If you have problems to print the paper please feel free to contact one of the authors of this contribution e.g. Yuri G. Kondratiev, Ludwig Streit, Werner Westerkamp at BiBoS, Universitaet Bielefeld, D 33615 Bielefeld, Germany BOBY: %!PS-Adobe-2.0 %%Creator: dvipsk 5.55a Copyright 1986, 1994 Radical Eye Software %%Title: nga5.dvi %%Pages: 42 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips nga5.dvi -o nga5.ps %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1995.11.23:0949 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls 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Fu(b)o(y)g(using)g(the)g(dual)h(family)g(of)e(spaces)h Fr(fH)1456 1059 y Fq(\000)p Fp(p)1516 1052 y Fu(:=)h Fr(H)1615 1036 y Fq(0)1615 1064 y Fp(p)1635 1052 y Fs(;)19 b(p)13 b Fr(2)f Fs(I)-12 b(N)5 b Fr(g)p Fu(.)59 1109 y(The)18 b(inductiv)o(e)i(limit)f(top)q(ology)f(\(w.r.t.)26 b(this)19 b(family\))f(is)h(the)f(\014nest)g(top)q(ology)g(on)g Fr(N)1613 1092 y Fq(0)1642 1109 y Fu(suc)o(h)g(that)59 1165 y(the)f(em)o(b)q(eddings)h Fr(H)426 1172 y Fq(\000)p Fp(p)489 1165 y Fs(,)-8 b Fr(!)15 b(N)598 1149 y Fq(0)626 1165 y Fu(are)i(con)o(tin)o(uous)g(for)f(all)h Fs(p)e Fr(2)h Fs(I)-12 b(N)t Fu(.)25 b(It)16 b(is)h(con)o(v)o(enien)o(t)h(to)e (denote)h(the)59 1222 y(norm)c(on)h Fr(H)276 1229 y Fq(\000)p Fp(p)337 1222 y Fu(b)o(y)g Fr(j)o(\001j)437 1235 y Fq(\000)p Fp(p)484 1222 y Fu(.)19 b(Let)14 b(us)g(men)o(tion)g(that)f(in)h(our)g (setting)f(the)h(top)q(ology)g Fs(\034)1485 1229 y Fp(ind)1552 1222 y Fu(coincides)h(with)59 1278 y(the)h(Mac)o(k)o(ey)f(top)q(ology)h Fs(\034)5 b Fu(\()p Fr(N)579 1262 y Fq(0)590 1278 y Fs(;)j Fr(N)f Fu(\))15 b(and)h(the)g(strong)e(top)q(ology)i Fs(\014)r Fu(\()p Fr(N)1273 1262 y Fq(0)1284 1278 y Fs(;)8 b Fr(N)f Fu(\).)21 b(F)l(urther)16 b(note)f(that)h(the)59 1335 y(dual)h(pair)g Fr(hN)318 1318 y Fq(0)329 1335 y Fs(;)8 b Fr(N)f(i)15 b Fu(is)i(re\015exiv)o(e)h(if)e Fr(N)740 1318 y Fq(0)768 1335 y Fu(is)h(equipp)q(ed)i(with)d Fs(\014)r Fu(\()p Fr(N)1205 1318 y Fq(0)1217 1335 y Fs(;)8 b Fr(N)f Fu(\).)22 b(In)17 b(addition)g(w)o(e)f(ha)o(v)o(e)g(that)59 1391 y(con)o(v)o(ergence)h(of)e(sequences)j(is)e(equiv)m(alen)o(t)i(in) g Fs(\014)r Fu(\()p Fr(N)980 1375 y Fq(0)991 1391 y Fs(;)8 b Fr(N)f Fu(\))15 b(and)i(the)f(w)o(eak)g(top)q(ology)f Fs(\033)r Fu(\()p Fr(N)1650 1375 y Fq(0)1661 1391 y Fs(;)8 b Fr(N)f Fu(\),)15 b(see)59 1447 y(e.g.,)f([HKPS93,)h(App)q(endix)i (5].)130 1504 y(F)l(urther)k(w)o(e)f(w)o(an)o(t)g(to)h(in)o(tro)q(duce) h(the)f(notion)g(of)g(tensor)g(p)q(o)o(w)o(er)f(of)h(a)g(n)o(uclear)h (space.)37 b(The)59 1560 y(simplest)21 b(w)o(a)o(y)e(to)g(do)h(this)g (is)g(to)f(start)g(from)g(usual)i(tensor)e(p)q(o)o(w)o(ers)g Fr(H)1353 1544 y Fq(\012)p Fp(n)1353 1572 y(p)1425 1560 y Fs(;)27 b(n)20 b Fr(2)h Fs(I)-12 b(N)25 b Fu(of)19 b(Hilb)q(ert)59 1617 y(spaces.)26 b(Since)18 b(there)g(is)f(no)g (danger)g(of)g(confusion)h(w)o(e)f(will)h(preserv)o(e)g(the)f(notation) f Fr(j\001j)1618 1630 y Fp(p)1655 1617 y Fu(and)i Fr(j)o(\001j)1783 1630 y Fq(\000)p Fp(p)59 1680 y Fu(for)d(the)g(norms)g(on)g Fr(H)445 1664 y Fq(\012)p Fp(n)445 1691 y(p)512 1680 y Fu(and)g Fr(H)638 1661 y Fq(\012)p Fp(n)638 1691 y Fq(\000)p Fp(p)705 1680 y Fu(resp)q(ectiv)o(ely)l(.)22 b(Using)15 b(the)h(de\014nition)748 1777 y Fr(N)792 1759 y Fq(\012)p Fp(n)855 1777 y Fu(:=)f(pr)h(lim)940 1816 y Fp(p)p Fq(2)p Fp(I)-9 b(N)1052 1777 y Fr(H)1090 1759 y Fq(\012)p Fp(n)1090 1789 y(p)59 1907 y Fu(one)15 b(can)f(pro)o(v)o(e) g([Sc)o(h71])g(that)g Fr(N)643 1890 y Fq(\012)p Fp(n)708 1907 y Fu(is)h(a)g(n)o(uclear)g(space)g(whic)o(h)g(is)g(called)h(the)f Fs(n)1476 1890 y Fp(th)1526 1907 y Fu(tensor)f(p)q(o)o(w)o(er)g(of)59 1963 y Fr(N)7 b Fs(:)15 b Fu(The)g(dual)h(space)g(of)f Fr(N)543 1947 y Fq(\012)p Fp(n)609 1963 y Fu(can)g(b)q(e)h(written)719 2022 y Fj(\000)738 2057 y Fr(N)782 2038 y Fq(\012)p Fp(n)833 2022 y Fj(\001)852 2031 y Fq(0)876 2057 y Fu(=)g(ind)g(lim)958 2086 y Fp(p)p Fq(2)p Fp(I)-9 b(N)1081 2057 y Fr(H)1119 2037 y Fq(\012)p Fp(n)1119 2068 y Fq(\000)p Fp(p)130 2169 y Fu(Most)12 b(imp)q(ortan)o(t)h(for)h(the)f(applications)j(w)o(e) d(ha)o(v)o(e)g(in)i(mind)f(is)h(the)e(follo)o(wing)i('k)o(ernel)f (theorem',)59 2226 y(see)i(e.g.,)d([BeKo88].)59 2323 y Fk(Theorem)18 b(2)23 b Fm(L)n(et)f Fs(\030)434 2330 y Fo(1)453 2323 y Fs(;)8 b(:::;)g(\030)554 2330 y Fp(n)600 2323 y Fr(7!)25 b Fs(F)699 2330 y Fp(n)730 2323 y Fu(\()p Fs(\030)768 2330 y Fo(1)787 2323 y Fs(;)8 b(:::;)g(\030)888 2330 y Fp(n)909 2323 y Fu(\))23 b Fm(b)n(e)f(an)h(n-line)n(ar)e(form)j (on)f Fr(N)1493 2307 y Fq(\012)p Fp(n)1566 2323 y Fm(which)h(is)e Fr(H)1794 2330 y Fp(p)1814 2323 y Fm(-)59 2379 y(c)n(ontinuous)16 b(,)g(i.e.,)669 2464 y Fr(j)p Fs(F)711 2471 y Fp(n)742 2464 y Fu(\()p Fs(\030)780 2471 y Fo(1)799 2464 y Fs(;)8 b(:::;)g(\030)900 2471 y Fp(n)921 2464 y Fu(\))f Fr(j)12 b(\024)h Fs(C)1084 2411 y Fp(n)1068 2423 y Fj(Y)1063 2516 y Fp(k)q Fo(=1)1135 2464 y Fr(j)p Fs(\030)1168 2471 y Fp(k)1188 2464 y Fr(j)1201 2471 y Fp(p)59 2578 y Fm(for)k(some)f Fs(p)c Fr(2)h Fs(I)-12 b(N)21 b Fm(and)c Fs(C)e(>)e Fu(0)p Fs(:)59 2634 y Fm(Then)h(for)h(al)r(l)g Fs(p)334 2618 y Fq(0)358 2634 y Fs(>)e(p)i Fm(such)g(that)g(the)g(emb)n(e)n(dding)f Fs(i)944 2642 y Fp(p)962 2633 y Fh(0)973 2642 y Fp(;p)1015 2634 y Fu(:)e Fr(H)1078 2642 y Fp(p)1096 2633 y Fh(0)1122 2634 y Fs(,)-8 b Fr(!)14 b(H)1224 2641 y Fp(p)1259 2634 y Fm(is)g(Hilb)n(ert-Schmidt)g(ther)n(e)h(exist)59 2691 y(a)i(unique)f Fu(\010)278 2674 y Fo(\()p Fp(n)p Fo(\))341 2691 y Fr(2)d(H)422 2672 y Fq(\012)p Fp(n)422 2705 y Fq(\000)p Fp(p)467 2696 y Fh(0)497 2691 y Fm(such)k(that)402 2796 y Fs(F)431 2803 y Fp(n)462 2796 y Fu(\()p Fs(\030)500 2803 y Fo(1)519 2796 y Fs(;)8 b(:)g(:)g(:)d(;)j(\030)641 2803 y Fp(n)664 2796 y Fu(\))k(=)h Fr(h)p Fu(\010)793 2777 y Fo(\()p Fp(n)p Fo(\))843 2796 y Fs(;)8 b(\030)884 2803 y Fo(1)913 2796 y Fr(\012)j(\001)d(\001)g(\001)g(\012)i Fs(\030)1087 2803 y Fp(n)1111 2796 y Fr(i)15 b Fs(;)70 b(\030)1247 2803 y Fo(1)1267 2796 y Fs(;)8 b(:::;)g(\030)1368 2803 y Fp(n)1401 2796 y Fr(2)13 b(N)p eop %%Page: 4 4 4 3 bop 59 50 a Fn(2)45 b(PRELIMINARIES)1311 b Fu(4)59 199 y Fm(and)16 b(the)h(fol)r(lowing)f(norm)g(estimate)g(holds)713 252 y Fj(\014)713 277 y(\014)713 302 y(\014)727 301 y Fu(\010)760 282 y Fo(\()p Fp(n)p Fo(\))811 252 y Fj(\014)811 277 y(\014)811 302 y(\014)824 329 y Fq(\000)p Fp(p)869 319 y Fh(0)896 301 y Fr(\024)d Fs(C)999 264 y Fj(\015)999 289 y(\015)1023 301 y Fs(i)1039 308 y Fp(p)1057 299 y Fh(0)1067 308 y Fp(;p)1097 264 y Fj(\015)1097 289 y(\015)1120 277 y Fp(n)1120 317 y(H)s(S)59 410 y Fm(using)j(the)g(Hilb)n (ert-Schmidt)g(norm)g(of)h Fs(i)770 418 y Fp(p)788 409 y Fh(0)798 418 y Fp(;p)828 410 y Fm(.)59 511 y Fk(Cor)o(ollar)m(y)h(3) 23 b Fm(L)n(et)18 b Fs(\030)472 518 y Fo(1)491 511 y Fs(;)8 b(:::;)g(\030)592 518 y Fp(n)631 511 y Fr(7!)17 b Fs(F)e Fu(\()o Fs(\030)774 518 y Fo(1)794 511 y Fs(;)8 b(:::;)g(\030)895 518 y Fp(n)916 511 y Fu(\))18 b Fm(b)n(e)h(an)f Fs(n)p Fm(-line)n(ar)h(form)g(on)g Fr(N)1477 494 y Fq(\012)p Fp(n)1547 511 y Fm(which)g(is)g Fr(H)1767 518 y Fq(\000)p Fp(p)1814 511 y Fm(-)59 567 y(c)n(ontinuous,)d(i.e.,)644 651 y Fr(j)p Fs(F)686 658 y Fp(n)717 651 y Fu(\()o Fs(\030)754 658 y Fo(1)774 651 y Fs(;)8 b(:)g(:)g(:)d(;)j(\030)896 658 y Fp(n)918 651 y Fu(\))g Fr(j)k(\024)h Fs(C)1082 599 y Fp(n)1066 611 y Fj(Y)1060 703 y Fp(k)q Fo(=1)1132 651 y Fr(j)p Fs(\030)1165 658 y Fp(k)1186 651 y Fr(j)1199 658 y Fq(\000)p Fp(p)59 767 y Fm(for)k(some)f Fs(p)c Fr(2)h Fs(I)-12 b(N)21 b Fm(and)c Fs(C)e(>)e Fu(0)p Fm(.)59 824 y(Then)h(for)h(al)r(l)g Fs(p)334 807 y Fq(0)358 824 y Fs(<)e(p)i Fm(such)g(that)g(the)g(emb)n(e)n(dding)f Fs(i)944 831 y Fp(p;p)990 822 y Fh(0)1015 824 y Fu(:)e Fr(H)1078 831 y Fp(p)1111 824 y Fs(,)-8 b Fr(!)13 b(H)1212 831 y Fp(p)1230 822 y Fh(0)1259 824 y Fm(is)h(Hilb)n(ert-Schmidt)g (ther)n(e)h(exist)59 880 y(a)i(unique)f Fu(\010)278 864 y Fo(\()p Fp(n)p Fo(\))341 880 y Fr(2)d(H)422 861 y Fq(\012)p Fp(n)422 894 y(p)440 885 y Fh(0)490 880 y Fm(such)k(that)421 988 y Fs(F)450 995 y Fp(n)481 988 y Fu(\()p Fs(\030)519 995 y Fo(1)539 988 y Fs(;)8 b(:::;)g(\030)640 995 y Fp(n)660 988 y Fu(\))13 b(=)g Fr(h)p Fu(\010)790 969 y Fo(\()p Fp(n)p Fo(\))840 988 y Fs(;)8 b(\030)881 995 y Fo(1)910 988 y Fr(\012)i(\001)e(\001)g(\001)g(\012)j Fs(\030)1084 995 y Fp(n)1107 988 y Fr(i)p Fs(;)70 b(\030)1228 995 y Fo(1)1248 988 y Fs(;)8 b(:::;)g(\030)1349 995 y Fp(n)1382 988 y Fr(2)13 b(N)59 1084 y Fm(and)j(the)h(fol)r(lowing)f(norm)g (estimate)g(holds)708 1137 y Fj(\014)708 1162 y(\014)708 1187 y(\014)722 1186 y Fu(\010)755 1167 y Fo(\()p Fp(n)p Fo(\))805 1137 y Fj(\014)805 1162 y(\014)805 1187 y(\014)819 1214 y Fp(p)837 1204 y Fh(0)863 1186 y Fr(\024)d Fs(C)967 1149 y Fj(\015)967 1174 y(\015)990 1186 y Fs(i)1006 1193 y Fp(p;p)1052 1184 y Fh(0)1064 1149 y Fj(\015)1064 1174 y(\015)1087 1163 y Fp(n)1087 1202 y(H)s(S)1168 1186 y Fm(.)130 1302 y Fu(If)i(in)h(Theorem)f(2)f(\(and)h(in)h(Corollary)f(3)g (resp)q(ectiv)o(ely)i(\))d(w)o(e)h(start)f(from)g(a)h(symmetric)g Fs(n)p Fu(-linear)59 1359 y(form)i Fs(F)198 1366 y Fp(n)239 1359 y Fu(on)g Fr(N)348 1342 y Fq(\012)p Fp(n)416 1359 y Fu(i.e.,)h Fs(F)534 1366 y Fp(n)558 1359 y Fu(\()p Fs(\030)596 1366 y Fp(\031)616 1371 y Fi(1)635 1359 y Fs(;)8 b(:)g(:)g(:)d(;)j(\030)757 1366 y Fp(\031)777 1370 y Fg(n)799 1359 y Fu(\))16 b(=)h Fs(F)914 1366 y Fp(n)945 1359 y Fu(\()p Fs(\030)983 1366 y Fo(1)1002 1359 y Fs(;)8 b(:)g(:)g(:)d(;)j(\030)1124 1366 y Fp(n)1147 1359 y Fu(\))17 b(for)f(an)o(y)h(p)q(erm)o(utation)h Fs(\031)r Fu(,)f(then)h(the)59 1422 y(corresp)q(onding)13 b(k)o(ernel)g(\010)512 1406 y Fo(\()p Fp(n)p Fo(\))574 1422 y Fu(can)f(b)q(e)h(lo)q(calized)h(in)f Fr(H)988 1397 y Fo(^)983 1403 y Fq(\012)q Fp(n)983 1436 y(p)1001 1427 y Fh(0)1047 1422 y Fr(\032)g(H)1133 1403 y Fq(\012)p Fp(n)1133 1436 y(p)1151 1427 y Fh(0)1197 1422 y Fu(\(the)e(n)1314 1406 y Fp(th)1362 1422 y Fu(symmetric)h(tensor)f(p)q(o)o(w)o(er)59 1486 y(of)j(the)h(Hilb)q(ert)i(space)e Fr(H)504 1493 y Fp(p)522 1484 y Fh(0)535 1486 y Fu(\).)k(F)l(or)c Fs(f)689 1493 y Fo(1)708 1486 y Fs(;)8 b(:)g(:)g(:)d(;)j(f)832 1493 y Fp(n)868 1486 y Fr(2)13 b(H)i Fu(let)1036 1479 y(^)1030 1486 y Fr(\012)g Fu(also)f(denote)h(the)g(symmetrization)g(of) g(the)59 1542 y(tensor)g(pro)q(duct)573 1615 y Fs(f)595 1622 y Fo(1)621 1608 y Fu(^)615 1615 y Fr(\012)8 b(\001)g(\001)g(\001) 724 1608 y Fu(^)718 1615 y Fr(\012)q Fs(f)776 1622 y Fp(n)812 1615 y Fu(:=)886 1584 y(1)p 877 1605 40 2 v 877 1646 a Fs(n)p Fu(!)930 1575 y Fj(X)949 1662 y Fp(\031)997 1615 y Fs(f)1019 1622 y Fp(\031)1039 1627 y Fi(1)1069 1615 y Fr(\012)j(\001)d(\001)g(\001)g(\012)i Fs(f)1245 1622 y Fp(\031)1265 1626 y Fg(n)1304 1615 y Fs(;)59 1726 y Fu(where)15 b(the)h(sum)f(extends)g(o)o(v)o(er)g(all)h(p)q(erm)o (utations)f(of)f Fs(n)i Fu(letters.)k(All)c(the)f(ab)q(o)o(v)o(e)g (quoted)g(theorems)59 1782 y(also)22 b(hold)h(for)f(complex)h(spaces,)h (in)f(particular)g(the)f(complexi\014ed)j(space)d Fr(N)p 1490 1789 2 18 v 1481 1789 a Fp(C)1511 1782 y Fu(.)41 b(By)23 b(de\014nition)59 1839 y(an)d(elemen)o(t)h Fs(\022)i Fr(2)d(N)p 437 1846 V 428 1846 a Fp(C)478 1839 y Fu(decomp)q(oses)h(in) o(to)f Fs(\022)j Fu(=)e Fs(\030)15 b Fu(+)f Fs(i\021)21 b(;)28 b(\030)r(;)8 b(\021)21 b Fr(2)g(N)7 b Fu(.)35 b(If)20 b(w)o(e)g(also)g(in)o(tro)q(duce)i(the)59 1895 y(corresp)q(onding)16 b(complexi\014ed)i(Hilb)q(ert)e(spaces)g Fr(H)947 1902 y Fp(p;)p 981 1902 V -3 w(C)1017 1895 y Fu(the)f(inner)h(pro)q(duct)g(b)q(ecomes)207 1991 y(\()p Fs(\022)246 1998 y Fo(1)266 1991 y Fs(;)8 b(\022)308 1998 y Fo(2)327 1991 y Fu(\))345 1998 y Fq(H)375 2004 y Fg(p;)p 405 2004 2 14 v -3 w(C)438 1991 y Fu(=)13 b(\()p Fs(\022)525 1998 y Fo(1)545 1991 y Fs(;)569 1979 y Fu(\026)566 1991 y Fs(\022)587 1998 y Fo(2)606 1991 y Fu(\))624 1998 y Fq(H)654 2002 y Fg(p)686 1991 y Fu(=)g(\()p Fs(\030)772 1998 y Fo(1)792 1991 y Fs(;)8 b(\030)833 1998 y Fo(2)851 1991 y Fu(\))869 1998 y Fq(H)899 2002 y Fg(p)929 1991 y Fu(+)j(\()p Fs(\021)1016 1998 y Fo(1)1035 1991 y Fs(;)d(\021)1079 1998 y Fo(2)1097 1991 y Fu(\))1115 1998 y Fq(H)1145 2002 y Fg(p)1175 1991 y Fu(+)j Fs(i)p Fu(\()p Fs(\021)1278 1998 y Fo(1)1296 1991 y Fs(;)d(\030)1337 1998 y Fo(2)1356 1991 y Fu(\))1374 1998 y Fq(H)1404 2002 y Fg(p)1434 1991 y Fr(\000)j Fs(i)p Fu(\()p Fs(\030)1534 1998 y Fo(1)1553 1991 y Fs(;)d(\021)1597 1998 y Fo(2)1615 1991 y Fu(\))1633 1998 y Fq(H)1663 2002 y Fg(p)59 2087 y Fu(for)k Fs(\022)147 2094 y Fo(1)167 2087 y Fs(;)c(\022)209 2094 y Fo(2)241 2087 y Fr(2)k(H)321 2094 y Fp(p;)p 355 2094 2 18 v -3 w(C)376 2087 y Fs(;)19 b(\022)429 2094 y Fo(1)462 2087 y Fu(=)13 b 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Fj(o)1359 2700 y Fr(2)13 b Fu(\000\()p Fr(H)p Fu(\))g Fs(:)p eop %%Page: 5 5 5 4 bop 59 50 a Fn(2)45 b(PRELIMINARIES)1311 b Fu(5)59 199 y Fl(2.2)56 b(Holomorph)n(y)17 b(on)i(lo)r(cally)d(con)n(v)n(ex)j (spaces)59 285 y Fu(W)l(e)h(shall)g(collect)h(some)e(facts)g(from)g (the)h(theory)f(of)g(holomorphic)i(functions)f(in)h(lo)q(cally)g(con)o (v)o(ex)59 342 y(top)q(ological)c(v)o(ector)d(spaces)i Fr(E)k Fu(\(o)o(v)o(er)14 b(the)i(complex)g(\014eld)p 1097 342 3 25 v 20 w Fs(C)s Fu(\),)f(see)h(e.g.,)f([Di81)o(].)21 b(Let)15 b Fr(L)p Fu(\()p Fr(E)1648 325 y Fp(n)1672 342 y Fu(\))g(b)q(e)i(the)59 398 y(space)d(of)e(n-linear)j(mappings)f(from) f Fr(E)730 381 y Fp(n)767 398 y Fu(in)o(to)p 872 398 V 16 w Fs(C)j Fu(and)d Fr(L)1026 405 y Fp(s)1045 398 y Fu(\()p Fr(E)1091 381 y Fp(n)1114 398 y Fu(\))g(the)h(subspace)g(of)f (symmetric)g(n-linear)59 454 y(forms.)33 b(Also)20 b(let)g Fn(P)427 438 y Fp(n)450 454 y Fu(\()p Fr(E)t Fu(\))g(denote)g(the)g (n-homogeneous)g(p)q(olynomials)h(on)f Fr(E)t 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(e\016cien)o(ts)h(in)g(a)f(con)o(v)o(enien)o(t)h(form.)j(Let)59 533 y Fs(')12 b Fr(2)h(E)172 517 y Fp(k)168 546 y Fo(min)229 533 y Fu(\()p Fr(N)291 517 y Fq(0)p 290 546 V 281 546 a Fp(C)311 533 y Fu(\))i(and)g Fs(z)g Fr(2)e(N)555 517 y Fq(0)p 554 546 V 545 546 a Fp(C)575 533 y Fu(,)i(then)g(there)g (exist)h(k)o(ernels)g Fs(')1112 517 y Fo(\()p Fp(n)p Fo(\))1175 533 y Fr(2)d(N)1267 508 y Fo(^)1262 514 y Fq(\012)q Fp(n)p 1261 547 V 1252 547 a(C)1328 533 y Fu(suc)o(h)j(that) 687 642 y Fr(h)p Fs(z)728 623 y Fq(\012)p Fp(n)779 642 y Fs(;)8 b(')830 623 y Fo(\()p Fp(n)p Fo(\))880 642 y Fr(i)k Fu(=)971 611 y(1)p 963 631 40 2 v 963 673 a Fs(n)p Fu(!)1046 626 y Fj(d)1008 642 y Fu(d)1033 629 y Fp(n)1056 642 y Fs(')p Fu(\(0\))o(\()p Fs(z)r Fu(\))59 738 y(i.e.,)718 823 y Fs(')p Fu(\()p Fs(z)r Fu(\))g(=)883 770 y Fq(1)870 782 y Fj(X)867 873 y Fp(n)p Fo(=0)934 823 y Fr(h)p Fs(z)975 804 y Fq(\012)p Fp(n)1025 823 y Fs(;)c(')1076 804 y Fo(\()p Fp(n)p Fo(\))1141 823 y Fr(i)p Fs(:)601 b Fu(\(1\))59 940 y(This)20 b(represen)o(tation)e(allo)o(ws)i(to)e(in)o(tro)q(duce)i (a)e(n)o(uclear)i(top)q(ology)e(on)h Fr(E)1360 924 y Fp(k)1356 952 y Fo(min)1417 940 y Fu(\()p Fr(N)1479 924 y Fq(0)p 1478 953 2 18 v 1469 953 a Fp(C)1499 940 y Fu(\),)g(see)g ([Ou91])f(for)59 997 y(details.)23 b(Let)16 b(E)336 980 y Fp(\014)336 1008 y(p;q)398 997 y Fu(denote)g(the)g(space)g(of)g(all)h (functions)f(of)g(the)g(form)f(\(1\))g(suc)o(h)h(that)f(the)h(follo)o (wing)59 1053 y(Hilb)q(ertian)h(norm)503 1128 y Fr(j)-8 b(j)g(j)o Fs(')p Fr(j)g(j)g(j)578 1108 y Fo(2)578 1142 y Fp(p;q)q(;\014)668 1128 y Fu(:=)745 1076 y Fq(1)732 1088 y Fj(X)729 1179 y Fp(n)p Fo(=0)796 1128 y Fu(\()p Fs(n)p Fu(!\))872 1110 y Fo(1+)p Fp(\014)939 1128 y Fu(2)962 1110 y Fp(nq)1010 1080 y Fj(\014)1010 1105 y(\014)1010 1130 y(\014)1024 1128 y Fs(')1054 1110 y Fo(\()p Fp(n)p Fo(\))1105 1080 y Fj(\014)1105 1105 y(\014)1105 1130 y(\014)1119 1093 y Fo(2)1119 1157 y Fp(p)1159 1128 y Fs(;)52 b(p;)8 b(q)14 b Fr(2)f Fs(I)-12 b(N)383 b Fu(\(2\))59 1255 y(is)18 b(\014nite)g(for)f Fs(\014)h Fr(2)e Fu([0)p Fs(;)8 b Fu(1].)24 b(\(By)608 1206 y Fj(\014)608 1231 y(\014)608 1256 y(\014)622 1255 y Fs(')652 1239 y Fo(\(0\))699 1206 y Fj(\014)699 1231 y(\014)699 1256 y(\014)713 1283 y Fp(p)750 1255 y Fu(w)o(e)17 b(simply)i(mean)e(the)g(complex)h(mo)q (dulus)h(for)e(all)h Fs(p)p Fu(.\))25 b(The)59 1339 y(space)16 b(E)212 1317 y Fq(\000)p Fp(\014)212 1350 y Fq(\000)p Fp(p)p Fq(\000)p Fp(;q)328 1339 y Fu(with)g(the)f(norm)g Fr(j)-8 b(j)g(j)o Fs(')p Fr(j)g(j)g(j)704 1352 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)q(;)p Fq(\000)p Fp(\014)879 1339 y Fu(is)16 b(de\014ned)h(analogously)l(.)59 1430 y Fk(Theorem)h(5)23 b Fm(The)16 b(fol)r(lowing)f(top)n(olo)n(gic)n(al)h(identity)g(holds:) 675 1537 y Fu(pr)f(lim)682 1576 y Fp(p;q)q Fq(2)p Fp(I)-9 b(N)821 1537 y Fu(E)852 1518 y Fp(\014)852 1548 y(p;q)911 1537 y Fu(=)13 b Fr(E)1013 1490 y Fi(2)p 992 1496 58 2 v 992 1517 a(1+)p Fg(\014)983 1550 y Fo(min)1057 1537 y Fu(\()p Fr(N)1119 1518 y Fq(0)p 1117 1548 2 18 v 1109 1548 a Fp(C)1138 1537 y Fu(\))46 b Fs(:)130 1657 y Fu(The)14 b(pro)q(of)f(is)h(an)g(immediate)g(consequence)h(of)f(the)g(follo)o (wing)g(t)o(w)o(o)e(lemmata)i(whic)o(h)g(sho)o(w)f(that)59 1713 y(the)i(t)o(w)o(o)f(systems)h(of)g(norms)f(are)h(in)h(fact)f (equiv)m(alen)o(t.)59 1824 y Fk(Lemma)i(6)23 b Fm(L)n(et)16 b Fs(')c Fr(2)p Fu(E)458 1808 y Fp(\014)458 1835 y(p;q)521 1824 y Fm(then)k Fs(')d Fr(2)g(E)762 1777 y Fi(2)p 741 1783 58 2 v 741 1804 a(1+)p Fg(\014)732 1842 y Fo(2)750 1832 y Fh(\000)p Fg(l)805 1824 y Fu(\()p Fr(H)861 1831 y Fq(\000)p Fp(p;)p 922 1831 2 18 v -3 w(C)943 1824 y Fu(\))j Fm(for)g Fs(l)d Fu(=)1154 1804 y Fp(q)p 1129 1814 67 2 v 1129 1840 a Fo(1+)p Fp(\014)1201 1824 y Fm(.)21 b(Mor)n(e)n(over)624 1920 y Fu(n)649 1927 y Fp(p;l;k)719 1920 y Fu(\()p Fs(')p Fu(\))12 b Fr(\024)h(j)-8 b(j)g(j)o Fs(')p Fr(j)g(j)g(j)920 1934 y Fp(p;q)q(;\014)1022 1920 y Fs(;)39 b(k)14 b Fu(=)1190 1902 y Fo(2)p 1165 1909 V 1165 1936 a(1+)p Fp(\014)1253 1920 y Fs(:)507 b Fu(\(3\))59 2023 y Fk(Pr)o(oof.)124 b Fu(W)l(e)20 b(lo)q(ok)g(at)g(the)g(con)o(v)o (ergence)h(of)e(the)i(series)g Fs(')p Fu(\()p Fs(z)r Fu(\))f(=)1359 1991 y Fj(P)1403 2004 y Fq(1)1403 2034 y Fp(n)p Fo(=0)1472 2023 y Fr(h)p Fs(z)1513 2006 y Fq(\012)p Fp(n)1563 2023 y Fs(;)8 b(')1614 2006 y Fo(\()p Fp(n)p Fo(\))1684 2023 y Fr(i)20 b Fu(,)h Fs(z)i Fr(2)59 2079 y(H)97 2086 y Fq(\000)p Fp(p;)p 158 2086 2 18 v -3 w(C)195 2079 y Fs(;)g(')261 2063 y Fo(\()p Fp(n)p Fo(\))325 2079 y Fr(2)14 b(H)407 2086 y Fp(p;)p 441 2086 V -3 w(C)477 2079 y Fu(if)519 2047 y Fj(P)563 2060 y Fq(1)563 2091 y Fp(n)p Fo(=0)632 2079 y Fu(\()p Fs(n)p Fu(!\))708 2063 y Fo(1+)p Fp(\014)776 2079 y Fu(2)799 2063 y Fp(nq)839 2079 y Fr(j)p Fs(')882 2063 y Fo(\()p Fp(n)p Fo(\))932 2079 y Fr(j)945 2063 y Fo(2)945 2091 y Fp(p)978 2079 y Fu(=)g Fr(j)-8 b(j)g(j)p Fs(')p Fr(j)f(j)h(j)1102 2058 y Fo(2)1102 2093 y Fp(p;q)q(;\014)1196 2079 y Fu(is)16 b(\014nite.)22 b(The)16 b(follo)o(wing)h(estimate)59 2136 y(holds:)204 2203 y Fq(1)191 2216 y Fj(X)188 2306 y Fp(n)p Fo(=0)262 2256 y Fr(jh)p Fs(z)316 2237 y Fq(\012)p Fp(n)367 2256 y Fs(;)8 b(')418 2237 y Fo(\()p Fp(n)p Fo(\))483 2256 y Fr(ij)40 b(\024)631 2184 y Fj( )680 2203 y Fq(1)667 2216 y Fj(X)664 2306 y Fp(n)p Fo(=0)731 2256 y Fu(\()p Fs(n)p Fu(!\))807 2237 y Fo(1+)p Fp(\014)875 2256 y Fu(2)898 2237 y Fp(nq)938 2256 y Fr(j)p Fs(')981 2237 y Fo(\()p Fp(n)p Fo(\))1031 2256 y Fr(j)1044 2237 y Fo(2)1044 2267 y Fp(p)1064 2184 y Fj(!)1096 2196 y Fo(1)p Fp(=)p Fo(2)1159 2184 y Fj( )1207 2203 y Fq(1)1195 2216 y Fj(X)1192 2306 y Fp(n)p Fo(=0)1332 2225 y Fu(1)p 1271 2245 144 2 v 1271 2287 a(\()p Fs(n)p Fu(!\))1347 2274 y Fo(1+)p Fp(\014)1420 2256 y Fu(2)1443 2237 y Fq(\000)p Fp(nq)1510 2256 y Fr(j)p Fs(z)r Fr(j)1559 2237 y Fo(2)p Fp(n)1559 2267 y Fq(\000)p Fp(p)1606 2184 y Fj(!)1639 2196 y Fo(1)p Fp(=)p Fo(2)554 2412 y Fr(\024)i(j)-8 b(j)g(j)p Fs(')p Fr(j)g(j)g(j)707 2426 y Fp(p;q)q(;\014)795 2412 y Fr(\001)817 2340 y Fj( )866 2359 y Fq(1)853 2372 y Fj(X)850 2462 y Fp(n)p Fo(=0)924 2353 y Fj(\032)969 2382 y Fu(1)p 960 2402 40 2 v 960 2443 a Fs(n)p Fu(!)1005 2412 y(2)1028 2390 y Fq(\000)1072 2374 y Fg(nq)p 1060 2382 58 2 v 1060 2403 a Fi(1+)p Fg(\014)1125 2412 y Fr(j)p Fs(z)r Fr(j)1189 2365 y Fi(2)p Fg(n)p 1178 2371 V 1178 2392 a Fi(1+)p Fg(\014)1174 2423 y Fq(\000)p Fp(p)1242 2353 y Fj(\033)1273 2362 y Fo(1+)p Fp(\014)1342 2340 y Fj(!)1375 2352 y Fo(1)p Fp(=)p Fo(2)554 2569 y Fr(\024)42 b(j)-8 b(j)g(j)p Fs(')p Fr(j)g(j)g(j)707 2582 y Fp(p;q)q(;\014)792 2497 y Fj( )840 2516 y Fq(1)828 2528 y Fj(X)825 2619 y Fp(n)p Fo(=0)913 2538 y Fu(1)p 904 2558 40 2 v 904 2600 a Fs(n)p Fu(!)949 2569 y(2)972 2546 y Fq(\000)1015 2530 y Fg(nq)p 1004 2538 58 2 v 1004 2559 a Fi(1+)p Fg(\014)1068 2569 y Fr(j)p Fs(z)r Fr(j)1133 2522 y Fi(2)p Fg(n)p 1122 2528 V 1122 2548 a Fi(1+)p Fg(\014)1117 2580 y Fq(\000)p Fp(p)1186 2497 y Fj(!)1219 2508 y Fo(\(1+)p Fp(\014)r Fo(\))p Fp(=)p Fo(2)554 2703 y Fr(\024)42 b(j)-8 b(j)g(j)p Fs(')p Fr(j)g(j)g(j)707 2716 y Fp(p;q)q(;\014)792 2703 y Fu(exp)869 2643 y Fj(\022)900 2703 y Fu(2)923 2680 y Fq(\000)976 2664 y Fg(q)p 955 2672 V 955 2693 a Fi(1+)p Fg(\014)1019 2703 y Fr(j)p Fs(z)r Fr(j)1093 2656 y Fi(2)p 1072 2662 V 1072 2682 a(1+)p Fg(\014)1068 2714 y Fq(\000)p Fp(p)1137 2643 y Fj(\023)1175 2703 y Fs(:)1797 2796 y Ff(2)p eop %%Page: 7 7 7 6 bop 59 50 a Fn(2)45 b(PRELIMINARIES)1311 b Fu(7)59 199 y Fk(Lemma)17 b(7)23 b Fm(F)m(or)16 b(any)g Fs(p)471 183 y Fq(0)483 199 y Fs(;)8 b(q)13 b Fr(2)g Fs(I)-12 b(N)22 b Fm(ther)n(e)16 b(exist)g Fs(p;)8 b(l)k Fr(2)h Fs(I)-12 b(N)21 b Fm(such)16 b(that)744 305 y Fr(E)798 258 y Fi(2)p 777 264 58 2 v 777 285 a(1+)p Fg(\014)768 323 y Fo(2)786 314 y Fh(\000)p Fg(l)841 305 y Fu(\()p Fr(H)897 312 y Fq(\000)p Fp(p;)p 958 312 2 18 v -3 w(C)979 305 y Fu(\))c Fr(\032)h Fu(E)1088 283 y Fp(\014)1088 320 y(p)1106 310 y Fh(0)1117 320 y Fp(;q)59 392 y Fm(i.e.,)j(ther)n(e)g (exists)g(a)g(c)n(onstant)f Fs(C)h(>)d Fu(0)j Fm(such)g(that)414 478 y Fr(j)-8 b(j)g(j)p Fs(')p Fr(j)f(j)h(j)489 492 y Fp(p)507 482 y Fh(0)518 492 y Fp(;q)q(;\014)591 478 y Fr(\024)13 b Fs(C)i Fu(n)712 485 y Fp(p;l;k)782 478 y Fu(\()p Fs(')p Fu(\))p Fs(;)53 b(')12 b Fr(2)h(E)1027 460 y Fp(k)1023 493 y Fo(2)1041 484 y Fh(\000)p Fg(l)1079 478 y Fu(\()p Fr(H)1135 485 y Fq(\000)p Fp(p;)p 1196 485 V -3 w(C)1217 478 y Fu(\))p Fs(;)53 b(k)14 b Fu(=)1416 460 y Fo(2)p 1392 467 67 2 v 1392 494 a(1+)p Fp(\014)1463 478 y Fs(:)59 584 y Fk(Remark.)73 b Fu(More)14 b(precisely)j(w)o(e)e (will)h(pro)o(v)o(e)e(the)h(follo)o(wing:)h(If)f Fs(')d Fr(2)h(E)1344 568 y Fp(k)1340 601 y Fo(2)1358 591 y Fh(\000)p Fg(l)1396 584 y Fu(\()p Fr(H)1452 591 y Fq(\000)p Fp(p;)p 1513 591 2 18 v -3 w(C)1534 584 y Fu(\))h(then)h Fs(')e Fr(2)p Fu(E)1773 562 y Fp(\014)1773 598 y(p)1791 589 y Fh(0)1802 598 y Fp(;q)59 653 y Fu(for)j Fs(k)f Fu(=)248 636 y Fo(2)p 224 643 67 2 v 224 669 a(1+)p Fp(\014)311 653 y Fu(and)i Fs(\032)d Fu(:=)g(2)524 637 y Fp(q)q Fq(\000)p Fo(2)p Fp(l=k)635 653 y Fs(k)660 637 y Fo(2)p Fp(=k)717 653 y Fs(e)738 637 y Fo(2)766 617 y Fj(\015)766 642 y(\015)789 653 y Fs(i)805 661 y Fp(p)823 652 y Fh(0)833 661 y Fp(;p)863 617 y Fj(\015)863 642 y(\015)886 630 y Fo(2)886 670 y Fp(H)s(S)957 653 y Fs(<)h Fu(1)h(\(in)g(particular)h(this)g(requires)g Fs(p)1613 