p Fu(G)p FB(\))33 b(and)f Fp(k)p Fu(')p Fp(k)1316 708 y Fy(B)s(M)7 b(O)1535 693 y Fp(\024)28 b FB(3)1689 616 y Fp(p)p 1772 616 143 4 v 77 x Fu(K)1855 708 y Fy(E)1914 693 y Fp(k)p Fu(')p Fp(k)2078 710 y Fy(H)2141 691 y Fl(2)2179 693 y FB(.)44 b(No)m(w)33 b(Theorem)g(3)f(yields,)h(that)748 910 y Fp(kH)882 925 y Fx(\010)p Fy(;A)1010 910 y Fu(')p Fp(k)1124 925 y Fy(B)s(M)7 b(O)1342 910 y Fp(\024)28 b(k)p FB(\010)p Fp(k)1617 926 y Fy(L)1665 907 y Fl(1)1704 910 y Fp(k)p Fu(')p Fp(k)1868 925 y Fy(B)s(M)7 b(O)2086 910 y Fp(\024)28 b FB(3)2240 818 y Fm(p)p 2340 818 V 92 x Fu(K)2423 925 y Fy(E)2482 910 y Fp(k)p FB(\010)p Fp(k)2652 926 y Fy(L)2700 907 y Fl(1)2739 910 y Fp(k)p Fu(')p Fp(k)2903 926 y Fy(H)2966 907 y Fl(2)257 1113 y FB(and)33 b(the)g(result)g(follo)m(ws.)404 1234 y(In)g(conclusion)f(to)g(this)g(section,)h(w)m(e)h(discuss)g(the)f (necessit)m(y)h(of)f(the)g(condition)e(in)257 1354 y(Corollary)g(2)i (of)f(Theorem)g(2)h(and)f(Theorem)h(3.)404 1474 y FC(Prop)s(osition)46 b(1.)141 b Fn(L)-5 b(et)42 b Fu(G)g Fn(b)-5 b(e)41 b(metrizable.)64 b(Assume)42 b(in)f(addition)g(to)h(the)g(as-)257 1595 y(sumptions)j(of)h(De\014nition)e(4)i(that)1633 1515 y Fm(R)1680 1629 y Fy(E)1756 1595 y FB(\010)p Fu(d\026)h Fp(6)p FB(=)h(0)d Fn(for)h(every)f(me)-5 b(asur)g(able)45 b Fu(E)54 b Fp(\032)48 b FB(\012)p Fn(,)257 1715 y Fu(\026)p FB(\()p Fu(E)6 b FB(\))28 b Fu(>)f FB(0)p Fn(.)43 b(If)31 b(the)g(Hausdor\013)g(op)-5 b(er)g(ator)30 b Fp(H)1889 1730 y Fx(\010)p Fy(;A)2048 1715 y Fn(acts)h(in)g Fu(H)2451 1679 y Fx(1)2490 1715 y FB(\()p Fu(G)p FB(\))f Fn(or)h Fu(B)5 b(M)10 b(O)s(A)p FB(\()p Fu(G)p FB(\))31 b Fn(then)257 1836 y Fu(A)p FB(\()p Fu(u)p FB(\))d Fp(2)g FB(Aut\()p Fu(G)p FB(\))902 1799 y Fx(+)996 1836 y Fn(for)34 b(a.)h(e.)44 b Fu(u)28 b Fp(2)g FB(\012)p Fn(.)404 1956 y FB(Pro)s(of.)44 b(Since)33 b Fu(X)1046 1971 y Fx(+)1134 1956 y Fp(\032)c Fu(B)5 b(M)10 b(O)s(A)p FB(\()p Fu(G)p FB(\))29 b Fp(\032)g Fu(H)1951 1920 y Fx(1)1990 1956 y FB(\()p Fu(G)p FB(\),)k(w)m(e)h(ha)m (v)m(e)h(for)d(ev)m(ery)j Fu(\037)29 b Fp(2)g Fu(X)3246 1971 y Fx(+)3338 1956 y FB(and)257 2076 y(ev)m(ery)35 b Fu(\030)d Fp(2)c Fu(X)765 2091 y Fs(\000)852 2076 y FB(=)f Fu(X)j Fp(n)22 b Fu(X)1219 2091 y Fx(+)1311 2076 y FB(b)m(y)33 b(Theorem)g(1)f(that)1123 2280 y(\()p Fp(H)1245 2295 y Fx(\010)p Fy(;A)1372 2280 y Fu(\037)p FB(\))1471 2239 y Fs(^)1523 2280 y FB(\()p Fu(\030)5 b FB(\))27 b(=)g(\()p Fp(H)1899 2297 y Fx(\010)p Fy(;)p Fx(\()p Fy(A)2050 2278 y Fi(\003)2086 2297 y Fx(\))2113 2278 y Fi(\000)p Fl(1)2209 2280 y Fm(b)-64 b Fu(\037)p FB(\)\()p Fu(\030)5 b FB(\))27 b(=)h(0)p Fu(:)257 2484 y FB(On)23 b(the)h(other)f(hand,)i(the)f(orthogonalit)m(y)d(of)h(c)m(haracters)j (of)d Fu(G)h FB(implies)e(that)31 b Fm(b)-64 b Fu(\037)28 b FB(=)g(1)3350 2500 y Fs(f)p Fy(\037)p Fs(g)3468 2484 y FB(,)257 2604 y(where)34 b(1)588 2619 y Fy(A)677 2604 y FB(stands)g(for)e(the)h(indicator)e(function)h(of)g(a)g(subset)j Fu(A)d FB(of)g Fu(X)8 b FB(.)44 b(Th)m(us,)920 2851 y(0)28 b(=)f(\()p Fp(H)1222 2868 y Fx(\010)p Fy(;)p Fx(\()p Fy(A)p Fx(\()p Fy(u)p Fx(\))1468 2849 y Fi(\003)1505 2868 y Fx(\))1532 2849 y Fi(\000)p Fl(1)1619 2851 y FB(1)1668 2867 y Fs(f)p Fy(\037)p Fs(g)1787 2851 y FB(\)\()p Fu(\030)5 b FB(\))26 b(=)2079 2715 y Fm(Z)2134 2941 y Fy(E)t Fx(\()p Fy(\037;\030)s Fx(\))2362 2851 y FB(\010\()p Fu(u)p FB(\))p Fu(d\026)p FB(\()p Fu(u)p FB(\))p Fu(;)257 3119 y FB(where)504 3322 y Fu(E)6 b FB(\()p Fu(\037;)17 b(\030)5 b FB(\))27 b(=)g Fp(f)p Fu(u)g Fp(2)h FB(\012)h(:)e(\()p Fu(A)p FB(\()p Fu(u)p FB(\))1564 3281 y Fs(\003)1603 3322 y FB(\))1641 3281 y Fs(\000)p Fx(1)1735 3322 y FB(\()p Fu(\030)5 b FB(\))27 b(=)h Fu(\037)p Fp(g)g FB(=)f Fp(f)p Fu(u)g Fp(2)h FB(\012)g(:)g Fu(A)p FB(\()p Fu(u)p FB(\))2817 3281 y Fs(\003)2856 3322 y FB(\()p Fu(\037)p FB(\))g(=)f Fu(\030)5 b Fp(g)p Fu(:)257 3526 y FB(Therefore)34 b Fu(\026)p FB(\()p Fu(E)6 b FB(\()p Fu(\037;)17 b(\030)5 b FB(\)\))26 b(=)i(0)k(for)g(an)h(arbitrary)e Fu(\037)d Fp(2)g Fu(X)2315 3541 y Fx(+)2407 3526 y FB(and)k Fu(\030)h Fp(2)28 b Fu(X)2847 3541 y Fs(\000)2906 3526 y FB(.)43 b(Moreo)m(v)m(er,)586 3730 y Fp(f)p Fu(u)27 b Fp(2)h FB(\012)h(:)e Fu(A)p FB(\()p Fu(u)p FB(\))1171 3689 y Fs(\003)1238 3730 y FB(:)h Fu(X)1374 3745 y Fx(+)1460 3730 y Ft(9)g Fu(X)1669 3745 y Fx(+)1728 3730 y Fp(g)f FB(=)h Fp([f)p Fu(E)6 b FB(\()p Fu(\037;)17 b(\030)5 b FB(\))27 b(:)h Fu(\037)f Fp(2)h Fu(X)2677 3745 y Fx(+)2736 3730 y Fu(;)17 b(\030)32 b Fp(2)c Fu(X)3030 3745 y Fs(\000)3089 3730 y Fp(g)p Fu(:)257 3934 y FB(Since)39 b