637 y Fq(0)1639 653 y Fs(>)d(p)i Fu(to)g(b)q(e)59 710 y(suc)o(h)g(that)e(the)h(em)o(b)q(edding)i Fs(i)584 718 y Fp(p)602 708 y Fh(0)613 718 y Fp(;p)655 710 y Fu(:)12 b Fr(H)718 718 y Fp(p)736 708 y Fh(0)762 710 y Fs(,)-8 b Fr(!)13 b(H)863 717 y Fp(p)899 710 y Fu(is)j(Hilb)q(ert-Sc)o (hmidt\).)59 766 y(Moreo)o(v)o(er)e(the)h(follo)o(wing)h(b)q(ound)g (holds)612 853 y Fr(j)-8 b(j)g(j)p Fs(')p Fr(j)g(j)g(j)688 866 y Fp(p)706 857 y Fh(0)717 866 y Fp(;q)q(;\014)789 853 y Fr(\024)13 b Fu(n)862 860 y Fp(p;l;k)932 853 y Fu(\()p Fs(')p Fu(\))c Fr(\001)h Fu(\(1)g Fr(\000)g Fs(\032)p Fu(\))1167 832 y Fq(\000)p Fo(1)p Fp(=)p Fo(2)1272 853 y Fs(:)488 b Fu(\(4\))59 946 y Fk(Pr)o(oof.)93 b Fu(The)15 b(assumption)h Fs(')c Fr(2)h(E)756 930 y Fp(k)752 963 y Fo(2)770 954 y Fh(\000)p Fg(l)808 946 y Fu(\()p Fr(H)864 953 y Fq(\000)p Fp(p;)p 925 953 2 18 v -3 w(C)945 946 y Fu(\))i(implies)j(a)c(b)q(ound)j(of)d(the)i(gro)o(wth)e(of)g Fs(')f Fu(:)628 1039 y Fr(j)p Fs(')p Fu(\()p Fs(z)r Fu(\))p Fr(j)e(\024)i Fu(n)827 1046 y Fp(p;l;k)897 1039 y Fu(\()p Fs(')p Fu(\))8 b(exp)o(\(2)1080 1021 y Fq(\000)p Fp(l)1120 1039 y Fr(j)p Fs(z)r Fr(j)1169 1021 y Fp(k)1169 1051 y Fq(\000)p Fp(p)1216 1039 y Fu(\))15 b Fs(:)59 1126 y Fu(F)l(or)g(eac)o(h)g Fs(\032)d(>)h Fu(0)i Fs(;)22 b(z)15 b Fr(2)e(H)517 1133 y Fq(\000)p Fp(p;)p 578 1133 V -3 w(C)614 1126 y Fu(the)i(Cauc)o(h)o(y)g(inequalit)o(y)i(from)e (complex)h(analysis)g([Di81)o(])f(giv)o(es)483 1173 y Fj(\014)483 1198 y(\014)483 1223 y(\014)483 1248 y(\014)510 1204 y Fu(1)p 502 1224 40 2 v 502 1266 a Fs(n)p Fu(!)585 1218 y Fj(d)547 1235 y Fu(d)572 1222 y Fp(n)595 1235 y Fs(')p Fu(\(0\))o(\()p Fs(z)r Fu(\))742 1173 y Fj(\014)742 1198 y(\014)742 1223 y(\014)742 1248 y(\014)768 1235 y Fr(\024)e Fu(n)841 1242 y Fp(p;l;k)911 1235 y Fu(\()p Fs(')p Fu(\))p Fs(\032)1001 1216 y Fq(\000)p Fp(n)1058 1235 y Fu(exp\()p Fs(\032)1169 1216 y Fp(k)1190 1235 y Fu(2)1213 1216 y Fq(\000)p Fp(l)1253 1235 y Fu(\))f Fr(j)p Fs(z)r Fr(j)1332 1216 y Fp(n)1332 1246 y Fq(\000)p Fp(p)1394 1235 y Fs(:)59 1341 y Fu(By)j(p)q(olarization)i([Di81)o(])e (it)g(follo)o(ws)h(for)e Fs(z)800 1348 y Fo(1)820 1341 y Fs(;)8 b(:)g(:)g(:)d(;)j(z)943 1348 y Fp(n)978 1341 y Fr(2)13 b(H)1059 1348 y Fq(\000)p Fp(p;)p 1121 1348 2 18 v -3 w(C)306 1399 y Fj(\014)306 1424 y(\014)306 1449 y(\014)306 1474 y(\014)333 1430 y Fu(1)p 324 1450 40 2 v 324 1492 a Fs(n)p Fu(!)369 1460 y(d)394 1442 y Fp(n)418 1460 y Fs(')p Fu(\(0\)\()p Fs(z)546 1467 y Fo(1)564 1460 y Fs(;)8 b(:)g(:)g(:)d(;)j(z)687 1467 y Fp(n)710 1460 y Fu(\))728 1399 y Fj(\014)728 1424 y(\014)728 1449 y(\014)728 1474 y(\014)754 1460 y Fr(\024)13 b Fu(n)827 1467 y Fp(p;l;k)897 1460 y Fu(\()p Fs(')p Fu(\))976 1430 y(1)p 968 1450 V 968 1492 a Fs(n)p Fu(!)1020 1401 y Fj(\022)1055 1430 y Fs(n)p 1055 1450 28 2 v 1057 1492 a(\032)1087 1401 y Fj(\023)1118 1412 y Fp(n)1149 1460 y Fu(exp\()p Fs(\032)1260 1442 y Fp(k)1281 1460 y Fu(2)1304 1442 y Fq(\000)p Fp(l)1344 1460 y Fu(\))1391 1407 y Fp(n)1375 1420 y Fj(Y)1369 1512 y Fp(k)q Fo(=1)1441 1460 y Fr(j)p Fs(z)1475 1467 y Fp(k)1496 1460 y Fr(j)1509 1467 y Fq(\000)p Fp(p)1571 1460 y Fs(:)59 1587 y Fu(F)l(or)k Fs(p)166 1570 y Fq(0)194 1587 y Fs(>)g(p)g Fu(suc)o(h)h(that)492 1551 y Fj(\015)492 1575 y(\015)515 1587 y Fs(i)531 1594 y Fp(p)549 1585 y Fh(0)560 1594 y Fp(;p)589 1551 y Fj(\015)589 1575 y(\015)612 1603 y Fp(H)s(S)687 1587 y Fu(is)g(\014nite,)h(an)e (application)i(of)e(the)h(k)o(ernel)g(theorem)f(guaran)o(tees)59 1655 y(the)e(existence)i(of)e(k)o(ernels)g Fs(')564 1639 y Fo(\()p Fp(n)p Fo(\))628 1655 y Fr(2)e(H)714 1630 y Fo(^)709 1636 y Fq(\012)q Fp(n)709 1669 y(p)727 1660 y Fh(0)738 1669 y Fp(;)p 754 1669 2 18 v -3 w(C)790 1655 y Fu(suc)o(h)i(that)724 1781 y Fs(')p Fu(\()p Fs(z)r Fu(\))d(=)889 1728 y Fq(1)877 1741 y Fj(X)873 1831 y Fp(n)p Fo(=0)940 1781 y Fr(h)p Fs(z)986 1757 y Fo(^)981 1763 y Fq(\012)p Fp(n)1032 1781 y Fs(;)c(')1083 1763 y Fo(\()p Fp(n)p Fo(\))1148 1781 y Fr(i)59 1901 y Fu(with)16 b(the)f(b)q(ound)442 1949 y Fj(\014)442 1974 y(\014)442 1999 y(\014)456 1998 y Fs(')486 1979 y Fo(\()p Fp(n)p Fo(\))537 1949 y Fj(\014)537 1974 y(\014)537 1999 y(\014)551 2026 y Fp(p)569 2016 y Fh(0)594 1998 y Fr(\024)e Fu(n)667 2005 y Fp(p;l;k)737 1998 y Fu(\()p Fs(')p Fu(\))816 1967 y(1)p 808 1987 40 2 v 808 2029 a Fs(n)p Fu(!)860 1938 y Fj(\022)895 1967 y Fs(n)p 895 1987 28 2 v 897 2029 a(\032)935 1962 y Fj(\015)935 1986 y(\015)958 1998 y Fs(i)974 2005 y Fp(p)992 1996 y Fh(0)1003 2005 y Fp(;p)1032 1962 y Fj(\015)1032 1986 y(\015)1055 2014 y Fp(H)s(S)1112 1938 y Fj(\023)1143 1950 y Fp(n)1174 1998 y Fu(exp\()p Fs(\032)1285 1979 y Fp(k)1316 1998 y Fr(\001)d Fu(2)1362 1979 y Fq(\000)p Fp(l)1402 1998 y Fu(\))15 b Fs(:)59 2112 y Fu(W)l(e)h(can)g(optimize)i(the)e(b)q(ound)h(with)f(the)h(c)o (hoice)f(of)g(an)g Fs(n)p Fu(-dep)q(enden)o(t)i Fs(\032)p Fu(.)k(Setting)17 b Fs(\032)1568 2095 y Fp(k)1603 2112 y Fu(=)d(2)1675 2095 y Fp(l)1688 2112 y Fs(n=k)j Fu(w)o(e)59 2168 y(obtain)380 2214 y Fj(\014)380 2239 y(\014)380 2264 y(\014)394 2263 y Fs(')424 2244 y Fo(\()p Fp(n)p Fo(\))475 2214 y Fj(\014)475 2239 y(\014)475 2264 y(\014)489 2291 y Fp(p)507 2281 y Fh(0)561 2263 y Fr(\024)42 b Fu(n)663 2270 y Fp(p;l;k)733 2263 y Fu(\()p Fs(')p Fu(\))811 2232 y(1)p 804 2252 40 2 v 804 2294 a Fs(n)p Fu(!)848 2263 y Fs(n)875 2244 y Fp(n)p Fo(\(1)p Fq(\000)p Fo(1)p Fp(=k)q Fo(\))1033 2216 y Fj(\020)1064 2245 y Fo(1)p 1063 2252 20 2 v 1063 2278 a Fp(k)1087 2263 y Fu(2)1110 2244 y Fp(l)1123 2216 y Fj(\021)1148 2227 y Fq(\000)p Fp(n=k)1243 2226 y Fj(\015)1243 2251 y(\015)1266 2263 y Fs(i)1282 2270 y Fp(p)1300 2261 y Fh(0)1311 2270 y Fp(;p)1340 2226 y Fj(\015)1340 2251 y(\015)1363 2240 y Fp(n)1363 2279 y(H)s(S)1428 2263 y Fs(e)1449 2244 y Fp(n=k)561 2364 y Fr(\024)g Fu(n)663 2371 y Fp(p;l;k)733 2364 y Fu(\()p Fs(')p Fu(\))12 b(\()p Fs(n)p Fu(!\))887 2345 y Fq(\000)p Fo(1)p Fp(=k)977 2317 y Fj(n)1005 2364 y Fu(\()p Fs(k)q Fu(2)1071 2345 y Fq(\000)p Fp(l)1111 2364 y Fu(\))1129 2345 y Fo(1)p Fp(=k)1185 2364 y Fs(e)1214 2327 y Fj(\015)1214 2352 y(\015)1237 2364 y Fs(i)1253 2371 y Fp(p)1271 2362 y Fh(0)1282 2371 y Fp(;p)1311 2327 y Fj(\015)1311 2352 y(\015)1334 2380 y Fp(H)s(S)1391 2317 y Fj(o)1419 2328 y Fp(n)1465 2364 y Fs(;)59 2463 y Fu(where)17 b(w)o(e)f(used)h Fs(n)392 2447 y Fp(n)431 2463 y Fr(\024)e Fs(n)p Fu(!)8 b Fs(e)550 2447 y Fp(n)590 2463 y Fu(in)17 b(the)f(last)h(estimate.)24 b(No)o(w)15 b(c)o(ho)q(ose)i Fs(\014)g Fr(2)e Fu([0)p Fs(;)8 b Fu(1])14 b(suc)o(h)j(that)f Fs(k)g Fu(=)1784 2445 y Fo(2)p 1759 2452 67 2 v 1759 2479 a(1+)p Fp(\014)59 2520 y Fu(to)f(estimate)g(the)g(follo)o(wing)h(norm:)307 2634 y Fr(j)-8 b(j)g(j)p Fs(')p Fr(j)f(j)h(j)382 2613 y Fo(2)382 2647 y Fp(p)400 2638 y Fh(0)411 2647 y Fp(;q)q(;\014)513 2634 y Fr(\024)41 b Fu(n)614 2615 y Fo(2)614 2645 y Fp(p;l;k)684 2634 y Fu(\()p Fs(')p Fu(\))773 2581 y Fq(1)760 2594 y Fj(X)757 2684 y Fp(n)p Fo(=0)824 2634 y Fu(\()p Fs(n)p Fu(!\))900 2615 y Fo(1+)p Fp(\014)r Fq(\000)999 2602 y Fi(2)p 998 2608 17 2 v 998 2628 a Fg(k)1022 2634 y Fu(2)1045 2615 y Fp(q)q(n)1093 2587 y Fj(n)1120 2634 y Fu(\()p Fs(k)q Fu(2)1186 2615 y Fq(\000)p Fp(l)1226 2634 y Fu(\))1244 2615 y Fo(1)p Fp(=k)1300 2634 y Fs(e)1329 2598 y Fj(\015)1329 2623 y(\015)1352 2634 y Fs(i)1368 2642 y Fp(p)1386 2632 y Fh(0)1397 2642 y Fp(;p)1427 2598 y Fj(\015)1427 2623 y(\015)1450 2650 y Fp(H)s(S)1507 2587 y Fj(o)1534 2598 y Fo(2)p Fp(n)513 2774 y Fr(\024)g Fu(n)614 2756 y Fo(2)614 2786 y Fp(p;l;k)684 2774 y Fu(\()p Fs(')p Fu(\))758 2715 y Fj(\022)788 2774 y Fu(1)9 b Fr(\000)i Fu(2)889 2756 y Fp(q)915 2727 y Fj(n)943 2774 y Fu(\()p Fs(k)q Fu(2)1009 2756 y Fq(\000)p Fp(l)1049 2774 y Fu(\))1067 2756 y Fo(1)p Fp(=k)1123 2774 y Fs(e)1152 2738 y Fj(\015)1152 2763 y(\015)1175 2774 y Fs(i)1191 2782 y Fp(p)1209 2773 y Fh(0)1220 2782 y Fp(;p)1249 2738 y Fj(\015)1249 2763 y(\015)1272 2791 y Fp(H)s(S)1329 2727 y Fj(o)1357 2739 y Fo(2)1377 2715 y Fj(\023)1407 2726 y Fq(\000)p Fo(1)p eop %%Page: 8 8 8 7 bop 59 50 a Fn(3)45 b(MEASURES)16 b(ON)g(LINEAR)g(TOPOLOGICAL)h(SP) l(A)o(CES)598 b Fu(8)59 199 y(for)15 b(su\016cien)o(tly)h(large)f Fs(l)q Fu(.)20 b(This)15 b(completes)h(the)g(pro)q(of.)773 b Ff(2)130 306 y Fu(Analogous)15 b(estimates)g(for)f(these)i(systems)f (of)f(norms)h(also)g(hold)h(if)g Fs(\014)r(;)8 b(p;)g(q)r(;)g(l)13 b Fu(b)q(ecome)j(negativ)o(e.)59 362 y(This)g(implies)h(the)e(follo)o (wing)h(theorem.)k(F)l(or)14 b(related)i(results)g(see)f(e.g.,)f ([Ou91,)h(Prop.8.6].)59 468 y Fk(Theorem)j(8)59 525 y Fm(If)e Fs(\014)f Fr(2)d Fu([0)p Fs(;)c Fu(1\))15 b Fm(then)h(the)g (fol)r(lowing)g(top)n(olo)n(gic)n(al)g(identity)g(holds:)627 631 y Fu(ind)h(lim)645 661 y Fp(p;q)q Fq(2)p Fp(I)-9 b(N)798 631 y Fu(E)829 609 y Fq(\000)p Fp(\014)829 642 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)943 631 y Fu(=)13 b Fr(E)1019 612 y Fo(2)p Fp(=)p Fo(\(1)p Fq(\000)p Fp(\014)r Fo(\))1015 642 y(max)1150 631 y Fu(\()p Fr(N)p 1211 638 2 18 v 1202 638 a Fp(C)1232 631 y Fu(\))p Fs(:)59 753 y Fm(If)j Fs(\014)f Fu(=)d(1)k Fm(we)h(have)655 809 y Fu(ind)f(lim)672 839 y Fp(p;q)q Fq(2)p Fp(I)-9 b(N)825 809 y Fu(E)856 790 y Fq(\000)p Fo(1)856 820 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)970 809 y Fu(=)13 b(Hol)1087 816 y Fo(0)1107 809 y Fu(\()p Fr(N)p 1168 816 V 1159 816 a Fp(C)1189 809 y Fu(\))j Fs(:)59 935 y Fu(This)h(theorem)f(and)h(its)g (pro)q(of)f(will)i(app)q(ear)f(in)g(the)g(con)o(text)f(of)g(section)h (8.)24 b(The)17 b(c)o(haracterization)59 991 y(of)f(distributions)h(in) g(in\014nite)g(dimensional)h(analysis)f(is)f(strongly)g(related)g(to)f (this)i(theorem.)k(F)l(rom)59 1048 y(this)16 b(p)q(oin)o(t)f(of)g(view) h(it)f(is)h(natural)f(to)g(p)q(ostp)q(one)h(its)f(pro)q(of)g(to)f (section)i(8.)59 1191 y Ft(3)67 b(Measures)21 b(on)h(linear)i(top)r (ological)f(spaces)59 1292 y Fu(T)l(o)e(in)o(tro)q(duce)g(probabilit)o (y)h(measures)f(on)g(the)f(v)o(ector)g(space)h Fr(N)1243 1276 y Fq(0)1255 1292 y Fu(,)h(w)o(e)e(consider)i Fr(C)1569 1299 y Fp(\033)1592 1292 y Fu(\()p Fr(N)1654 1276 y Fq(0)1666 1292 y Fu(\))e(the)h Fs(\033)r Fu(-)59 1349 y(algebra)11 b(generated)g(b)o(y)g(cylinder)h(sets)f(on)g Fr(N)832 1332 y Fq(0)843 1349 y Fu(,)h(whic)o(h)f(coincides)i(with)f(the)e (Borel)i Fs(\033)r Fu(-algebras)e Fr(B)1716 1356 y Fp(\033)1740 1349 y Fu(\()p Fr(N)1802 1332 y Fq(0)1813 1349 y Fu(\))59 1405 y(and)15 b Fr(B)177 1412 y Fp(\014)201 1405 y Fu(\()p Fr(N)263 1389 y Fq(0)274 1405 y Fu(\))g(generated)g(b)o(y)g(the)g(w)o (eak)g(and)g(strong)f(top)q(ology)h(on)g Fr(N)1290 1389 y Fq(0)1316 1405 y Fu(resp)q(ectiv)o(ely)l(.)22 b(Th)o(us)15 b(w)o(e)g(will)59 1462 y(consider)i(this)g Fs(\033)r Fu(-algebra)f(as)g(the)g Fm(natur)n(al)h Fs(\033)r Fu(-algebra)f(on)g Fr(N)1134 1445 y Fq(0)1145 1462 y Fu(.)24 b(Detailed)17 b(de\014nitions)h(of)e(the)g(ab)q(o)o(v)o(e)59 1518 y(notions)f(and)h (pro)q(ofs)f(of)f(the)i(men)o(tioned)g(relations)f(can)h(b)q(e)f(found) h(in)g(e.g.,)e([BeKo88].)130 1575 y(W)l(e)j(will)i(restrict)f(our)f(in) o(v)o(estigations)h(to)f(a)g(sp)q(ecial)i(class)f(of)f(measures)h Fs(\026)f Fu(on)h Fr(C)1571 1582 y Fp(\033)1594 1575 y Fu(\()p Fr(N)1656 1558 y Fq(0)1667 1575 y Fu(\),)g(whic)o(h)59 1631 y(satisfy)13 b(t)o(w)o(o)f(additional)i(assumptions.)19 b(The)14 b(\014rst)e(one)i(concerns)f(some)g(analyticit)o(y)h(of)e(the) i(Laplace)59 1688 y(transformation)392 1795 y Fs(l)406 1802 y Fp(\026)429 1795 y Fu(\()p Fs(\022)q Fu(\))e(=)547 1737 y Fj(Z)570 1832 y Fq(N)604 1822 y Fh(0)625 1795 y Fu(exp)c Fr(h)p Fs(x;)g(\022)q Fr(i)22 b Fu(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))12 b(=:)h Fs(I)-12 b(E)1061 1802 y Fp(\026)1084 1795 y Fu(\(exp)8 b Fr(h)o Fs(x;)g(\022)q Fr(i)p Fu(\))14 b(,)h Fs(\022)f Fr(2)f(N)p 1464 1802 V 1455 1802 a Fp(C)1485 1795 y Fu(.)59 1913 y(Here)k(w)o(e)g(also)g(ha) o(v)o(e)f(in)o(tro)q(duced)i(the)f(con)o(v)o(enien)o(t)h(notion)f(of)f (exp)q(ectation)i Fs(I)-12 b(E)1459 1920 y Fp(\026)1499 1913 y Fu(of)17 b(a)f Fs(\026)p Fu(-in)o(tegrable)59 1970 y(function.)59 2026 y Fk(Assumption)k Fu(1)64 b(The)18 b(measure)h Fs(\026)g Fu(has)f(an)h(analytic)g(Laplace)h(transform)d (in)i(a)f(neigh)o(b)q(orho)q(o)q(d)59 2083 y(of)h(zero.)32 b(That)19 b(means)g(there)h(exists)g(an)f(op)q(en)h(neigh)o(b)q(orho)q (o)q(d)h Fr(U)j(\032)c(N)p 1374 2090 V 1365 2090 a Fp(C)1415 2083 y Fu(of)f(zero,)g(suc)o(h)h(that)f Fs(l)1808 2090 y Fp(\026)59 2139 y Fu(is)e(holomorphic)h(on)f Fr(U)5 b Fu(,)17 b(i.e.,)g Fs(l)598 2146 y Fp(\026)637 2139 y Fr(2)e Fu(Hol)752 2146 y Fo(0)772 2139 y Fu(\()p Fr(N)p 833 2146 V 824 2146 a Fp(C)853 2139 y Fu(\))i(.)25 b(This)17 b(class)h(of)e Fm(analytic)i(me)n(asur)n(es)e Fu(is)h(denoted)h(b)o(y) 59 2195 y Fr(M)114 2202 y Fp(a)134 2195 y Fu(\()p Fr(N)196 2179 y Fq(0)208 2195 y Fu(\))p Fs(:)59 2302 y Fu(An)e(equiv)m(alen)o(t) g(description)h(of)e(analytic)h(measures)f(is)h(giv)o(en)f(b)o(y)h(the) f(follo)o(wing)h(lemma.)59 2408 y Fk(Lemma)h(9)23 b Fm(The)16 b(fol)r(lowing)g(statements)g(ar)n(e)g(e)n(quivalent)59 2489 y Fv(1\))47 b Fs(\026)13 b Fr(2)g(M)290 2496 y Fp(a)310 2489 y Fu(\()p Fr(N)372 2473 y Fq(0)384 2489 y Fu(\))59 2578 y Fv(2\))47 b Fr(9)p Fs(p)200 2585 y Fp(\026)236 2578 y Fr(2)13 b Fs(I)-12 b(N)5 b(;)54 b Fr(9)p Fs(C)15 b(>)e Fu(0)f(:)671 2517 y Fj(\014)671 2542 y(\014)671 2567 y(\014)671 2592 y(\014)684 2521 y(Z)708 2615 y Fq(N)742 2606 y Fh(0)771 2578 y Fr(h)p Fs(x;)c(\022)q Fr(i)876 2560 y Fp(n)911 2578 y Fu(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))1025 2517 y Fj(\014)1025 2542 y(\014)1025 2567 y(\014)1025 2592 y(\014)1052 2578 y Fr(\024)13 b Fs(n)p Fu(!)8 b Fs(C)1184 2560 y Fp(n)1214 2578 y Fr(j)o Fs(\022)q Fr(j)1262 2558 y Fp(n)1262 2592 y(p)1280 2596 y Fg(\026)1322 2578 y Fs(;)54 b(\022)14 b Fr(2)f(H)1505 2585 y Fp(p)1523 2589 y Fg(\026)1544 2585 y Fp(;)p 1560 2585 V -3 w(C)59 2691 y Fv(3\))47 b Fr(9)p Fs(p)200 2672 y Fq(0)200 2702 y Fp(\026)236 2691 y Fr(2)13 b Fs(I)-12 b(N)5 b(;)54 b Fr(9)p Fs(")442 2698 y Fp(\026)478 2691 y Fs(>)13 b Fu(0)f(:)696 2634 y Fj(Z)719 2728 y Fq(N)753 2718 y Fh(0)790 2691 y Fu(exp\()p Fs(")898 2698 y Fp(\026)929 2691 y Fr(j)o Fs(x)p Fr(j)980 2704 y Fq(\000)p Fp(p)1025 2695 y Fh(0)1025 2713 y Fg(\026)1048 2691 y Fu(\))c(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))k Fs(<)h Fr(1)p eop %%Page: 9 9 9 8 bop 59 50 a Fn(3)45 b(MEASURES)16 b(ON)g(LINEAR)g(TOPOLOGICAL)h(SP) l(A)o(CES)598 b Fu(9)59 199 y Fk(Pr)o(oof.)83 b Fu(The)19 b(pro)q(of)g(can)g(b)q(e)g(found)g(in)g([KoSW95].)29 b(W)l(e)19 b(giv)o(e)g(its)g(outline)h(in)g(the)e(follo)o(wing.)59 256 y(The)d(only)h(non-trivial)h(step)e(is)h(the)f(pro)q(of)g(of)g(2\)) p Fr(\))p Fu(3\).)59 312 y(By)g(p)q(olarization)i([Di81)o(])e(2\))g (implies)413 373 y Fj(\014)413 398 y(\014)413 423 y(\014)413 448 y(\014)413 473 y(\014)427 389 y(Z)450 484 y Fq(N)484 474 y Fh(0)497 447 y Fr(h)p Fs(x)541 428 y Fq(\012)p Fp(n)592 447 y Fs(;)632 407 y Fp(n)620 415 y Fj(N)612 485 y Fp(j)r Fo(=1)681 447 y Fs(\030)701 454 y Fp(j)719 447 y Fr(i)g Fu(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))866 373 y Fj(\014)866 398 y(\014)866 423 y(\014)866 448 y(\014)866 473 y(\014)892 447 y Fr(\024)e Fs(n)p Fu(!)g Fs(C)1029 428 y Fp(n)1080 394 y(n)1064 407 y Fj(Y)1060 498 y Fp(j)r Fo(=1)1129 447 y Fr(j)o Fs(\030)1161 454 y Fp(j)1179 447 y Fr(j)1192 462 y Fp(p)1210 466 y Fg(\026)1255 447 y Fs(;)53 b(\030)1341 454 y Fp(j)1372 447 y Fr(2)13 b(H)1453 455 y Fp(p)1471 445 y Fh(0)1773 447 y Fu(\(5\))59 593 y(for)j(a)f(\(new\))h(constan)o(t)f Fs(C)i(>)e Fu(0.)22 b(Cho)q(ose)16 b Fs(p)822 577 y Fq(0)848 593 y Fs(>)e(p)920 600 y Fp(\026)960 593 y Fu(suc)o(h)i(that)g(the)g(em)o(b)q(edding)i Fs(i)1489 601 y Fp(p)1507 591 y Fh(0)1517 601 y Fp(;p)1545 605 y Fg(\026)1582 593 y Fu(:)c Fr(H)1647 601 y Fp(p)1665 591 y Fh(0)1692 593 y Fr(!)h(H)1790 600 y Fp(p)1808 604 y Fg(\026)59 650 y Fu(is)h(of)f(Hilb)q(ert-Sc)o(hmidt)k(t)o(yp)q(e.)i (Let)16 b Fr(f)p Fs(e)740 657 y Fp(k)761 650 y Fs(;)22 b(k)14 b Fr(2)f Fs(I)-12 b(N)5 b Fr(g)13 b(\032)h(N)22 b Fu(b)q(e)17 b(an)e(orthonormal)g(basis)h(in)h Fr(H)1662 657 y Fp(p)1680 648 y Fh(0)1693 650 y Fu(.)k(Then)59 727 y Fr(j)p Fs(x)p Fr(j)110 706 y Fo(2)110 740 y Fq(\000)p Fp(p)155 731 y Fh(0)181 727 y Fu(=)244 686 y Fq(1)240 695 y Fj(P)229 766 y Fp(k)q Fo(=1)301 727 y Fr(h)p Fs(x;)8 b(e)387 734 y Fp(k)408 727 y Fr(i)425 706 y Fo(2)445 727 y Fu(,)15 b Fs(x)d Fr(2)h(H)592 734 y Fq(\000)p Fp(p)637 725 y Fh(0)651 727 y Fu(.)20 b(W)l(e)15 b(will)i(\014rst)e(estimate)g (the)g(momen)o(ts)g(of)g(ev)o(en)g(order)307 834 y Fj(Z)330 928 y Fq(N)364 919 y Fh(0)385 891 y Fr(j)o Fs(x)p Fr(j)436 870 y Fo(2)p Fp(n)436 905 y Fq(\000)p Fp(p)481 895 y Fh(0)517 891 y Fu(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))e(=)714 838 y Fq(1)702 851 y Fj(X)692 943 y Fp(k)710 948 y Fi(1)727 943 y Fo(=1)780 891 y Fr(\001)8 b(\001)g(\001)865 838 y Fq(1)853 851 y Fj(X)841 943 y Fp(k)859 947 y Fg(n)880 943 y Fo(=1)933 834 y Fj(Z)956 928 y Fq(N)990 919 y Fh(0)1011 891 y Fr(h)o Fs(x;)g(e)1096 898 y Fp(k)1114 903 y Fi(1)1133 891 y Fr(i)1151 870 y Fo(2)1194 891 y Fr(\001)g(\001)g(\001)d(h)p Fs(x;)j(e)1340 898 y Fp(k)1358 902 y Fg(n)1381 891 y Fr(i)1399 870 y Fo(2)1441 891 y Fu(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))15 b(,)59 1033 y(where)k(w)o(e)g(c)o(hanged)h(the)f(order) g(of)g(summation)g(and)g(in)o(tegration)g(b)o(y)g(a)g(monotone)f(con)o (v)o(ergence)59 1090 y(argumen)o(t.)h(Using)d(the)f(b)q(ound)h(\(5\))f (w)o(e)g(ha)o(v)o(e)332 1165 y Fj(Z)355 1259 y Fq(N)389 1250 y Fh(0)410 1222 y Fr(j)o Fs(x)p Fr(j)461 1201 y Fo(2)p Fp(n)461 1236 y Fq(\000)p Fp(p)506 1226 y Fh(0)542 1222 y Fu(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))42 b Fr(\024)57 b Fs(C)826 1203 y Fo(2)p Fp(n)882 1222 y Fu(\(2)p Fs(n)p Fu(\)!)1010 1169 y Fq(1)998 1182 y Fj(X)987 1274 y Fp(k)1005 1279 y Fi(1)1023 1274 y Fo(=1)1076 1222 y Fr(\001)8 b(\001)g(\001)1161 1169 y Fq(1)1149 1182 y Fj(X)1136 1274 y Fp(k)1154 1278 y Fg(n)1176 1274 y Fo(=1)1228 1222 y Fr(j)p Fs(e)1262 1229 y Fp(k)1280 1234 y Fi(1)1300 1222 y Fr(j)1313 1201 y Fo(2)1313 1236 y Fp(p)1331 1240 y Fg(\026)1361 1222 y Fr(\001)g(\001)g(\001)d(j)p Fs(e)1455 1229 y Fp(k)1473 1233 y Fg(n)1497 1222 y Fr(j)1510 1201 y Fo(2)1510 1236 y Fp(p)1528 1240 y Fg(\026)698 1374 y Fu(=)57 b Fs(C)826 1355 y Fo(2)p Fp(n)882 1374 y Fu(\(2)p Fs(n)p Fu(\)!)989 1302 y Fj( )1035 1321 y Fq(1)1022 1334 y Fj(X)1020 1426 y Fp(k)q Fo(=1)1092 1374 y Fr(j)p Fs(e)1126 1381 y Fp(k)1147 1374 y Fr(j)1160 1353 y Fo(2)1160 1388 y Fp(p)1178 1392 y Fg(\026)1200 1302 y Fj(!)1233 1314 y Fp(n)698 1504 y Fu(=)790 1457 y Fj(\020)814 1504 y Fs(C)13 b Fr(\001)883 1455 y Fj(\015)883 1480 y(\015)883 1505 y(\015)906 1504 y Fs(i)922 1511 y Fp(p)940 1502 y Fh(0)951 1511 y Fp(;p)979 1515 y Fg(\026)1001 1455 y Fj(\015)1001 1480 y(\015)1001 1505 y(\015)1024 1532 y Fp(H)s(S)1081 1457 y Fj(\021)1106 1468 y Fo(2)p Fp(n)1155 1504 y Fu(\(2)p Fs(n)p Fu(\)!)59 1613 y(b)q(ecause)722 1636 y Fq(1)710 1648 y Fj(X)708 1740 y Fp(k)q Fo(=1)780 1689 y Fr(j)o Fs(e)813 1696 y Fp(k)835 1689 y Fr(j)847 1668 y Fo(2)847 1702 y Fp(p)865 1706 y Fg(\026)901 1689 y Fu(=)949 1640 y Fj(\015)949 1665 y(\015)949 1690 y(\015)972 1689 y Fs(i)988 1696 y Fp(p)1006 1687 y Fh(0)1016 1696 y Fp(;p)1044 1700 y Fg(\026)1067 1640 y Fj(\015)1067 1665 y(\015)1067 1690 y(\015)1090 1653 y Fo(2)1090 1717 y Fp(H)s(S)1169 1689 y Fu(.)59 1807 y(The)i(momen)o(ts)g(of)g(arbitrary)f(order)h(can)h(no)o (w)e(b)q(e)i(estimated)f(b)o(y)h(the)f(Sc)o(h)o(w)o(arz)f(inequalit)o (y)423 1886 y Fj(Z)473 1944 y Fr(j)o Fs(x)p Fr(j)524 1923 y Fp(n)524 1957 y Fq(\000)p Fp(p)569 1948 y Fh(0)605 1944 y Fu(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))41 b Fr(\024)837 1892 y Fj(q)p 879 1892 119 2 v 52 x Fs(\026)p Fu(\()p Fr(N)968 1931 y Fq(0)980 1944 y Fu(\))1005 1884 y Fj(\022)1035 1886 y(Z)1084 1944 y Fr(j)p Fs(x)p Fr(j)1136 1923 y Fo(2)p Fp(n)1136 1957 y Fq(\000)p Fp(p)1206 1944 y Fu(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))1320 1884 y Fj(\023)1355 1882 y Fi(1)p 1355 1888 16 2 v 1355 1909 a(2)760 2053 y Fr(\024)837 2002 y Fj(q)p 879 2002 119 2 v 51 x Fs(\026)p Fu(\()p Fr(N)968 2040 y Fq(0)980 2053 y Fu(\))1005 2006 y Fj(\020)1030 2053 y Fs(C)1073 2004 y Fj(\015)1073 2029 y(\015)1073 2054 y(\015)1096 2053 y Fs(i)1112 2061 y Fp(p)1130 2051 y Fh(0)1141 2061 y Fp(;p)1169 2065 y Fg(\026)1191 2004 y Fj(\015)1191 2029 y(\015)1191 2054 y(\015)1214 2081 y Fp(H)s(S)1271 2006 y Fj(\021)1296 2018 y Fp(n)1327 2002 y Fj(q)p 1368 2002 98 2 v 1368 2053 a Fu(\(2)p Fs(n)p Fu(\)!)760 2152 y Fr(\024)837 2101 y Fj(q)p 879 2101 119 2 v 51 x Fs(\026)p Fu(\()p Fr(N)968 2139 y Fq(0)980 2152 y Fu(\))1005 2105 y Fj(\020)1030 2152 y Fu(2)p Fs(C)1096 2104 y Fj(\015)1096 2129 y(\015)1096 2153 y(\015)1119 2152 y Fs(i)1135 2160 y Fp(p)1153 2151 y Fh(0)1163 2160 y Fp(;p)1191 2164 y Fg(\026)1214 2104 y Fj(\015)1214 2129 y(\015)1214 2153 y(\015)1237 2180 y Fp(H)s(S)1294 2105 y Fj(\021)1318 2117 y Fp(n)1350 2152 y Fs(n)p Fu(!)59 2268 y(since)16 b(\(2)p Fs(n)p Fu(\)!)c Fr(\024)h Fu(4)352 2252 y Fp(n)375 2268 y Fu(\()p Fs(n)p Fu(!\))451 2252 y Fo(2)485 2268 y Fu(.)59 2341 y(Cho)q(ose)i Fs(")e(<)301 2294 y Fj(\020)341 2341 y Fu(2)p Fs(C)407 2292 y Fj(\015)407 2317 y(\015)407 2342 y(\015)431 2341 y Fs(i)447 2348 y Fp(p)465 2339 y Fh(0)475 2348 y Fp(;p)503 2352 y Fg(\026)526 2292 y Fj(\015)526 2317 y(\015)526 2342 y(\015)549 2369 y Fp(H)s(S)606 2294 y Fj(\021)630 2305 y Fq(\000)p Fo(1)693 2341 y Fu(then)374 2432 y Fj(Z)423 2490 y Fs(e)444 2470 y Fp(")p Fq(j)p Fp(x)p Fq(j)500 2480 y Fh(\000)p Fg(p)540 2473 y Fh(0)555 2490 y Fu(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))42 b(=)803 2437 y Fq(1)791 2449 y Fj(X)787 2540 y Fp(n)p Fo(=0)866 2459 y Fs(")887 2443 y Fp(n)p 866 2479 45 2 v 868 2521 a Fs(n)p Fu(!)924 2432 y Fj(Z)973 2490 y Fr(j)o Fs(x)p Fr(j)1024 2469 y Fp(n)1024 2503 y Fq(\000)p Fp(p)1069 2494 y Fh(0)1105 2490 y Fu(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))711 2632 y Fr(\024)787 2581 y Fj(q)p 829 2581 119 2 v 51 x Fs(\026)p Fu(\()p Fr(N)918 2619 y Fq(0)930 2632 y Fu(\))986 2579 y Fq(1)973 2592 y Fj(X)970 2682 y Fp(n)p Fo(=0)1044 2585 y Fj(\020)1069 2632 y Fs(")15 b Fu(2)p Fs(C)1171 2583 y Fj(\015)1171 2608 y(\015)1171 2633 y(\015)1194 2632 y Fs(i)1210 2640 y Fp(p)1228 2630 y Fh(0)1239 2640 y Fp(;p)1267 2644 y Fg(\026)1289 2583 y Fj(\015)1289 2608 y(\015)1289 2633 y(\015)1312 2660 y Fp(H)s(S)1369 2585 y Fj(\021)1394 2597 y Fp(n)1430 2632 y Fs(<)e Fr(1)250 b Fu(\(6\))59 2768 y(Hence)16 b(the)f(lemma)h(is)g(pro)o(v)o(en.)1171 b Ff(2)p eop %%Page: 10 10 10 9 bop 59 50 a Fn(3)45 b(MEASURES)16 b(ON)g(LINEAR)g(TOPOLOGICAL)h (SP)l(A)o(CES)575 b Fu(10)130 199 y(F)l(or)14 b Fs(\026)g Fr(2)f(M)350 206 y Fp(a)370 199 y Fu(\()p Fr(N)432 183 y Fq(0)443 199 y Fu(\))i(the)h(estimate)f(in)h(statemen)o(t)e(2)h(of)g (the)h(ab)q(o)o(v)o(e)e(lemma)i(allo)o(ws)f(to)g(de\014ne)i(the)59 260 y(momen)o(t)e(k)o(ernels)h(M)429 244 y Fp(\026)429 271 y(n)466 260 y Fr(2)e Fu(\()p Fr(N)576 238 y Fo(^)571 244 y Fq(\012)q Fp(n)622 260 y Fu(\))640 244 y Fq(0)652 260 y Fs(:)h Fu(This)h(is)g(done)g(b)o(y)g(extending)h(the)e(ab)q(o)o (v)o(e)h(estimate)f(b)o(y)h(a)f(simple)59 317 y(p)q(olarization)h (argumen)o(t)f(and)g(applying)i(the)e(k)o(ernel)h(theorem.)k(The)15 b(k)o(ernels)h(are)f(determined)h(b)o(y)708 442 y Fs(l)722 449 y Fp(\026)745 442 y Fu(\()p Fs(\022)q Fu(\))d(=)879 390 y Fq(1)867 402 y Fj(X)864 493 y Fp(n)p Fo(=0)951 412 y Fu(1)p 943 432 40 2 v 943 474 a Fs(n)p Fu(!)987 442 y Fr(h)p Fu(M)1047 424 y Fp(\026)1047 454 y(n)1070 442 y Fs(;)8 b(\022)1113 424 y Fq(\012)p Fp(n)1164 442 y Fr(i)59 574 y Fu(or)15 b(equiv)m(alen)o(tly)314 691 y Fr(h)p Fu(M)374 672 y Fp(\026)374 702 y(n)397 691 y Fs(;)8 b(\022)439 698 y Fo(1)465 684 y Fu(^)458 691 y Fr(\012)g(\001)g(\001)g(\001)568 684 y Fu(^)562 691 y Fr(\012)p Fs(\022)618 698 y Fp(n)642 691 y Fr(i)k Fu(=)804 660 y Fs(@)831 644 y Fp(n)p 730 680 198 2 v 730 722 a Fs(@)s(t)773 729 y Fo(1)801 722 y Fr(\001)c(\001)g(\001)d Fs(@)s(t)904 729 y Fp(n)933 691 y Fs(l)947 698 y Fp(\026)970 691 y Fu(\()p Fs(t)1004 698 y Fo(1)1023 691 y Fs(\022)1044 698 y Fo(1)1075 691 y Fu(+)10 b Fr(\001)e(\001)g(\001)g Fu(+)j Fs(t)1245 698 y Fp(n)1268 691 y Fs(\022)1289 698 y Fp(n)1313 691 y Fu(\))1331 630 y Fj(\014)1331 655 y(\014)1331 680 y(\014)1331 705 y(\014)1345 732 y Fp(t)1358 737 y Fi(1)1375 732 y Fo(=)p Fq(\001\001\001)p Fo(=)p Fp(t)1472 736 y Fg(n)1493 732 y Fo(=0)1563 691 y Fs(:)59 813 y Fu(Moreo)o(v)o(er,)i(if)j Fs(p)d(>)g(p)420 820 y Fp(\026)458 813 y Fu(is)j(suc)o(h)f(that)g(em)o(b)q(edding)h Fs(i)950 820 y Fp(p;p)996 824 y Fg(\026)1031 813 y Fu(:)c Fr(H)1094 820 y Fp(p)1127 813 y Fs(,)-8 b Fr(!)13 b(H)1228 820 y Fp(p)1246 824 y Fg(\026)1284 813 y Fu(is)j(Hilb)q(ert-Sc)o(hmidt)i (then)443 925 y Fr(j)o Fu(M)497 906 y Fp(\026)497 936 y(n)521 925 y Fr(j)533 938 y Fq(\000)p Fp(p)593 925 y Fr(\024)641 878 y Fj(\020)666 925 y Fs(nC)736 889 y Fj(\015)736 914 y(\015)759 925 y Fs(i)775 932 y Fp(p;p)821 936 y Fg(\026)843 889 y Fj(\015)843 914 y(\015)866 943 y Fp(H)s(S)923 878 y Fj(\021)948 889 y Fp(n)984 925 y Fr(\024)13 b Fs(n)p Fu(!)1095 878 y Fj(\020)1120 925 y Fs(eC)1184 889 y Fj(\015)1184 914 y(\015)1207 925 y Fs(i)1223 932 y Fp(p;p)1269 936 y Fg(\026)1291 889 y Fj(\015)1291 914 y(\015)1314 943 y Fp(H)s(S)1371 878 y Fj(\021)1396 889 y Fp(n)1442 925 y Fs(:)318 b Fu(\(7\))59 1042 y Fk(Definition)16 b(10)23 b Fm(A)c(function)g Fs(')f Fu(:)g Fr(N)747 1026 y Fq(0)776 1042 y Fr(!)p 856 1042 3 25 v 23 w Fs(C)k Fm(of)e(the)f(form)h Fs(')p Fu(\()p Fs(x)p Fu(\))d(=)1309 1010 y Fj(P)1353 1023 y Fp(N)1353 1054 y(n)p Fo(=0)1422 1042 y Fr(h)p Fs(x)1466 1026 y Fq(\012)p Fp(n)1516 1042 y Fs(;)8 b(')1567 1026 y Fo(\()p Fp(n)p Fo(\))1617 1042 y Fr(i)p Fm(,)20 b Fs(x)e Fr(2)g(N)1805 1026 y Fq(0)1817 1042 y Fm(,)59 1106 y Fs(N)f Fr(2)c Fs(I)-12 b(N)5 b(;)14 b Fm(is)g(c)n(al)r(le)n(d)g (a)h(c)n(ontinuous)f(p)n(olynomial)h(\(short)f Fs(')e Fr(2)h(P)t Fu(\()p Fr(N)1212 1090 y Fq(0)1223 1106 y Fu(\))h Fm(\))h(i\013)f Fs(')1375 1090 y Fo(\()p Fp(n)p Fo(\))1438 1106 y Fr(2)f(N)1530 1081 y Fo(^)1525 1087 y Fq(\012)q Fp(n)p 1524 1120 2 18 v 1515 1120 a(C)1576 1106 y Fm(,)i Fr(8)p Fs(n)e Fr(2)g Fs(I)-12 b(N)1763 1113 y Fo(0)1795 1106 y Fu(=)59 1162 y Fs(I)g(N)15 b Fr([)10 b(f)p Fu(0)p Fr(g)p Fm(.)130 1265 y Fu(No)o(w)k(w)o(e)h(are)g (ready)g(to)g(form)o(ulate)f(the)i(second)f(assumption:)59 1368 y Fk(Assumption)i Fu(2)61 b(F)l(or)16 b(all)h Fs(')c Fr(2)i(P)t Fu(\()p Fr(N)757 1352 y Fq(0)767 1368 y Fu(\))h(with)g Fs(')e Fu(=)h(0)g Fs(\026)p Fu(-almost)h(ev)o(erywhere)h(w)o(e)e(ha)o (v)o(e)h Fs(')e Fr(\021)g Fu(0.)22 b(In)59 1425 y(the)15 b(follo)o(wing)h(a)f(measure)g(with)h(this)f(prop)q(ert)o(y)g(will)i(b) q(e)f(called)h Fm(non-de)n(gener)n(ate)p Fu(.)59 1528 y Fk(Note.)72 b Fu(Assumption)16 b(2)f(is)h(equiv)m(alen)o(t)h(to:)59 1585 y(Let)e Fs(')e Fr(2)g(P)t Fu(\()p Fr(N)324 1568 y Fq(0)334 1585 y Fu(\))i(with)471 1550 y Fj(R)490 1598 y Fp(A)527 1585 y Fs(')8 b Fu(d)p Fs(\026)k Fu(=)h(0)i(for)g(all)h Fs(A)c Fr(2)h(C)961 1592 y Fp(\033)985 1585 y Fu(\()p Fr(N)1047 1568 y Fq(0)1058 1585 y Fu(\))i(then)g Fs(')e Fr(\021)g Fu(0.)59 1641 y(A)19 b(su\016cien)o(t)g(condition)h(can)f(b)q (e)h(obtained)f(b)o(y)g(regarding)g(admissible)i(shifts)e(of)f(the)h (measure)g Fs(\026)p Fu(.)59 1698 y(If)e Fs(\026)p Fu(\()p Fr(\001)12 b Fu(+)f Fs(\030)r Fu(\))17 b(is)h(absolutely)g(con)o(tin)o (uous)f(with)h(resp)q(ect)f(to)g Fs(\026)g Fu(for)g(all)h Fs(\030)g Fr(2)e(N)7 b Fs(;)17 b Fu(i.e.,)g(there)g(exists)h(the)59 1754 y(Radon-Nik)o(o)q(dym)e(deriv)m(ativ)o(e)617 1874 y Fs(\032)641 1881 y Fp(\026)664 1874 y Fu(\()p Fs(\030)r(;)8 b(x)p Fu(\))j(=)833 1843 y(d)p Fs(\026)p Fu(\()p Fs(x)f Fu(+)h Fs(\030)r Fu(\))p 833 1863 192 2 v 872 1905 a(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))1045 1874 y Fs(;)52 b(x)13 b Fr(2)g(N)1236 1855 y Fq(0)1260 1874 y Fs(;)59 1996 y Fu(Then)f(w)o(e)g(sa)o(y)f(that)g Fs(\026)h Fu(is)h Fr(N)7 b Fm({quasi-invariant)12 b Fu(see)g(e.g.,)f([GV68)o(,)g(Sk74].) 19 b(This)12 b(is)g(su\016cien)o(t)h(to)e(ensure)59 2052 y(Assumption)16 b(2,)e(see)i(e.g.,)e([KoTs91)o(,)g(BeKo88].)59 2156 y Fk(Example)g(1)g Fu(In)e(Gaussian)g(Analysis)h(\(esp)q(ecially)h (White)e(Noise)g(Analysis\))h(the)f(Gaussian)g(measure)59 2212 y Fs(\015)83 2219 y Fq(H)130 2212 y Fu(corresp)q(onding)k(to)f (the)g(Hilb)q(ert)i(space)e Fr(H)h Fu(is)g(considered.)21 b(Its)15 b(Laplace)h(transform)e(is)i(giv)o(en)g(b)o(y)646 2315 y Fs(l)660 2322 y Fp(\015)678 2328 y Fh(H)708 2315 y Fu(\()p Fs(\022)q Fu(\))d(=)g Fs(e)853 2283 y Fi(1)p 853 2289 16 2 v 853 2310 a(2)873 2297 y Fq(h)p Fp(\022)q(;\022)q Fq(i)963 2315 y Fs(;)98 b(\022)14 b Fr(2)f(N)p 1195 2322 2 18 v 1186 2322 a Fp(C)1231 2315 y Fs(;)59 2413 y Fu(hence)18 b Fs(\015)211 2420 y Fq(H)257 2413 y Fr(2)d(M)357 2420 y Fp(a)377 2413 y Fu(\()p Fr(N)439 2397 y Fq(0)451 2413 y Fu(\).)23 b(It)17 b(is)g(w)o(ell)g(kno)o(wn)f(that)g Fs(\015)965 2420 y Fq(H)1013 2413 y Fu(is)h Fr(N)7 b Fu({quasi-in)o(v)m(arian)o(t)18 b(\(moreo)o(v)o(er)c Fr(H)p Fu({quasi-)59 2470 y(in)o(v)m(arian)o(t\))g(see)g(e.g.,)e ([Sk74,)h(BeKo88].)19 b(Due)13 b(to)g(the)h(previous)g(note)g Fs(\015)1294 2477 y Fq(H)1339 2470 y Fu(satis\014es)g(also)f (Assumption)59 2526 y(2.)59 2630 y Fk(Example)18 b(2)f Fm(\(Poisson)e(me)n(asur)n(es\))59 2698 y Fu(Let)g(use)h(consider)g (the)f(classical)i(\(real\))e(Sc)o(h)o(w)o(artz)f(triple)694 2796 y Fr(S)s Fu(\()p Fs(I)-12 b(R)p Fu(\))12 b Fr(\032)h Fs(L)895 2777 y Fo(2)915 2796 y Fu(\()p Fs(I)-12 b(R)p Fu(\))12 b Fr(\032)h(S)1085 2777 y Fq(0)1097 2796 y Fu(\()p Fs(I)-12 b(R)p Fu(\))8 b Fs(:)p eop %%Page: 11 11 11 10 bop 59 50 a Fn(3)45 b(MEASURES)16 b(ON)g(LINEAR)g(TOPOLOGICAL)h (SP)l(A)o(CES)575 b Fu(11)59 199 y(The)13 b(P)o(oisson)f(white)h(noise) g(measure)f Fs(\026)744 206 y Fp(p)777 199 y Fu(is)h(de\014ned)h(as)e (a)g(probabilit)o(y)h(measure)g(on)f Fr(C)1550 206 y Fp(\033)1573 199 y Fu(\()p Fr(S)1622 183 y Fq(0)1634 199 y Fu(\()p Fs(I)-12 b(R)p Fu(\)\))11 b(with)59 256 y(the)k(Laplace)i(transform)265 372 y Fs(l)279 379 y Fp(\026)300 383 y Fg(p)320 372 y Fu(\()p 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b(\030)18 b Fr(6)p Fu(=)f(0)g(the)h(measures)f Fs(\026)1526 610 y Fp(p)1564 603 y Fu(and)h Fs(\026)1682 610 y Fp(p)1702 603 y Fu(\()p Fr(\001)11 b Fu(+)h Fs(\030)r Fu(\))59 659 y(are)i(orthogonal)f(\(see)h([V)o(GG75)n(])g(for)f(a)h (detailed)h(analysis\).)20 b(It)14 b(means)g(that)f Fs(\026)1430 666 y Fp(p)1464 659 y Fu(is)i(not)e Fr(S)s Fu(\()p Fs(I)-12 b(R)p Fu(\)-quasi-)59 716 y(in)o(v)m(arian)o(t)16 b(and)f(the)h(note)f (after)f(Assumption)i(2)f(is)g(not)g(applicable)j(no)o(w.)130 772 y(Let)e(some)g Fs(')d Fr(2)i(P)t Fu(\()p Fr(S)500 756 y Fq(0)510 772 y Fu(\()p Fs(I)-12 b(R)p Fu(\)\))8 b Fs(;)14 b(')g Fu(=)g(0)32 b Fs(\026)817 779 y Fp(p)837 772 y Fu(-a.s.)22 b(b)q(e)17 b(giv)o(en.)23 b(W)l(e)16 b(need)h(to)e(sho)o(w)g(that)h(then)g Fs(')e Fr(\021)g Fu(0.)59 829 y(T)l(o)20 b(this)g(end)h(w)o(e)f(will)i(in)o(tro)q(duce)f (a)f(system)f(of)h(orthogonal)f(p)q(olynomials)j(in)f(the)f(space)g Fs(L)1728 812 y Fo(2)1748 829 y Fu(\()p Fs(\026)1793 836 y Fp(p)1813 829 y Fu(\))59 885 y(whic)o(h)c(can)f(b)q(e)h (constructed)g(in)g(the)f(follo)o(wing)h(w)o(a)o(y)l(.)j(The)c(mapping) 419 983 y Fs(\022)q Fu(\()p Fr(\001)p Fu(\))d Fr(7!)h Fs(\013)p Fu(\()p Fs(\022)q Fu(\)\()p Fr(\001)p Fu(\))f(=)h(log\(1)d(+) g Fs(\022)q Fu(\()p Fr(\001)p Fu(\)\))i Fr(2)h(S)p 1088 990 V 1079 990 a Fp(C)1108 983 y Fu(\()p Fs(I)-12 b(R)p Fu(\))8 b Fs(;)52 b(\022)14 b Fr(2)f(S)p 1372 990 V 1363 990 a Fp(C)1393 983 y Fu(\()p Fs(I)-12 b(R)p Fu(\))59 1081 y(is)16 b(holomorphic)g(on)f(a)g(neigh)o(b)q(orho)q(o)q(d)h Fr(U)i(\032)13 b(S)p 879 1088 V 870 1088 a Fp(C)899 1081 y Fu(\()p Fs(I)-12 b(R)p Fu(\))8 b Fs(;)14 b Fu(0)f Fr(2)g(U)5 b Fu(.)19 b(Then)272 1211 y Fs(e)293 1192 y Fp(\013)293 1222 y(\026)314 1226 y Fg(p)335 1211 y Fu(\()p Fs(\022)q Fu(;)8 b Fs(x)p Fu(\))j(=)519 1180 y Fs(e)540 1163 y Fq(h)p Fp(\013)p Fo(\()p Fp(\022)q Fo(\))p Fp(;x)p Fq(i)p 504 1200 178 2 v 504 1242 a Fs(l)518 1249 y Fp(\026)539 1253 y Fg(p)559 1242 y Fu(\()p Fs(\013)p Fu(\()p Fs(\022)q Fu(\)\))700 1211 y(=)i(exp)p Fr(fh)p Fs(\013)p Fu(\()p Fs(\022)q Fu(\))p Fs(;)8 b(x)p Fr(i)h(\000)h(h)p Fs(\022)q(;)e Fu(1)p Fr(ig)g Fs(;)51 b(\022)14 b Fr(2)f(U)g Fs(;)22 b(x)12 b Fr(2)h(S)1527 1192 y Fq(0)1539 1211 y Fu(\()p Fs(I)-12 b(R)p Fu(\))59 1338 y(is)19 b(a)e(holomorphic)j(function)e(on) g Fr(U)23 b Fu(for)18 b(an)o(y)25 b Fs(x)18 b Fr(2)f(S)1000 1322 y Fq(0)1012 1338 y Fu(\()p Fs(I)-12 b(R)p Fu(\).)28 b(The)18 b(T)l(a)o(ylor)g(decomp)q(osition)h(and)g(the)59 1395 y(k)o(ernel)d(theorem)f(\(just)f(as)h(in)h(subsection)h(4.1)d(b)q (elo)o(w\))h(giv)o(e)636 1523 y Fs(e)657 1504 y Fp(\013)657 1534 y(\026)678 1538 y Fg(p)699 1523 y Fu(\()p Fs(\022)q Fu(;)8 b Fs(x)p Fu(\))j(=)879 1470 y Fq(1)867 1483 y Fj(X)863 1573 y Fp(n)p Fo(=0)951 1492 y Fu(1)p 942 1513 40 2 v 942 1554 a Fs(n)p Fu(!)987 1523 y Fr(h)p Fs(\022)1027 1504 y Fq(\012)p Fp(n)1078 1523 y Fs(;)d(C)1132 1530 y Fp(n)1155 1523 y Fu(\()p Fs(x)p Fu(\))p Fr(i)g Fs(;)59 1669 y Fu(where)17 b Fs(C)225 1676 y Fp(n)263 1669 y Fu(:)e Fr(S)322 1652 y Fq(0)333 1669 y Fu(\()p Fs(I)-12 b(R)p Fu(\))15 b Fr(!)h(S)519 1652 y Fq(0)530 1669 y Fu(\()p Fs(I)-12 b(R)p Fu(\))614 1646 y Fo(^)609 1652 y Fq(\012)p Fp(n)677 1669 y Fu(are)16 b(p)q(olynomial)i(mappings.)25 b(F)l(or)16 b Fs(')1330 1652 y Fo(\()p Fp(n)p Fo(\))1396 1669 y Fr(2)f(S)p 1475 1676 2 18 v 1466 1676 a Fp(C)1496 1669 y Fu(\()p Fs(I)-12 b(R)p Fu(\))1580 1646 y Fo(^)1575 1652 y Fq(\012)p Fp(n)1633 1669 y Fs(;)15 b(n)g Fr(2)g Fs(I)-12 b(N)1798 1676 y Fo(0)1818 1669 y Fu(,)59 1725 y(w)o(e)15 b(de\014ne)h(Charlier)g(p)q(olynomials)446 1823 y Fs(x)d Fr(7!)g Fs(C)576 1830 y Fp(n)599 1823 y Fu(\()p Fs(')647 1804 y Fo(\()p Fp(n)p Fo(\))697 1823 y Fu(;)8 b Fs(x)p Fu(\))k(=)h Fr(h)p Fs(')870 1804 y Fo(\()p Fp(n)p Fo(\))920 1823 y Fs(;)8 b(C)974 1830 y Fp(n)996 1823 y Fu(\()p Fs(x)p Fu(\))p Fr(i)k(2)p 1146 1823 3 25 v 16 w Fs(C)e(;)30 b(x)13 b Fr(2)g(S)1333 1804 y Fq(0)1344 1823 y Fu(\()p Fs(I)-12 b(R)p Fu(\))8 b Fs(:)59 1921 y Fu(Due)15 b(to)g([Ito88)o(,)g(IK88])g(w)o(e)g(ha)o(v)o(e)f(the)i (follo)o(wing)g(orthogonalit)o(y)e(prop)q(ert)o(y:)576 2026 y Fr(8)p Fs(')631 2007 y Fo(\()p Fp(n)p Fo(\))695 2026 y Fr(2)f(S)p 772 2033 2 18 v 763 2033 a Fp(C)792 2026 y Fu(\()p Fs(I)-12 b(R)p Fu(\))876 2001 y Fo(^)871 2007 y Fq(\012)p Fp(n)929 2026 y Fs(;)15 b Fr(8)p Fs( )1014 2007 y Fo(\()p Fp(m)p Fo(\))1087 2026 y Fr(2)e(S)p 1164 2033 V 1155 2033 a Fp(C)1184 2026 y Fu(\()p Fs(I)-12 b(R)p Fu(\))1268 2001 y Fo(^)1263 2007 y Fq(\012)p Fp(n)488 2083 y Fj(Z)537 2141 y Fs(C)570 2148 y Fp(n)593 2141 y Fu(\()p Fs(')641 2122 y Fo(\()p Fp(n)p Fo(\))692 2141 y Fu(\))p Fs(C)743 2148 y Fp(m)775 2141 y Fu(\()p Fs( )825 2122 y Fo(\()p Fp(m)p Fo(\))885 2141 y Fu(\))12 b(d)p Fs(\026)967 2148 y Fp(p)1000 2141 y Fu(=)h Fs(\016)1068 2148 y Fp(nm)1123 2141 y Fs(n)p Fu(!)p Fr(h)p Fs(')1211 2122 y Fo(\()p Fp(n)p Fo(\))1261 2141 y Fs(;)8 b( )1314 2122 y Fo(\()p Fp(n)p Fo(\))1364 2141 y Fr(i)g Fs(:)59 2236 y Fu(No)o(w)i(the)h(rest)g(is)g(simple.)20 b(An)o(y)11 b(con)o(tin)o(uous)g(p)q(olynomial)i Fs(')e 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w(')1122 2641 y Fo(\()p Fp(n)p Fo(\))1171 2655 y Fr(i)13 b Fu(=)g(0)p Fs(:)59 2796 y Fu(Hence)j Fs(')224 2780 y Fo(\()p Fp(n)p Fo(\))287 2796 y Fu(=)d(0)8 b Fs(;)15 b(n)d Fu(=)h(0)8 b Fs(;)14 b(:)8 b(:)g(:)e(;)15 b(N)5 b Fu(,)14 b(i.e.,)h Fs(')d Fr(\021)h Fu(0.)20 b(So)15 b(Assumption)h(2)f(is)g(satis\014ed.)p eop %%Page: 12 12 12 11 bop 59 50 a Fn(4)45 b(CONCEPT)15 b(OF)g(DISTRIBUTIONS)j(IN)d (INFINITE)i(DIMENSIONAL)f(ANAL)l(YSIS)84 b Fu(12)59 199 y Ft(4)67 b(Concept)22 b(of)g(distributions)j(in)e(in\014nite)i (dimensional)f(analysis)59 301 y Fu(In)16 b(this)g(section)g(w)o(e)g (will)h(in)o(tro)q(duce)g(a)e(preliminary)i(distribution)h(theory)d(in) h(in\014nite)i(dimensional)59 357 y(non-Gaussian)23 b(analysis.)41 b(W)l(e)23 b(w)o(an)o(t)e(to)g(p)q(oin)o(t)i(out)f(in)h(adv)m(ance)g (that)e(the)i(distribution)g(space)59 414 y(constructed)18 b(here)g(is)g(in)h(some)e(sense)i(to)q(o)e(big)h(for)f(practical)i (purp)q(oses.)28 b(In)18 b(this)g(sense)h(section)f(4)59 470 y(ma)o(y)d(b)q(e)g(view)o(ed)h(as)f(a)g(stepping)h(stone)f(to)g(in) o(tro)q(duce)h(the)f(more)g(useful)h(structures)f(in)h Fr(x)p Fu(5)f(and)h Fr(x)p Fu(6.)130 527 y(W)l(e)h(will)i(c)o(ho)q(ose) e Fr(P)t Fu(\()p Fr(N)542 510 y Fq(0)552 527 y Fu(\))g(as)g(our)g (\(minimal\))h(test)f(function)h(space.)26 b(\(The)17 b(idea)g(to)g(use)g(spaces)59 583 y(of)f(this)g(t)o(yp)q(e)h(as)e (appropriate)h(spaces)h(of)e(test)h(functions)h(is)f(rather)g(old)h (see)f([KMP65)o(].)22 b(They)17 b(also)59 639 y(discussed)g(in)g(whic)o (h)g(sense)f(this)g(space)g(is)h(\\minimal".\))22 b(First)15 b(w)o(e)h(ha)o(v)o(e)f(to)h(ensure)g(that)f Fr(P)t Fu(\()p Fr(N)1756 623 y Fq(0)1767 639 y Fu(\))g(is)59 696 y(densely)20 b(em)o(b)q(edded)f(in)h Fs(L)528 679 y Fo(2)547 696 y Fu(\()p Fs(\026)p Fu(\).)29 b(This)19 b(is)g(ful\014lled)i(b)q(ecause)e (of)f(our)g(assumption)g(1)g([Sk74)o(,)h(Sec.10)59 752 y(Th.1].)34 b(The)21 b(space)f Fr(P)t Fu(\()p Fr(N)536 736 y Fq(0)547 752 y Fu(\))g(ma)o(y)f(b)q(e)i(equipp)q(ed)i(with)d(v)m (arious)h(di\013eren)o(t)g(top)q(ologies,)g(but)g(there)59 809 y(exists)14 b(a)g(natural)g(one)g(suc)o(h)g(that)f Fr(P)t Fu(\()p Fr(N)756 792 y Fq(0)767 809 y Fu(\))g(b)q(ecomes)i(a)f (n)o(uclear)g(space)g([BeKo88].)19 b(The)14 b(top)q(ology)g(on)59 865 y Fr(P)t Fu(\()p Fr(N)157 849 y Fq(0)168 865 y Fu(\))i(is)h(c)o (hosen)g(suc)o(h)f(that)g(is)h(b)q(ecomes)g(isomorphic)h(to)d(the)i (top)q(ological)g(direct)h(sum)e(of)g(tensor)59 929 y(p)q(o)o(w)o(ers)f Fr(N)259 904 y Fo(^)254 910 y Fq(\012)p Fp(n)p 253 942 2 18 v 244 942 a(C)320 929 y Fu(see)g(e.g.,)f([Sc)o(h71,)h(Ch)g(I)q(I)h (6.1,)e(Ch)h(I)q(I)q(I)i(7.4])753 1064 y Fr(P)t Fu(\()p Fr(N)851 1045 y Fq(0)862 1064 y Fu(\))12 b Fr(')956 1011 y Fq(1)942 1023 y Fj(M)940 1114 y Fp(n)p Fo(=0)1014 1064 y Fr(N)1063 1038 y Fo(^)1058 1044 y Fq(\012)p Fp(n)p 1057 1077 V 1048 1077 a(C)1124 1064 y Fu(.)59 1198 y(via)444 1274 y Fs(')p Fu(\()p Fs(x)p Fu(\))g(=)612 1221 y Fq(1)599 1233 y Fj(X)596 1324 y Fp(n)p Fo(=0)670 1226 y Fj(D)696 1274 y Fs(x)722 1255 y Fq(\012)p Fp(n)772 1274 y Fs(;)c(')823 1255 y Fo(\()p Fp(n)p Fo(\))873 1226 y Fj(E)911 1274 y Fr( )-8 b(!)18 b Fs(~)-27 b(')13 b Fu(=)1098 1226 y Fj(n)1125 1274 y Fs(')1155 1255 y Fo(\()p Fp(n)p Fo(\))1219 1225 y Fj(\014)1219 1250 y(\014)1219 1275 y(\014)1245 1274 y Fs(n)g Fr(2)g Fs(I)-12 b(N)1378 1281 y Fo(0)1397 1226 y Fj(o)1433 1274 y Fs(:)59 1400 y Fu(Note)18 b(that)f(only)i(a)e (\014nite)i(n)o(um)o(b)q(er)g(of)e Fs(')788 1384 y Fo(\()p Fp(n)p Fo(\))857 1400 y Fu(is)i(non-zero.)28 b(W)l(e)19 b(will)g(not)f(repro)q(duce)h(the)f(full)h(con-)59 1457 y(struction)d(here,)h(but)g(w)o(e)f(will)i(describ)q(e)g(the)e(notion)h (of)e(con)o(v)o(ergence)i(of)f(sequences)h(this)g(top)q(ology)59 1519 y(on)f Fr(P)t Fu(\()p Fr(N)221 1502 y Fq(0)232 1519 y Fu(\).)24 b(F)l(or)15 b Fs(')g Fr(2)g(P)t Fu(\()p Fr(N)557 1502 y Fq(0)568 1519 y Fu(\),)h Fs(')p Fu(\()p Fs(x)p Fu(\))e(=)771 1487 y Fj(P)815 1495 y Fp(N)t Fo(\()p Fp(')p Fo(\))815 1531 y Fp(n)p Fo(=0)906 1472 y Fj(D)932 1519 y Fs(x)958 1502 y Fq(\012)p Fp(n)1009 1519 y Fs(;)8 b(')1060 1502 y Fo(\()p Fp(n)p Fo(\))1110 1472 y Fj(E)1151 1519 y Fu(let)17 b Fs(p)1241 1526 y Fp(n)1280 1519 y Fu(:)d Fr(P)t Fu(\()p Fr(N)1405 1502 y Fq(0)1416 1519 y Fu(\))g Fr(!)h(N)1557 1494 y Fo(^)1552 1500 y Fq(\012)q Fp(n)p 1552 1532 V 1542 1532 a(C)1620 1519 y Fu(denote)i(the)59 1590 y(mapping)i Fs(p)272 1597 y Fp(n)313 1590 y Fu(is)g(de\014ned)g(b) o(y)f Fs(p)610 1597 y Fp(n)634 1590 y Fs(')f Fu(:=)g Fs(')776 1574 y Fo(\()p Fp(n)p Fo(\))827 1590 y Fs(:)h Fu(A)g(sequence)h Fr(f)p Fs(')1154 1597 y Fp(j)1172 1590 y Fs(;)j(j)15 b Fr(2)e Fs(I)-12 b(N)5 b Fr(g)18 b Fu(of)g(smo)q(oth)f (p)q(olynomials)59 1656 y(con)o(v)o(erges)i(to)f Fs(')i Fr(2)f(P)t Fu(\()p Fr(N)523 1639 y Fq(0)534 1656 y Fu(\))g(i\013)g(the) g Fs(N)5 b Fu(\()p Fs(')801 1663 y Fp(j)819 1656 y Fu(\))18 b(are)h(b)q(ounded)i(and)e Fs(p)1240 1663 y Fp(n)1264 1656 y Fs(')1294 1663 y Fp(j)1341 1656 y Fr(\000)-8 b(!)1331 1680 y Fp(n)p Fq(!1)1443 1656 y Fs(p)1466 1663 y Fp(n)1489 1656 y Fs(')19 b Fu(in)h Fr(N)1644 1630 y Fo(^)1639 1636 y Fq(\012)p Fp(n)p 1638 1669 V 1629 1669 a(C)1709 1656 y Fu(for)f(all)59 1720 y Fs(n)13 b Fr(2)g Fs(I)-12 b(N)5 b Fu(.)130 1776 y(No)o(w)16 b(w)o(e)i(can)f(in)o(tro)q(duce)i(the)e (dual)i(space)e Fr(P)938 1760 y Fq(0)934 1788 y Fp(\026)957 1776 y Fu(\()p Fr(N)1019 1760 y Fq(0)1031 1776 y Fu(\))g(of)g Fr(P)t Fu(\()p Fr(N)1218 1760 y Fq(0)1229 1776 y Fu(\))g(with)h(resp)q (ect)g(to)e Fs(L)1616 1760 y Fo(2)1636 1776 y Fu(\()p Fs(\026)p Fu(\).)27 b(As)17 b(a)59 1833 y(result)f(w)o(e)f(ha)o(v)o(e)f (constructed)i(the)f(triple)691 1935 y Fr(P)t Fu(\()p Fr(N)789 1916 y Fq(0)800 1935 y Fu(\))d Fr(\032)h Fs(L)909 1916 y Fo(2)929 1935 y Fu(\()p Fs(\026)p Fu(\))f Fr(\032)h(P)1088 1916 y Fq(0)1084 1946 y Fp(\026)1107 1935 y Fu(\()p Fr(N)1169 1916 y Fq(0)1181 1935 y Fu(\))59 2037 y(The)h(\(bilinear\))g(dual)g (pairing)h Fr(h)-8 b(h\001)p Fs(;)8 b Fr(\001i)-8 b(i)703 2044 y Fp(\026)737 2037 y Fu(b)q(et)o(w)o(een)14 b Fr(P)947 2021 y Fq(0)943 2048 y Fp(\026)966 2037 y Fu(\()p Fr(N)1028 2021 y Fq(0)1039 2037 y Fu(\))f(and)g Fr(P)t Fu(\()p Fr(N)1254 2021 y Fq(0)1265 2037 y Fu(\))g(is)h(connected)g(to)f(the)g (\(sesqui-)59 2093 y(linear\))j(inner)g(pro)q(duct)g(on)f Fs(L)584 2077 y Fo(2)604 2093 y Fu(\()p Fs(\026)p Fu(\))g(b)o(y)440 2196 y Fr(h)-8 b(h)p Fs(';)19 b( )r Fr(i)-8 b(i)590 2203 y Fp(\026)625 2196 y Fu(=)13 b(\()p Fs(';)p 753 2159 32 2 v 19 w( )r Fu(\))803 2205 y Fp(L)827 2196 y Fi(2)843 2205 y Fo(\()p Fp(\026)p Fo(\))909 2196 y Fs(;)53 b(')12 b Fr(2)h Fs(L)1091 2177 y Fo(2)1111 2196 y Fu(\()p Fs(\026)p Fu(\))p Fs(;)22 b( )14 b Fr(2)f(P)t Fu(\()p Fr(N)1394 2177 y Fq(0)1405 2196 y Fu(\))h(.)59 2298 y(Since)i(the)e(constan)o(t)g (function)h(1)f(is)h(in)g Fr(P)t Fu(\()p Fr(N)846 2281 y Fq(0)857 2298 y Fu(\))f(w)o(e)g(ma)o(y)g(extend)h(the)f(concept)h(of) f(exp)q(ectation)h(from)59 2354 y(random)g(v)m(ariables)h(to)f (distributions;)h(for)f(\010)d Fr(2)h(P)942 2338 y Fq(0)938 2365 y Fp(\026)961 2354 y Fu(\()p Fr(N)1023 2338 y Fq(0)1035 2354 y Fu(\))747 2456 y Fs(I)-12 b(E)791 2463 y Fp(\026)814 2456 y Fu(\(\010\))12 b(:=)h Fr(h)-8 b(h)p Fu(\010)p Fs(;)8 b Fu(1)p Fr(i)-10 b(i)1087 2470 y Fp(\026)1131 2456 y Fs(:)59 2558 y Fu(The)21 b(main)g(goal)g(of)f(this)i(section)f (is)g(to)f(pro)o(vide)i(a)e(description)j(of)d Fr(P)1343 2542 y Fq(0)1339 2570 y Fp(\026)1362 2558 y Fu(\()p Fr(N)1424 2542 y Fq(0)1435 2558 y Fu(\))h(,)g(see)h(Theorem)e(18)59 2615 y(b)q(elo)o(w.)25 b(The)17 b(simplest)g(approac)o(h)f(to)g(this)h (problem)h(seems)e(to)g(b)q(e)i(the)e(use)h(of)f(so)h(called)h Fs(\026)p Fu(-App)q(ell)59 2671 y(p)q(olynomials.)p eop %%Page: 13 13 13 12 bop 59 50 a Fn(4)45 b(CONCEPT)15 b(OF)g(DISTRIBUTIONS)j(IN)d (INFINITE)i(DIMENSIONAL)f(ANAL)l(YSIS)84 b Fu(13)59 199 y Fl(4.1)56 b(App)r(ell)17 b(p)r(olynomials)f(asso)r(ciated)i(to)h(the) f(measure)f Fe(\026)59 285 y Fu(Because)f(of)f(the)g(holomorph)o(y)g (of)g Fs(l)679 292 y Fp(\026)717 285 y Fu(and)g Fs(l)819 292 y Fp(\026)842 285 y Fu(\(0\))d(=)h(1)i(there)g(exists)g(a)g(neigh)o (b)q(orho)q(o)q(d)i(of)e(zero)653 393 y Fr(U)681 400 y Fo(0)714 393 y Fu(=)762 346 y Fj(n)789 393 y Fs(\022)f Fr(2)f(N)p 910 400 2 18 v 901 400 a Fp(C)947 344 y Fj(\014)947 369 y(\014)947 394 y(\014)976 393 y Fu(2)999 374 y Fp(q)1015 379 y Fi(0)1041 393 y Fr(j)p Fs(\022)q Fr(j)1089 406 y Fp(p)1107 411 y Fi(0)1139 393 y Fs(<)g Fu(1)1210 346 y Fj(o)59 501 y Fs(p)82 508 y Fo(0)102 501 y Fs(;)8 b(q)143 508 y Fo(0)174 501 y Fr(2)13 b Fs(I)-12 b(N)5 b(;)14 b(p)317 508 y Fo(0)350 501 y Fr(\025)e Fs(p)420 484 y Fq(0)420 512 y Fp(\026)458 501 y Fu(,)i(2)508 484 y Fq(\000)p Fp(q)551 489 y Fi(0)583 501 y Fr(\024)f Fs(")652 508 y Fp(\026)690 501 y Fu(\()p Fs(p)731 484 y Fq(0)731 512 y Fp(\026)754 501 y Fs(;)8 b(")796 508 y Fp(\026)833 501 y Fu(from)14 b(Lemma)g(9\))g(suc)o(h)h(that)e Fs(l)1369 508 y Fp(\026)1392 501 y Fu(\()p Fs(\022)q Fu(\))g Fr(6)p Fu(=)g(0)h(for)g Fs(\022)g Fr(2)f(U)1723 508 y Fo(0)1757 501 y Fu(and)59 557 y(the)i(normalized)i(exp)q(onen)o(tial)544 684 y Fs(e)565 691 y Fp(\026)588 684 y Fu(\()p Fs(\022)q Fu(;)8 b Fs(z)r Fu(\))k(=)755 653 y Fs(e)776 636 y Fq(h)p Fp(z)q(;\022)q Fq(i)p 755 673 96 2 v 755 715 a Fs(l)769 722 y Fp(\026)792 715 y Fu(\()p Fs(\022)q Fu(\))916 684 y(for)j Fs(\022)f Fr(2)f(U)1092 691 y Fo(0)1112 684 y Fs(;)53 b(z)15 b Fr(2)e(N)1301 665 y Fq(0)p 1300 695 2 18 v 1291 695 a Fp(C)1333 684 y Fs(;)427 b Fu(\(8\))59 808 y(is)20 b(w)o(ell)g(de\014ned.)33 b(W)l(e)19 b(use)g(the)g (holomorph)o(y)g(of)g Fs(\022)i Fr(7!)e Fs(e)1074 815 y Fp(\026)1098 808 y Fu(\()p Fs(\022)q Fu(;)8 b Fs(z)r Fu(\))18 b(to)h(expand)g(it)h(in)g(a)e(p)q(o)o(w)o(er)h(series)59 864 y(in)24 b Fs(\022)g Fu(similar)g(to)e(the)h(case)g(corresp)q (onding)g(to)f(the)h(construction)g(of)g(one)g(dimensional)h(App)q(ell) 59 921 y(p)q(olynomials)17 b([Bo76)o(].)i(W)l(e)d(ha)o(v)o(e)e(in)i (analogy)f(to)g([AKS93,)f(ADKS94])653 1049 y Fs(e)674 1056 y Fp(\026)698 1049 y Fu(\()p Fs(\022)q Fu(;)8 b Fs(z)r Fu(\))k(=)875 996 y Fq(1)863 1008 y Fj(X)860 1099 y Fp(n)p Fo(=0)947 1018 y Fu(1)p 939 1038 40 2 v 939 1080 a Fs(n)p Fu(!)1051 1033 y Fj(d)984 1049 y Fu(d)1009 1036 y Fp(n)1032 1049 y Fs(e)1053 1056 y Fp(\026)1077 1049 y Fu(\(0)p Fs(;)c(z)r Fu(\))n(\()p Fs(\022)q Fu(\))59 1196 y(where)262 1180 y Fj(d)194 1196 y Fu(d)219 1183 y Fp(n)243 1196 y Fs(e)264 1203 y Fp(\026)287 1196 y Fu(\(0;)g Fs(z)r Fu(\))18 b(is)h(an)g(n-homogeneous)g(con)o(tin)o(uous) h(p)q(olynomial.)33 b(But)19 b(since)h Fs(e)1572 1203 y Fp(\026)1595 1196 y Fu(\()p Fs(\022)q Fu(;)8 b Fs(z)r Fu(\))19 b(is)g(not)59 1253 y(only)e(G-holomorphic)g(but)f(holomorphic) h(w)o(e)f(kno)o(w)g(that)f Fs(\022)h Fr(!)g Fs(e)1223 1260 y Fp(\026)1247 1253 y Fu(\()p Fs(\022)q Fu(;)8 b Fs(z)r Fu(\))15 b(is)i(also)f(lo)q(cally)i(b)q(ounded.)59 1309 y(Th)o(us)d(Cauc)o(h)o(y's)g(inequalit)o(y)h(for)f(T)l(a)o(ylor)g (series)h([Di81)o(])f(ma)o(y)f(b)q(e)i(applied,)h