Fu(G)f FB(is)g(metrizable,) h Fu(X)46 b FB(is)38 b(coun)m(table)g(\(see,)j(e.g.,)g([)2398 3934 y SDict begin H.S end 2398 3934 a FB(25)2495 3870 y SDict begin H.R end 2495 3870 a 2495 3934 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Morris) cvn H.B /ANN pdfmark end 2495 3934 a FB(,)f(Corollary)d(of)h(Theorem)257 4054 y(29]\).)61 b(It)39 b(follo)m(ws)e(that)h Fu(A)p FB(\()p Fu(u)p FB(\))1368 4018 y Fs(\003)1445 4054 y FB(:)g Fu(X)1591 4069 y Fx(+)1688 4054 y Fp(!)f Fu(X)1906 4069 y Fx(+)2004 4054 y FB(for)h Fu(\026)p FB(-a.)f(e.)62 b Fu(u)p FB(,)39 b(whic)m(h)g(completes)g(the)257 4175 y(pro)s(of.)257 4443 y SDict begin H.S end 257 4443 a 257 4443 a SDict begin 14.5 H.A end 257 4443 a 257 4443 a SDict begin [/View [/XYZ H.V]/Dest (section.5) cvn /DEST pdfmark end 257 4443 a 182 x Fv(5)161 b(Hausdor\013)36 b(Op)t(erators)f(on)g(Spaces)g Fh(B)7 b(M)15 b(O)t Fg(\()p Fh(G)p Fg(\))35 b Fv(and)499 4808 y Fh(H)626 4756 y FB(1)615 4847 y Ft(R)692 4808 y Fg(\()p Fh(G)p Fg(\))257 5027 y FC(Theorem)55 b(4.)87 b Fn(L)-5 b(et)49 b Fu(A)p FB(\()p Fu(u)p FB(\))j Fp(2)g FB(Aut)q(\()p Fu(G)p FB(\))1799 4991 y Fx(+)1906 5027 y Fn(for)c Fu(\026)p Fn(-a.)f(e.)85 b Fu(u)52 b Fp(2)h FB(\012)p Fn(.)85 b(The)47 b(Hausdor\013)257 5147 y(op)-5 b(er)g(ator)44 b Fp(H)734 5162 y Fx(\010)p Fy(;A)906 5147 y Fn(is)g(b)-5 b(ounde)g(d)44 b(on)g Fu(B)5 b(M)10 b(O)s FB(\()p Fu(G)p FB(\))44 b Fn(if)g(and)g(only)g(if)g FB(\010)i Fp(2)f Fu(L)2929 5111 y Fx(1)2969 5147 y FB(\()p Fu(\026)p FB(\))p Fn(.)73 b(In)43 b(this)257 5268 y(c)-5 b(ase,)1316 5388 y Fp(kH)1450 5403 y Fx(\010)p Fy(;A)1578 5388 y Fp(k)1628 5404 y Fs(L)p Fx(\()p Fy(B)s(M)7 b(O)r Fx(\))1950 5388 y Fp(\024)28 b(k)p FB(\010)p Fp(k)2225 5405 y Fy(L)2273 5386 y Fl(1)2308 5405 y Fx(\()p Fy(\026)p Fx(\))2409 5388 y Fu(:)1828 5637 y FB(10)p eop end end %%Page: 11 11 TeXDict begin HPSdict begin 11 10 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 249 158 a SDict begin H.S end 249 158 a 249 158 a SDict begin H.R end 249 158 a 249 158 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Fu(')29 b Fp(2)g Fu(B)5 b(M)10 b(O)s FB(\()p Fu(G;)17 b Ft(R)5 b FB(\))41 b Fn(the)36 b(line)-5 b(ar)257 1902 y(functional)1527 2065 y Fu(F)1590 2080 y Fy(')1640 2065 y FB(\()p Fu(q)t FB(\))27 