Fs(\032)12 b Fr(\024)h Fu(2)1485 1293 y Fq(\000)p Fp(q)1528 1298 y Fi(0)1563 1309 y Fu(,)h Fs(p)f Fr(\025)g Fs(p)1697 1316 y Fo(0)273 1366 y Fj(\014)273 1391 y(\014)273 1416 y(\014)273 1441 y(\014)300 1396 y Fu(1)p 292 1417 V 292 1458 a Fs(n)p Fu(!)404 1411 y Fj(d)337 1427 y Fu(d)362 1414 y Fp(n)386 1427 y Fs(e)407 1434 y Fp(\026)430 1427 y Fu(\(0;)8 b Fs(z)r Fu(\))n(\()p Fs(\022)q Fu(\))589 1366 y Fj(\014)589 1391 y(\014)589 1416 y(\014)589 1441 y(\014)616 1427 y Fr(\024)681 1396 y Fu(1)p 669 1417 47 2 v 669 1458 a Fs(\032)693 1445 y Fp(n)744 1427 y Fu(sup)728 1468 y Fq(j)p Fp(\022)q Fq(j)765 1478 y Fg(p)783 1468 y Fo(=)p Fp(\032)837 1427 y Fr(j)o Fs(e)870 1434 y Fp(\026)894 1427 y Fu(\()p Fs(\022)q Fu(;)g Fs(z)r Fu(\))p Fr(j)e(j)p Fs(\022)q Fr(j)1063 1406 y Fp(n)1063 1441 y(p)1099 1427 y Fr(\024)1164 1396 y Fu(1)p 1152 1417 V 1152 1458 a Fs(\032)1176 1445 y Fp(n)1228 1427 y Fu(sup)1212 1468 y Fq(j)o Fp(\022)q Fq(j)1249 1478 y Fg(p)1267 1468 y Fo(=)p Fp(\032)1361 1396 y Fu(1)p 1325 1417 95 2 v 1325 1458 a Fs(l)1339 1465 y Fp(\026)1362 1458 y Fu(\()p Fs(\022)q Fu(\))1425 1427 y Fs(e)1446 1407 y Fp(\032)p Fq(j)p Fp(z)q Fq(j)1501 1417 y Fh(\000)p Fg(p)1553 1427 y Fr(j)p Fs(\022)q Fr(j)1601 1406 y Fp(n)1601 1441 y(p)1773 1427 y Fu(\(9\))59 1567 y(if)17 b Fs(z)g Fr(2)e(H)223 1574 y Fq(\000)p Fp(p;)p 285 1574 2 18 v -3 w(C)306 1567 y Fu(.)24 b(This)17 b(inequalit)o(y)h(extends)f(b)o(y)g (p)q(olarization)g([Di81])f(to)g(an)h(estimate)f(su\016cien)o(t)h(for) 59 1632 y(the)i(k)o(ernel)h(theorem.)30 b(Th)o(us)19 b(w)o(e)f(ha)o(v)o(e)h(a)f(represen)o(tation)1191 1616 y Fj(d)1124 1632 y Fu(d)1149 1619 y Fp(n)1173 1632 y Fs(e)1194 1639 y Fp(\026)1217 1632 y Fu(\(0;)8 b Fs(z)r Fu(\))o(\()p Fs(\022)q Fu(\))18 b(=)h Fr(h)p Fs(P)1502 1615 y Fp(\026)1496 1643 y(n)1526 1632 y Fu(\()p Fs(z)r Fu(\))p Fs(;)8 b(\022)1628 1615 y Fq(\012)p Fp(n)1678 1632 y Fr(i)18 b Fu(where)59 1703 y Fs(P)94 1687 y Fp(\026)88 1714 y(n)118 1703 y Fu(\()p Fs(z)r Fu(\))f Fr(2)243 1656 y Fj(\020)267 1703 y Fr(N)316 1678 y Fo(^)311 1684 y Fq(\012)q Fp(n)p 311 1717 V 301 1717 a(C)362 1656 y Fj(\021)387 1668 y Fq(0)399 1703 y Fu(.)29 b(The)19 b(k)o(ernel)g(theorem)f(really) h(giv)o(es)g(a)f(little)i(more:)26 b Fs(P)1424 1687 y Fp(\026)1418 1714 y(n)1447 1703 y Fu(\()p Fs(z)r Fu(\))18 b Fr(2)g(H)1615 1678 y Fo(^)1610 1684 y Fq(\012)q Fp(n)1610 1717 y Fq(\000)p Fp(p)1655 1708 y Fh(0)1687 1703 y Fu(for)g(an)o(y)59 1768 y Fs(p)82 1752 y Fq(0)94 1768 y Fu(\()p Fs(>)12 b(p)h Fr(\025)g Fs(p)266 1775 y Fo(0)285 1768 y Fu(\))h(suc)o(h)h(that) f(the)h(em)o(b)q(edding)h(op)q(erator)d Fs(i)1022 1776 y Fp(p)1040 1766 y Fh(0)1051 1776 y Fp(;p)1093 1768 y Fu(:)f Fr(H)1156 1776 y Fp(p)1174 1766 y Fh(0)1200 1768 y Fs(,)-8 b Fr(!)14 b(H)1302 1775 y Fp(p)1334 1768 y Fu(is)h(Hilb)q(ert-Sc)o(hmidt.)22 b(Th)o(us)59 1825 y(w)o(e)15 b(ha)o(v)o(e)423 1900 y Fs(e)444 1907 y Fp(\026)468 1900 y Fu(\()p Fs(\022)q Fu(;)8 b Fs(z)r Fu(\))k(=)645 1847 y Fq(1)633 1860 y Fj(X)630 1950 y Fp(n)p Fo(=0)717 1869 y Fu(1)p 709 1890 40 2 v 709 1931 a Fs(n)p Fu(!)761 1866 y Fj(\012)781 1900 y Fs(P)816 1881 y Fp(\026)810 1911 y(n)840 1900 y Fu(\()p Fs(z)r Fu(\))p Fs(;)c(\022)942 1881 y Fq(\012)p Fp(n)992 1866 y Fj(\013)1065 1900 y Fu(for)14 b Fs(\022)g Fr(2)f(U)1240 1907 y Fo(0)1260 1900 y Fs(;)23 b(z)15 b Fr(2)e(N)1419 1881 y Fq(0)p 1418 1911 2 18 v 1409 1911 a Fp(C)1454 1900 y Fu(.)283 b(\(10\))59 2015 y(W)l(e)15 b(will)i(also)e(use)h(the)f(notation)377 2113 y Fs(P)412 2094 y Fp(\026)406 2124 y(n)436 2113 y Fu(\()p Fs(')484 2094 y Fo(\()p Fp(n)p Fo(\))534 2113 y Fu(\)\()p Fs(z)r Fu(\))d(:=)683 2066 y Fj(D)709 2113 y Fs(P)744 2094 y Fp(\026)738 2124 y(n)767 2113 y Fu(\()p Fs(z)r Fu(\))p Fs(;)c(')877 2094 y Fo(\()p Fp(n)p Fo(\))927 2066 y Fj(E)960 2113 y Fs(;)98 b(')1101 2094 y Fo(\()p Fp(n)p Fo(\))1164 2113 y Fr(2)13 b(N)1256 2088 y Fo(^)1251 2094 y Fq(\012)q Fp(n)p 1250 2127 V 1241 2127 a(C)1302 2113 y Fs(;)53 b(n)13 b Fr(2)g Fs(I)-12 b(N)t(:)59 2217 y Fu(Th)o(us)15 b(for)g(an)o(y)g(measure)g(satisfying)g(Assumption)h(1) f(w)o(e)g(ha)o(v)o(e)g(de\014ned)h(the)g Fs(I)-12 b(P)1449 2199 y Fp(\026)1473 2217 y Fu(-system)485 2338 y Fs(I)g(P)529 2320 y Fp(\026)565 2338 y Fu(=)613 2279 y Fj(\032)644 2291 y(D)669 2338 y Fs(P)704 2320 y Fp(\026)698 2350 y(n)728 2338 y Fu(\()p Fr(\001)p Fu(\))p Fs(;)8 b(')828 2320 y Fo(\()p Fp(n)p Fo(\))877 2291 y Fj(E)925 2277 y(\014)925 2302 y(\014)925 2327 y(\014)925 2352 y(\014)954 2338 y Fs(')984 2320 y Fo(\()p Fp(n)p Fo(\))1047 2338 y Fr(2)13 b(N)1139 2313 y Fo(^)1134 2319 y Fq(\012)q Fp(n)p 1134 2352 V 1124 2352 a(C)1185 2338 y Fs(;)23 b(n)13 b Fr(2)f Fs(I)-12 b(N)1353 2279 y Fj(\033)1392 2338 y Fs(:)130 2523 y Fu(Let)15 b(us)g(collect)i(some)e(prop)q(erties) h(of)e(the)i(p)q(olynomials)g Fs(P)1156 2507 y Fp(\026)1150 2534 y(n)1180 2523 y Fu(\()p Fs(z)r Fu(\))p Fs(:)59 2625 y Fk(Pr)o(oposition)g(11)23 b Fm(F)m(or)16 b Fs(x;)8 b(y)14 b Fr(2)f(N)679 2609 y Fq(0)707 2625 y Fs(;)23 b(n)13 b Fr(2)g Fs(I)-12 b(N)21 b Fm(the)c(fol)r(lowing)f(holds)59 2752 y(\(P1\))632 2757 y Fs(P)667 2738 y Fp(\026)661 2768 y(n)691 2757 y Fu(\()p Fs(x)p Fu(\))c(=)834 2704 y Fp(n)815 2716 y Fj(X)813 2809 y Fp(k)q Fo(=0)885 2685 y Fj( )917 2726 y Fs(n)919 2788 y(k)945 2685 y Fj(!)978 2757 y Fs(x)1004 2738 y Fq(\012)p Fp(k)1059 2750 y Fu(^)1052 2757 y Fr(\012)q Fs(P)1123 2735 y Fp(\026)1117 2771 y(n)p Fq(\000)p Fp(k)1187 2757 y Fu(\(0\))p Fs(;)491 b Fu(\(11\))p eop %%Page: 14 14 14 13 bop 59 50 a Fn(4)45 b(CONCEPT)15 b(OF)g(DISTRIBUTIONS)j(IN)d (INFINITE)i(DIMENSIONAL)f(ANAL)l(YSIS)84 b Fu(14)59 315 y Fm(\(P2\))660 319 y Fs(x)686 300 y Fq(\012)p Fp(n)750 319 y Fu(=)819 266 y Fp(n)800 279 y Fj(X)798 371 y Fp(k)q Fo(=0)869 247 y Fj( )902 288 y Fs(n)903 350 y(k)930 247 y Fj(!)962 319 y Fs(P)997 297 y Fp(\026)991 333 y(k)1021 319 y Fu(\()p Fs(x)p Fu(\))1089 312 y(^)1083 319 y Fr(\012)p Fu(M)1160 297 y Fp(\026)1160 333 y(n)p Fq(\000)p Fp(k)1750 319 y Fu(\(12\))59 518 y Fm(\(P3\))465 511 y Fs(P)500 492 y Fp(\026)494 522 y(n)524 511 y Fu(\()p Fs(x)10 b Fu(+)g Fs(y)r Fu(\))i(=)778 471 y Fj(X)725 563 y Fp(k)q Fo(+)p Fp(l)p Fo(+)p Fp(m)p Fo(=)p Fp(n)949 480 y Fs(n)p Fu(!)p 903 500 133 2 v 903 542 a Fs(k)q Fu(!)c Fs(l)q Fu(!)g Fs(m)p Fu(!)1041 511 y Fs(P)1076 489 y Fp(\026)1070 525 y(k)1099 511 y Fu(\()p Fs(x)p Fu(\))1167 504 y(^)1161 511 y Fr(\012)p Fs(P)1231 489 y Fp(\026)1225 525 y(l)1255 511 y Fu(\()p Fs(y)r Fu(\))1321 504 y(^)1315 511 y Fr(\012)p Fu(M)1392 492 y Fp(\026)1392 522 y(m)693 675 y Fu(=)762 622 y Fp(n)743 635 y Fj(X)740 727 y Fp(k)q Fo(=0)812 603 y Fj( )845 644 y Fs(n)846 706 y(k)873 603 y Fj(!)905 675 y Fs(P)940 653 y Fp(\026)934 689 y(k)964 675 y Fu(\()p Fs(x)p Fu(\))1032 668 y(^)1026 675 y Fr(\012)p Fs(y)1085 656 y Fq(\012)p Fo(\()p Fp(n)p Fq(\000)p Fp(k)q Fo(\))1750 675 y Fu(\(13\))59 796 y Fm(\(P4\))16 b(F)m(urther)g(we)g(observe)421 906 y Fs(I)-12 b(E)466 913 y Fp(\026)489 906 y Fu(\()p Fr(h)p Fs(P)560 887 y Fp(\026)554 917 y(m)587 906 y Fu(\()p Fr(\001)p Fu(\))p Fs(;)8 b(')687 887 y Fo(\()p Fp(m)p Fo(\))745 906 y Fr(i)p Fu(\))k(=)h(0)93 b Fm(for)16 b Fs(m)d Fr(6)p Fu(=)g(0)j Fs(;)8 b(')1220 887 y Fo(\()p Fp(m)p Fo(\))1292 906 y Fr(2)13 b(N)1384 880 y Fo(^)1379 886 y Fq(\012)q Fp(m)p 1378 919 2 18 v 1369 919 a(C)1456 906 y Fs(:)281 b Fu(\(14\))59 1008 y Fm(\(P5\))19 b(F)m(or)g(al)r(l)g Fs(p)g(>)f(p)444 1015 y Fo(0)484 1008 y Fm(such)h(that)h(the)g(emb)n(e) n(dding)f Fr(H)1025 1015 y Fp(p)1063 1008 y Fs(,)-8 b Fr(!)19 b(H)1170 1015 y Fp(p)1188 1020 y Fi(0)1228 1008 y Fm(is)g(Hilb)n(ert{Schmidt)g(and)g(for)h(al)r(l)59 1086 y Fs(")13 b(>)g Fu(0)j Fm(smal)r(l)g(enough)455 1027 y Fj(\022)486 1086 y Fs(")d Fr(\024)627 1068 y Fo(2)645 1057 y Fh(\000)p Fg(q)683 1064 y Fi(0)p 572 1075 184 2 v 572 1108 a Fp(e)588 1111 y Fr(k)612 1108 y Fp(i)624 1112 y Fg(p;p)665 1119 y Fi(0)684 1111 y Fr(k)707 1125 y Fg(H)r(S)761 1027 y Fj(\023)808 1086 y Fm(ther)n(e)j(exists)g(a)g(c)n (onstant)f Fs(C)1299 1093 y Fp(p;")1357 1086 y Fs(>)e Fu(0)j Fm(with)525 1218 y Fr(j)o Fs(P)572 1200 y Fp(\026)566 1230 y(n)596 1218 y Fu(\()p Fs(z)r Fu(\))p Fr(j)667 1232 y Fq(\000)p Fp(p)727 1218 y Fr(\024)d Fs(C)808 1225 y Fp(p;")861 1218 y Fs(n)p Fu(!)8 b Fs(")930 1200 y Fq(\000)p Fp(n)988 1218 y Fs(e)1009 1200 y Fp(")p Fq(j)p Fp(z)q Fq(j)1062 1204 y Fh(\000)p Fg(p)1107 1218 y Fs(;)54 b(z)15 b Fr(2)e(H)1291 1225 y Fq(\000)p Fp(p;)p 1352 1225 2 18 v -3 w(C)1750 1218 y Fu(\(15\))59 1325 y Fk(Pr)o(oof.)72 b Fu(W)l(e)16 b(restrict)f(ourselv)o(es)g(to)g(a)g(sk)o(etc)o(h)g(of)f (pro)q(of,)h(details)h(can)f(b)q(e)h(found)g(in)g([ADKS94)o(].)59 1381 y(\(P1\))e(This)i(form)o(ula)f(can)g(b)q(e)h(obtained)g(simply)g (b)o(y)f(substituting)520 1483 y(1)p 484 1503 95 2 v 484 1545 a Fs(l)498 1552 y Fp(\026)521 1545 y Fu(\()p Fs(\022)q Fu(\))596 1514 y(=)660 1461 y Fq(1)648 1473 y Fj(X)644 1564 y Fp(n)p Fo(=0)732 1483 y Fu(1)p 723 1503 40 2 v 723 1545 a Fs(n)p Fu(!)776 1479 y Fj(\012)796 1514 y Fs(P)831 1495 y Fp(\026)825 1525 y(n)854 1514 y Fu(\(0\))p Fs(;)8 b(\022)956 1495 y Fq(\012)p Fp(n)1006 1479 y Fj(\013)1033 1514 y Fs(;)53 b(\022)14 b Fr(2)f(N)p 1220 1521 2 18 v 1211 1521 a Fp(C)1241 1514 y Fs(;)8 b Fr(j)o Fs(\022)q Fr(j)1309 1527 y Fp(q)1341 1514 y Fs(<)13 b(\016)341 b Fu(\(16\))59 1649 y(and)509 1725 y Fs(e)530 1706 y Fq(h)o Fp(x;\022)q Fq(i)619 1725 y Fu(=)683 1672 y Fq(1)670 1684 y Fj(X)667 1775 y Fp(n)p Fo(=0)755 1694 y Fu(1)p 746 1714 40 2 v 746 1756 a Fs(n)p Fu(!)799 1690 y Fj(\012)818 1725 y Fs(x)844 1706 y Fq(\012)p Fp(n)895 1725 y Fs(;)8 b(\022)938 1706 y Fq(\012)p Fp(n)989 1690 y Fj(\013)1016 1725 y Fs(;)53 b(\022)14 b Fr(2)f(N)p 1203 1732 2 18 v 1194 1732 a Fp(C)1224 1725 y Fs(;)8 b(x)k Fr(2)h(N)1370 1706 y Fq(0)59 1852 y Fu(in)j(the)g(equalit)o(y)h Fs(e)386 1859 y Fp(\026)409 1852 y Fu(\()p Fs(\022)q Fu(;)8 b Fs(x)p Fu(\))k(=)i Fs(e)596 1835 y Fq(h)p Fp(x;\022)q Fq(i)673 1852 y Fs(l)688 1835 y Fq(\000)p Fo(1)687 1863 y Fp(\026)734 1852 y Fu(\()p Fs(\022)q Fu(\).)21 b(A)16 b(comparison)g(with)g(\(10\))e(pro)o(v)o(es)h(\(P1\).)20 b(The)c(pro)q(of)f(of)59 1908 y(\(P2\))f(is)i(completely)g(analogous)f (to)g(the)g(pro)q(of)g(of)g(\(P1\).)59 1965 y(\(P3\))f(W)l(e)i(start)d (from)i(the)g(follo)o(wing)h(ob)o(vious)g(equation)f(of)g(the)g (generating)h(functions)598 2067 y Fs(e)619 2074 y Fp(\026)642 2067 y Fu(\()p Fs(\022)q Fu(;)8 b Fs(x)i Fu(+)g Fs(y)r Fu(\))j(=)g Fs(e)908 2074 y Fp(\026)931 2067 y Fu(\()p Fs(\022)q Fu(;)8 b Fs(x)p Fu(\))g Fs(e)1065 2074 y Fp(\026)1087 2067 y Fu(\()p Fs(\022)q Fu(;)g Fs(y)r Fu(\))g Fs(l)1212 2074 y Fp(\026)1234 2067 y Fu(\()p Fs(\022)q Fu(\))59 2169 y(This)16 b(implies)251 2246 y Fq(1)238 2258 y Fj(X)235 2349 y Fp(n)p Fo(=0)323 2268 y Fu(1)p 314 2288 40 2 v 314 2330 a Fs(n)p Fu(!)359 2299 y Fr(h)p Fs(P)412 2280 y Fp(\026)406 2310 y(n)435 2299 y Fu(\()p Fs(x)10 b Fu(+)h Fs(y)r Fu(\))p Fs(;)d(\022)620 2280 y Fq(\012)p Fp(n)670 2299 y Fr(i)k Fu(=)794 2246 y Fq(1)781 2258 y Fj(X)748 2351 y Fp(k)q(;l;m)p Fo(=0)942 2268 y Fu(1)p 887 2288 133 2 v 887 2330 a Fs(k)q Fu(!)c Fs(l)q Fu(!)g Fs(m)p Fu(!)1032 2299 y Fr(h)p Fs(P)1079 2306 y Fp(k)1100 2299 y Fu(\()p Fs(x)p Fu(\))1168 2292 y(^)1162 2299 y Fr(\012)p Fs(P)1226 2306 y Fp(l)1239 2299 y Fu(\()p Fs(y)r Fu(\))1305 2292 y(^)1299 2299 y Fr(\012)p Fu(M)1376 2306 y Fp(m)1408 2299 y Fs(;)20 b(\022)1463 2280 y Fq(\012)p Fo(\()p Fp(k)q Fo(+)p Fp(l)p Fo(+)p Fp(m)p Fo(\))1637 2299 y Fr(i)59 2445 y Fu(from)15 b(this)g(\(P3\))f(follo)o(ws)i(immediately)l(.)59 2502 y(\(P4\))e(T)l(o)h(see)h(this)f(w)o(e)g(use,)g Fs(\022)g Fr(2)d(N)p 671 2509 2 18 v 662 2509 a Fp(C)693 2502 y Fu(,)380 2582 y Fq(1)367 2594 y Fj(X)364 2685 y Fp(n)p Fo(=0)452 2604 y Fu(1)p 443 2624 40 2 v 443 2666 a Fs(n)p Fu(!)488 2635 y Fs(I)-12 b(E)532 2642 y Fp(\026)556 2635 y Fu(\()p Fr(h)p Fs(P)627 2616 y Fp(\026)621 2646 y(m)654 2635 y Fu(\()p Fr(\001)p Fu(\))p Fs(;)8 b(\022)746 2616 y Fq(\012)p Fp(n)795 2635 y Fr(i)p Fu(\))k(=)h Fs(I)-12 b(E)936 2642 y Fp(\026)959 2635 y Fu(\()p Fs(e)998 2642 y Fp(\026)1021 2635 y Fu(\()p Fs(\022)q Fu(;)8 b Fs(x)p Fu(\)\))k(=)1209 2604 y Fs(I)-12 b(E)1253 2611 y Fp(\026)1277 2604 y Fu(\()p Fs(e)1316 2588 y Fq(h)p Fp(x;\022)q Fq(i)1392 2604 y Fu(\))p 1209 2624 202 2 v 1262 2666 a Fs(l)1276 2673 y Fp(\026)1299 2666 y Fu(\()p Fs(\022)q Fu(\))1427 2635 y(=)13 b(1)i Fs(:)59 2771 y Fu(Then)h(a)f(comparison)g(of)g(co)q (e\016cien)o(ts)h(and)f(the)g(p)q(olarization)i(iden)o(tit)o(y)f(giv)o (es)f(the)g(ab)q(o)o(v)o(e)g(result.)p eop %%Page: 15 15 15 14 bop 59 50 a Fn(4)45 b(CONCEPT)15 b(OF)g(DISTRIBUTIONS)j(IN)d (INFINITE)i(DIMENSIONAL)f(ANAL)l(YSIS)84 b Fu(15)59 199 y(\(P5\))14 b(W)l(e)i(can)f(use)285 334 y Fr(j)p Fs(P)333 315 y Fp(\026)327 345 y(n)357 334 y Fu(\()p Fs(z)r Fu(\))p Fr(j)429 342 y Fq(\000)p Fp(p)474 332 y Fh(0)499 334 y Fr(\024)e Fs(n)p Fu(!)595 262 y Fj( )643 334 y Fu(sup)627 375 y Fq(j)p Fp(\022)q Fq(j)664 379 y Fg(p)682 375 y Fo(=)p Fp(\032)777 303 y Fu(1)p 741 324 95 2 v 741 365 a Fs(l)755 372 y Fp(\026)777 365 y Fu(\()p Fs(\022)q Fu(\))840 262 y Fj(!)881 334 y Fs(e)902 315 y Fp(\032)p Fq(j)p Fp(z)q Fq(j)957 319 y Fh(\000)p Fg(p)1009 275 y Fj(\022)1046 303 y Fs(e)p 1045 324 24 2 v 1045 365 a(\032)1088 298 y Fj(\015)1088 323 y(\015)1111 334 y Fs(i)1127 342 y Fp(p)1145 332 y Fh(0)1156 342 y Fp(;p)1186 298 y Fj(\015)1186 323 y(\015)1209 350 y Fp(H)s(S)1266 275 y Fj(\023)1296 286 y Fp(n)1340 334 y Fs(;)53 b(z)14 b Fr(2)f(H)1522 341 y Fq(\000)p Fp(p;)p 1584 341 2 18 v -3 w(C)1750 334 y Fu(\(17\))59 473 y Fs(p)h(>)g(p)168 480 y Fo(0)187 473 y Fs(;)8 b(p)231 456 y Fq(0)242 473 y Fs(;)g(\032)15 b Fu(de\014ned)i(ab)q(o)o(v)o(e.)k(\(17\))15 b(is)h(a)g(simple)h(consequence)h(of)d(the)h(k)o(ernel)h(theorem)e(b)o (y)h(\(9\).)21 b(In)59 529 y(particular)16 b(w)o(e)f(ha)o(v)o(e)490 661 y Fr(j)p Fs(P)538 643 y Fp(\026)532 673 y(n)562 661 y Fu(\(0\))p Fr(j)632 675 y Fq(\000)p Fp(p)692 661 y Fr(\024)e Fs(n)p Fu(!)788 589 y Fj( )845 661 y Fu(sup)821 702 y Fq(j)p Fp(\022)q Fq(j)858 706 y Fg(p)874 713 y Fi(0)893 702 y Fo(=)p Fp(\032)987 631 y Fu(1)p 951 651 95 2 v 951 693 a Fs(l)965 700 y Fp(\026)988 693 y Fu(\()p Fs(\022)q Fu(\))1051 589 y Fj(!)1091 602 y(\022)1128 631 y Fs(e)p 1127 651 24 2 v 1127 693 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y Fu(\).)59 886 y Fk(Lemma)i(12)23 b Fm(F)m(or)16 b(any)g Fs(')d Fr(2)g(P)t Fu(\()p Fr(N)657 869 y Fq(0)667 886 y Fu(\))j Fm(ther)n(e)h(exists)e(a)h(unique)h(r)n(epr)n(esentation) 527 1027 y Fs(')p Fu(\()p Fs(x)p Fu(\))12 b(=)696 974 y Fp(N)682 986 y Fj(X)679 1077 y Fp(n)p Fo(=0)753 980 y Fj(D)778 1027 y Fs(P)813 1008 y Fp(\026)807 1038 y(n)837 1027 y Fu(\()p Fs(x)p Fu(\))p Fs(;)c(')950 1008 y Fo(\()p Fp(n)p Fo(\))999 980 y Fj(E)1049 1027 y Fs(;)69 b(')1161 1008 y Fo(\()p Fp(n)p Fo(\))1225 1027 y Fr(2)13 b(N)1317 1002 y Fo(^)1312 1008 y Fq(\012)p Fp(n)p 1311 1040 2 18 v 1302 1040 a(C)1750 1027 y Fu(\(19\))59 1165 y Fm(and)j(vic)n(e)g (versa,)g(any)g(functional)g(of)g(the)h(form)g(\(19\))e(is)h(a)h(smo)n (oth)f(p)n(olynomial.)59 1271 y Fk(Pr)o(oof.)72 b Fu(The)16 b(represen)o(tations)f(from)f(De\014nition)j(10)e(and)g(equation)g (\(19\))f(can)i(b)q(e)g(transformed)59 1328 y(in)o(to)f(one)h(another)e (using)i(\(11\))e(and)i(\(12\).)997 b Ff(2)59 1449 y Fl(4.2)56 b(The)18 b(dual)h(App)r(ell)e(system)g(and)i(the)f(represen)n (tation)g(theorem)e(for)j Fd(P)1714 1431 y Fq(0)1710 1462 y Fp(\026)1733 1449 y Fz(\()p Fd(N)1800 1431 y Fq(0)1812 1449 y Fz(\))59 1535 y Fu(T)l(o)k(giv)o(e)g(an)f(in)o(ternal)i (description)g(of)e(the)h(t)o(yp)q(e)g(\(19\))f(for)g Fr(P)1193 1519 y Fq(0)1189 1546 y Fp(\026)1212 1535 y Fu(\()p Fr(N)1274 1519 y Fq(0)1285 1535 y Fu(\))g(w)o(e)h(ha)o(v)o(e)f (to)g(construct)h(an)59 1592 y(appropriate)13 b(system)g(of)g (generalized)i(functions,)f(the)p 1032 1592 3 25 v 16 w Fs(Q)1056 1574 y Fp(\026)1080 1592 y Fu(-system.)19 b(The)13 b(construction)h(w)o(e)f(prop)q(ose)59 1648 y(here)i(is)f(di\013eren)o(t)h(from)e(that)h(of)f([ADKS94])g(where)i (smo)q(othness)f(of)f(the)i(logarithmic)g(deriv)m(ativ)o(e)g(of)59 1705 y Fs(\026)f Fu(w)o(as)e(demanded)i(and)f(used)h(for)f(the)g (construction)g(of)g(the)p 1132 1705 V 17 w Fs(Q)1156 1687 y Fp(\026)1179 1705 y Fu(-system.)19 b(T)l(o)13 b(a)o(v)o(oid)g(this)g(additional)59 1761 y(assumption)e(\(whic)o(h)g (excludes)h(e.g.,)e(P)o(oisson)g(measures\))g(w)o(e)g(prop)q(ose)g(to)g (construct)g(the)p 1633 1761 V 14 w Fs(Q)1657 1743 y Fp(\026)1680 1761 y Fu(-system)59 1818 y(using)16 b(di\013eren)o(tial)g (op)q(erators.)130 1885 y(F)l(or)f(\010)245 1868 y Fo(\()p Fp(n)p Fo(\))310 1885 y Fr(2)355 1837 y Fj(\020)380 1885 y Fr(N)428 1859 y Fo(^)424 1865 y Fq(\012)p Fp(n)p 423 1898 2 18 v 414 1898 a(C)474 1837 y Fj(\021)499 1849 y Fq(0)527 1885 y Fu(de\014ne)i(a)f(di\013eren)o(tial)i(op)q(erator)d (of)h(order)g Fs(n)g Fu(with)g(constan)o(t)g(co)q(e\016cien)o(ts)59 1963 y Fs(')89 1946 y Fo(\()p Fp(m)p Fo(\))162 1963 y Fr(2)d(N)254 1938 y Fo(^)249 1944 y Fq(\012)q Fp(m)p 248 1976 V 239 1976 a(C)310 1963 y Fs(;)8 b(m)k Fr(2)g Fs(I)-12 b(N)265 2131 y(D)q Fu(\(\010)355 2112 y Fo(\()p Fp(n)p Fo(\))405 2131 y Fu(\))p Fr(h)p Fs(x)467 2112 y Fq(\012)p Fp(m)528 2131 y Fs(;)8 b(')579 2112 y Fo(\()p Fp(m)p Fo(\))638 2131 y Fr(i)13 b Fu(=)716 2059 y Fj(\()815 2058 y Fs(m)p Fu(!)p 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2 v 780 642 a(\()p Fs(m)10 b Fr(\000)g Fs(n)p Fu(\)!)956 611 y(\(\010)1007 592 y Fo(\()p Fp(n)p Fo(\))1057 611 y Fs(;)20 b(')1120 592 y Fo(\()p Fp(m)p Fo(\))1181 611 y Fu(\))1199 627 y Fq(H)1233 613 y Fi(^)1229 618 y Fh(\012)o Fg(n)59 752 y Fu(where)15 b(the)h(con)o(traction)f(\(\010)559 736 y Fo(\()p Fp(n)p Fo(\))609 752 y Fs(;)20 b(')672 736 y Fo(\()p Fp(m)p Fo(\))732 752 y Fu(\))750 769 y Fq(H)784 754 y Fi(^)780 759 y Fh(\012)p Fg(n)842 752 y Fr(2)13 b(N)929 728 y Fq(\012)p Fo(\()p Fp(m)p Fq(\000)p Fp(n)p Fo(\))p 929 766 2 18 v 919 766 a Fp(C)1082 752 y Fu(is)i(de\014ned)i(b)o(y)419 862 y Fr(h)p Fs(x)463 843 y Fq(\012)p Fo(\()p Fp(m)p Fq(\000)p Fp(n)p Fo(\))599 862 y Fs(;)j Fu(\(\010)683 843 y Fo(\()p Fp(n)p Fo(\))734 862 y Fs(;)f(')796 843 y Fo(\()p Fp(m)p Fo(\))857 862 y Fu(\))875 878 y Fq(H)909 864 y Fi(^)905 869 y Fh(\012)p Fg(n)952 862 y Fr(i)12 b Fu(:=)h Fr(h)p Fs(x)1087 843 y Fq(\012)p Fo(\()p Fp(m)p Fq(\000)p Fp(n)p Fo(\))1230 855 y Fu(^)1224 862 y Fr(\012)p Fu(\010)1292 843 y Fo(\()p Fp(n)p Fo(\))1343 862 y Fs(;)8 b(')1394 843 y Fo(\()p Fp(m)p Fo(\))1453 862 y Fr(i)279 b Fu(\(20\))59 964 y(for)15 b(all)h Fs(x)c Fr(2)h(N)317 948 y Fq(0)329 964 y Fu(.)20 b(It)15 b(is)h(easy)f(to)f(v)o(erify)i(that)464 1066 y Fr(j)p Fu(\(\010)528 1048 y Fo(\()p Fp(n)p Fo(\))578 1066 y Fs(;)k(')641 1048 y Fo(\()p Fp(m)p Fo(\))702 1066 y Fu(\))720 1083 y Fq(H)754 1068 y Fi(^)750 1073 y Fh(\012)p Fg(n)797 1066 y Fr(j)810 1073 y Fp(q)841 1066 y Fr(\024)13 b(j)p Fu(\010)935 1048 y Fo(\()p Fp(n)p Fo(\))985 1066 y Fr(j)998 1073 y Fq(\000)p Fp(q)1045 1066 y Fr(j)p Fs(')1088 1048 y Fo(\()p Fp(m)p Fo(\))1148 1066 y Fr(j)1161 1073 y Fp(q)1194 1066 y Fu(,)91 b Fs(q)15 b Fr(2)d Fs(I)-12 b(N)59 1181 y Fu(whic)o(h)22 b(guaran)o(tees)f(that)f(\(\010)583 1165 y Fo(\()p Fp(n)p Fo(\))634 1181 y Fs(;)30 b(')707 1165 y Fo(\()p Fp(m)p Fo(\))767 1181 y Fu(\))785 1198 y Fq(H)819 1183 y Fi(^)815 1188 y Fh(\012)p Fg(n)885 1181 y Fr(2)23 b(N)982 1157 y Fq(\012)p Fo(\()p Fp(m)p Fq(\000)p Fp(n)p Fo(\))p 981 1195 V 972 1195 a Fp(C)1140 1181 y Fu(and)f(sho)o(ws)f(at)f(the)i(same)f(time)g(that)59 1243 y Fs(D)q Fu(\(\010)149 1227 y Fo(\()p Fp(n)p Fo(\))199 1243 y Fu(\))12 b(is)g(con)o(tin)o(uous)g(on)g(eac)o(h)g(comp)q(onen)o (t.)19 b(This)12 b(is)g(su\016cien)o(t)h(to)e(ensure)h(the)g(stated)f (con)o(tin)o(uit)o(y)59 1300 y(of)k Fs(D)q Fu(\(\010)201 1283 y Fo(\()p Fp(n)p Fo(\))251 1300 y Fu(\))g(on)g Fr(P)t Fu(\()p Fr(N)445 1283 y Fq(0)456 1300 y Fu(\))p Fs(:)1310 b Ff(2)59 1463 y Fk(Lemma)17 b(14)23 b Fm(F)m(or)16 b Fu(\010)419 1446 y Fo(\()p Fp(n)p Fo(\))483 1463 y Fr(2)d(N)570 1444 y Fq(0)585 1438 y Fo(^)580 1444 y Fq(\012)p Fp(n)p 569 1476 V 560 1476 a(C)646 1463 y Fm(,)j Fs(')706 1446 y Fo(\()p Fp(m)p Fo(\))780 1463 y Fr(2)d(N)871 1438 y Fo(^)867 1444 y Fq(\012)p Fp(m)p 866 1476 V 857 1476 a(C)943 1463 y Fm(we)k(have)59 1625 y(\(P6\))231 1626 y Fs(D)q Fu(\(\010)321 1607 y Fo(\()p Fp(n)p Fo(\))371 1626 y Fu(\))p Fr(h)p Fs(P)442 1607 y Fp(\026)436 1637 y(m)469 1626 y Fu(\()p Fs(x)p Fu(\))p Fs(;)8 b(')582 1607 y Fo(\()p Fp(m)p Fo(\))641 1626 y Fr(i)13 b Fu(=)719 1554 y Fj(\()828 1552 y Fs(m)p Fu(!)p 769 1572 171 2 v 769 1614 a(\()p Fs(m)d Fr(\000)g Fs(n)p Fu(\)!)952 1536 y Fj(D)978 1583 y Fs(P)1013 1561 y Fp(\026)1007 1594 y(m)p Fq(\000)p Fp(n)1089 1583 y Fu(\()p Fs(x)p Fu(\))1157 1576 y(^)1151 1583 y Fr(\012)p Fu(\010)1219 1566 y Fo(\()p Fp(n)p Fo(\))1270 1583 y Fs(;)19 b(')1332 1566 y Fo(\()p Fp(m)p Fo(\))1393 1536 y Fj(E)1442 1583 y Fm(for)e Fs(m)12 b Fr(\025)h Fs(n)930 1669 y Fu(0)500 b Fm(for)17 b Fs(m)12 b(<)h(n)1750 1626 y Fu(\(21\))59 1763 y Fk(Pr)o(oof.)83 b Fu(This)19 b(follo)o(ws)g(from)e(the)i (general)g(prop)q(ert)o(y)f(of)g(App)q(ell)j(p)q(olynomials)f(whic)o(h) f(b)q(eha)o(v)o(e)59 1820 y(lik)o(e)d(ordinary)g(p)q(o)o(w)o(ers)e (under)i(di\013eren)o(tiation.)21 b(More)15 b(precisely)l(,)h(b)o(y)g (using)484 1942 y Fr(h)p Fs(P)537 1923 y Fp(\026)531 1953 y(m)564 1942 y Fs(;)8 b(\022)607 1923 y Fq(\012)p Fp(m)668 1942 y Fr(i)k Fu(=)751 1882 y Fj(\022)795 1911 y Fu(d)p 786 1931 42 2 v 786 1973 a(d)p Fs(t)833 1882 y Fj(\023)864 1892 y Fp(m)904 1942 y Fs(e)925 1949 y Fp(\026)949 1942 y Fu(\()p Fs(t\022)q Fu(;)c Fr(\001)p Fu(\))1057 1880 y Fj(\014)1057 1905 y(\014)1057 1930 y(\014)1057 1955 y(\014)1070 1982 y Fp(t)p Fo(=0)1153 1942 y Fs(;)98 b(\022)14 b Fr(2)f(N)p 1385 1949 2 18 v 1376 1949 a Fp(C)59 2064 y Fu(w)o(e)i(ha)o(v)o(e)202 2176 y Fs(D)q Fu(\(\010)292 2157 y Fo(\(1\))338 2176 y Fu(\))p Fr(h)p Fs(P)403 2183 y Fp(m)436 2176 y Fu(\()p Fs(x)p Fu(\))p Fs(;)8 b(\022)541 2157 y Fq(\012)p Fp(m)601 2176 y Fr(i)41 b Fu(=)760 2145 y(d)p 747 2165 52 2 v 747 2207 a(d)p Fs(\025)804 2176 y Fr(h)p Fs(P)857 2157 y Fp(\026)851 2187 y(m)884 2176 y Fu(\()p Fs(x)10 b Fu(+)g Fs(\025)p Fu(\010)1043 2157 y Fo(\(1\))1090 2176 y Fu(\))p Fs(;)e(\022)1151 2157 y Fq(\012)p Fp(m)1211 2176 y Fr(i)1229 2114 y Fj(\014)1229 2139 y(\014)1229 2164 y(\014)1229 2189 y(\014)1242 2216 y Fp(\025)p Fo(=0)660 2296 y Fu(=)742 2237 y Fj(\022)786 2266 y Fs(@)p 777 2286 44 2 v 777 2327 a(@)s(t)826 2237 y Fj(\023)856 2247 y Fp(m)915 2266 y Fs(@)p 902 2286 54 2 v 902 2327 a(@)s(\025)960 2296 y(e)981 2303 y Fp(\026)1005 2296 y Fu(\()p Fs(t\022)q Fu(;)g Fs(x)i Fu(+)g Fs(\025)p Fu(\010)1223 2277 y Fo(\(1\))1269 2296 y Fu(\))1287 2235 y Fj(\014)1287 2260 y(\014)1287 2285 y(\014)1287 2310 y(\014)1310 2317 y Fp(t)p Fo(=0)1306 2352 y Fp(\025)p Fo(=0)660 2419 y Fu(=)42 b Fr(h)p Fu(\010)788 2400 y Fo(\(1\))835 2419 y Fs(;)8 b(\022)q Fr(i)908 2372 y Fj(\020)944 2401 y Fp(@)p 937 2408 34 2 v 937 2435 a(@)r(t)976 2372 y Fj(\021)1000 2383 y Fp(m)1041 2419 y Fs(t)13 b(e)1091 2426 y Fp(\026)1115 2419 y Fu(\()p Fs(t\022)q Fu(;)8 b Fs(x)p Fu(\))1236 2370 y Fj(\014)1236 2395 y(\014)1236 2420 y(\014)1249 2447 y Fp(t)p Fo(=0)660 2537 y Fu(=)47 b Fr(h)p Fu(\010)793 2518 y Fo(\(1\))840 2537 y Fs(;)8 b(\022)q Fr(i)924 2484 y Fp(m)910 2497 y Fj(X)908 2589 y Fp(k)q Fo(=0)980 2503 y Fj(\000)999 2517 y Fp(m)1005 2553 y(k)1030 2503 y Fj(\001)1056 2478 y(\022)1087 2490 y(\020)1123 2519 y Fo(d)p 1117 2526 33 2 v 1117 2553 a(d)p Fp(t)1154 2490 y Fj(\021)1179 2502 y Fp(k)1208 2537 y Fs(t)1224 2478 y Fj(\023)1262 2490 y(\020)1298 2519 y Fo(d)p 1292 2526 V 1292 2553 a(d)p Fp(t)1329 2490 y Fj(\021)1354 2502 y Fp(m)p Fq(\000)p Fp(k)1442 2537 y Fs(e)1463 2544 y Fp(\026)1486 2537 y Fu(\()p Fs(t\022)q Fu(;)g Fs(x)p Fu(\))1607 2464 y Fj(\014)1607 2489 y(\014)1607 2513 y(\014)1607 2538 y(\014)1607 2563 y(\014)1621 2595 y Fp(t)p Fo(=0)660 2673 y Fu(=)47 b Fs(m)8 b Fr(h)p Fu(\010)841 2654 y Fo(\(1\))887 2673 y Fs(;)g(\022)q Fr(i)956 2626 y Fj(\020)991 2655 y Fo(d)p 985 2662 V 985 2689 a(d)p Fp(t)1022 2626 y Fj(\021)1047 2638 y Fp(m)p Fq(\000)p Fo(1)1133 2673 y Fs(e)1154 2680 y Fp(\026)1177 2673 y Fu(\()p Fs(t\022)q Fu(;)g Fs(x)p Fu(\))1298 2612 y Fj(\014)1298 2637 y(\014)1298 2662 y(\014)1298 2687 y(\014)1312 2714 y Fp(t)p Fo(=0)660 2779 y Fu(=)42 b Fs(m)8 b Fr(h)p Fu(\010)836 2760 y Fo(\(1\))882 2779 y Fs(;)g(\022)q Fr(i)951 2732 y Fj(D)976 2779 y Fs(P)1005 2786 y Fp(m)p Fq(\000)p Fo(1)1083 2779 y Fu(\()p Fs(x)p Fu(\))p Fs(;)g(\022)1188 2760 y Fq(\012)p Fo(\()p Fp(m)p Fq(\000)p Fo(1\))1320 2732 y Fj(E)1369 2779 y Fu(.)p eop %%Page: 18 18 18 17 bop 59 50 a Fn(4)45 b(CONCEPT)15 b(OF)g(DISTRIBUTIONS)j(IN)d (INFINITE)i(DIMENSIONAL)f(ANAL)l(YSIS)84 b Fu(18)59 199 y(This)16 b(pro)o(v)o(es)501 261 y Fs(D)q Fu(\(\010)591 242 y Fo(\(1\))637 261 y Fu(\))p Fr(h)p Fs(P)708 242 y Fp(\026)702 272 y(m)735 261 y Fs(;)8 b(')786 242 y Fo(\()p Fp(m)p Fo(\))846 261 y Fr(i)k Fu(=)h Fs(m)972 214 y Fj(D)997 261 y Fs(P)1032 239 y Fp(\026)1026 273 y(m)p Fq(\000)p Fo(1)1111 254 y Fu(^)1105 261 y Fr(\012)p Fu(\010)1173 242 y Fo(\(1\))1220 261 y Fs(;)20 b(')1283 242 y Fo(\()p Fp(m)p Fo(\))1343 214 y Fj(E)1376 261 y Fs(:)59 352 y Fu(The)15 b(prop)q(ert)o(y)g(\(21\),)f(then)h(follo)o(ws) h(b)o(y)f(induction.)836 b Ff(2)130 458 y Fu(In)14 b(view)g(of)g(Lemma) f(16)g(it)h(is)h(p)q(ossible)g(to)e(de\014ne)i(the)f(adjoin)o(t)f(op)q (erator)g Fs(D)q Fu(\(\010)1501 442 y Fo(\()p Fp(n)p Fo(\))1551 458 y Fu(\))1569 442 y Fq(\003)1601 458 y Fu(:)g Fr(P)1663 442 y Fq(0)1659 470 y Fp(\026)1682 458 y Fu(\()p Fr(N)1744 442 y Fq(0)1755 458 y Fu(\))f Fr(!)59 528 y(P)95 512 y Fq(0)91 539 y Fp(\026)114 528 y Fu(\()p Fr(N)176 512 y Fq(0)187 528 y Fu(\))18 b(for)g(\010)329 512 y Fo(\()p Fp(n)p Fo(\))397 528 y Fr(2)g(N)489 509 y Fq(0)504 503 y Fo(^)499 509 y Fq(\012)p Fp(n)p 489 542 2 18 v 479 542 a(C)568 528 y Fu(.)29 b(F)l(urther)18 b(w)o(e)g(can)h(in)o(tro)q(duce)g(the)f(constan)o(t)g(function)h(1)-15 b(1)17 b Fr(2)h(P)1721 512 y Fq(0)1717 539 y Fp(\026)1740 528 y Fu(\()p Fr(N)1802 512 y Fq(0)1813 528 y Fu(\))59 585 y(suc)o(h)e(that)e(1)-15 b(1)o(\()p Fs(x)p Fu(\))12 b Fr(\021)h Fu(1)i(for)g(all)h Fs(x)c Fr(2)h(N)708 568 y Fq(0)735 585 y Fu(,)i(so)580 703 y Fr(h)-8 b(h)p Fu(1)-15 b(1)p Fs(;)19 b(')p Fr(i)-8 b(i)729 710 y Fp(\026)765 703 y Fu(=)813 646 y Fj(Z)836 740 y Fq(N)870 731 y Fh(0)890 703 y Fs(')p Fu(\()p Fs(x)p Fu(\))8 b(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))k(=)h Fs(I)-12 b(E)1208 710 y Fp(\026)1232 703 y Fu(\()p Fs(')p Fu(\))p Fs(:)59 822 y Fu(No)o(w)15 b(w)o(e)g(are)f(ready)i(to)e(de\014ne)i(our)p 713 822 3 25 v 18 w Fs(Q)p Fu(-system.)59 938 y Fk(Definition)g(15)23 b Fm(F)m(or)16 b(any)g Fu(\010)594 922 y Fo(\()p Fp(n)p Fo(\))658 938 y Fr(2)701 891 y Fj(\020)725 938 y Fr(N)774 913 y Fo(^)769 919 y Fq(\012)q Fp(n)p 769 952 2 18 v 759 952 a(C)820 891 y Fj(\021)845 903 y Fq(0)873 938 y Fm(we)g(de\014ne)g Fs(Q)1109 922 y Fp(\026)1109 950 y(n)1132 938 y Fu(\(\010)1183 922 y Fo(\()p Fp(n)p Fo(\))1234 938 y Fu(\))c Fr(2)h(P)1343 922 y Fq(0)1339 950 y Fp(\026)1362 938 y Fu(\()p Fr(N)1424 922 y Fq(0)1435 938 y Fu(\))j Fm(by)707 1058 y Fs(Q)743 1039 y Fp(\026)743 1069 y(n)766 1058 y Fu(\(\010)817 1039 y Fo(\()p Fp(n)p Fo(\))868 1058 y Fu(\))c(=)h Fs(D)q Fu(\(\010)1036 1039 y Fo(\()p Fp(n)p Fo(\))1086 1058 y Fu(\))1104 1039 y Fq(\003)1124 1058 y Fu(1)-15 b(1)15 b Fs(:)130 1164 y Fu(W)l(e)e(w)o(an)o(t)g(to)g (in)o(tro)q(duce)h(an)g(additional)h(formal)e(notation)g Fs(Q)1202 1148 y Fp(\026)1202 1175 y(n)1225 1164 y Fu(\()p Fs(x)p Fu(\))g(whic)o(h)i(stresses)e(the)g(linearit)o(y)59 1221 y(of)i(\010)144 1204 y Fo(\()p Fp(n)p Fo(\))207 1221 y Fr(7!)e Fs(Q)301 1204 y Fp(\026)301 1232 y(n)325 1221 y Fu(\(\010)376 1204 y Fo(\()p Fp(n)p Fo(\))426 1221 y Fu(\))f Fr(2)h Fs(P)534 1204 y Fq(0)528 1232 y Fp(\026)552 1221 y Fu(\()p Fr(N)614 1204 y Fq(0)625 1221 y Fu(\))f(:)706 1332 y Fr(h)p Fs(Q)760 1313 y Fp(\026)760 1343 y(n)783 1332 y Fs(;)c Fu(\010)837 1313 y Fo(\()p Fp(n)p Fo(\))887 1332 y Fr(i)k Fu(:=)h Fs(Q)1014 1313 y Fp(\026)1014 1343 y(n)1037 1332 y Fu(\(\010)1088 1313 y Fo(\()p Fp(n)p Fo(\))1139 1332 y Fu(\))h Fs(:)59 1434 y Fk(Example)i(4)g Fu(It)e(is)g(p)q(ossible)i(to)d(put)h(further)g (assumptions)g(on)g(the)g(measure)g Fs(\026)g Fu(to)f(ensure)h(that)g (the)59 1490 y(expression)19 b(is)g(more)f(than)g(formal.)29 b(Let)18 b(us)h(assume)f(a)g(smo)q(oth)g(measure)g(\(i.e.,)g(the)h (logarithmic)59 1547 y(deriv)m(ativ)o(e)e(of)d Fs(\026)i Fu(is)g(in\014nitely)h(di\013eren)o(tiable,)g(see)e([ADKS94])f(for)h (details\))h(with)f(the)h(prop)q(ert)o(y)563 1649 y Fr(9)p Fs(q)f Fr(2)e Fs(I)-12 b(N)20 b(;)i Fr(9f)p Fs(C)847 1656 y Fp(n)883 1649 y Fr(\025)13 b Fu(0)p Fs(;)19 b(n)13 b Fr(2)g Fs(I)-12 b(N)5 b Fr(g)13 b Fu(:)f Fr(8)p Fs(\030)j Fr(2)e(N)612 1710 y Fj(\014)612 1735 y(\014)612 1760 y(\014)612 1785 y(\014)626 1714 y(Z)675 1771 y Fs(D)714 1753 y Fp(n)713 1783 y(\030)738 1771 y Fs(')f Fu(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))894 1710 y Fj(\014)894 1735 y(\014)894 1760 y(\014)894 1785 y(\014)920 1771 y Fr(\024)h Fs(C)1001 1778 y Fp(n)1032 1771 y Fr(k)p Fs(')p Fr(k)1107 1785 y Fp(L)1131 1775 y Fi(2)1148 1785 y Fo(\()p Fp(\026)p Fo(\))1207 1771 y Fr(j)p Fs(\030)r Fr(j)1255 1753 y Fp(n)1255 1783 y(q)59 1876 y Fu(where)18 b Fs(')f Fu(is)i(an)o(y)e(\014nitely)i(based)f(b)q(ounded)h Fr(C)876 1859 y Fq(1)913 1876 y Fu(-function)g(on)e Fr(N)1218 1859 y Fq(0)1230 1876 y Fu(.)27 b(This)19 b(ob)o(viously)f(establishes) h(a)59 1932 y(b)q(ound)d(of)f(the)g(t)o(yp)q(e)271 2062 y Fr(k)p Fs(Q)330 2043 y Fp(\026)330 2073 y(n)353 2062 y Fu(\()p Fs(\030)391 2069 y Fo(1)421 2062 y Fr(\012)10 b(\001)e(\001)g(\001)g(\012)j Fs(\030)595 2069 y Fp(n)618 2062 y Fu(\))p Fr(k)658 2076 y Fp(L)682 2066 y Fi(2)700 2076 y Fo(\()p Fp(\026)p Fo(\))763 2062 y Fr(\024)i Fs(C)847 2043 y Fq(0)844 2073 y Fp(n)894 2009 y(n)879 2022 y Fj(Y)875 2113 y Fp(j)r Fo(=1)943 2062 y Fr(j)p Fs(\030)976 2069 y Fp(j)994 2062 y Fr(j)1007 2069 y Fp(q)1041 2062 y Fs(;)98 b(\030)1172 2069 y Fo(1)1192 2062 y Fs(;)8 b(:)g(:)g(:)t(;)g(\030)1313 2069 y Fp(n)1349 2062 y Fr(2)13 b(N)22 b Fs(;)g(n)13 b Fr(2)g Fs(I)-12 b(N)59 2227 y Fu(whic)o(h)13 b(is)g(su\016cien)o(t)g (to)f(sho)o(w)g(\(b)o(y)g(means)g(of)g(k)o(ernel)i(theorem\))d(that)h (there)h(exists)f Fs(Q)1534 2211 y Fp(\026)1534 2239 y(n)1558 2227 y Fu(\()p Fs(x)p Fu(\))g Fr(2)1675 2180 y Fj(\020)1699 2227 y Fr(N)1748 2202 y Fo(^)1743 2208 y Fq(\012)q Fp(n)p 1743 2241 V 1733 2241 a(C)1794 2180 y Fj(\021)1819 2192 y Fq(0)59 2293 y Fu(for)j(almost)f(all)i Fs(x)d Fr(2)g(N)464 2276 y Fq(0)491 2293 y Fu(suc)o(h)i(that)g(w)o(e)g (ha)o(v)o(e)g(the)g(represen)o(tation)489 2399 y Fs(Q)525 2380 y Fp(\026)525 2410 y(n)548 2399 y Fu(\()p Fs(')596 2380 y Fo(\()p Fp(n)p Fo(\))647 2399 y Fu(\)\()p Fs(x)p Fu(\))c(=)i Fr(h)p Fs(Q)840 2380 y Fp(\026)840 2410 y(n)863 2399 y Fu(\()p Fs(x)p Fu(\))p Fs(;)8 b(')976 2380 y Fo(\()p Fp(n)p Fo(\))1026 2399 y Fr(i)15 b Fs(;)98 b(')1200 2380 y Fo(\()p Fp(n)p Fo(\))1263 2399 y Fr(2)13 b(N)1355 2374 y Fo(^)1350 2380 y Fq(\012)p Fp(n)p 1349 2413 V 1340 2413 a(C)59 2511 y Fu(for)i(almost)f(all)i Fs(x)d Fr(2)g(N)464 2494 y Fq(0)475 2511 y Fu(.)20 b(F)l(or)15 b(an)o(y)g(smo)q(oth)f(k)o (ernel)i Fs(')1000 2494 y Fo(\()p Fp(n)p Fo(\))1063 2511 y Fr(2)d(N)1155 2485 y Fo(^)1150 2491 y Fq(\012)q Fp(n)p 1149 2524 V 1140 2524 a(C)1216 2511 y Fu(w)o(e)i(ha)o(v)o(e)g(then)g (that)g(the)g(function)590 2623 y Fs(x)d Fr(7!)h(h)p Fs(Q)740 2604 y Fp(\026)740 2634 y(n)763 2623 y Fu(\()p Fs(x)p Fu(\))p Fs(;)8 b(')876 2604 y Fo(\()p Fp(n)p Fo(\))926 2623 y Fr(i)27 b Fu(=)13 b Fs(Q)1055 2604 y Fp(\026)1055 2634 y(n)1086 2576 y Fj(\020)1111 2623 y Fs(')1141 2604 y Fo(\()p Fp(n)p Fo(\))1191 2576 y Fj(\021)1224 2623 y Fu(\()p Fs(x)p Fu(\))59 2736 y(b)q(elongs)j(to)f Fs(L)309 2720 y Fo(2)328 2736 y Fu(\()p Fs(\026)p Fu(\))p Fs(:)p eop %%Page: 19 19 19 18 bop 59 50 a Fn(4)45 b(CONCEPT)15 b(OF)g(DISTRIBUTIONS)j(IN)d (INFINITE)i(DIMENSIONAL)f(ANAL)l(YSIS)84 b Fu(19)59 199 y Fk(Example)23 b(5)f Fu(The)e(simplest)h(non)f(trivial)h(case)f(can)g (b)q(e)g(studied)h(using)g(\014nite)f(dimensional)i(real)59 256 y(analysis.)e(W)l(e)14 b(consider)h Fs(I)-12 b(R)14 b Fu(as)g(our)f(basic)i(Hilb)q(ert)g(space)f(and)g(as)g(our)g(n)o (uclear)g(space)g Fr(N)7 b Fu(.)20 b(Th)o(us)14 b(the)59 312 y(n)o(uclear)i(\\triple")g(is)g(simply)819 369 y Fs(I)-12 b(R)13 b Fr(\022)g Fs(I)-12 b(R)13 b Fr(\022)g Fs(I)-12 b(R)59 452 y Fu(and)14 b(the)g(dual)h(pairing)g(b)q(et)o(w)o (een)f(a)g(\\test)f(function")h(and)h(a)e(\\distribution")i (degenerates)f(to)g(m)o(ulti-)59 509 y(plication.)21 b(On)13 b Fs(I)-12 b(R)13 b Fu(w)o(e)f(consider)i(a)e(measure)h(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))g(=)g Fs(\032)p Fu(\()p Fs(x)p Fu(\))8 b(d)p Fs(x)j Fu(where)i Fs(\032)f Fu(is)h(a)f(p)q (ositiv)o(e)i Fr(C)1608 492 y Fq(1)1645 509 y Fu({function)59 565 y(on)22 b Fs(I)-12 b(R)22 b Fu(suc)o(h)g(that)f(Assumptions)h(1)f (and)h(2)f(are)h(ful\014lled.)42 b(In)22 b(this)g(setting)g(the)g (adjoin)o(t)f(of)h(the)59 621 y(di\013eren)o(tiation)16 b(op)q(erator)e(is)i(giv)o(en)g(b)o(y)437 685 y Fj(\022)486 714 y Fu(d)p 473 734 52 2 v 473 776 a(d)p Fs(x)529 685 y Fj(\023)559 696 y Fq(\003)587 745 y Fs(f)5 b Fu(\()p Fs(x)p Fu(\))12 b(=)h Fr(\000)779 698 y Fj(\020)818 727 y Fo(d)p 808 734 40 2 v 808 760 a(d)p Fp(x)863 745 y Fu(+)d Fs(\014)r Fu(\()p Fs(x)p Fu(\))998 698 y Fj(\021)1030 745 y Fs(f)5 b Fu(\()p Fs(x)p Fu(\))15 b Fs(;)98 b(f)18 b Fr(2)13 b(C)1355 726 y Fo(1)1374 745 y Fu(\()p Fs(I)-12 b(R)p Fu(\))59 865 y(where)15 b(the)h(logarithmic)g(deriv)m(ativ)o(e)g Fs(\014)i Fu(of)c(the)i(measure)f Fs(\026)g Fu(is)h(giv)o(en)g(b)o(y) 878 989 y Fs(\014)f Fu(=)972 958 y Fs(\032)996 942 y Fq(0)p 972 978 36 2 v 977 1020 a Fs(\032)59 1111 y Fu(This)h(enables)g (us)g(to)e(calculate)i(the)p 719 1111 3 25 v 19 w Fs(Q)743 1093 y Fp(\026)766 1111 y Fu(-system.)k(One)c(has)494 1222 y Fs(Q)530 1203 y Fp(\026)530 1233 y(n)554 1222 y Fu(\()p Fs(x)p Fu(\))c(=)675 1175 y Fj(\020\020)740 1204 y Fo(d)p 730 1211 40 2 v 730 1237 a(d)p Fp(x)774 1175 y Fj(\021)799 1186 y Fq(\003)819 1175 y Fj(\021)844 1184 y Fp(n)875 1222 y Fu(1)-15 b(1)12 b(=)h(\()p Fr(\000)p Fu(1\))1060 1203 y Fp(n)1090 1175 y Fj(\020)1130 1204 y Fo(d)p 1120 1211 V 1120 1237 a(d)p Fp(x)1175 1222 y Fu(+)d Fs(\014)r Fu(\()p Fs(x)p Fu(\))1310 1175 y Fj(\021)1334 1186 y Fp(n)1365 1222 y Fu(1)-15 b(1)776 1370 y(=)13 b(\()p Fr(\000)p Fu(1\))918 1351 y Fp(n)946 1339 y Fs(\032)970 1322 y Fo(\()p Fp(n)p Fo(\))1020 1339 y Fu(\()p Fs(x)p Fu(\))p 946 1359 136 2 v 971 1401 a Fs(\032)p Fu(\()p Fs(x)p Fu(\))1101 1370 y Fs(:)59 1475 y Fu(The)i(last)h(equalit)o(y)g (can)f(b)q(e)h(seen)g(b)o(y)f(simple)h(induction.)59 1531 y(If)g Fs(\032)c Fu(=)219 1514 y Fo(1)p 194 1521 69 2 v 194 1526 a Fq(p)p 223 1526 40 2 v 27 x Fo(2)p Fp(\031)267 1531 y Fs(e)288 1513 y Fq(\000)320 1500 y Fi(1)p 321 1506 16 2 v 321 1527 a(2)341 1513 y Fp(x)361 1502 y Fi(2)395 1531 y Fu(is)k(the)f(Gaussian)h(densit)o(y)f Fs(Q)907 1515 y Fp(\026)907 1543 y(n)946 1531 y Fu(is)h(related)f(to)g (the)g(n)1302 1515 y Fp(th)1353 1531 y Fu(Hermite)g(p)q(olynomial:)694 1656 y Fs(Q)730 1637 y Fp(\026)730 1667 y(n)754 1656 y Fu(\()p Fs(x)p Fu(\))d(=)h(2)899 1637 y Fq(\000)p Fp(n=)p Fo(2)985 1656 y Fs(H)1023 1663 y Fp(n)1053 1609 y Fj(\020)1097 1638 y Fp(x)p 1083 1645 48 2 v 1083 1650 a Fq(p)p 1113 1650 18 2 v 1113 1677 a Fo(2)1135 1609 y Fj(\021)1183 1656 y Fs(:)59 1914 y Fk(Definition)j(16)23 b Fm(We)17 b(de\014ne)e(the)p 696 1914 3 25 v 21 w Fs(Q)720 1896 y Fp(\026)743 1914 y Fm(-system)h(in)g Fr(P)1000 1897 y Fq(0)996 1925 y Fp(\026)1019 1914 y Fu(\()p Fr(N)1081 1897 y Fq(0)1092 1914 y Fu(\))g Fm(by)p 437 2043 V 425 2043 a Fs(Q)461 2024 y Fp(\026)497 2043 y Fu(=)545 1983 y Fj(\032)576 2043 y Fs(Q)612 2024 y Fp(\026)612 2054 y(n)635 2043 y Fu(\(\010)686 2024 y Fo(\()p Fp(n)p Fo(\))737 2043 y Fu(\))770 1994 y Fj(\014)770 2019 y(\014)770 2044 y(\014)893 2043 y Fu(\010)926 2024 y Fo(\()p Fp(n)p Fo(\))990 2043 y Fr(2)1033 1996 y Fj(\020)1058 2043 y Fr(N)1106 2018 y Fo(^)1102 2024 y Fq(\012)p Fp(n)p 1101 2056 2 18 v 1092 2056 a(C)1153 1996 y Fj(\021)1177 2007 y Fq(0)1197 2043 y Fs(;)23 b(n)13 b Fr(2)g Fs(I)-12 b(N)1366 2050 y Fo(0)1402 1983 y Fj(\033)1457 2043 y Fs(;)59 2165 y Fm(and)16 b(the)h(p)n(air)f Fu(\()p Fs(I)-12 b(P)378 2147 y Fp(\026)401 2165 y Fs(;)p 438 2165 3 25 v 12 w(Q)462 2147 y Fp(\026)485 2165 y Fu(\))16 b Fm(wil)r(l)g(b)n(e)g(c)n(al)r(le)n (d)f(the)h(App)n(el)r(l)g(system)g(I)-8 b Fu(A)1194 2147 y Fp(\026)1234 2165 y Fm(gener)n(ate)n(d)16 b(by)g(the)g(me)n(asur)n(e) g Fs(\026)p Fm(.)130 2271 y Fu(No)o(w)e(w)o(e)h(are)g(going)g(to)g (discuss)h(the)f(cen)o(tral)h(prop)q(ert)o(y)f(of)f(the)i(App)q(ell)h (system)e Fm(I)-8 b Fu(A)1590 2253 y Fp(\026)1614 2271 y Fu(.)59 2378 y Fk(Theorem)18 b(17)23 b Fu(\(Biorthogonalit)o(y)15 b(w.r.t.)j Fs(\026)p Fu(\))466 2440 y Fj(D)-15 b(D)501 2487 y Fr(h)p Fs(Q)555 2469 y Fp(\026)555 2499 y(n)578 2487 y Fu(\(\010)629 2469 y Fo(\()p Fp(n)p Fo(\))680 2487 y Fu(\))p Fs(;)23 b Fr(h)p Fs(P)787 2469 y Fp(\026)781 2499 y(m)814 2487 y Fs(;)8 b(')865 2469 y Fo(\()p Fp(m)p Fo(\))925 2487 y Fr(i)943 2440 y Fj(E)-15 b(E)978 2516 y Fp(\026)1014 2487 y Fu(=)13 b Fs(\016)1082 2494 y Fp(m;n)1159 2487 y Fs(n)p Fu(!)g Fr(h)p Fu(\010)1263 2469 y Fo(\()p Fp(n)p Fo(\))1313 2487 y Fs(;)8 b(')1364 2469 y Fo(\()p Fp(n)p Fo(\))1414 2487 y Fr(i)318 b Fu(\(22\))59 2626 y Fm(for)17 b Fu(\010)165 2609 y Fo(\()p Fp(n)p Fo(\))228 2626 y Fr(2)271 2579 y Fj(\020)296 2626 y Fr(N)345 2601 y Fo(^)340 2607 y Fq(\012)p Fp(n)p 339 2639 2 18 v 330 2639 a(C)391 2579 y Fj(\021)415 2590 y Fq(0)443 2626 y Fm(and)g Fs(')562 2609 y Fo(\()p Fp(m)p Fo(\))635 2626 y Fr(2)c(N)727 2601 y Fo(^)722 2607 y Fq(\012)p Fp(m)p 721 2639 V 712 2639 a(C)799 2626 y Fm(.)p eop %%Page: 20 20 20 19 bop 59 50 a Fn(4)45 b(CONCEPT)15 b(OF)g(DISTRIBUTIONS)j(IN)d (INFINITE)i(DIMENSIONAL)f(ANAL)l(YSIS)84 b Fu(20)59 199 y Fk(Pr)o(oof.)72 b Fu(It)16 b(follo)o(ws)f(from)f(\(14\))g(and)i (\(21\))e(that)262 262 y Fj(D)-14 b(D)298 309 y Fs(Q)334 290 y Fp(\026)334 320 y(n)357 309 y Fu(\(\010)408 290 y Fo(\()p Fp(n)p Fo(\))459 309 y Fu(\))p Fs(;)22 b Fr(h)p Fs(P)565 290 y Fp(\026)559 320 y(m)592 309 y Fs(;)8 b(')643 290 y Fo(\()p Fp(m)p Fo(\))703 309 y Fr(i)721 262 y Fj(E)-15 b(E)756 337 y Fp(\026)821 309 y Fu(=)898 262 y Fj(D)g(D)933 309 y Fu(1)g(1)o Fs(;)8 b(D)q Fu(\(\010)1074 290 y Fo(\()p Fp(n)p Fo(\))1124 309 y Fu(\))p Fr(h)p Fs(P)1195 290 y Fp(\026)1189 320 y(m)1222 309 y Fs(;)g(')1273 290 y Fo(\()p Fp(m)p Fo(\))1332 309 y Fr(i)1350 262 y Fj(E)-15 b(E)1386 337 y Fp(\026)821 423 y Fu(=)962 392 y Fs(m)p Fu(!)p 903 412 171 2 v 903 454 a(\()p Fs(m)9 b Fr(\000)i Fs(n)p Fu(\)!)1078 423 y Fs(I)-12 b(E)1123 430 y Fp(\026)1154 376 y Fj(\020)1179 423 y Fr(h)p Fs(P)1232 400 y Fp(\026)1226 438 y Fo(\()p Fp(m)p Fq(\000)p Fp(n)p Fo(\))1341 416 y Fu(^)1335 423 y Fr(\012)p Fu(\010)1403 404 y Fo(\()p Fp(n)p Fo(\))1454 423 y Fs(;)20 b(')1517 404 y Fo(\()p Fp(m)p Fo(\))1577 423 y Fr(i)1595 376 y Fj(\021)821 525 y Fu(=)42 b Fs(m)p Fu(!)12 b Fs(\016)983 532 y Fp(m;n)1060 525 y Fr(h)p Fu(\010)1111 507 y Fo(\()p Fp(m)p Fo(\))1171 525 y Fs(;)c(')1222 507 y Fo(\()p Fp(m)p Fo(\))1282 525 y Fr(i)15 b Fs(:)1797 628 y Ff(2)130 734 y Fu(No)o(w)f(w)o(e)h(are)g (going)g(to)g(c)o(haracterize)g(the)h(space)f Fr(P)1040 717 y Fq(0)1036 745 y Fp(\026)1059 734 y Fu(\()p Fr(N)1121 717 y Fq(0)1132 734 y Fu(\))59 847 y Fk(Theorem)j(18)23 b Fm(F)m(or)15 b(al)r(l)h Fu(\010)c Fr(2)h(P)627 830 y Fq(0)623 858 y Fp(\026)646 847 y Fu(\()p Fr(N)708 830 y Fq(0)719 847 y Fu(\))j Fm(ther)n(e)f(exists)g(a)h(unique)g(se)n (quenc)n(e)e Fr(f)p Fu(\010)1414 830 y Fo(\()p Fp(n)p Fo(\))1465 847 y Fr(j)h Fs(n)e Fr(2)g Fs(I)-12 b(N)1626 854 y Fo(0)1645 847 y Fr(g)p Fs(;)23 b Fu(\010)1737 830 y Fo(\()p Fp(n)p Fo(\))1800 847 y Fr(2)59 876 y Fj(\020)84 923 y Fr(N)133 898 y Fo(^)128 904 y Fq(\012)p Fp(n)p 127 936 2 18 v 118 936 a(C)179 876 y Fj(\021)204 887 y Fq(0)231 923 y Fm(such)17 b(that)609 1024 y Fu(\010)12 b(=)718 971 y Fq(1)706 984 y Fj(X)702 1074 y Fp(n)p Fo(=0)776 1024 y Fs(Q)812 1005 y Fp(\026)812 1035 y(n)836 1024 y Fu(\(\010)887 1005 y Fo(\()p Fp(n)p Fo(\))937 1024 y Fu(\))h Fr(\021)1031 971 y Fq(1)1019 984 y Fj(X)1016 1074 y Fp(n)p Fo(=0)1082 1024 y Fr(h)p Fs(Q)1136 1005 y Fp(\026)1136 1035 y(n)1159 1024 y Fs(;)8 b Fu(\010)1213 1005 y Fo(\()p Fp(n)p Fo(\))1263 1024 y Fr(i)469 b Fu(\(23\))59 1145 y Fm(and)16 b(vic)n(e)g(versa,)g(every)g(series)g(of)g(the)g(form) h(\(23\))f(gener)n(ates)f(a)i(gener)n(alize)n(d)d(function)i(in)g Fr(P)1698 1128 y Fq(0)1694 1156 y Fp(\026)1717 1145 y Fu(\()p Fr(N)1779 1128 y Fq(0)1790 1145 y Fu(\))p Fs(:)59 1270 y Fk(Pr)o(oof.)72 b Fu(F)l(or)15 b(\010)e Fr(2)f(P)494 1254 y Fq(0)490 1282 y Fp(\026)513 1270 y Fu(\()p Fr(N)575 1254 y Fq(0)587 1270 y Fu(\))j(w)o(e)f(can)i(uniquely)h(de\014ne)f (\010)1118 1254 y Fo(\()p Fp(n)p Fo(\))1182 1270 y Fr(2)1225 1223 y Fj(\020)1250 1270 y Fr(N)1298 1245 y Fo(^)1294 1251 y Fq(\012)p Fp(n)p 1293 1284 V 1284 1284 a(C)1344 1223 y Fj(\021)1369 1235 y Fq(0)1396 1270 y Fu(b)o(y)427 1406 y Fr(h)p Fu(\010)478 1387 y Fo(\()p Fp(n)p Fo(\))529 1406 y Fs(;)8 b(')580 1387 y Fo(\()p Fp(n)p Fo(\))630 1406 y Fr(i)k Fu(=)722 1375 y(1)p 713 1396 40 2 v 713 1437 a Fs(n)p Fu(!)765 1406 y Fr(h)-8 b(h)p Fu(\010)p Fs(;)20 b Fr(h)p Fs(P)912 1387 y Fp(\026)906 1417 y(n)935 1406 y Fs(;)8 b(')986 1387 y Fo(\()p Fp(n)p Fo(\))1036 1406 y Fr(ii)-8 b(i)1082 1413 y Fp(\026)1120 1406 y Fs(;)98 b(')1261 1387 y Fo(\()p Fp(n)p Fo(\))1325 1406 y Fr(2)12 b(N)1416 1381 y Fo(^)1411 1387 y Fq(\012)q Fp(n)p 1411 1420 2 18 v 1401 1420 a(C)59 1529 y Fu(This)f(de\014nition)h(is)e(p)q (ossible)i(b)q(ecause)f Fr(h)p Fs(P)777 1512 y Fp(\026)771 1540 y(n)801 1529 y Fs(;)d(')852 1512 y Fo(\()p Fp(n)p Fo(\))902 1529 y Fr(i)k(2)h(P)t Fu(\()p Fr(N)1073 1512 y Fq(0)1084 1529 y Fu(\).)18 b(The)10 b(con)o(tin)o(uit)o(y)h(of)e Fs(')1507 1512 y Fo(\()p Fp(n)p Fo(\))1571 1529 y Fr(7!)k(h)p Fu(\010)1680 1512 y Fo(\()p Fp(n)p Fo(\))1730 1529 y Fs(;)8 b(')1781 1512 y Fo(\()p Fp(n)p Fo(\))1831 1529 y Fr(i)59 1585 y Fu(follo)o(ws)24 b(from)g(the)h(con)o(tin)o(uit)o(y)f (of)g Fs(')k Fr(7!)g(h)-8 b(h)p Fu(\010)p Fs(;)8 b(')p Fr(i)-8 b(i)23 b Fs(;)32 b(')27 b Fr(2)e(P)t Fu(\()p Fr(N)1255 1569 y Fq(0)1265 1585 y Fu(\).)48 b(This)24 b(implies)j(that)d Fs(')j Fr(7!)59 1609 y Fj(P)103 1622 y Fq(1)103 1653 y Fp(n)p Fo(=0)179 1641 y Fs(n)p Fu(!)13 b Fr(h)p Fu(\010)283 1625 y Fo(\()p Fp(n)p Fo(\))333 1641 y Fs(;)8 b(')384 1625 y Fo(\()p Fp(n)p Fo(\))434 1641 y Fr(i)i Fu(is)h(con)o(tin)o(uous)g(on)g Fr(P)t Fu(\()p Fr(N)883 1625 y Fq(0)894 1641 y Fu(\).)18 b(This)11 b(de\014nes)g(a)g(generalized)h(function)g(in)f Fr(P)1708 1625 y Fq(0)1704 1653 y Fp(\026)1727 1641 y Fu(\()p Fr(N)1789 1625 y Fq(0)1800 1641 y Fu(\),)59 1705 y(whic)o(h)16 b(w)o(e)f(denote)g(b)o(y)466 1673 y Fj(P)509 1686 y Fq(1)509 1716 y Fp(n)p Fo(=0)586 1705 y Fs(Q)622 1688 y Fp(\026)622 1716 y(n)645 1705 y Fu(\(\010)696 1688 y Fo(\()p Fp(n)p Fo(\))746 1705 y Fu(\).)20 b(In)c(view)g(of)e(Theorem)h(17)g(it)g(is)h (ob)o(vious)g(that)758 1837 y(\010)c(=)867 1784 y Fq(1)855 1797 y Fj(X)851 1887 y Fp(n)p Fo(=0)925 1837 y Fs(Q)961 1819 y Fp(\026)961 1849 y(n)985 1837 y Fu(\(\010)1036 1819 y Fo(\()p Fp(n)p Fo(\))1086 1837 y Fu(\))j Fs(:)130 1976 y Fu(T)l(o)e(see)g(the)g(con)o(v)o(erse)g(consider)h(a)f(series)h (of)f(the)g(form)f(\(23\))g(and)i Fs(')e Fr(2)h(P)t Fu(\()p Fr(N)1447 1960 y Fq(0)1458 1976 y Fu(\).)18 b(Then)c(there)f(exist)59 2040 y Fs(')89 2023 y Fo(\()p Fp(n)p Fo(\))152 2040 y Fr(2)g(N)244 2015 y Fo(^)239 2021 y Fq(\012)q Fp(n)p 238 2054 V 229 2054 a(C)305 2040 y Fs(;)23 b(n)13 b Fr(2)g Fs(I)-12 b(N)20 b Fu(and)15 b Fs(N)i Fr(2)c Fs(I)-12 b(N)20 b Fu(suc)o(h)c(that)e(w)o(e)h(ha)o(v)o(e)g(the)g(represen)o (tation)761 2183 y Fs(')e Fu(=)869 2130 y Fp(N)855 2143 y Fj(X)852 2233 y Fp(n)p Fo(=0)926 2183 y Fs(P)961 2164 y Fp(\026)955 2194 y(n)985 2183 y Fu(\()p Fs(')1033 2164 y Fo(\()p Fp(n)p Fo(\))1083 2183 y Fu(\))i Fs(:)59 2319 y Fu(So)g(w)o(e)g(ha)o(v)o(e)530 2331 y Fj(*)-22 b(*)590 2350 y Fq(1)577 2362 y Fj(X)574 2453 y Fp(n)p Fo(=0)648 2403 y Fs(Q)684 2384 y Fp(\026)684 2414 y(n)708 2403 y Fu(\(\010)759 2384 y Fo(\()p Fp(n)p Fo(\))809 2403 y Fu(\))p Fs(;)8 b(')878 2331 y Fj(+)-24 b(+)909 2459 y Fp(\026)945 2403 y Fu(:=)1023 2350 y Fp(N)1009 2362 y Fj(X)1006 2453 y Fp(n)p Fo(=0)1080 2403 y Fs(n)p Fu(!)12 b Fr(h)p Fu(\010)1183 2384 y Fo(\()p Fp(n)p Fo(\))1234 2403 y Fs(;)c(')1285 2384 y Fo(\()p Fp(n)p Fo(\))1335 2403 y Fr(i)59 2538 y Fu(b)q(ecause)22 b(of)e(Theorem)g(17.)35 b(The)21 b(con)o(tin)o(uit)o(y)g(of)f Fs(')h Fr(7!)h(h)-8 b(h)1103 2506 y Fj(P)1147 2519 y Fq(1)1147 2549 y Fp(n)p Fo(=0)1223 2538 y Fs(Q)1259 2522 y Fp(\026)1259 2549 y(n)1283 2538 y Fu(\(\010)1334 2522 y Fo(\()p Fp(n)p Fo(\))1384 2538 y Fu(\))p Fs(;)8 b(')p Fr(i)-8 b(i)1481 2545 y Fp(\026)1523 2538 y Fu(follo)o(ws)21 b(b)q(ecause)59 2594 y Fs(')89 2578 y Fo(\()p Fp(n)p Fo(\))152 2594 y Fr(7!)13 b(h)p Fu(\010)261 2578 y Fo(\()p Fp(n)p Fo(\))312 2594 y Fs(;)8 b(')363 2578 y Fo(\()p Fp(n)p Fo(\))413 2594 y Fr(i)15 b Fu(is)g(con)o(tin)o(uous)h(for)e(all)i Fs(n)d Fr(2)g Fs(I)-12 b(N)20 b Fu(.)770 b Ff(2)p eop %%Page: 21 21 21 20 bop 59 50 a Fn(5)45 b(TEST)15 b(FUNCTIONS)h(ON)g(A)f(LINEAR)h(SP) l(A)o(CE)g(WITH)f(MEASURE)365 b Fu(21)59 199 y Ft(5)67 b(T)-6 b(est)22 b(functions)h(on)f(a)g(linear)i(space)e(with)h(measure) 59 301 y Fu(In)14 b(this)g(section)f(w)o(e)g(will)i(construct)e(the)h (test)e(function)i(space)g(\()p Fr(N)7 b Fu(\))1236 284 y Fo(1)1268 301 y Fu(and)13 b(study)h(its)f(prop)q(erties.)20 b(On)59 357 y(the)d(space)g Fr(P)t Fu(\()p Fr(N)360 341 y Fq(0)371 357 y Fu(\))f(w)o(e)h(can)g(de\014ne)h(a)e(system)g(of)h (norms)f(using)i(the)f(represen)o(tation)g(from)f(Lemma)59 414 y(12.)j(Let)672 497 y Fs(')12 b Fu(=)780 445 y Fp(N)765 457 y Fj(X)762 548 y Fp(n)p Fo(=0)829 497 y Fr(h)p Fs(P)882 479 y Fp(\026)876 509 y(n)905 497 y Fs(;)20 b(')968 479 y Fo(\()p Fp(n)p Fo(\))1018 497 y Fr(i)13 b(2)g(P)t Fu(\()p Fr(N)1190 479 y Fq(0)1200 497 y Fu(\))59 629 y(b)q(e)i(giv)o(en,)f (then)h Fs(')383 613 y Fo(\()p Fp(n)p Fo(\))446 629 y Fr(2)e(H)532 604 y Fo(^)527 610 y Fq(\012)q Fp(n)p 533 643 2 18 v 524 643 a(C)s(;p)596 629 y Fu(for)g(eac)o(h)h Fs(p)f Fr(\025)g Fu(0)27 b(\()p Fs(n)13 b Fr(2)g Fs(I)-12 b(N)5 b Fu(\).)19 b(Th)o(us)14 b(w)o(e)g(ma)o(y)f(de\014ne)i(for)f(an)o (y)f Fs(p;)8 b(q)14 b Fr(2)f Fs(I)-12 b(N)59 686 y Fu(a)15 b(Hilb)q(ertian)i(norm)e(on)g Fr(P)t Fu(\()p Fr(N)594 669 y Fq(0)605 686 y Fu(\))g(b)o(y)650 823 y Fr(k)p Fs(')p Fr(k)726 802 y Fo(2)726 836 y Fp(p;q)q(;\026)816 823 y Fu(=)881 770 y Fp(N)867 782 y Fj(X)864 873 y Fp(n)p Fo(=0)930 823 y Fu(\()p Fs(n)p Fu(!