b(=)1894 1930 y Fm(Z)1894 2213 y Fy(G)2010 2065 y Fu(q)t('d\027)1151 b FB(\(2\))257 2361 y Fn(on)25 b FB(P)m(ol\()p Fu(G;)17 b Ft(R)t FB(\))31 b Fn(extends)25 b(uniquely)g(to)h(a)f(c)-5 b(ontinuous)25 b(line)-5 b(ar)25 b(functional)f Fu(F)2974 2376 y Fy(')3050 2361 y Fn(on)h Fu(H)3269 2325 y Fx(1)3261 2386 y Fw(R)3313 2361 y FB(\()p Fu(G)p FB(\))p Fn(.)257 2481 y(Mor)-5 b(e)g(over,)54 b(the)d(c)-5 b(orr)g(esp)g(ondenc)g(e)48 b Fu(')57 b Fp(7!)f Fu(F)1938 2496 y Fy(')2038 2481 y Fn(is)50 b(an)g(isometric)-5 b(al)50 b(isomorphism)f(of)257 2602 y FB(\()p Fu(B)5 b(M)10 b(O)s FB(\()p Fu(G;)17 b Ft(R)5 b FB(\))p Fu(;)17 b Fp(k\001k)991 2617 y Fs(\003)1035 2602 y FB(\))25 b Fn(and)g Fu(H)1367 2565 y Fx(1)1359 2627 y Fw(R)1411 2602 y FB(\()p Fu(G)p FB(\))1564 2565 y Fs(\003)1603 2602 y Fn(,)i(and)d(a)h(top)-5 b(olo)g(gic)g(al)23 b(isomorphism)h(of)g FB(\()p Fu(B)5 b(M)10 b(O)s FB(\()p Fu(G;)17 b Ft(R)5 b FB(\))p Fn(,)257 2722 y Fp(k)22 b(\001)g(k)429 2737 y Fy(B)s(M)7 b(O)620 2722 y FB(\))35 b Fn(and)f Fu(H)971 2686 y Fx(1)963 2748 y Fw(R)1015 2722 y FB(\()p Fu(G)p FB(\))1168 2686 y Fs(\003)1207 2722 y Fn(.)404 2842 y FC(Corollary)51 b(4.)78 b Fn(The)-5 b(or)g(em)45 b(B)g(is)h(valid)f(with)h Fu(q)51 b Fp(2)e Fu(L)2469 2806 y Fx(2)2509 2842 y FB(\()p Fu(G;)17 b Ft(R)t FB(\))52 b Fn(in)45 b(plac)-5 b(e)45 b(of)h Fu(q)52 b Fp(2)257 2963 y FB(P)m(ol\()p Fu(G;)17 b Ft(R)t FB(\))p Fn(.)404 3083 y FB(Pro)s(of.)79 b(Since)45 b(P)m(ol)o(\()p Fu(G;)17 b Ft(R)5 b FB(\))54 b Fp(\032)49 b Fu(L)1660 3047 y Fx(2)1700 3083 y FB(\()p Fu(G;)17 b Ft(R)5 b FB(\))50 b(and)45 b(P)m(ol\()p Fu(G;)17 b Ft(R)t FB(\))51 b(is)44 b(dense)i(in)e Fu(H)3271 3047 y Fx(1)3263 3109 y Fw(R)3315 3083 y FB(\()p Fu(G)p FB(\),)257 3204 y(it)c(su\016ces)j(to)d(sho)m(w)i(that)e(the)h 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2224 3478 a 2224 3542 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.DS2019) cvn H.B /ANN pdfmark end 2224 3542 a FB(],)i([)2338 3542 y SDict begin H.S end 2338 3542 a FB(29)2435 3478 y SDict begin H.R end 2435 3478 a 2435 3542 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.I.