\))1006 804 y Fo(2)1038 823 y Fu(2)1061 804 y Fp(nq)1114 823 y Fr(j)p Fs(')1157 804 y Fo(\()p Fp(n)p Fo(\))1207 823 y Fr(j)1220 804 y Fo(2)1220 834 y Fp(p)59 961 y Fu(The)g(completion)i(of)e Fr(P)t Fu(\()p Fr(N)535 945 y Fq(0)545 961 y Fu(\))g(w.r.t.)j Fr(k\001k)763 975 y Fp(p;q)q(;\026)855 961 y Fu(is)e(called)h(\()p Fr(H)1086 968 y Fp(p)1106 961 y Fu(\))1124 945 y Fo(1)1124 973 y Fp(q)q(;\026)1189 961 y Fu(.)59 1064 y Fk(Definition)f(19)23 b Fm(We)17 b(de\014ne)702 1121 y Fu(\()p Fr(N)7 b Fu(\))782 1102 y Fo(1)782 1132 y Fp(\026)817 1121 y Fu(:=)16 b(pr)g(lim)889 1159 y Fp(p;q)q Fq(2)p Fp(I)-9 b(N)1015 1121 y Fu(\()p Fr(H)1071 1128 y Fp(p)1091 1121 y Fu(\))1109 1102 y Fo(1)1109 1132 y Fp(q)q(;\026)1175 1121 y Fs(:)130 1252 y Fu(This)15 b(space)h(has)f(the)g(follo)o(wing)h(prop)q(erties)59 1354 y Fk(Theorem)i(20)23 b Fu(\()p Fr(N)7 b Fu(\))434 1338 y Fo(1)434 1365 y Fp(\026)479 1354 y Fm(is)22 b(a)g(nucle)n(ar)g (sp)n(ac)n(e.)39 b(The)22 b(top)n(olo)n(gy)g Fu(\()p Fr(N)7 b Fu(\))1260 1338 y Fo(1)1260 1365 y Fp(\026)1305 1354 y Fm(is)22 b(uniquely)h(de\014ne)n(d)e(by)h(the)59 1411 y(top)n(olo)n(gy)16 b(on)g Fr(N)7 b Fm(:)21 b(It)16 b(do)n(es)g(not)g(dep)n(end)g(on)g(the)h(choic)n(e)f(of)g(the)h(family) f(of)g(norms)g Fr(fj)10 b(\001)g(j)1577 1418 y Fp(p)1596 1411 y Fr(g)p Fm(.)59 1513 y Fk(Pr)o(oof.)77 b Fu(Nuclearit)o(y)18 b(of)e(\()p Fr(N)7 b Fu(\))649 1497 y Fo(1)649 1525 y Fp(\026)688 1513 y Fu(follo)o(ws)17 b(essen)o(tially)h(from)e(that)f (of)i Fr(N)7 b Fs(:)16 b Fu(F)l(or)g(\014xed)h Fs(p;)8 b(q)18 b Fu(consider)59 1570 y(the)d(em)o(b)q(edding)663 1626 y Fs(I)683 1634 y Fp(p)701 1624 y Fh(0)712 1634 y Fp(;q)739 1624 y Fh(0)750 1634 y Fp(;p;q)819 1626 y Fu(:)844 1592 y Fj(\000)863 1626 y Fr(H)901 1634 y Fp(p)919 1624 y Fh(0)932 1592 y Fj(\001)951 1603 y Fo(1)951 1642 y Fp(q)968 1633 y Fh(0)980 1642 y Fp(;\026)1025 1626 y Fr(!)e Fu(\()p Fr(H)1139 1633 y Fp(p)1159 1626 y Fu(\))1177 1605 y Fo(1)1177 1642 y Fp(q)q(;\026)59 1713 y Fu(where)i Fs(p)213 1697 y Fq(0)240 1713 y Fu(is)h(c)o(hosen)f(suc)o(h)h(that)e (the)i(em)o(b)q(edding)786 1812 y Fs(i)802 1819 y Fp(p)820 1810 y Fh(0)830 1819 y Fp(;p)873 1812 y Fu(:)k Fr(H)944 1819 y Fp(p)962 1810 y Fh(0)988 1812 y Fr(!)13 b(H)1084 1819 y Fp(p)59 1910 y Fu(is)j(Hilb)q(ert{Sc)o(hmidt.)22 b(Then)15 b Fs(I)603 1918 y Fp(p)621 1909 y Fh(0)632 1918 y Fp(;q)659 1909 y Fh(0)670 1918 y Fp(;p;q)742 1910 y Fu(is)h(induced)h(b)o(y)274 2042 y Fs(I)294 2050 y Fp(p)312 2040 y Fh(0)323 2050 y Fp(;q)350 2040 y Fh(0)361 2050 y Fp(;p;q)417 2042 y Fs(')c Fu(=)523 1989 y Fq(1)511 2002 y Fj(X)508 2092 y Fp(n)p Fo(=0)574 2042 y Fr(h)p Fs(P)627 2023 y Fp(\026)621 2053 y(n)651 2042 y Fs(;)8 b(i)688 2023 y Fq(\012)p Fp(n)688 2056 y(p)706 2047 y Fh(0)716 2056 y Fp(;p)745 2042 y Fs(')775 2023 y Fo(\()p Fp(n)p Fo(\))826 2042 y Fr(i)60 b Fu(for)g Fs(')13 b Fu(=)1125 1989 y Fq(1)1113 2002 y Fj(X)1110 2092 y Fp(n)p Fo(=0)1176 2042 y Fr(h)p Fs(P)1229 2023 y Fp(\026)1223 2053 y(n)1253 2042 y Fs(;)8 b(')1304 2023 y Fo(\()p Fp(n)p Fo(\))1353 2042 y Fr(i)13 b(2)1427 2007 y Fj(\000)1446 2042 y Fr(H)1484 2050 y Fp(p)1502 2040 y Fh(0)1515 2007 y Fj(\001)1534 2019 y Fo(1)1534 2058 y Fp(q)1551 2049 y Fh(0)1562 2058 y Fp(;\026)1603 2042 y Fs(:)59 2187 y Fu(Its)j(Hilb)q(ert{Sc)o(hmidt)j(norm)d(is)h(easily)g(estimated)g(b)o (y)f(using)i(an)e(orthonormal)g(basis)g(of)1649 2152 y Fj(\000)1668 2187 y Fr(H)1706 2195 y Fp(p)1724 2185 y Fh(0)1738 2152 y Fj(\001)1757 2164 y Fo(1)1757 2203 y Fp(q)1774 2194 y Fh(0)1785 2203 y Fp(;\026)1818 2187 y Fu(.)59 2244 y(The)f(result)h(is)g(the)f(b)q(ound)589 2325 y Fj(\015)589 2350 y(\015)612 2361 y Fs(I)632 2369 y Fp(p)650 2359 y Fh(0)661 2369 y Fp(;q)688 2359 y Fh(0)699 2369 y Fp(;p;q)755 2325 y Fj(\015)755 2350 y(\015)778 2338 y Fo(2)778 2377 y Fp(H)s(S)848 2361 y Fr(\024)911 2308 y Fq(1)899 2321 y Fj(X)896 2411 y Fp(n)p Fo(=0)970 2361 y Fu(2)993 2342 y Fp(n)p Fo(\()p Fp(q)q Fq(\000)p Fp(q)1089 2331 y Fh(0)1100 2342 y Fo(\))1124 2325 y Fj(\015)1124 2350 y(\015)1147 2361 y Fs(i)1163 2369 y Fp(p)1181 2359 y Fh(0)1191 2369 y Fp(;p)1221 2325 y Fj(\015)1221 2350 y(\015)1244 2338 y Fo(2)p Fp(n)1244 2377 y(H)s(S)59 2497 y Fu(whic)o(h)h(is)g(\014nite)g(for)f(suitably)h(c)o(hosen)f Fs(q)761 2480 y Fq(0)773 2497 y Fu(.)130 2553 y(Let)j(us)h(assume)f (that)f(w)o(e)h(are)g(giv)o(en)h(t)o(w)o(o)e(di\013eren)o(t)i(systems)e (of)h(Hilb)q(ertian)j(norms)c Fr(j)8 b(\001)g(j)1719 2567 y Fp(p)1757 2553 y Fu(and)59 2617 y Fr(j)f(\001)h(j)112 2596 y Fq(0)112 2630 y Fp(k)133 2617 y Fu(,)13 b(suc)o(h)f(that)g(they) g(induce)i(the)e(same)g(top)q(ology)f(on)i Fr(N)19 b Fu(.)f(F)l(or)12 b(\014xed)h Fs(k)g Fu(and)f Fs(l)h Fu(w)o(e)f(ha)o(v)o (e)f(to)h(estimate)59 2673 y Fr(k)7 b(\001)h(k)132 2652 y Fq(0)132 2686 y Fp(k)q(;l;\026)224 2673 y Fu(b)o(y)18 b Fr(k)8 b(\001)g(k)364 2686 y Fp(p;q)q(;\026)460 2673 y Fu(for)17 b(some)i Fs(p;)8 b(q)19 b Fu(\(and)f(vice)i(v)o(ersa)e (whic)o(h)h(is)g(completely)h(analogous\).)29 b(Since)59 2736 y Fr(j)7 b(\001)h(j)112 2715 y Fq(0)112 2750 y Fp(k)148 2736 y Fu(has)15 b(to)f(b)q(e)i(con)o(tin)o(uous)f(with)g(resp)q(ect)g (to)g(the)g(pro)s(jectiv)o(e)f(limit)j(top)q(ology)d(on)h Fr(N)7 b Fu(,)14 b(there)h(exists)59 2793 y Fs(p)e Fu(and)h(a)f (constan)o(t)f Fs(C)k Fu(suc)o(h)e(that)e Fr(j)p Fs(f)5 b Fr(j)698 2772 y Fq(0)698 2806 y Fp(k)732 2793 y Fr(\024)13 b Fs(C)d Fr(j)p Fs(f)5 b Fr(j)875 2806 y Fp(p)895 2793 y Fu(,)13 b(for)g(all)h Fs(f)k Fr(2)13 b(N)7 b Fu(,)13 b(i.e.,)h(the)f(injection)i Fs(i)e Fu(from)f Fr(H)1720 2800 y Fp(p)1754 2793 y Fu(in)o(to)p eop %%Page: 22 22 22 21 bop 59 50 a Fn(5)45 b(TEST)15 b(FUNCTIONS)h(ON)g(A)f(LINEAR)h(SP) l(A)o(CE)g(WITH)f(MEASURE)365 b Fu(22)59 199 y(the)15 b(completion)h Fr(K)404 206 y Fp(k)440 199 y Fu(of)e Fr(N)22 b Fu(with)15 b(resp)q(ect)h(to)e Fr(j)j(\001)g(j)937 183 y Fq(0)937 212 y Fp(k)972 199 y Fu(is)f(a)e(mapping)i(b)q(ounded)g (b)o(y)f Fs(C)s Fu(.)20 b(W)l(e)15 b(denote)g(b)o(y)59 256 y Fs(i)e Fu(also)h(its)g(linear)h(extension)g(from)e Fr(H)p 721 263 2 18 v 712 263 a Fp(C)r(;p)783 256 y Fu(in)o(to)g Fr(K)p 920 263 V 911 263 a Fp(C)s(;k)970 256 y Fu(.)19 b(It)14 b(follo)o(ws)g(that)f Fs(\023)1311 239 y Fq(\012)p Fp(n)1375 256 y Fu(is)i(b)q(ounded)g(b)o(y)e Fs(C)1701 239 y Fp(n)1738 256 y Fu(from)59 325 y Fr(H)97 305 y Fq(\012)p Fp(n)p 109 338 V 100 338 a(C)s(;p)173 325 y Fu(in)o(to)i Fr(K)301 305 y Fq(\012)p Fp(n)p 311 339 V 303 339 a(C)r(;k)361 325 y Fu(.)20 b(No)o(w)15 b(w)o(e)g(c)o(ho)q (ose)g Fs(q)i Fu(suc)o(h)e(that)g(2)975 291 y Fg(q)q Fh(\000)p Fg(l)p 975 299 50 2 v 991 320 a Fi(2)1043 325 y Fr(\025)e Fs(C)s Fu(.)20 b(Then)617 489 y Fr(k)7 b(\001)h(k)690 468 y Fq(0)p Fo(2)690 502 y Fp(k)q(;l;\026)805 489 y Fu(=)897 436 y Fq(1)885 448 y Fj(X)882 539 y Fp(n)p Fo(=0)948 489 y Fu(\()p Fs(n)p Fu(!\))1024 470 y Fo(2)1051 489 y Fu(2)1074 470 y Fp(nl)1115 489 y Fr(j)g(\001)g(j)1168 468 y Fq(0)p Fo(2)1168 502 y Fp(k)805 631 y Fr(\024)897 578 y Fq(1)885 590 y Fj(X)882 681 y Fp(n)p Fo(=0)948 631 y Fu(\()p Fs(n)p Fu(!\))1024 612 y Fo(2)1051 631 y Fu(2)1074 612 y Fp(nl)1108 631 y Fs(C)1144 612 y Fo(2)p Fp(n)1192 631 y Fr(j)g(\001)g(j)1245 610 y Fo(2)1245 644 y Fp(p)805 744 y Fr(\024)42 b(k)7 b(\001)h(k)955 723 y Fo(2)955 758 y Fp(p;q)q(;\026)1055 744 y Fs(;)59 833 y Fu(whic)o(h)16 b(had)f(to)g(b)q(e)h(pro)o(v)o(ed.)1256 b Ff(2)59 966 y Fk(Lemma)17 b(21)23 b Fm(Ther)n(e)16 b(exist)g Fs(p;)8 b(C)q(;)g(K)13 b(>)g Fu(0)j Fm(such)g(that)h(for)g (al)r(l)f Fs(n)629 1011 y Fj(Z)678 1069 y Fr(j)p Fs(P)726 1050 y Fp(\026)720 1080 y(n)749 1069 y Fu(\()p Fs(x)p Fu(\))p Fr(j)824 1050 y Fo(2)824 1080 y Fq(\000)p Fp(p)883 1069 y Fu(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))d Fr(\024)g Fu(\()p Fs(n)p Fu(!\))1134 1050 y Fo(2)1160 1069 y Fs(C)1196 1050 y Fp(n)1227 1069 y Fs(K)484 b Fu(\(24\))59 1181 y Fk(Pr)o(oof.)92 b Fu(The)13 b(estimate)h(\(17\))e(ma)o(y)h(b)q(e)h (used)g(for)f Fs(\032)f Fr(\024)h Fu(2)1108 1164 y Fq(\000)p Fp(q)1151 1169 y Fi(0)1184 1181 y Fu(and)h Fs(\032)e Fr(\024)h Fu(2)p Fs(")1399 1188 y Fp(\026)1436 1181 y Fu(\()p Fs(")1475 1188 y Fp(\026)1511 1181 y Fu(from)g(Lemma)h(9\).)59 1237 y(This)i(giv)o(es)352 1262 y Fj(Z)401 1320 y Fr(j)p Fs(P)449 1301 y Fp(\026)443 1331 y(n)472 1320 y Fu(\()p Fs(x)p Fu(\))p Fr(j)547 1301 y Fo(2)547 1331 y Fq(\000)p Fp(p)606 1320 y Fu(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))d Fr(\024)g Fu(\()p Fs(n)p Fu(!\))857 1301 y Fo(2)883 1260 y Fj(\022)920 1289 y Fs(e)p 919 1309 24 2 v 919 1351 a(\032)955 1320 y Fr(k)p Fs(i)994 1327 y Fp(p;p)1040 1332 y Fi(0)1058 1320 y Fr(k)1081 1335 y Fp(H)s(S)1138 1260 y Fj(\023)1168 1272 y Fo(2)p Fp(n)1217 1262 y Fj(Z)1266 1320 y Fs(e)1287 1301 y Fo(2)p Fp(\032)p Fq(j)p Fp(x)p Fq(j)1363 1305 y Fh(\000)p Fg(p)1403 1312 y Fi(0)1424 1320 y Fu(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))59 1416 y(whic)o(h)j(is)g(\014nite)g(b)q(ecause)g(of)f(Lemma)g(9.)1028 b Ff(2)59 1508 y Fk(Theorem)18 b(22)23 b Fm(Ther)n(e)15 b(exist)h Fs(p)613 1492 y Fq(0)625 1508 y Fs(;)8 b(q)668 1492 y Fq(0)692 1508 y Fs(<)13 b Fu(0)j Fm(such)g(that)h(for)g(al)r(l)f Fs(p)d Fr(\025)g Fs(p)1220 1492 y Fq(0)1231 1508 y Fs(;)24 b(q)15 b Fr(\025)e Fs(q)1373 1492 y Fq(0)1401 1508 y Fm(the)j(top)n(olo)n(gic)n(al)g(emb)n(e)n(d-)59 1565 y(ding)g Fu(\()p Fr(H)215 1572 y Fp(p)235 1565 y Fu(\))253 1548 y Fo(1)253 1576 y Fp(q)285 1565 y Fr(\032)d Fs(L)364 1548 y Fo(2)384 1565 y Fu(\()p Fs(\026)p Fu(\))j Fm(holds.)59 1658 y Fk(Pr)o(oof.)94 b Fu(Elemen)o(ts)23 b(of)e(the)h(space)h(\()p Fr(N)7 b Fu(\))866 1641 y Fo(1)866 1669 y Fp(\026)910 1658 y Fu(are)22 b(de\014ned)i(as)d(series)i(con)o(v)o(ergen)o(t)e(in)i (the)f(giv)o(en)59 1714 y(top)q(ology)l(.)e(No)o(w)14 b(w)o(e)h(need)h(to)f(study)g(the)h(con)o(v)o(ergence)f(of)g(these)g (series)h(in)g Fs(L)1421 1697 y Fo(2)1441 1714 y Fu(\()p Fs(\026)p Fu(\).)j(Cho)q(ose)d Fs(q)1717 1697 y Fq(0)1743 1714 y Fu(suc)o(h)59 1770 y(that)f Fs(C)g(>)e Fu(2)277 1754 y Fp(q)294 1742 y Fh(0)334 1770 y Fu(\()p Fs(C)k Fu(from)e(estimate)g(\(24\)\).)j(Let)e(us)f(tak)o(e)f(an)i(arbitrary) 678 1889 y Fs(')c Fu(=)784 1837 y Fq(1)772 1849 y Fj(X)768 1940 y Fp(n)p Fo(=0)835 1889 y Fr(h)p Fs(P)888 1871 y Fp(\026)882 1901 y(n)911 1889 y Fs(;)c(')962 1871 y Fo(\()p Fp(n)p Fo(\))1012 1889 y Fr(i)k(2)h(P)t Fu(\()p Fr(N)1183 1871 y Fq(0)1194 1889 y Fu(\))59 2015 y(F)l(or)i Fs(p)d(>)h(p)247 1998 y Fq(0)274 2015 y Fu(\()p Fs(p)315 1998 y Fq(0)341 2015 y Fu(as)i(in)h(Lemma)f(21)g(\))g(and)g Fs(q)g(>)e(q)899 1998 y Fq(0)925 2015 y Fu(the)j(follo)o(wing)f(estimates)g(hold)360 2134 y Fr(k)p Fs(')p Fr(k)435 2147 y Fp(L)459 2138 y Fi(2)477 2147 y Fo(\()p Fp(\026)p Fo(\))569 2134 y Fr(\024)661 2081 y Fq(1)649 2094 y Fj(X)646 2184 y Fp(n)p Fo(=0)720 2085 y Fj(\015)720 2110 y(\015)720 2135 y(\015)743 2134 y Fr(h)p Fs(P)796 2115 y Fp(\026)790 2145 y(n)819 2134 y Fs(;)8 b(')870 2115 y Fo(\()p Fp(n)p Fo(\))920 2134 y Fr(i)938 2085 y Fj(\015)938 2110 y(\015)938 2135 y(\015)961 2162 y Fp(L)985 2153 y Fi(2)1002 2162 y Fo(\()p Fp(\026)p Fo(\))569 2276 y Fr(\024)661 2223 y Fq(1)649 2236 y Fj(X)646 2326 y Fp(n)p Fo(=0)720 2276 y Fr(j)p Fs(')763 2257 y Fo(\()p Fp(n)p Fo(\))813 2276 y Fr(j)826 2283 y Fq(\000)p Fp(p)881 2276 y Fr(k)f(j)p Fs(P)959 2257 y Fp(\026)953 2287 y(n)982 2276 y Fr(j)995 2283 y Fq(\000)p Fp(p)1042 2276 y Fr(k)1065 2292 y Fp(L)1089 2282 y Fi(2)1106 2292 y Fo(\()p Fp(\026)p Fo(\))569 2418 y Fr(\024)42 b Fs(K)711 2366 y Fq(1)698 2378 y Fj(X)695 2469 y Fp(n)p Fo(=0)769 2418 y Fs(n)p Fu(!)8 b(2)840 2400 y Fp(nq)q(=)p Fo(2)923 2370 y Fj(\014)923 2395 y(\014)923 2420 y(\014)937 2418 y Fs(')967 2400 y Fo(\()p Fp(n)p Fo(\))1017 2370 y Fj(\014)1017 2395 y(\014)1017 2420 y(\014)1031 2447 y Fq(\000)p Fp(p)1086 2418 y Fu(\()p Fs(C)s Fu(2)1163 2400 y Fq(\000)p Fp(q)1209 2418 y Fu(\))1227 2400 y Fp(n=)p Fo(2)569 2579 y Fr(\024)42 b Fs(K)695 2508 y Fj( )744 2527 y Fq(1)731 2539 y Fj(X)728 2630 y Fp(n)p Fo(=0)795 2579 y Fu(\()p Fs(C)10 b Fu(2)879 2561 y Fq(\000)p Fp(q)925 2579 y Fu(\))943 2561 y Fp(n)966 2508 y Fj(!)1004 2505 y Fi(1)p 1004 2511 16 2 v 1004 2532 a(2)1034 2508 y Fj( )1082 2527 y Fq(1)1070 2539 y Fj(X)1066 2630 y Fp(n)p Fo(=0)1133 2579 y Fu(\()p Fs(n)p Fu(!\))1209 2561 y Fo(2)1236 2579 y Fu(2)1259 2561 y Fp(q)q(n)1306 2531 y Fj(\014)1306 2556 y(\014)1306 2581 y(\014)1320 2579 y Fs(')1350 2561 y Fo(\()p Fp(n)p Fo(\))1401 2531 y Fj(\014)1401 2556 y(\014)1401 2581 y(\014)1415 2544 y Fo(2)1415 2608 y Fq(\000)p Fp(p)1462 2508 y Fj(!)1500 2505 y Fi(1)p 1500 2511 V 1500 2532 a(2)569 2701 y Fu(=)42 b Fs(K)695 2666 y Fj(\000)714 2701 y Fu(1)10 b Fr(\000)h Fs(C)f Fu(2)859 2682 y Fq(\000)p Fp(q)905 2666 y Fj(\001)924 2675 y Fq(\000)p Fo(1)p Fp(=)p Fo(2)1014 2701 y Fr(k)p Fs(')p Fr(k)1089 2714 y Fp(p;q)q(;\026)1174 2701 y Fu(.)59 2795 y(T)l(aking)16 b(the)f(closure)h(the)f(inequalit)o(y)i(extends)f (to)e(the)h(whole)h(space)g(\()p Fr(H)1336 2802 y Fp(p)1356 2795 y Fu(\))1374 2779 y Fo(1)1374 2806 y Fp(q)1393 2795 y Fu(.)391 b Ff(2)p eop %%Page: 23 23 23 22 bop 59 50 a Fn(5)45 b(TEST)15 b(FUNCTIONS)h(ON)g(A)f(LINEAR)h(SP) l(A)o(CE)g(WITH)f(MEASURE)365 b Fu(23)59 199 y Fk(Cor)o(ollar)m(y)18 b(23)23 b Fu(\()p Fr(N)7 b Fu(\))476 183 y Fo(1)476 210 y Fp(\026)515 199 y Fm(is)16 b(c)n(ontinuously)f(and)h(densely)f(emb)n (e)n(dde)n(d)h(in)g Fs(L)1360 183 y Fo(2)1380 199 y Fu(\()p Fs(\026)p Fu(\))p Fm(.)59 306 y Fk(Example)i(6)47 b Fm(\()p Fs(\026)p Fm(-exp)n(onentials)15 b(as)h(test)g(functions\))59 374 y Fu(The)f Fs(\026)p Fu(-exp)q(onen)o(tial)j(giv)o(en)d(in)h (\(10\))e(has)h(the)h(follo)o(wing)g(norm)532 507 y Fr(jj)p Fs(e)579 514 y Fp(\026)601 507 y Fu(\()p Fs(\022)q Fu(;)8 b Fr(\001)p Fu(\))p Fr(jj)719 488 y Fo(2)719 518 y Fp(p;q)q(;\026)808 507 y Fu(=)871 454 y Fq(1)859 467 y Fj(X)856 557 y Fp(n)p Fo(=0)930 507 y Fu(2)953 488 y Fp(nq)1001 507 y Fr(j)p Fs(\022)q Fr(j)1049 488 y Fo(2)p Fp(n)1049 518 y(p)1105 507 y Fs(;)98 b(\022)14 b Fr(2)f(N)p 1337 514 2 18 v 1328 514 a Fp(C)59 649 y Fu(This)k(expression)h(is)f(\014nite)g(if)g (and)g(only)g(if)g(2)849 632 y Fp(q)868 649 y Fr(j)p Fs(\022)q Fr(j)916 632 y Fo(2)916 660 y Fp(p)950 649 y Fs(<)e Fu(1.)24 b(Th)o(us)16 b(w)o(e)g(ha)o(v)o(e)g Fs(e)1371 656 y Fp(\026)1395 649 y Fu(\()p Fs(\022)q Fu(;)8 b Fr(\001)p Fu(\))19 b Fs(=)-28 b Fr(2)15 b Fu(\()p Fr(N)7 b Fu(\))1626 632 y Fo(1)1626 660 y Fp(\026)1665 649 y Fu(if)17 b Fs(\022)f Fr(6)p Fu(=)f(0.)59 705 y(But)e(w)o(e)g(ha)o (v)o(e)g(that)g Fs(e)433 712 y Fp(\026)456 705 y Fu(\()p Fs(\022)q Fu(;)8 b Fr(\001)p Fu(\))k(is)i(a)f(test)g(function)h(of)f (\014nite)h(order)f(i.e.,)g Fs(e)1289 712 y Fp(\026)1313 705 y Fu(\()p Fs(\022)q Fu(;)8 b Fr(\001)p Fu(\))j Fr(2)i Fu(\()p Fr(H)1515 712 y Fp(p)1535 705 y Fu(\))1553 689 y Fo(1)1553 717 y Fp(q)1586 705 y Fu(if)g(2)1648 689 y Fp(q)1667 705 y Fr(j)p Fs(\022)q Fr(j)1715 689 y Fo(2)1715 717 y Fp(p)1747 705 y Fs(<)g Fu(1.)59 762 y(This)k(is)g(in)h(con)o (trast)d(to)h(some)g(useful)h(spaces)g(of)f(test)g(functions)i(in)f (Gaussian)f(Analysis,)i(see)f(e.g.,)59 818 y([BeKo88,)d(HKPS93].)130 875 y(The)k(set)g(of)g(all)h Fs(\026)p Fu({exp)q(onen)o(tials)g Fr(f)p Fs(e)779 882 y Fp(\026)802 875 y Fu(\()p Fs(\022)q Fu(;)8 b Fr(\001)p Fu(\))17 b Fr(j)g Fu(2)964 858 y Fp(q)982 875 y Fr(j)p Fs(\022)q Fr(j)1030 858 y Fo(2)1030 886 y Fp(p)1068 875 y Fs(<)g Fu(1)p Fs(;)25 b(\022)20 b Fr(2)d(N)p 1312 882 V 1303 882 a Fp(C)1333 875 y Fr(g)h Fu(is)h(a)e(total)h(set)g (in)h(\()p Fr(H)1761 882 y Fp(p)1781 875 y Fu(\))1799 858 y Fo(1)1799 886 y Fp(q)1818 875 y Fu(.)59 931 y(This)d(can)f(b)q (een)h(sho)o(wn)f(using)h(the)g(relation)f(d)880 915 y Fp(n)904 931 y Fs(e)925 938 y Fp(\026)949 931 y Fu(\(0;)8 b Fr(\001)p Fu(\)\()p Fs(\022)1081 938 y Fo(1)1098 931 y Fs(;)g(:::;)g(\022)1200 938 y Fp(n)1221 931 y Fu(\))13 b(=)g Fr(h)p Fs(P)1353 915 y Fp(\026)1347 942 y(n)1376 931 y Fs(;)8 b(\022)1418 938 y Fo(1)1444 924 y Fu(^)1437 931 y Fr(\012)g(\001)g(\001)g(\001)1547 924 y Fu(^)1541 931 y Fr(\012)p Fs(\022)1597 938 y Fp(n)1621 931 y Fr(i)p Fs(:)59 1037 y Fk(Pr)o(oposition)16 b(24)23 b Fm(A)o(ny)15 b(test)g(function)f Fs(')h Fm(in)g Fu(\()p Fr(N)7 b Fu(\))962 1021 y Fo(1)962 1049 y Fp(\026)1000 1037 y Fm(has)15 b(a)g(uniquely)g(de\014ne)n(d)f(extension)g(to)i Fr(N)1747 1021 y Fq(0)p 1746 1050 V 1737 1050 a Fp(C)1782 1037 y Fm(an)59 1094 y(element)f(of)i Fr(E)309 1077 y Fo(1)305 1106 y(min)373 1094 y Fu(\()p Fr(N)435 1077 y Fq(0)p 434 1107 V 425 1107 a Fp(C)455 1094 y Fu(\))59 1200 y Fk(Pr)o(oof.)72 b Fu(An)o(y)16 b(elemen)o(t)g Fs(')f Fu(in)h(\()p Fr(N)7 b Fu(\))732 1184 y Fo(1)732 1211 y Fp(\026)769 1200 y Fu(is)16 b(de\014ned)h(as)d(a)h(series)h(of)f(the) g(follo)o(wing)h(t)o(yp)q(e)590 1339 y Fs(')d Fu(=)696 1286 y Fq(1)684 1299 y Fj(X)681 1389 y Fp(n)p Fo(=0)747 1339 y Fr(h)p Fs(P)800 1320 y Fp(\026)794 1350 y(n)824 1339 y Fs(;)8 b(')875 1320 y Fo(\()p Fp(n)p Fo(\))924 1339 y Fr(i)15 b Fs(;)98 b(')1098 1320 y Fo(\()p Fp(n)p Fo(\))1162 1339 y Fr(2)13 b(N)1253 1314 y Fo(^)1249 1320 y Fq(\012)p Fp(n)p 1248 1353 V 1239 1353 a(C)59 1475 y Fu(suc)o(h)j(that)655 1550 y Fr(k)p Fs(')p Fr(k)731 1529 y Fo(2)731 1564 y Fp(p;q)q(;\026)821 1550 y Fu(=)884 1497 y Fq(1)872 1510 y Fj(X)869 1600 y Fp(n)p Fo(=0)935 1550 y Fu(\()p Fs(n)p Fu(!\))1011 1531 y Fo(2)1038 1550 y Fu(2)1061 1531 y Fp(nq)1109 1550 y Fr(j)p Fs(')1152 1531 y Fo(\()p Fp(n)p Fo(\))1202 1550 y Fr(j)1215 1531 y Fo(2)1215 1561 y Fp(p)59 1667 y Fu(is)g(\014nite)g(for)e(eac)o(h)i Fs(p;)8 b(q)13 b Fr(2)g Fs(I)-12 b(N)20 b Fu(.)g(In)c(this)g(pro)q(of)e (w)o(e)h(will)i(sho)o(w)e(the)g(con)o(v)o(ergence)h(of)e(the)i(series) 668 1744 y Fq(1)656 1757 y Fj(X)653 1847 y Fp(n)p Fo(=0)719 1797 y Fr(h)p Fs(P)772 1778 y Fp(\026)766 1808 y(n)796 1797 y Fu(\()p Fs(z)r Fu(\))p Fs(;)8 b(')906 1778 y Fo(\()p Fp(n)p Fo(\))955 1797 y Fr(i)p Fs(;)52 b(z)15 b Fr(2)e(H)1155 1804 y Fq(\000)p Fp(p;)p 1216 1804 V -3 w(C)59 1933 y Fu(to)i(an)g(en)o(tire)g(function)h(in)g Fs(z)r Fu(.)130 1989 y(Let)e Fs(p)f(>)g(p)317 1996 y Fo(0)351 1989 y Fu(suc)o(h)i(that)e(the)i(em)o(b)q(edding)h Fs(i)873 1996 y Fp(p;p)919 2001 y Fi(0)950 1989 y Fu(:)c Fr(H)1013 1996 y Fp(p)1046 1989 y Fs(,)-8 b Fr(!)13 b(H)1147 1996 y Fp(p)1165 2001 y Fi(0)1199 1989 y Fu(is)i(Hilb)q(ert-Sc)o(hmidt.)22 b(Then)15 b(for)f(all)59 2046 y(0)e Fs(<)h(")g Fr(\024)g Fu(2)247 2029 y Fq(\000)p Fp(q)290 2034 y Fi(0)309 2046 y Fs(=e)8 b Fr(k)p Fs(i)400 2053 y Fp(p;p)446 2058 y Fi(0)464 2046 y Fr(k)487 2061 y Fp(H)s(S)559 2046 y Fu(w)o(e)15 b(can)g(use)h(\(15\))e(and)h(estimate)g(as)g(follo)o(ws)180 2129 y Fq(1)168 2142 y Fj(X)165 2232 y Fp(n)p Fo(=0)239 2182 y Fr(jh)p Fs(P)305 2164 y Fp(\026)299 2194 y(n)328 2182 y Fu(\()p Fs(z)r Fu(\))p Fs(;)8 b(')438 2164 y Fo(\()p Fp(n)p Fo(\))487 2182 y Fr(ij)41 b(\024)652 2129 y Fq(1)639 2142 y Fj(X)636 2232 y Fp(n)p Fo(=0)710 2182 y Fr(j)p Fs(P)758 2164 y Fp(\026)752 2194 y(n)781 2182 y Fu(\()p Fs(z)r Fu(\))p Fr(j)853 2189 y Fq(\000)p Fp(p)900 2182 y Fr(j)p Fs(')943 2164 y Fo(\()p Fp(n)p Fo(\))993 2182 y Fr(j)1006 2189 y Fp(p)559 2325 y Fr(\024)h Fs(C)669 2332 y Fp(p;")722 2325 y Fs(e)743 2306 y Fp(")p Fq(j)p Fp(z)q Fq(j)796 2310 y Fh(\000)p Fg(p)864 2272 y Fq(1)852 2284 y Fj(X)848 2375 y Fp(n)p Fo(=0)923 2325 y Fs(n)p Fu(!)8 b Fr(j)p Fs(')1014 2306 y Fo(\()p Fp(n)p Fo(\))1063 2325 y Fr(j)1076 2332 y Fp(p)1103 2325 y Fs(")1124 2306 y Fq(\000)p Fp(n)559 2481 y Fr(\024)42 b Fs(C)669 2488 y Fp(p;")729 2481 y Fs(e)750 2462 y Fp(")p Fq(j)p Fp(z)q Fq(j)803 2466 y Fh(\000)p Fg(p)864 2409 y Fj( )912 2428 y Fq(1)900 2440 y Fj(X)896 2531 y Fp(n)p Fo(=0)963 2481 y Fu(\()p Fs(n)p Fu(!\))1039 2462 y Fo(2)1058 2481 y Fu(2)1081 2462 y Fp(nq)1121 2481 y Fr(j)p Fs(')1164 2462 y Fo(\()p Fp(n)p Fo(\))1214 2481 y Fr(j)1227 2462 y Fo(2)1227 2492 y Fp(p)1247 2409 y Fj(!)1280 2420 y Fo(1)p Fp(=)p Fo(2)1342 2409 y Fj( )1391 2428 y Fq(1)1378 2440 y Fj(X)1375 2531 y Fp(n)p Fo(=0)1449 2481 y Fu(2)1472 2462 y Fq(\000)p Fp(nq)1540 2481 y Fs(")1561 2462 y Fq(\000)p Fo(2)p Fp(n)1630 2409 y Fj(!)1662 2420 y Fo(1)p Fp(=)p Fo(2)559 2612 y Fu(=)g Fs(C)669 2619 y Fp(p;")737 2565 y Fj(\020)762 2612 y Fu(1)10 b Fr(\000)g Fu(2)863 2594 y Fq(\000)p Fp(q)909 2612 y Fs(")930 2594 y Fq(\000)p Fo(2)978 2565 y Fj(\021)1002 2577 y Fq(\000)p Fo(1)p Fp(=)p Fo(2)1100 2612 y Fr(k)p Fs(')p Fr(k)1175 2626 y Fp(p;q)q(;\026)1281 2612 y Fs(e)1302 2594 y Fp(\032)p Fq(j)p Fp(z)q Fq(j)1357 2598 y Fh(\000)p Fg(p)59 2729 y Fu(if)15 b(2)123 2713 y Fp(q)154 2729 y Fs(>)e(")223 2713 y Fq(\000)p Fo(2)271 2729 y Fu(.)19 b(That)14 b(means)g(the)h(series)753 2697 y Fj(P)797 2711 y Fq(1)797 2741 y Fp(n)p Fo(=0)865 2729 y Fr(h)p Fs(P)918 2713 y Fp(\026)912 2741 y(n)942 2729 y Fu(\()p Fs(z)r Fu(\))p Fs(;)8 b(')1052 2713 y Fo(\()p Fp(n)p Fo(\))1101 2729 y Fr(i)14 b Fu(con)o(v)o(erges)g(uniformly)h (and)g(absolutely)59 2786 y(in)i(an)o(y)f(neigh)o(b)q(orho)q(o)q(d)h (of)f(zero)g(of)g(an)o(y)f(space)i Fr(H)936 2793 y Fq(\000)p Fp(p;)p 997 2793 V -3 w(C)1034 2786 y Fu(.)23 b(Since)17 b(eac)o(h)f(term)g Fr(h)p Fs(P)1455 2769 y Fp(\026)1449 2797 y(n)1478 2786 y Fu(\()p Fs(z)r Fu(\))p Fs(;)8 b(')1588 2769 y Fo(\()p Fp(n)p Fo(\))1638 2786 y Fr(i)16 b Fu(is)g(en)o(tire)p eop %%Page: 24 24 24 23 bop 59 50 a Fn(5)45 b(TEST)15 b(FUNCTIONS)h(ON)g(A)f(LINEAR)h(SP) l(A)o(CE)g(WITH)f(MEASURE)365 b Fu(24)59 199 y(in)16 b Fs(z)g Fu(the)f(uniform)g(con)o(v)o(ergence)g(implies)i(that)d Fs(z)g Fr(7!)993 167 y Fj(P)1037 180 y Fq(1)1037 211 y Fp(n)p Fo(=0)1106 199 y Fr(h)p Fs(P)1159 183 y Fp(\026)1153 210 y(n)1182 199 y Fu(\()p Fs(z)r Fu(\))p Fs(;)8 b(')1292 183 y Fo(\()p Fp(n)p Fo(\))1342 199 y Fr(i)14 b Fu(is)h(en)o(tire)g(on) g(eac)o(h)g Fr(H)1749 206 y Fq(\000)p Fp(p;)p 1810 206 2 18 v -3 w(C)59 256 y Fu(and)g(hence)i(on)e Fr(N)381 239 y Fq(0)p 380 269 V 371 269 a Fp(C)401 256 y Fu(.)20 b(This)15 b(completes)h(the)g(pro)q(of.)854 b Ff(2)130 357 y Fu(The)16 b(follo)o(wing)h(corollary)g(is)f(an)h(immediate)g (consequence)h(of)e(the)g(ab)q(o)o(v)o(e)g(pro)q(of)g(and)g(giv)o(es)h (an)59 413 y(explicit)g(estimate)e(on)h(the)f(gro)o(wth)f(of)h(the)g (test)f(functions.)59 511 y Fk(Cor)o(ollar)m(y)k(25)23 b Fm(F)m(or)c(al)r(l)h Fs(p)e(>)h(p)671 518 y Fo(0)711 511 y Fm(such)g(that)i(the)f(norm)f Fr(k)p Fs(i)1156 518 y Fp(p;p)1202 523 y Fi(0)1221 511 y Fr(k)1243 527 y Fp(H)s(S)1320 511 y Fm(of)h(the)g(emb)n(e)n(dding)e(is)i(\014nite)59 568 y(and)e(for)g(al)r(l)g Fu(0)e Fs(<)g(")g Fr(\024)g Fu(2)491 551 y Fq(\000)p Fp(q)534 556 y Fi(0)553 568 y Fs(=e)8 b Fr(k)p Fs(i)644 575 y Fp(p;p)690 580 y Fi(0)708 568 y Fr(k)731 583 y Fp(H)s(S)806 568 y Fm(we)18 b(c)n(an)f(cho)n(ose)h Fs(q)g Fr(2)e Fs(I)-12 b(N)23 b Fm(such)18 b(that)g Fu(2)1480 551 y Fp(q)1515 568 y Fs(>)e(")1587 551 y Fq(\000)p Fo(2)1652 568 y Fm(to)j(obtain)59 624 y(the)e(fol)r(lowing)e(b)n(ound.)399 718 y Fr(j)p Fs(')p Fu(\()p Fs(z)r Fu(\))p Fr(j)c(\024)i Fs(C)18 b Fr(k)p Fs(')p Fr(k)699 732 y Fp(p;q)q(;\026)792 718 y Fs(e)813 699 y Fp(")p Fq(j)p Fp(z)q Fq(j)866 703 y Fh(\000)p Fg(p)927 718 y Fs(;)101 b(')12 b Fr(2)h Fu(\()p Fr(N)7 b Fu(\))1206 699 y Fo(1)1225 718 y Fs(;)23 b(z)15 b Fr(2)e(H)1378 725 y Fq(\000)p Fp(p;)p 1439 725 V -3 w(C)1477 718 y Fm(,)59 812 y(wher)n(e)666 877 y Fs(C)i Fu(=)e Fs(C)795 884 y Fp(p;")856 830 y Fj(\020)881 877 y Fu(1)c Fr(\000)i Fu(2)982 858 y Fq(\000)p Fp(q)1028 877 y Fs(")1049 858 y Fq(\000)p Fo(2)1096 830 y Fj(\021)1121 841 y Fq(\000)p Fo(1)p Fp(=)p Fo(2)1211 877 y Fs(:)149 987 y Fu(Let)19 b(us)g(lo)q(ok)g(at)g(Prop)q(osition)g(24)g(again.)31 b(On)20 b(one)f(hand)g(an)o(y)g(function)h Fs(')f Fr(2)g Fu(\()p Fr(N)7 b Fu(\))1655 970 y Fo(1)1655 998 y Fp(\026)1697 987 y Fu(can)19 b(b)q(e)59 1043 y(written)c(in)h(the)g(form)516 1156 y Fs(')p Fu(\()p Fs(z)r Fu(\))c(=)681 1103 y Fq(1)669 1116 y Fj(X)665 1206 y Fp(n)p Fo(=0)732 1156 y Fr(h)p Fs(P)785 1137 y Fp(\026)779 1167 y(n)808 1156 y Fu(\()p Fs(x)p Fu(\))p Fs(;)c(')921 1137 y Fo(\()p Fp(n)p Fo(\))970 1156 y Fr(i)15 b Fs(;)98 b(')1144 1137 y Fo(\()p Fp(n)p Fo(\))1208 1156 y Fr(2)13 b(N)1300 1131 y Fo(^)1295 1137 y Fq(\012)p Fp(n)p 1294 1170 V 1285 1170 a(C)1361 1156 y Fs(;)376 b Fu(\(25\))59 1284 y(on)15 b(the)g(other)g(hand)h(it)f(is)h (en)o(tire,)f(i.e.,)g(it)h(has)f(the)g(represen)o(tation)539 1405 y Fs(')p Fu(\()p Fs(z)r Fu(\))d(=)704 1353 y Fq(1)692 1365 y Fj(X)688 1456 y Fp(n)p Fo(=0)755 1405 y Fr(h)p Fs(z)796 1387 y Fq(\012)p Fp(n)847 1405 y Fs(;)i Fu(~)-29 b Fs(')898 1387 y Fo(\()p Fp(n)p Fo(\))947 1405 y Fr(i)15 b Fs(;)106 b Fu(~)-31 b Fs(')1121 1387 y Fo(\()p Fp(n)p Fo(\))1185 1405 y Fr(2)13 b(N)1276 1380 y Fo(^)1272 1386 y Fq(\012)p Fp(n)p 1271 1419 V 1262 1419 a(C)1338 1405 y Fs(;)399 b Fu(\(26\))59 1550 y(T)l(o)11 b(pro)q(ceed)h(further)f(w)o (e)g(need)h(the)f(explicit)j(corresp)q(ondence)1144 1503 y Fj(n)1172 1550 y Fs(')1202 1534 y Fo(\()p Fp(n)p Fo(\))1253 1550 y Fs(;)8 b(n)k Fr(2)h Fs(I)-12 b(N)1406 1503 y Fj(o)1446 1550 y Fr( )k(!)1542 1503 y Fj(n)1577 1550 y Fu(~)-31 b Fs(')1599 1534 y Fo(\()p Fp(n)p Fo(\))1650 1550 y Fs(;)8 b(n)k Fr(2)h Fs(I)-12 b(N)1803 1503 y Fj(o)59 1612 y Fu(whic)o(h)16 b(is)g(giv)o(en)f(in)i(the)e(next)g(lemma.)59 1710 y Fk(Lemma)i(26)23 b Fv(\(Reordering\))59 1778 y Fm(Equations)16 b(\(25\))g(and)g(\(26\))g(hold)h(i\013)559 1904 y Fu(~)-30 b Fs(')582 1886 y Fo(\()p Fp(k)q Fo(\))643 1904 y Fu(=)707 1851 y Fq(1)694 1864 y Fj(X)691 1954 y Fp(n)p Fo(=0)765 1832 y Fj( )798 1874 y Fs(n)10 b Fu(+)h Fs(k)839 1936 y(k)906 1832 y Fj(!)946 1857 y(\020)971 1904 y Fs(P)1006 1886 y Fp(\026)1000 1916 y(n)1030 1904 y Fu(\(0\))p Fs(;)d(')1140 1886 y Fo(\()p Fp(n)p Fo(+)p Fp(k)q Fo(\))1236 1857 y Fj(\021)1260 1932 y Fq(H)1294 1918 y Fi(^)1290 1923 y Fh(\012)p Fg(n)59 2032 y Fm(or)17 b(e)n(quivalently)578 2119 y Fs(')608 2100 y Fo(\()p Fp(k)q Fo(\))669 2119 y Fu(=)733 2066 y Fq(1)720 2078 y Fj(X)717 2169 y Fp(n)p Fo(=0)791 2047 y Fj( )824 2088 y Fs(n)10 b Fu(+)h Fs(k)865 2150 y(k)932 2047 y Fj(!)972 2071 y(\020)997 2119 y Fu(M)1039 2100 y Fp(\026)1039 2130 y(n)1062 2119 y Fs(;)k Fu(~)-30 b Fs(')1113 2100 y Fo(\()p Fp(n)p Fo(+)p Fp(k)q Fo(\))1210 2071 y Fj(\021)1235 2147 y Fq(H)1269 2132 y Fi(^)1265 2137 y Fh(\012)p Fg(n)59 2249 y Fm(wher)n(e)187 2202 y Fj(\020)212 2249 y Fs(P)247 2232 y Fp(\026)241 2260 y(n)271 2249 y Fu(\(0\))p Fs(;)8 b(')381 2232 y Fo(\()p Fp(n)p Fo(+)p Fp(k)q Fo(\))477 2202 y Fj(\021)501 2277 y Fq(H)535 2262 y Fi(^)531 2267 y Fh(\012)p Fg(n)595 2249 y Fm(and)683 2202 y Fj(\020)708 2249 y Fu(M)750 2232 y Fp(\026)750 2260 y(n)773 2249 y Fs(;)15 b Fu(~)-30 b Fs(')824 2232 y Fo(\()p Fp(n)p Fo(+)p Fp(k)q Fo(\))921 2202 y Fj(\021)946 2277 y Fq(H)980 2262 y Fi(^)976 2267 y Fh(\012)p Fg(n)1039 2249 y Fm(denote)16 b(c)n(ontr)n(actions)g(de\014ne)n(d)f(by)h(\(20\).)59 2357 y Fu(This)g(is)g(a)e(consequence)j(of)e(\(11\))f(and)h(\(12\).)k (W)l(e)c(omit)g(the)g(simple)i(pro)q(of.)130 2458 y(Prop)q(osition)e (24)g(states)775 2514 y(\()p Fr(N)7 b Fu(\))855 2495 y Fo(1)855 2525 y Fp(\026)891 2514 y Fr(\022)13 b(E)967 2495 y Fo(1)963 2525 y(min)1023 2514 y Fu(\()p Fr(N)1085 2495 y Fq(0)1097 2514 y Fu(\))59 2593 y(as)i(sets,)f(where)582 2655 y Fr(E)610 2636 y Fo(1)606 2666 y(min)667 2655 y Fu(\()p Fr(N)729 2636 y Fq(0)740 2655 y Fu(\))f(=)819 2608 y Fj(n)846 2655 y Fs(')p Fr(j)889 2663 y Fq(N)923 2653 y Fh(0)948 2606 y Fj(\014)948 2631 y(\014)948 2656 y(\014)975 2655 y Fs(')f Fr(2)h(E)1088 2636 y Fo(1)1084 2666 y(min)1145 2655 y Fu(\()p Fr(N)1207 2636 y Fq(0)p 1206 2666 V 1197 2666 a Fp(C)1227 2655 y Fu(\))1245 2608 y Fj(o)1295 2655 y Fs(:)59 2740 y Fu(Corollary)19 b(25)f(then)h (implies)h(that)e(the)h(em)o(b)q(edding)h(is)g(also)e(con)o(tin)o (uous.)31 b(No)o(w)18 b(w)o(e)g(are)h(going)f(to)59 2796 y(sho)o(w)d(that)f(the)h(con)o(v)o(erse)g(also)h(holds.)p eop %%Page: 25 25 25 24 bop 59 50 a Fn(5)45 b(TEST)15 b(FUNCTIONS)h(ON)g(A)f(LINEAR)h(SP) l(A)o(CE)g(WITH)f(MEASURE)365 b Fu(25)59 199 y Fk(Theorem)18 b(27)23 b Fm(F)m(or)16 b(al)r(l)g(me)n(asur)n(es)f Fs(\026)e Fr(2)g(M)836 206 y Fp(a)857 199 y Fu(\()p Fr(N)919 183 y Fq(0)930 199 y Fu(\))j Fm(we)g(have)h(the)f(top)n(olo)n(gic)n(al)g (identity)761 284 y Fu(\()p Fr(N)7 b Fu(\))841 265 y Fo(1)841 295 y Fp(\026)876 284 y Fu(=)13 b Fr(E)952 265 y Fo(1)948 295 y(min)1009 284 y Fu(\()p Fr(N)1071 265 y Fq(0)1082 284 y Fu(\))j Fs(:)59 372 y Fu(T)l(o)f(pro)o(v)o(e)g(the)g (missing)h(top)q(ological)g(inclusion)i(it)d(is)h(con)o(v)o(enien)o(t)g (to)e(use)i(the)f(n)o(uclear)h(top)q(ology)f(on)59 429 y Fr(E)87 412 y Fo(1)83 441 y(min)144 429 y Fu(\()p Fr(N)206 412 y Fq(0)p 205 441 2 18 v 196 441 a Fp(C)226 429 y Fu(\))i(\(giv)o(en)g(b)o(y)g(the)h(norms)f Fr(j)-8 b(j)g(j)o(\001j)g(j) g(j)742 442 y Fp(p;q)q(;)p Fo(1)816 429 y Fu(\))17 b(in)o(tro)q(duced)h (in)g(section)g(2.)26 b(Theorem)18 b(5)f(ensures)g(that)59 485 y(this)g(top)q(ology)g(is)g(equiv)m(alen)o(t)i(to)d(the)h(pro)s (jectiv)o(e)f(top)q(ology)h(induced)i(b)o(y)d(the)h(norms)g(n)1620 492 y Fp(p;l;k)1689 485 y Fu(.)25 b(Then)59 542 y(the)15 b(ab)q(o)o(v)o(e)g(theorem)g(is)h(an)f(immediate)h(consequence)h(of)d (the)i(follo)o(wing)g(norm)e(estimate.)59 630 y Fk(Pr)o(oposition)i(28) 23 b Fm(L)n(et)15 b Fs(p)d(>)h(p)608 637 y Fp(\026)647 630 y Fm(\()p Fs(p)689 637 y Fp(\026)727 630 y Fm(as)i(in)g(L)n(emma)f (9\))i(such)f(that)1250 594 y Fj(\015)1250 619 y(\015)1273 630 y Fs(i)1289 637 y Fp(p;p)1335 641 y Fg(\026)1357 594 y Fj(\015)1357 619 y(\015)1380 648 y Fp(H)s(S)1452 630 y Fm(is)g(\014nite)f(and)h Fs(q)g Fr(2)e Fs(I)-12 b(N)59 693 y Fm(such)15 b(that)g Fu(2)275 677 y Fp(q)q(=)p Fo(2)342 693 y Fs(>)e(K)429 700 y Fp(p)463 693 y Fm(\()p Fs(K)521 700 y Fp(p)553 693 y Fu(:=)f Fs(eC)678 657 y Fj(\015)678 682 y(\015)701 693 y Fs(i)717 700 y Fp(p;p)763 704 y Fg(\026)785 657 y Fj(\015)785 682 y(\015)808 711 y Fp(H)s(S)879 693 y Fm(as)j(in)f(\(7\)\).)19 b(F)m(or)14 b(any)h Fs(')d Fr(2)h Fu(E)1388 677 y Fo(1)1388 705 y Fp(p;q)1449 693 y Fm(the)i(r)n(estriction)f Fs(')p Fr(j)1784 701 y Fq(N)1818 692 y Fh(0)59 750 y Fm(is)i(a)g(function)g(fr)n(om)h Fu(\()p Fr(H)490 757 y Fp(p)510 750 y Fu(\))528 733 y Fo(1)528 763 y Fp(q)545 754 y Fh(0)555 763 y Fp(;\026)605 750 y Fs(;)23 b(q)663 733 y Fq(0)687 750 y Fs(<)13 b(q)r Fm(.)21 b(Mor)n(e)n(over)16 b(the)h(fol)r(lowing)e(estimate)i(holds)442 846 y Fr(jj)p Fs(')p Fr(jj)524 854 y Fp(p;q)569 844 y Fh(0)578 854 y Fp(;\026)624 846 y Fr(\024)c(j)-8 b(j)g(j)p Fs(')p Fr(j)g(j)g(j)747 860 y Fp(p;q)q(;)p Fo(1)829 846 y Fu(\(1)9 b Fr(\000)i Fu(2)948 827 y Fp(q)q(=)p Fo(2)1002 846 y Fs(K)1041 853 y Fp(p)1060 846 y Fu(\))1078 827 y Fq(\000)p Fo(1)1125 846 y Fu(\(1)f Fr(\000)g Fu(2)1244 827 y Fp(q)1261 816 y Fh(0)1272 827 y Fq(\000)p Fp(q)1318 846 y Fu(\))1336 827 y Fq(\000)p Fo(1)p Fp(=)p Fo(2)1435 846 y Fs(:)59 940 y Fk(Pr)o(oof.)76 b Fu(Let)16 b Fs(p;)8 b(q)15 b Fr(2)g Fs(I)-12 b(N)5 b Fu(,)16 b Fs(K)616 947 y Fp(p)651 940 y Fu(b)q(e)h(de\014ned)h(as)e(ab)q(o)o(v)o(e.)22 b(A)16 b(function)h Fs(')d Fr(2)h Fu(E)1427 924 y Fo(1)1427 951 y Fp(p;q)1490 940 y Fu(has)h(the)g(represen-)59 997 y(tation)f(\(25\).)k(Using)c(the)h(Reordering)g(lemma)f(com)o(bined)i (with)e(\(7\))f(and)680 1051 y Fj(\014)680 1076 y(\014)680 1101 y(\014)702 1100 y Fu(~)-31 b Fs(')724 1081 y Fo(\()p Fp(n)p Fo(\))775 1051 y Fj(\014)775 1076 y(\014)775 1101 y(\014)789 1128 y Fp(p)821 1100 y Fr(\024)883 1069 y Fu(1)p 874 1090 40 2 v 874 1131 a Fs(n)p Fu(!)927 1100 y(2)950 1081 y Fq(\000)p Fp(nq)q(=)p Fo(2)1060 1100 y Fr(j)-8 b(j)g(j)p Fs(')p Fr(j)g(j)g(j)1135 1114 y Fp(p;q)q(;)p Fo(1)59 1198 y Fu(w)o(e)15 b(obtain)402 1255 y Fj(\014)402 1280 y(\014)402 1305 y(\014)416 1304 y Fs(')446 1285 y Fo(\()p Fp(n)p Fo(\))497 1255 y Fj(\014)497 1280 y(\014)497 1305 y(\014)510 1332 y Fp(p)572 1304 y Fr(\024)663 1251 y Fq(1)651 1264 y Fj(X)649 1356 y Fp(k)q Fo(=0)721 1232 y Fj( )753 1273 y Fs(n)c Fu(+)f Fs(k)795 1335 y(k)861 1232 y Fj(!)902 1268 y(\014)902 1293 y(\014)916 1304 y Fu(M)958 1282 y Fp(\026)958 1318 y(k)981 1268 y Fj(\014)981 1293 y(\014)994 1320 y Fq(\000)p Fp(p)1049 1255 y Fj(\014)1049 1280 y(\014)1049 1305 y(\014)1070 1304 y Fu(~)-30 b Fs(')1093 1285 y Fo(\()p Fp(n)p Fo(+)p Fp(k)q Fo(\))1190 1255 y Fj(\014)1190 1280 y(\014)1190 1305 y(\014)1204 1332 y Fp(p)572 1450 y Fr(\024)42 b(j)-8 b(j)g(j)o Fs(')p Fr(j)g(j)g(j)724 1464 y Fp(p;q)q(;)p Fo(1)820 1397 y Fq(1)808 1410 y Fj(X)805 1502 y Fp(k)q Fo(=0)877 1378 y Fj( )910 1419 y Fs(n)11 b Fu(+)f Fs(k)952 1481 y(k)1018 1378 y Fj(!)1115 1419 y Fs(k)q Fu(!)p 1056 1440 156 2 v 1056 1481 a(\()p Fs(n)g Fu(+)h Fs(k)q Fu(\)!)1217 1450 y Fs(K)1259 1431 y Fp(k)1256 1461 y(p)1288 1450 y Fu(2)1311 1431 y Fq(\000)p Fo(\()p Fp(n)p Fo(+)p Fp(k)q Fo(\))p Fp(q)q(=)p Fo(2)572 1594 y Fr(\024)42 b(j)-8 b(j)g(j)o Fs(')p Fr(j)g(j)g(j)724 1608 y Fp(p;q)q(;)p Fo(1)819 1563 y Fu(1)p 810 1584 40 2 v 810 1625 a Fs(n)p Fu(!)855 1594 y(2)878 1575 y Fq(\000)p Fp(nq)q(=)p Fo(2)1003 1541 y Fq(1)991 1554 y Fj(X)989 1646 y Fp(k)q Fo(=0)1053 1594 y Fu(\(2)1094 1575 y Fq(\000)p Fp(q)q(=)p Fo(2)1175 1594 y Fs(K)1214 1601 y Fp(p)1234 1594 y Fu(\))1252 1575 y Fp(k)572 1727 y Fr(\024)42 b(j)-8 b(j)g(j)o Fs(')p Fr(j)g(j)g(j)724 1740 y Fp(p;q)q(;)p Fo(1)819 1696 y Fu(1)p 810 1716 V 810 1758 a Fs(n)p Fu(!)855 1727 y(2)878 1708 y Fq(\000)p Fp(nq)q(=)p Fo(2)981 1727 y Fu(\(1)10 b Fr(\000)g Fu(2)1100 1708 y Fq(\000)p Fp(q)q(=)p Fo(2)1182 1727 y Fs(K)1221 1734 y Fp(p)1240 1727 y Fu(\))1258 1708 y Fq(\000)p Fo(1)59 1825 y Fu(F)l(or)15 b Fs(q)163 1808 y Fq(0)187 1825 y Fs(<)e(q)k Fu(this)f(allo)o(ws)f(the)g(follo)o (wing)h(estimate)395 1937 y Fr(jj)p Fs(')p Fr(jj)477 1918 y Fo(2)477 1948 y Fp(p;q)522 1939 y Fh(0)531 1948 y Fp(;\026)606 1937 y Fu(=)698 1884 y Fq(1)686 1897 y Fj(X)682 1987 y Fp(n)p Fo(=0)749 1937 y Fu(\()p Fs(n)p Fu(!\))825 1918 y Fo(2)852 1937 y Fu(2)875 1918 y Fp(q)892 1907 y Fh(0)902 1918 y Fp(n)933 1937 y Fr(j)p Fs(')976 1918 y Fo(\()p Fp(n)p Fo(\))1027 1937 y Fr(j)1040 1918 y Fo(2)1040 1948 y Fp(p)606 2079 y Fr(\024)41 b(j)-7 b(j)f(j)o Fs(')p Fr(j)g(j)g(j)758 2058 y Fo(2)758 2093 y Fp(p;q)q(;)p Fo(1)839 2079 y Fu(\(1)10 b 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Fk(Cor)o(ollar)m(y)j(31)23 b Fm(F)m(or)17 b(al)r(l)g Fs(z)g Fr(2)e(N)675 2712 y Fq(0)p 674 2741 2 18 v 665 2741 a Fp(C)713 2728 y Fm(the)i(sp)n(ac)n(e)g Fu(\()p Fr(N)7 b Fu(\))989 2712 y Fo(1)1026 2728 y Fm(is)17 b(invariant)g (under)g(the)h(shift)f(op)n(er)n(ator)i Fs(\034)1784 2735 y Fp(z)1818 2728 y Fu(:)59 2785 y Fs(')12 b Fr(7!)h Fs(')p Fu(\()p Fr(\001)d Fu(+)g Fs(z)r Fu(\))p Fm(.)p eop %%Page: 26 26 26 25 bop 59 50 a Fn(6)45 b(DISTRIBUTIONS)1291 b Fu(26)59 199 y Ft(6)67 b(Distributions)59 301 y Fu(In)14 b(this)f(section)g(w)o (e)g(will)h(in)o(tro)q(duce)g(and)f(study)g(the)g(space)g(\()p Fr(N)7 b Fu(\))1181 284 y Fq(\000)p Fo(1)1181 312 y Fp(\026)1241 301 y Fu(of)12 b(distributions)j(corresp)q(onding)59 357 y(to)g(the)h(space)g(of)f(test)g(functions)h(\()p Fr(N)7 b Fu(\))733 341 y Fo(1)752 357 y Fu(.)21 b(Since)c Fr(P)t Fu(\()p Fr(N)1003 341 y Fq(0)1014 357 y Fu(\))c Fr(\032)h Fu(\()p Fr(N)7 b Fu(\))1174 341 y Fo(1)1208 357 y Fu(the)16 b(space)g(\()p Fr(N)7 b Fu(\))1489 341 y Fq(\000)p Fo(1)1489 368 y Fp(\026)1551 357 y Fu(can)16 b(b)q(e)g(view)o(ed)59 414 y(as)f(a)g(subspace)h(of)e Fr(P)430 397 y Fq(0)426 425 y Fp(\026)449 414 y Fu(\()p Fr(N)511 397 y Fq(0)523 414 y Fu(\))778 476 y(\()p Fr(N)7 b Fu(\))858 457 y Fq(\000)p Fo(1)858 487 y Fp(\026)918 476 y Fr(\032)12 b(P)1001 457 y Fq(0)997 487 y Fp(\026)1020 476 y Fu(\()p Fr(N)1082 457 y Fq(0)1094 476 y Fu(\))59 565 y(Let)j(us)h(no)o(w)f(in)o(tro)q(duce)h(the)f(Hilb)q(ertian)i (subspace)f(\()p Fr(H)1037 572 y Fq(\000)p Fp(p)1084 565 y Fu(\))1102 546 y Fq(\000)p Fo(1)1102 576 y Fq(\000)p Fp(q)q(;\026)1195 565 y Fu(of)f Fr(P)1283 548 y Fq(0)1279 576 y Fp(\026)1302 565 y Fu(\()p Fr(N)1364 548 y Fq(0)1375 565 y Fu(\))g(for)f(whic)o(h)i(the)g(norm)626 704 y Fr(k)p Fu(\010)p Fr(k)704 683 y Fo(2)704 717 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)q(;\026)849 704 y Fu(:=)926 651 y Fq(1)913 663 y Fj(X)910 754 y Fp(n)p Fo(=0)984 704 y Fu(2)1007 685 y Fq(\000)p Fp(q)q(n)1082 655 y Fj(\014)1082 680 y(\014)1082 705 y(\014)1096 704 y Fu(\010)1129 685 y Fo(\()p Fp(n)p Fo(\))1180 655 y Fj(\014)1180 680 y(\014)1180 705 y(\014)1194 668 y Fo(2)1194 732 y Fq(\000)p Fp(p)59 840 y Fu(is)g(\014nite.)21 b(Here)15 b(w)o(e)g(used)h(the)f(canonical)i (represen)o(tation)664 969 y(\010)12 b(=)773 917 y Fq(1)760 929 y Fj(X)757 1020 y Fp(n)p Fo(=0)831 969 y Fs(Q)867 951 y Fp(\026)867 981 y(n)890 969 y Fu(\(\010)941 951 y Fo(\()p Fp(n)p Fo(\))992 969 y Fu(\))g Fr(2)h(P)1101 951 y Fq(0)1097 981 y Fp(\026)1120 969 y Fu(\()p Fr(N)1182 951 y Fq(0)1193 969 y Fu(\))59 1114 y(from)i(Theorem)h(18.)21 b(The)16 b(space)g(\()p Fr(H)712 1121 y Fq(\000)p Fp(p)759 1114 y Fu(\))777 1095 y Fq(\000)p Fo(1)777 1125 y Fq(\000)p Fp(q)q(;\026)870 1114 y Fu(is)h(the)f(dual)g(space)g(of)g(\()p Fr(H)1328 1121 y Fp(p)1348 1114 y Fu(\))1366 1098 y Fo(1)1366 1125 y Fp(q)1401 1114 y Fu(with)g(resp)q(ect)g(to)f Fs(L)1748 1098 y Fo(2)1768 1114 y Fu(\()p Fs(\026)p Fu(\))59 1171 y(\(b)q(ecause)h(of)f(the)g(biorthogonalit)o(y)g(of)g Fs(I)-12 b(P)7 b Fr(\000)p 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b(w)o(e)g(ha)o(v)o(e)g(the)59 1592 y(n)o(uclear)g(triple) 701 1648 y(\()p Fr(N)7 b Fu(\))781 1630 y Fo(1)813 1648 y Fr(\032)13 b Fs(L)892 1630 y Fo(2)911 1648 y Fu(\()p Fs(\026)p Fu(\))g Fr(\032)g Fu(\()p Fr(N)7 b Fu(\))1115 1630 y Fq(\000)p Fo(1)1115 1660 y Fp(\026)1176 1648 y Fs(:)59 1739 y Fu(The)13 b(action)g(of)f(\010)h(=)427 1706 y Fj(P)471 1720 y Fq(1)471 1750 y Fp(n)p Fo(=0)547 1739 y Fs(Q)583 1722 y Fp(\026)583 1750 y(n)606 1739 y Fu(\(\010)657 1722 y Fo(\()p Fp(n)p Fo(\))708 1739 y Fu(\))f Fr(2)h Fu(\()p Fr(N)7 b Fu(\))861 1722 y Fq(\000)p Fo(1)861 1750 y Fp(\026)920 1739 y Fu(on)13 b(a)f(test)h(function)g Fs(')g Fu(=)1369 1706 y Fj(P)1413 1720 y Fq(1)1413 1750 y Fp(n)p Fo(=0)1481 1739 y Fr(h)p Fs(P)1534 1722 y Fp(\026)1528 1750 y(n)1558 1739 y Fs(;)8 b(')1609 1722 y Fo(\()p Fp(n)p Fo(\))1658 1739 y Fr(i)13 b(2)g Fu(\()p Fr(N)7 b Fu(\))1812 1722 y Fo(1)59 1795 y Fu(is)16 b(giv)o(en)f(b)o(y)653 1879 y Fr(h)-8 b(h)p Fu(\010)p Fs(;)8 b(')p Fr(i)-8 b(i)793 1886 y Fp(\026)827 1879 y Fu(=)891 1826 y Fq(1)878 1839 y Fj(X)875 1929 y Fp(n)p Fo(=0)949 1879 y Fs(n)p Fu(!)p Fr(h)p Fu(\010)1040 1861 y Fo(\()p Fp(n)p Fo(\))1091 1879 y Fs(;)8 b(')1142 1861 y Fo(\()p Fp(n)p Fo(\))1191 1879 y Fr(i)15 b Fs(:)130 2069 y Fu(F)l(or)j(a)h(more)f(detailed)i(c)o (haracterization)f(of)g(the)g(singularit)o(y)h(of)e(distributions)j(in) f(\()p Fr(N)7 b Fu(\))1714 2053 y Fq(\000)p Fo(1)1714 2080 y Fp(\026)1779 2069 y Fu(w)o(e)59 2125 y(will)17 b(in)o(tro)q(duce)f(some)f(subspaces)h(in)g(this)f(distribution)i (space.)j(F)l(or)15 b Fs(\014)g Fr(2)e Fu([0)p Fs(;)8 b Fu(1])13 b(w)o(e)i(de\014ne)139 2260 y(\()p Fr(H)195 2267 y Fq(\000)p Fp(p)243 2260 y Fu(\))261 2238 y Fq(\000)p Fp(\014)261 2271 y Fq(\000)p Fp(q)q(;\026)350 2260 y Fu(=)398 2188 y Fj(\()432 2260 y Fu(\010)d Fr(2)h(P)556 2242 y Fq(0)552 2272 y Fp(\026)575 2260 y Fu(\()p Fr(N)637 2242 y Fq(0)649 2260 y Fu(\))679 2199 y Fj(\014)679 2224 y(\014)679 2249 y(\014)679 2274 y(\014)728 2207 y Fq(1)716 2220 y Fj(X)713 2310 y Fp(n)p Fo(=0)779 2260 y Fu(\()p Fs(n)p Fu(!\))855 2242 y Fo(1)p Fq(\000)p Fp(\014)923 2260 y Fu(2)946 2242 y Fq(\000)p Fp(q)q(n)1021 2212 y Fj(\014)1021 2236 y(\014)1021 2261 y(\014)1035 2260 y Fu(\010)1068 2242 y Fo(\()p Fp(n)p Fo(\))1119 2212 y Fj(\014)1119 2236 y(\014)1119 2261 y(\014)1133 2225 y Fo(2)1133 2288 y Fq(\000)p Fp(p)1193 2260 y Fs(<)g Fr(1)i Fu(for)g(\010)d(=)1480 2207 y Fq(1)1468 2220 y Fj(X)1464 2310 y Fp(n)p Fo(=0)1538 2260 y Fs(Q)1574 2242 y Fp(\026)1574 2272 y(n)1598 2260 y Fu(\(\010)1649 2242 y Fo(\()p Fp(n)p Fo(\))1699 2260 y Fu(\))1717 2188 y Fj(\))59 2396 y Fu(and)683 2452 y(\()p Fr(N)7 b Fu(\))763 2434 y Fq(\000)p Fp(\014)763 2464 y(\026)826 2452 y Fu(=)892 2412 y Fj([)861 2504 y Fp(p;q)q Fq(2)p Fp(I)-9 b(N)980 2452 y Fu(\()p Fr(H)1036 2459 y Fq(\000)p Fp(p)1083 2452 y Fu(\))1101 2430 y Fq(\000)p Fp(\014)1101 2463 y Fq(\000)p Fp(q)q(;\026)1194 2452 y Fs(;)59 2577 y Fu(It)15 b(is)h(clear)g(that)e(the)h(singularit)o(y)i (increases)f(with)f(increasing)i Fs(\014)r Fu(:)582 2679 y(\()p Fr(N)7 b Fu(\))662 2661 y Fq(\000)p Fo(0)721 2679 y Fr(\032)13 b Fu(\()p Fr(N)7 b Fu(\))849 2661 y Fq(\000)p Fp(\014)896 2666 y Fi(1)927 2679 y Fr(\032)13 b Fu(\()p Fr(N)7 b Fu(\))1055 2661 y Fq(\000)p Fp(\014)1102 2666 y Fi(2)1134 2679 y Fr(\032)13 b Fu(\()p Fr(N)7 b Fu(\))1262 2661 y Fq(\000)p Fo(1)59 2781 y Fu(if)16 b Fs(\014)127 2788 y Fo(1)159 2781 y Fr(\024)d Fs(\014)233 2788 y Fo(2)252 2781 y Fu(.W)l(e)i(will)i(also)e(consider)h(\()p Fr(N)7 b Fu(\))777 2765 y Fp(\014)777 2793 y(\026)815 2781 y Fu(as)15 b(equipp)q(ed)j(with)d(the)g(natural)h(top)q(ology)l(.)p eop %%Page: 27 27 27 26 bop 59 50 a Fn(6)45 b(DISTRIBUTIONS)1291 b Fu(27)59 199 y Fk(Example)18 b(7)32 b Fm(\(Gener)n(alize)n(d)15 b(R)n(adon{Niko)n(dym)h(derivative\))59 268 y Fu(W)l(e)j(w)o(an)o(t)e (to)h(de\014ne)i(a)e(generalized)j(function)e Fs(\032)934 275 y Fp(\026)957 268 y Fu(\()p Fs(z)r(;)8 b Fr(\001)p Fu(\))17 b Fr(2)h Fu(\()p Fr(N)7 b Fu(\))1195 251 y Fq(\000)p Fo(1)1195 279 y Fp(\026)1260 268 y Fu(,)20 b Fs(z)g Fr(2)f(N)1427 251 y Fq(0)p 1426 280 2 18 v 1417 280 a Fp(C)1465 268 y Fu(with)g(the)g(follo)o(wing)59 324 y(prop)q(ert)o(y)425 395 y Fr(h)-8 b(h)p Fs(\032)477 402 y Fp(\026)499 395 y Fu(\()p Fs(z)r(;)8 b Fr(\001)p Fu(\))p Fs(;)g(')p Fr(i)-8 b(i)671 402 y Fp(\026)704 395 y Fu(=)752 337 y Fj(Z)775 431 y Fq(N)809 422 y Fh(0)830 395 y Fs(')p Fu(\()p Fs(x)10 b Fr(\000)g Fs(z)r Fu(\))j(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))i Fs(;)98 b(')12 b Fr(2)h Fu(\()p Fr(N)7 b Fu(\))1418 376 y Fo(1)1452 395 y Fs(:)59 496 y Fu(That)18 b(means)g(w)o(e)g(ha)o (v)o(e)g(to)g(establish)h(the)g(con)o(tin)o(uit)o(y)g(of)f Fs(\032)1124 503 y Fp(\026)1146 496 y Fu(\()p Fs(z)r(;)8 b Fr(\001)p Fu(\).)28 b(Let)19 b Fs(z)h Fr(2)e(H)1492 503 y Fq(\000)p Fp(p;)p 1553 503 V -3 w(C)1574 496 y Fu(.)g(If)g Fs(p)1676 479 y Fq(0)1706 496 y Fr(\025)g Fs(p)g Fu(is)59 552 y(su\016cien)o(tly)h(large)f(and)h Fs(")e(>)h Fu(0)f(small)i(enough,)g(Corollary)e(25)h(applies)h(i.e.,)g Fr(9)p Fs(q)g Fr(2)e Fs(I)-12 b(N)23 b Fu(and)18 b Fs(C)i(>)e Fu(0)59 609 y(suc)o(h)e(that)345 655 y Fj(\014)345 680 y(\014)345 705 y(\014)345 730 y(\014)359 659 y(Z)382 753 y Fq(N)416 743 y Fh(0)437 716 y Fs(')p Fu(\()p Fs(x)9 b Fr(\000)i Fs(z)r Fu(\)d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))736 655 y Fj(\014)736 680 y(\014)736 705 y(\014)736 730 y(\014)791 716 y Fr(\024)42 b Fs(C)10 b Fr(k)p Fs(')p Fr(k)986 729 y Fp(p)1004 720 y Fh(0)1015 729 y Fp(;q)q(;\026)1083 659 y Fj(Z)1106 753 y Fq(N)1140 743 y Fh(0)1161 716 y Fs(e)1182 697 y Fp(")p Fq(j)p Fp(x)p Fq(\000)p Fp(z)q Fq(j)1282 706 y Fh(\000)p Fg(p)1322 699 y Fh(0)1338 716 y Fu(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))791 829 y Fr(\024)42 b Fs(C)10 b Fr(k)p Fs(')p Fr(k)986 842 y Fp(p)1004 833 y Fh(0)1015 842 y Fp(;q)q(;\026)1083 829 y Fs(e)1104 809 y Fp(")p Fq(j)p Fp(z)q Fq(j)1157 819 y Fh(\000)p Fg(p)1197 812 y Fh(0)1221 771 y Fj(Z)1244 865 y Fq(N)1278 856 y Fh(0)1298 829 y Fs(e)1319 809 y Fp(")p Fq(j)p Fp(x)p Fq(j)1375 819 y Fh(\000)p Fg(p)1415 812 y Fh(0)1431 829 y Fu(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))59 950 y(If)h Fs(")g Fu(is)h(c)o(hosen)f(su\016cien)o(tly)i(small)e(the)g(last)g(in)o (tegral)h(exists.)19 b(Th)o(us)11 b(w)o(e)f(ha)o(v)o(e)h(in)h(fact)e Fs(\032)p Fu(\()p Fs(z)r(;)e Fr(\001)p Fu(\))j Fr(2)i Fu(\()p Fr(N)7 b Fu(\))1772 933 y Fq(\000)p Fo(1)1772 961 y Fp(\026)1818 950 y Fu(.)59 1019 y(It)13 b(is)g(clear)g(that)f (whenev)o(er)h(the)g(Radon{Nik)o(o)q(dym)g(deriv)m(ativ)o(e)1180 996 y Fo(d)p Fp(\026)p Fo(\()p Fp(x)p Fo(+)p Fp(\030)q Fo(\))p 1180 1008 133 2 v 1202 1035 a(d)p Fp(\026)p Fo(\()p Fp(x)p Fo(\))1330 1019 y Fu(exists)g(\(e.g.,)f Fs(\030)j Fr(2)d(N)20 b Fu(in)14 b(case)59 1081 y Fs(\026)h Fu(is)g Fr(N)7 b Fu(-quasi-in)o(v)m(arian)o(t\))16 b(it)f(coincides)i(with)e Fs(\032)895 1088 y Fp(\026)918 1081 y Fu(\()p Fs(\030)r(;)8 b Fr(\001)p Fu(\))13 b(de\014ned)j(ab)q(o)o(v)o(e.)j(W)l(e)c(will)h(no) o(w)f(sho)o(w)f(that)g(in)59 1138 y(\()p Fr(N)7 b Fu(\))139 1121 y Fq(\000)p Fo(1)139 1149 y Fp(\026)201 1138 y Fu(w)o(e)15 b(ha)o(v)o(e)f(the)i(canonical)g(expansion)634 1273 y Fs(\032)658 1280 y Fp(\026)681 1273 y Fu(\()p Fs(z)r(;)8 b Fr(\001)p Fu(\))j(=)849 1220 y Fq(1)836 1232 y Fj(X)833 1323 y Fp(n)p Fo(=0)921 1242 y Fu(1)p 912 1262 40 2 v 912 1304 a Fs(n)p Fu(!)957 1273 y(\()p Fr(\000)p Fu(1\))1051 1254 y Fp(n)1074 1273 y Fs(Q)1110 1254 y Fp(\026)1110 1284 y(n)1133 1273 y Fu(\()p Fs(z)1174 1254 y Fq(\012)p Fp(n)1225 1273 y Fu(\))p Fs(:)59 1411 y Fu(It)i(is)h(easy)f(to)g(see)h (that)e(the)i(r.h.s.)k(de\014nes)c(an)g(elemen)o(t)g(in)g(\()p Fr(N)7 b Fu(\))1171 1395 y Fq(\000)p Fo(1)1171 1423 y Fp(\026)1217 1411 y Fu(.)20 b(Since)14 b(b)q(oth)g(sides)g(are)f(in)h (\()p Fr(N)7 b Fu(\))1784 1395 y Fq(\000)p Fo(1)1784 1423 y Fp(\026)59 1481 y Fu(it)16 b(is)f(su\016cien)o(t)h(to)f(compare) g(their)g(action)h(on)f(a)g(total)f(set)h(from)g(\()p Fr(N)7 b Fu(\))1292 1465 y Fo(1)1311 1481 y Fu(.)20 b(F)l(or)14 b Fs(')1455 1465 y Fo(\()p Fp(n)p Fo(\))1518 1481 y Fr(2)f(N)1610 1456 y Fo(^)1605 1462 y Fq(\012)q Fp(n)p 1605 1495 2 18 v 1595 1495 a(C)1671 1481 y Fu(w)o(e)i(ha)o(v)o(e)188 1552 y Fj(D)-15 b(D)224 1599 y Fs(\032)248 1606 y Fp(\026)270 1599 y Fu(\()p Fs(z)r(;)8 b Fr(\001)p Fu(\))p Fs(;)g Fr(h)p Fs(P)437 1580 y Fp(\026)431 1610 y(n)458 1599 y Fs(;)g(')509 1580 y Fo(\()p Fp(n)p Fo(\))559 1599 y Fr(i)577 1552 y Fj(E)-15 b(E)613 1627 y Fp(\026)677 1599 y Fu(=)754 1541 y Fj(Z)777 1635 y Fq(N)811 1626 y Fh(0)824 1599 y Fr(h)p Fs(P)877 1580 y Fp(\026)871 1610 y(n)901 1599 y Fu(\()p Fs(x)10 