\040Schoolmann) cvn H.B /ANN pdfmark end 2435 3542 a FB(],)g(and)g([)2739 3542 y SDict begin H.S end 2739 3542 a FB(7)2788 3478 y SDict begin H.R end 2788 3478 a 2788 3542 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.DSservey) cvn H.B /ANN pdfmark end 2788 3542 a FB(].)404 3662 y(Let)g Fu(\025)d FB(=)f(\()p Fu(\025)866 3677 y Fy(n)913 3662 y FB(\))k(b)s(e)h(a)f(non-negativ)m(e)h(strictly)f(increasing)g (sequence)j(of)d(real)g(n)m(um-)257 3783 y(b)s(ers)c(tending)e(to)g Fp(1)h FB(\("a)f(frequency"\).)44 b(The)28 b(v)-5 b(alue)27 b 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y Fp(2)g Fu(X)47 b FB(\(whic)m(h)257 4519 y(then)34 b(is)e(unique\))h (suc)m(h)h(that)e Fu(h)1420 4534 y Fy(\025)1461 4542 y Fj(n)1530 4519 y Fp(\016)22 b Fu(\014)33 b FB(=)1796 4493 y Fm(c)1794 4519 y Fu(\025)1851 4534 y Fy(n)1898 4519 y FB(.)404 4639 y(W)-8 b(e)33 b(consider)g(formal)d(general)i (Diric)m(hlet)e(series)1415 4907 y Fu(D)1496 4922 y Fy(\025)1569 4907 y FB(=)1709 4782 y Fs(1)1672 4812 y Fm(X)1678 5021 y Fy(n)p Fx(=1)1833 4907 y Fu(a)1884 4922 y Fy(n)1931 4907 y FB(\()p Fu(D)s FB(\))p Fu(e)2136 4866 y Fs(\000)p Fy(\025)2232 4874 y Fj(n)2274 4866 y Fy(s)2311 4907 y Fu(:)404 5191 y FB(In)j([)553 5191 y SDict begin H.S end 553 5191 a FB(6)601 5127 y SDict begin H.R end 601 5127 a 601 5191 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.DS2019) cvn H.B /ANN pdfmark end 601 5191 a FB(])g(the)g(next)h(t)m(w)m(o)f (spaces)h(w)m(ere)g(in)m(tro)s(duced)485 5388 y Fp(D)562 5403 y Fs(1)637 5388 y FB(\()p Fu(\025)p FB(\))27 b(:=)h Fp(f)p Fu(D)1059 5403 y Fy(\025)1132 5388 y FB(:)f Fu(D)1267 5403 y Fy(\025)1345 5388 y FB(con)m(v)m(erge)35 b(to)d(a)g(function)g (from)g Fu(H)2646 5347 y Fs(1)2720 5388 y FB(\()p Fp(f)p FB(Re)27 b Fu(>)h FB(0)p Fp(g)p FB(\))p Fp(g)p Fu(;)1828 5637 y FB(16)p eop end end %%Page: 17 17 TeXDict begin HPSdict begin 17 16 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 249 158 a SDict begin H.S end 249 158 a 249 158 a SDict begin H.R end 249 158 a 249 158 a SDict begin [/View [/XYZ H.V]/Dest (page.17) cvn /DEST pdfmark end 249 158 a 257 573 a FB(and)47 b Fp(D)541 537 y Fy(ext)538 597 y Fs(1)642 573 y FB(\()p Fu(\025)p FB(\))f(of)g(all)e(somewhere)k(con)m (v)m(ergen)m(t)g Fu(\025)p FB(-Diric)m(hlet)c(series,)50 b(whic)m(h)d(ha)m(v)m(e)g(a)257 693 y(holomorphic)40 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