b Fr(\000)g Fs(z)r Fu(\))p Fs(;)e(')1092 1580 y Fo(\()p Fp(n)p Fo(\))1142 1599 y Fr(i)k Fu(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))677 1726 y(=)770 1673 y Fq(1)758 1685 y Fj(X)754 1776 y Fp(n)p Fo(=0)828 1654 y Fj( )861 1695 y Fs(n)862 1757 y(k)888 1654 y Fj(!)921 1726 y Fu(\()p Fr(\000)p Fu(1\))1015 1707 y Fp(n)p Fq(\000)p Fp(k)1093 1668 y Fj(Z)1116 1763 y Fq(N)1150 1753 y Fh(0)1163 1726 y Fr(h)p Fs(P)1216 1704 y Fp(\026)1210 1740 y(k)1239 1726 y Fu(\()p Fs(x)p Fu(\))1307 1719 y(^)1301 1726 y Fr(\012)p Fs(z)1359 1707 y Fq(\012)p Fp(n)p Fq(\000)p Fp(k)1457 1726 y Fs(;)c(')1508 1707 y Fo(\()p Fp(n)p Fo(\))1557 1726 y Fr(i)13 b Fu(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))677 1840 y(=)42 b(\()p Fr(\000)p Fu(1\))848 1822 y Fp(n)871 1840 y Fr(h)p Fs(z)912 1822 y Fq(\012)p Fp(n)963 1840 y Fs(;)8 b(')1014 1822 y Fo(\()p Fp(n)p Fo(\))1064 1840 y Fr(i)677 1942 y Fu(=)754 1870 y Fj(*)-14 b(*)820 1889 y Fq(1)808 1901 y Fj(X)806 1994 y Fp(k)q Fo(=0)890 1911 y Fu(1)p 883 1931 38 2 v 883 1973 a Fs(k)q Fu(!)926 1942 y(\()p Fr(\000)p Fu(1\))1020 1923 y Fp(k)1040 1942 y Fs(Q)1076 1920 y Fp(\026)1076 1956 y(k)1100 1942 y Fu(\()p Fs(z)1141 1923 y Fq(\012)p Fp(k)1189 1942 y Fu(\))p Fs(;)8 b Fr(h)p Fs(P)1281 1923 y Fp(\026)1275 1953 y(n)1304 1942 y Fs(;)g(')1355 1923 y Fo(\()p Fp(n)p Fo(\))1404 1942 y Fr(i)1422 1870 y Fj(+)-15 b(+)1474 2000 y Fp(\026)1520 1942 y Fs(;)59 2087 y Fu(where)15 b(w)o(e)f(ha)o(v)o(e)g(used)h(\(13\),)d(\(14\))h(and)i(the)f (biorthogonalit)o(y)h(of)f Fs(I)-12 b(P)7 b Fu(-)14 b(and)p 1375 2087 3 25 v 18 w Fs(Q)p Fu(-systems.)19 b(This)c(had)f(to)59 2143 y(b)q(e)h(sho)o(wn.)k(In)d(other)e(w)o(ords,)f(w)o(e)h(ha)o(v)o(e) g(pro)o(v)o(en)g(that)g Fs(\032)1031 2150 y Fp(\026)1054 2143 y Fu(\()p Fr(\000)p Fs(z)r(;)8 b Fr(\001)p Fu(\))13 b(is)i(the)g(generating)f(function)i(of)e(the)p 74 2200 V 62 2200 a Fs(Q)p Fu(-functions)668 2284 y Fs(\032)692 2291 y Fp(\026)714 2284 y Fu(\()p Fr(\000)p Fs(z)r(;)8 b Fr(\001)p Fu(\))k(=)917 2231 y Fq(1)905 2244 y Fj(X)902 2334 y Fp(n)p Fo(=0)989 2253 y Fu(1)p 981 2274 40 2 v 981 2315 a Fs(n)p Fu(!)1026 2284 y Fs(Q)1062 2265 y Fp(\026)1062 2295 y(n)1085 2284 y Fu(\()p Fs(z)1126 2265 y Fq(\012)p Fp(n)1177 2284 y Fu(\))j Fs(:)527 b Fu(\(27\))59 2399 y(Let)16 b(use)h(\014nally)g(remark)f(that)f(the)h(ab)q(o)o(v)o(e)g (expansion)h(allo)o(ws)f(for)g(more)f(detailed)j(estimates.)k(It)16 b(is)59 2456 y(easy)f(to)g(see)g(that)g Fs(\032)411 2463 y Fp(\026)446 2456 y Fr(2)e Fu(\()p Fr(N)7 b Fu(\))569 2439 y Fq(\000)p Fo(0)569 2467 y Fp(\026)615 2456 y Fu(.)59 2560 y Fk(Example)18 b(8)f Fm(\(Delta)f(distribution\))59 2628 y Fu(F)l(or)f Fs(z)f Fr(2)f(N)263 2612 y Fq(0)p 262 2641 2 18 v 253 2641 a Fp(C)298 2628 y Fu(w)o(e)i(de\014ne)i(a)e (distribution)h(b)o(y)g(the)f(follo)o(wing)p 1131 2628 3 25 v 19 w Fs(Q)p Fu(-decomp)q(osition:)727 2758 y Fs(\016)747 2765 y Fp(z)779 2758 y Fu(=)843 2706 y Fq(1)830 2718 y Fj(X)827 2809 y Fp(n)p Fo(=0)915 2728 y Fu(1)p 906 2748 40 2 v 906 2790 a Fs(n)p Fu(!)951 2758 y Fs(Q)987 2740 y Fp(\026)987 2770 y(n)1011 2758 y Fu(\()p Fs(P)1064 2740 y Fp(\026)1058 2770 y(n)1087 2758 y Fu(\()p Fs(z)r Fu(\)\))p eop %%Page: 28 28 28 27 bop 59 50 a Fn(7)45 b(INTEGRAL)16 b(TRANSF)o(ORMA)l(TIONS)926 b Fu(28)59 199 y(If)18 b Fs(p)e Fr(2)h Fs(I)-12 b(N)22 b Fu(is)c(large)f(enough)h(and)g Fs(")e(>)h Fu(0)g(su\016cien)o(tly)i (small)f(there)f(exists)h Fs(C)1430 206 y Fp(p;")1492 199 y Fs(>)e Fu(0)h(according)h(to)59 256 y(\(15\))c(suc)o(h)i(that)359 388 y Fr(k)p Fs(\016)402 395 y Fp(z)422 388 y Fr(k)445 367 y Fo(2)445 402 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)q(;\026)619 388 y Fu(=)711 335 y Fq(1)699 348 y Fj(X)696 438 y Fp(n)p Fo(=0)762 388 y Fu(\()p Fs(n)p Fu(!\))838 369 y Fq(\000)p Fo(2)885 388 y Fu(2)908 369 y Fq(\000)p Fp(nq)983 388 y Fr(j)o Fs(P)1030 369 y Fp(\026)1024 399 y(n)1054 388 y Fu(\()p Fs(z)r Fu(\))p Fr(j)1125 367 y Fo(2)1125 402 y Fq(\000)p Fp(p)619 530 y Fr(\024)42 b Fs(C)732 512 y Fo(2)729 542 y Fp(p;")782 530 y Fs(e)803 512 y Fo(2)p Fp(")p Fq(j)p Fp(z)q Fq(j)874 516 y Fh(\000)p Fg(p)941 478 y Fq(1)929 490 y Fj(X)926 581 y Fp(n)p Fo(=0)1000 530 y Fu(2)1023 512 y Fq(\000)p Fp(nq)1090 530 y Fs(")1111 512 y Fq(\000)p Fo(2)p Fp(n)1195 530 y Fs(;)99 b(z)14 b Fr(2)f(H)1423 537 y Fq(\000)p Fp(p;)p 1484 537 2 18 v -3 w(C)1518 530 y Fs(;)59 672 y Fu(whic)o(h)j(is)g(\014nite)g(for)f (su\016cien)o(tly)h(large)f Fs(q)g Fr(2)e Fs(I)-12 b(N)5 b Fu(.)19 b(Th)o(us)d Fs(\016)1055 679 y Fp(z)1087 672 y Fr(2)d Fu(\()p Fr(N)7 b Fu(\))1210 656 y Fq(\000)p Fo(1)1210 684 y Fp(\026)1257 672 y Fu(.)130 736 y(F)l(or)14 b Fs(')f Fu(=)302 704 y Fj(P)345 717 y Fq(1)345 747 y Fp(n)p Fo(=0)414 736 y Fr(h)p Fs(P)467 719 y Fp(\026)461 747 y(n)490 736 y Fs(;)8 b(')541 719 y Fo(\()p Fp(n)p Fo(\))591 736 y Fr(i)13 b(2)f Fu(\()p Fr(N)7 b Fu(\))744 719 y Fo(1)779 736 y Fu(the)15 b(action)g(of)g Fs(\016)1065 743 y Fp(z)1100 736 y Fu(is)h(giv)o(en)g(b)o(y)596 868 y Fr(h)-8 b(h)p Fs(\016)644 875 y Fp(z)664 868 y Fs(;)8 b(')p Fr(i)-8 b(i)743 875 y Fp(\026)777 868 y Fu(=)841 815 y Fq(1)828 828 y Fj(X)825 918 y Fp(n)p Fo(=0)892 868 y Fr(h)p Fs(P)945 850 y Fp(\026)939 880 y(n)968 868 y Fu(\()p Fs(z)r Fu(\))p Fs(;)8 b(')1078 850 y Fo(\()p Fp(n)p Fo(\))1128 868 y Fr(i)k Fu(=)h Fs(')p Fu(\()p Fs(z)r Fu(\))59 1007 y(b)q(ecause)f(of)e(\(22\).)17 b(This)12 b(means)f(that)f Fs(\016)731 1014 y Fp(z)762 1007 y Fu(\(in)h (particular)g(for)g Fs(z)i Fu(real\))d(pla)o(ys)h(the)g(role)g(of)g(a)f (\\)p Fs(\016)r Fu(-function")59 1063 y(\(ev)m(aluation)16 b(map\))f(in)h(the)f(calculus)i(w)o(e)e(discuss.)59 1206 y Ft(7)67 b(In)n(tegral)23 b(transformations)59 1308 y Fu(W)l(e)h(will)h(\014rst)e(in)o(tro)q(duce)i(the)f(Laplace)g (transform)e(of)i(a)f(function)h Fs(')j Fr(2)g Fs(L)1470 1291 y Fo(2)1489 1308 y Fu(\()p Fs(\026)p Fu(\).)45 b(The)24 b(global)59 1364 y(assumption)h Fs(\026)k Fr(2)g(M)479 1371 y Fp(a)499 1364 y Fu(\()p Fr(N)561 1348 y Fq(0)573 1364 y Fu(\))24 b(guaran)o(tees)g(the)h(existence)h(of)e Fs(p)1226 1348 y Fq(0)1226 1375 y Fp(\026)1278 1364 y Fr(2)29 b Fs(I)-12 b(N)29 b Fu(,)e Fs(")1472 1371 y Fp(\026)1524 1364 y Fs(>)i Fu(0)24 b(suc)o(h)h(that)59 1386 y Fj(R)79 1434 y Fq(N)113 1425 y Fh(0)133 1421 y Fu(exp)q(\()p Fs(")242 1428 y Fp(\026)265 1421 y Fr(j)p Fs(x)p Fr(j)317 1428 y Fq(\000)p Fp(p)362 1419 y Fh(0)362 1437 y Fg(\026)384 1421 y Fu(\))8 b(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))15 b Fs(<)h Fr(1)h Fu(b)o(y)g(Lemma)g(9.)25 b(Th)o(us)17 b(exp\()p Fr(h)p Fs(x;)8 b(\022)q Fr(i)p Fu(\))14 b Fr(2)i Fs(L)1360 1404 y Fo(2)1380 1421 y Fu(\()p Fs(\026)p Fu(\))h(if)g(2)p Fr(j)p Fs(\022)q Fr(j)1574 1428 y Fp(p)1592 1419 y Fh(0)1592 1437 y Fg(\026)1630 1421 y Fr(\024)f Fs(")1702 1428 y Fp(\026)1742 1421 y Fs(;)8 b(\022)16 b Fr(2)59 1477 y(H)97 1485 y Fp(p)115 1475 y Fh(0)115 1493 y Fg(\026)136 1485 y Fp(;)p 152 1485 V -3 w(C)172 1477 y Fu(.)k(Then)c(b)o(y)f(Cauc)o(h)o (y{Sc)o(h)o(w)o(arz)f(inequalit)o(y)j(the)e(Laplace)i(transform)d (de\014ned)i(b)o(y)602 1605 y Fs(L)633 1612 y Fp(\026)656 1605 y Fs(')p Fu(\()p Fs(\022)q Fu(\))c(:=)817 1548 y Fj(Z)840 1642 y Fq(N)874 1633 y Fh(0)894 1605 y Fs(')p Fu(\()p Fs(x)p Fu(\))c(exp)p Fr(h)p Fs(x;)g(\022)q Fr(i)g Fu(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))59 1730 y(is)13 b(w)o(ell)h(de\014ned)g(for)f Fs(')f Fr(2)h Fs(L)530 1713 y Fo(2)550 1730 y Fu(\()p Fs(\026)p Fu(\))f Fs(;)c(\022)14 b Fr(2)f(H)762 1737 y Fp(p)780 1728 y Fh(0)780 1746 y Fg(\026)800 1737 y Fp(;)p 816 1737 V -3 w(C)850 1730 y Fu(with)g(2)p Fr(j)p Fs(\022)q Fr(j)1022 1737 y Fp(p)1040 1728 y Fh(0)1040 1746 y Fg(\026)1075 1730 y Fr(\024)g Fs(")1144 1737 y Fp(\026)1167 1730 y Fu(.)19 b(No)o(w)12 b(w)o(e)h(are)g(in)o(terested)g(to)f(extend)59 1793 y(this)k(in)o (tegral)f(transform)f(from)h Fs(L)661 1776 y Fo(2)680 1793 y Fu(\()p Fs(\026)p Fu(\))g(to)g(the)g(space)g(of)g(distributions) i(\()p Fr(N)7 b Fu(\))1411 1776 y Fq(\000)p Fo(1)1411 1804 y Fp(\026)1457 1793 y Fu(.)130 1849 y(Since)16 b(our)e (construction)g(of)g(test)g(function)h(and)g(distribution)h(spaces)f (is)f(closely)i(related)f(to)f Fs(I)-12 b(P)7 b Fu(-)59 1905 y(and)p 163 1905 3 25 v 19 w Fs(Q)o Fu(-systems)15 b(it)h(is)f(useful)i(to)d(in)o(tro)q(duce)i(the)g(so)f(called)h Fs(S)1140 1912 y Fp(\026)1164 1905 y Fu(-transform)749 2033 y Fs(S)777 2040 y Fp(\026)800 2033 y Fs(')p Fu(\()p Fs(\022)q Fu(\))c(:=)966 2003 y Fs(L)997 2010 y Fp(\026)1020 2003 y Fs(')p Fu(\()p Fs(\022)q Fu(\))p 966 2023 142 2 v 990 2065 a Fs(l)1004 2072 y Fp(\026)1026 2065 y Fu(\()p Fs(\022)q Fu(\))1128 2033 y Fs(:)59 2169 y Fu(Since)17 b Fs(e)199 2176 y Fp(\026)222 2169 y Fu(\()p Fs(\022)q Fu(;)8 b Fs(x)p Fu(\))k(=)h Fs(e)408 2153 y Fq(h)p Fp(x;\022)q Fq(i)485 2169 y Fs(=l)522 2176 y Fp(\026)544 2169 y Fu(\()p Fs(\022)q Fu(\))i(w)o(e)g(ma)o(y)g(also)g(write)608 2290 y Fs(S)636 2297 y Fp(\026)659 2290 y Fs(')p Fu(\()p Fs(\022)q Fu(\))d(=)807 2232 y Fj(Z)831 2326 y Fq(N)865 2317 y Fh(0)885 2290 y Fs(')p Fu(\()p Fs(x)p Fu(\))c Fs(e)1006 2297 y Fp(\026)1028 2290 y Fu(\()p Fs(\022)q Fu(;)g Fs(x)p Fu(\))g(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))14 b Fs(:)59 2414 y Fu(The)f Fs(\026)p Fu(-exp)q(onen)o(tial)j Fs(e)456 2421 y Fp(\026)479 2414 y Fu(\()p Fs(\022)q(;)8 b Fr(\001)p Fu(\))k(is)i(not)f(a)f(test)h(function)h(in)g(\()p Fr(N)7 b Fu(\))1135 2398 y Fo(1)1154 2414 y Fu(,)13 b(see)h(Example)f(6)g(.)19 b(So)14 b(the)f(de\014nition)59 2471 y(of)k(the)g Fs(S)221 2478 y Fp(\026)244 2471 y Fu(-transform)e(of)i(a)g(distribution)h(\010) d Fr(2)h Fu(\()p Fr(N)7 b Fu(\))986 2454 y Fq(\000)p Fo(1)986 2482 y Fp(\026)1050 2471 y Fu(m)o(ust)16 b(b)q(e)i(more)e (careful.)26 b(Ev)o(ery)16 b(suc)o(h)h(\010)g(is)59 2533 y(of)e(\014nite)i(order)e(i.e.,)h Fr(9)p Fs(p;)8 b(q)14 b Fr(2)g Fs(I)-12 b(N)21 b Fu(suc)o(h)16 b(that)e(\010)g Fr(2)f Fu(\()p Fr(H)994 2540 y Fq(\000)p Fp(p)1042 2533 y Fu(\))1060 2514 y Fq(\000)p Fo(1)1060 2544 y Fq(\000)p Fp(q)q(;\026:)1162 2533 y Fu(As)j(sho)o(wn)f(in)i(Example)f(6)f Fs(e)1670 2540 y Fp(\026)1694 2533 y Fu(\()p Fs(\022)q(;)8 b Fr(\001)p Fu(\))14 b(is)59 2589 y(in)j(the)f(corresp)q(onding)g(dual) h(space)f(\()p Fr(H)764 2596 y Fp(p)784 2589 y Fu(\))802 2573 y Fo(1)802 2601 y Fp(q)q(;\026)867 2589 y Fu(if)h Fs(\022)e Fr(2)f(H)1028 2596 y Fp(p;)p 1061 2596 2 18 v -3 w(C)1098 2589 y Fu(is)i(suc)o(h)g(that)f(2)1369 2573 y Fp(q)1388 2589 y Fr(j)p Fs(\022)q Fr(j)1436 2573 y Fo(2)1436 2601 y Fp(p)1470 2589 y Fs(<)e Fu(1.)22 b(Then)16 b(w)o(e)g(can)59 2646 y(de\014ne)g(a)f(consisten)o(t)h(extension)g(of)e Fs(S)720 2653 y Fp(\026)744 2646 y Fu(-transform.)697 2748 y Fs(S)725 2755 y Fp(\026)749 2748 y Fu(\010\()p Fs(\022)q Fu(\))e(:=)h Fr(h)-8 b(h)p Fu(\010)p Fs(;)19 b(e)1027 2755 y Fp(\026)1050 2748 y Fu(\()p Fs(\022)q(;)8 b Fr(\001)p Fu(\))p Fr(i)-8 b(i)1170 2755 y Fp(\026)p eop %%Page: 29 29 29 28 bop 59 50 a Fn(8)45 b(CHARA)o(CTERIZA)l(TION)17 b(THEOREMS)888 b Fu(29)59 199 y(if)16 b Fs(\022)g Fu(is)g(c)o(hosen)g (in)g(the)f(ab)q(o)o(v)o(e)g(w)o(a)o(y)l(.)k(The)c(biorthogonalit)o(y)h (of)e Fs(I)-12 b(P)7 b Fu(-)16 b(and)p 1337 199 3 25 v 18 w Fs(Q)p Fu(-system)f(implies)690 329 y Fs(S)718 336 y Fp(\026)741 329 y Fu(\010\()p Fs(\022)q Fu(\))e(=)908 276 y Fq(1)896 289 y Fj(X)893 379 y Fp(n)p Fo(=0)959 329 y Fr(h)p Fu(\010)1010 310 y Fo(\()p Fp(n)p Fo(\))1061 329 y Fs(;)8 b(\022)1104 310 y Fq(\012)p Fp(n)1154 329 y Fr(i)15 b Fs(:)59 465 y Fu(It)20 b(is)h(easy)f(to)g(see)h(that)e(the) i(series)g(con)o(v)o(erges)f(uniformly)h(and)g(absolutely)g(on)f(an)o (y)g(closed)h(ball)59 480 y Fj(n)92 527 y Fs(\022)14 b Fr(2)f(H)208 534 y Fp(p;)p 242 534 2 18 v -3 w(C)263 527 y Fr(j)19 b(j)p Fs(\022)q Fr(j)343 510 y Fo(2)343 538 y Fp(p)376 527 y Fr(\024)13 b Fs(r)o(;)21 b(r)14 b(<)f Fu(2)584 510 y Fq(\000)p Fp(q)630 480 y Fj(o)658 527 y Fu(,)19 b(see)h(the)f(pro)q(of)f(of)h(Theorem)g(34.)31 b(Th)o(us)19 b Fs(S)1461 534 y Fp(\026)1484 527 y Fu(\010)g(is)h (holomorphic)59 591 y(a)f(neigh)o(b)q(orho)q(o)q(d)h(of)f(zero,)g (i.e.,)h Fs(S)678 598 y Fp(\026)701 591 y Fu(\010)f Fr(2)h Fu(Hol)872 598 y Fo(0)892 591 y Fu(\()p Fr(N)p 953 598 V 944 598 a Fp(C)974 591 y Fu(\).)31 b(In)19 b(the)h(next)f(section)g (w)o(e)g(will)i(discuss)f(this)59 647 y(relation)c(to)e(the)i(theory)e (of)h(holomorphic)h(functions)g(in)g(more)f(detail.)130 704 y(The)20 b(third)g(in)o(tegral)g(transform)f(w)o(e)g(are)h(going)g (to)f(in)o(tro)q(duce)i(is)f(more)g(appropriate)f(for)h(the)59 760 y(test)c(function)g(space)h(\()p Fr(N)7 b Fu(\))530 744 y Fo(1)549 760 y Fu(.)22 b(W)l(e)16 b(in)o(tro)q(duce)h(the)f(con)o (v)o(olution)h(of)e(a)h(function)h Fs(')d Fr(2)g Fu(\()p Fr(N)7 b Fu(\))1628 744 y Fo(1)1663 760 y Fu(with)17 b(the)59 817 y(measure)e Fs(\026)h Fu(b)o(y)544 887 y Fs(C)577 894 y Fp(\026)600 887 y Fs(')p Fu(\()p Fs(y)r Fu(\))c(:=)762 830 y Fj(Z)785 924 y Fq(N)819 915 y Fh(0)840 887 y Fs(')p Fu(\()p Fs(x)e Fu(+)g Fs(y)r Fu(\))e(d)p Fs(\026)p Fu(\()p Fs(x)p Fu(\))p Fs(;)52 b(y)14 b Fr(2)f(N)1321 868 y Fq(0)1333 887 y Fs(:)59 989 y Fu(F)l(rom)18 b(Example)h(7)f(the)g (existence)i(of)e(a)g(generalized)i(Radon{Nik)o(o)q(dym)f(deriv)m(ativ) o(e)g Fs(\032)1613 996 y Fp(\026)1636 989 y Fu(\()p Fs(z)r(;)8 b Fr(\001)p Fu(\),)17 b Fs(z)j Fr(2)59 1046 y(N)103 1029 y Fq(0)p 102 1059 V 93 1059 a Fp(C)145 1046 y Fu(in)j(\()p Fr(N)7 b Fu(\))285 1029 y Fq(\000)p Fo(1)285 1057 y Fp(\026)353 1046 y Fu(is)22 b(guaran)o(teed.)40 b(So)21 b(for)h(an)o(y)f Fs(')j Fr(2)g Fu(\()p Fr(N)7 b Fu(\))1103 1029 y Fo(1)1122 1046 y Fu(,)23 b Fs(z)j Fr(2)e(N)1303 1029 y Fq(0)p 1302 1059 V 1293 1059 a Fp(C)1345 1046 y Fu(the)e(con)o(v)o(olution)g(has)g (the)59 1102 y(represen)o(tation)672 1159 y Fs(C)705 1166 y Fp(\026)728 1159 y Fs(')p Fu(\()p Fs(z)r Fu(\))12 b(=)h Fr(h)-8 b(h)p Fs(\032)929 1166 y Fp(\026)952 1159 y Fu(\()p Fr(\000)p Fs(z)r(;)8 b Fr(\001)p Fu(\))p Fs(;)18 b(')p Fr(i)-8 b(i)1169 1166 y Fp(\026)1205 1159 y Fs(:)59 1242 y Fu(If)16 b Fs(')c Fr(2)h Fu(\()p Fr(N)7 b Fu(\))270 1226 y Fo(1)304 1242 y Fu(has)15 b(the)g(canonical)i(represen)o(tation) 763 1375 y Fs(')12 b Fu(=)869 1322 y Fq(1)856 1334 y Fj(X)853 1425 y Fp(n)p Fo(=0)919 1375 y Fr(h)p Fs(P)972 1356 y Fp(\026)966 1386 y(n)996 1375 y Fs(;)20 b(')1059 1356 y Fo(\()p Fp(n)p Fo(\))1109 1375 y Fr(i)59 1513 y Fu(w)o(e)15 b(ha)o(v)o(e)g(b)o(y)g(equation)h(\(27\))690 1600 y Fs(C)723 1607 y Fp(\026)746 1600 y Fs(')p Fu(\()p Fs(z)r Fu(\))c(=)911 1547 y Fq(1)898 1559 y Fj(X)895 1650 y Fp(n)p Fo(=0)961 1600 y Fr(h)p Fs(z)1002 1581 y Fq(\012)p Fp(n)1053 1600 y Fs(;)c(')1104 1581 y Fo(\()p Fp(n)p Fo(\))1154 1600 y Fr(i)15 b Fs(:)130 1717 y Fu(In)21 b(Gaussian)g(Analysis)h Fs(C)614 1724 y Fp(\026)637 1717 y Fu(-)f(and)g Fs(S)795 1724 y Fp(\026)818 1717 y Fu(-transform)f (coincide.)39 b(It)21 b(is)h(a)e(t)o(ypical)i(non-Gaussian)59 1773 y(e\013ect)15 b(that)f(these)i(t)o(w)o(o)e(transformations)g (di\013er)h(from)g(eac)o(h)g(other.)59 1916 y Ft(8)67 b(Characterization)23 b(theorems)59 2018 y Fu(Gaussian)c(Analysis)i (has)e(sho)o(wn)g(that)f(for)h(applications)i(it)e(is)h(v)o(ery)f (useful)h(to)f(c)o(haracterize)g(test)59 2074 y(and)d(distribution)i (spaces)e(b)o(y)g(the)g(in)o(tegral)h(transforms)d(in)o(tro)q(duced)j (in)g(the)f(previous)h(section.)23 b(In)59 2131 y(the)e(non-Gaussian)g (setting)f(\014rst)g(results)h(in)g(this)g(direction)h(ha)o(v)o(e)e(b)q (een)i(obtained)f(b)o(y)f([AKS93,)59 2187 y(ADKS94].)130 2293 y(W)l(e)15 b(will)i(start)d(to)g(c)o(haracterize)i(the)f(space)h (\()p Fr(N)7 b Fu(\))990 2277 y Fo(1)1024 2293 y Fu(in)16 b(terms)e(of)h(the)g(con)o(v)o(olution)h Fs(C)1610 2300 y Fp(\026)1633 2293 y Fu(.)59 2400 y Fk(Theorem)i(32)23 b Fm(The)16 b(c)n(onvolution)f Fs(C)721 2407 y Fp(\026)760 2400 y 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y(generalized)h(k)o(ernels) 446 1191 y Fj(n)474 1238 y Fs(')504 1222 y Fo(\()p Fp(n)p Fo(\))567 1238 y Fr(2)c(N)654 1222 y Fq(0)p 653 1251 V 644 1251 a Fp(C)686 1238 y Fr(j)g Fs(n)g Fr(2)g Fs(I)-12 b(N)844 1245 y Fo(0)864 1191 y Fj(o)907 1238 y Fu(suc)o(h)16 b(that)715 1382 y Fs(F)6 b Fu(\()p Fs(z)r Fu(\))13 b(=)886 1329 y Fq(1)873 1342 y Fj(X)870 1432 y Fp(n)p Fo(=0)936 1382 y Fr(h)p Fs(z)977 1363 y Fq(\012)p Fp(n)1028 1382 y Fs(;)8 b(')1079 1363 y Fo(\()p Fp(n)p Fo(\))1129 1382 y Fr(i)15 b Fs(:)59 1515 y Fu(Moreo)o(v)o(er)f(for)g(all)j Fs(p;)8 b(q)13 b Fr(2)g Fs(I)-12 b(N)562 1522 y Fo(0)653 1642 y Fr(j)k(j)g(j)p Fs(F)6 b Fr(j)-8 b(j)g(j)734 1621 y Fo(2)734 1655 y Fp(p;q)q(;)p Fo(1)821 1642 y Fu(=)884 1589 y Fq(1)872 1601 y Fj(X)869 1692 y Fp(n)p Fo(=0)935 1642 y Fu(\()p Fs(n)p Fu(!\))1011 1623 y Fo(2)1038 1642 y Fu(2)1061 1623 y Fp(nq)1109 1593 y Fj(\014)1109 1618 y(\014)1109 1643 y(\014)1122 1642 y Fs(')1152 1623 y Fo(\()p Fp(n)p Fo(\))1203 1593 y Fj(\014)1203 1618 y(\014)1203 1643 y(\014)1217 1606 y Fo(2)1217 1670 y Fp(p)59 1774 y Fu(is)16 b(\014nite.)21 b(Cho)q(osing)769 1858 y Fs(')12 b Fu(=)875 1806 y Fq(1)863 1818 y Fj(X)859 1909 y Fp(n)p Fo(=0)926 1858 y Fr(h)p Fs(P)979 1840 y Fp(\026)973 1870 y(n)1002 1858 y Fs(;)c(')1053 1840 y Fo(\()p Fp(n)p Fo(\))1103 1858 y Fr(i)59 1980 y Fu(w)o(e)13 b(ha)o(v)o(e)h Fr(k)o Fs(')p Fr(k)303 1994 y Fp(p;q)q(;\026)393 1980 y Fu(=)f Fr(j)-8 b(j)g(j)p Fs(F)6 b Fr(j)-8 b(j)g(j)522 1994 y Fp(p;q)q(;)p Fo(1)596 1980 y Fu(.)19 b(Th)o(us)14 b Fs(')e Fr(2)h Fu(\()p Fr(N)7 b Fu(\))907 1964 y Fo(1)926 1980 y Fu(.)20 b(Since)15 b Fs(C)1109 1987 y Fp(\026)1131 1980 y Fs(')e Fu(=)g Fs(F)20 b Fu(w)o(e)13 b(ha)o(v)o(e)g(sho)o(wn)h (the)f(existence)59 2037 y(and)i(con)o(tin)o(uit)o(y)h(of)f(the)g(in)o (v)o(erse)h(of)f Fs(C)728 2044 y Fp(\026)750 2037 y Fu(.)1034 b Ff(2)130 2141 y Fu(T)l(o)15 b(illustrate)h(the)g(ab)q(o)o(v)o(e)e (theorem)h(in)i(terms)d(of)h(the)h(natural)f(top)q(ology)g(on)g Fr(E)1521 2124 y Fo(1)1517 2153 y(min)1578 2141 y Fu(\()p Fr(N)1640 2124 y Fq(0)p 1639 2154 V 1630 2154 a Fp(C)1660 2141 y Fu(\))g(w)o(e)g(will)59 2197 y(reform)o(ulate)21 b(the)h(ab)q(o)o(v)o(e)f(theorem)g(and)h(add)g(some)f(useful)i (estimates)e(whic)o(h)i(relate)f(gro)o(wth)e(in)59 2254 y Fr(E)87 2237 y Fo(1)83 2266 y(min)144 2254 y Fu(\()p Fr(N)206 2237 y Fq(0)p 205 2267 V 196 2267 a Fp(C)226 2254 y Fu(\))14 b(to)h(norms)g(on)g(\()p Fr(N)7 b Fu(\))594 2237 y Fo(1)613 2254 y Fu(.)59 2357 y Fk(Cor)o(ollar)m(y)18 b(33)59 2413 y Fm(1\))e(L)n(et)g Fs(')c Fr(2)h Fu(\()p Fr(N)7 b Fu(\))361 2397 y Fo(1)396 2413 y Fm(then)16 b(for)h(al)r(l)f Fs(p;)8 b(l)k Fr(2)h Fs(I)-12 b(N)798 2420 y Fo(0)834 2413 y Fm(and)16 b Fs(z)f Fr(2)e(H)1039 2420 y Fq(\000)p Fp(p;)p 1100 2420 V -3 w(C)1137 2413 y Fm(the)k(fol)r(lowing)f(estimate)g(holds)612 2512 y Fr(j)p Fs(C)658 2519 y Fp(\026)681 2512 y Fs(')p Fu(\()p Fs(z)r Fu(\))p Fr(j)11 b(\024)i(k)p Fs(')p Fr(k)917 2526 y Fp(p;)p Fo(2)p Fp(l;\026)1014 2512 y Fu(exp\(2)1124 2494 y Fq(\000)p Fp(l)1164 2512 y Fr(j)p Fs(z)r Fr(j)1213 2519 y Fq(\000)p Fp(p)1260 2512 y Fu(\))59 2617 y Fm(i.e.,)j(C)185 2624 y Fp(\026)208 2617 y Fs(')c Fr(2)h(E)321 2600 y Fo(1)317 2629 y(min)378 2617 y Fu(\()p Fr(N)440 2600 y Fq(0)p 439 2630 V 430 2630 a Fp(C)460 2617 y Fu(\))p Fm(.)59 2697 y(2\))j(L)n(et)g Fs(F)j Fr(2)13 b(E)315 2681 y Fo(1)311 2710 y(min)372 2697 y Fu(\()p Fr(N)434 2681 y Fq(0)p 432 2710 V 424 2710 a Fp(C)453 2697 y Fu(\))p Fm(.)21 b(Then)15 b(ther)n(e)i(exists)e Fs(')d Fr(2)h Fu(\()p Fr(N)7 b Fu(\))1027 2681 y Fo(1)1063 2697 y Fm(with)16 b Fs(C)1194 2704 y Fp(\026)1217 2697 y Fs(')d Fu(=)f Fs(F)6 b Fm(.)22 b(The)16 b(estimate)701 2796 y Fr(j)p Fs(F)6 b Fu(\()p Fs(z)r Fu(\))p Fr(j)12 b(\024)h Fs(C)e Fu(exp\(2)1035 2777 y Fq(\000)p Fp(l)1075 2796 y Fr(j)p Fs(z)r Fr(j)1124 2803 y Fq(\000)p Fp(p)1171 2796 y Fu(\))p eop %%Page: 31 31 31 30 bop 59 50 a Fn(8)45 b(CHARA)o(CTERIZA)l(TION)17 b(THEOREMS)888 b Fu(31)59 199 y Fm(for)17 b Fs(C)e(>)e Fu(0)p Fs(;)23 b(p;)8 b(q)14 b Fr(2)f Fs(I)-12 b(N)458 206 y Fo(0)494 199 y Fm(implies)544 318 y Fr(k)p Fs(')p Fr(k)619 332 y Fp(p)637 322 y Fh(0)648 332 y Fp(;q)q(;\026)721 318 y Fr(\024)13 b Fs(C)812 271 y Fj(\020)837 318 y Fu(1)d Fr(\000)g Fu(2)938 300 y Fp(q)q Fq(\000)p Fo(2)p Fp(l)1013 318 y Fs(e)1034 300 y Fo(2)1061 282 y Fj(\015)1061 307 y(\015)1084 318 y Fs(i)1100 326 y Fp(p)1118 317 y Fh(0)1129 326 y Fp(;p)1159 282 y Fj(\015)1159 307 y(\015)1182 295 y Fo(2)1182 335 y Fp(H)s(S)1239 271 y Fj(\021)1263 283 y Fq(\000)p Fo(1)p Fp(=)p Fo(2)59 436 y Fm(if)16 b(the)h(emb)n(e)n (dding)e Fs(i)413 443 y Fp(p)431 434 y Fh(0)442 443 y Fp(;p)484 436 y Fu(:)d Fr(H)547 443 y Fp(p)565 434 y Fh(0)591 436 y Fs(,)-8 b Fr(!)13 b(H)692 443 y Fp(p)729 436 y Fm(is)j(Hilb)n(ert-Schmidt)f(and)i Fu(2)1214 419 y Fp(l)p Fq(\000)p Fp(q)q(=)p Fo(2)1319 436 y Fs(>)c(e)1396 399 y Fj(\015)1395 424 y(\015)1419 436 y Fs(i)1435 443 y Fp(p)1453 434 y Fh(0)1463 443 y Fp(;p)1493 399 y Fj(\015)1493 424 y(\015)1516 452 y Fp(H)s(S)1573 436 y Fm(.)59 542 y Fk(Pr)o(oof.)72 b Fu(The)16 b(\014rst)f(statemen)o(t)f(follo)o(ws)h (from)575 644 y Fr(j)o Fs(C)620 651 y Fp(\026)643 644 y Fs(')p Fu(\()p Fs(z)r Fu(\))p Fr(j)d(\024)h Fu(n)830 651 y Fp(p;l;)p Fo(1)898 644 y Fu(\()p Fs(C)949 651 y Fp(\026)971 644 y Fs(')p Fu(\))d Fr(\001)f Fu(exp)q(\(2)1162 625 y Fq(\000)p Fp(l)1202 644 y Fr(j)p Fs(z)r Fr(j)1251 651 y Fq(\000)p Fp(p)1297 644 y Fu(\))59 746 y(whic)o(h)18 b(follo)o(ws)g(from)f(the)h(de\014nition)h(of)e(n)817 753 y Fp(p;l;)p Fo(1)903 746 y Fu(and)h(estimate)f(\(3\).)26 b(The)18 b(second)g(statemen)o(t)f(is)h(an)59 803 y(immediate)e (consequence)h(of)e(Lemma)g(7.)1009 b Ff(2)130 909 y Fu(The)15 b(next)g(theorem)g(c)o(haracterizes)h(distributions)g(from)f (\()p Fr(N)7 b Fu(\))1225 892 y Fq(\000)p Fo(1)1225 920 y Fp(\026)1287 909 y Fu(in)16 b(terms)e(of)h Fs(S)1546 916 y Fp(\026)1569 909 y Fu(-transform.)59 1015 y Fk(Theorem)j(34)23 b Fm(The)16 b Fs(S)475 1022 y Fp(\026)498 1015 y Fm(-tr)n(ansform)g(is) g(a)g(top)n(olo)n(gic)n(al)g(isomorphism)h(fr)n(om)f Fu(\()p Fr(N)7 b Fu(\))1486 999 y Fq(\000)p Fo(1)1486 1026 y Fp(\026)1549 1015 y Fm(on)16 b Fu(Hol)1683 1022 y Fo(0)1703 1015 y Fu(\()p Fr(N)p 1764 1022 2 18 v 1755 1022 a Fp(C)1785 1015 y Fu(\))p Fm(.)59 1121 y Fk(Remark.)73 b Fu(The)13 b(ab)q(o)o(v)o(e)h(theorem)f(is)h(closely)h(related)f(to)f (the)h(second)g(part)f(of)h(Theorem)f(8.)19 b(Since)59 1178 y(w)o(e)c(left)h(the)f(pro)q(of)g(op)q(en)g(w)o(e)g(will)i(giv)o (e)f(a)f(detailed)h(pro)q(of)f(here.)59 1234 y Fk(Pr)o(oof.)72 b Fu(Let)16 b(\010)c Fr(2)h Fu(\()p Fr(N)7 b Fu(\))538 1218 y Fq(\000)p Fo(1)538 1246 y Fp(\026)600 1234 y Fu(.)20 b(Then)15 b(there)h(exists)f Fs(p;)8 b(q)14 b Fr(2)f Fs(I)-12 b(N)20 b Fu(suc)o(h)15 b(that)649 1373 y Fr(k)o Fu(\010)p Fr(k)727 1352 y Fo(2)727 1387 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)q(;\026)872 1373 y Fu(=)936 1320 y Fq(1)923 1333 y Fj(X)920 1423 y Fp(n)p Fo(=0)994 1373 y Fu(2)1017 1354 y Fq(\000)p Fp(nq)1085 1373 y Fr(j)p Fu(\010)1131 1354 y Fo(\()p Fp(n)p Fo(\))1181 1373 y Fr(j)1194 1354 y Fo(2)1194 1384 y Fq(\000)p Fp(p)59 1509 y Fu(is)h(\014nite.)21 b(F)l(rom)14 b(the)h(previous)h(section)g(w)o(e)f(ha)o(v)o(e)691 1639 y Fs(S)719 1646 y Fp(\026)742 1639 y Fu(\010\()p Fs(\022)q Fu(\))e(=)909 1586 y Fq(1)897 1598 y Fj(X)894 1689 y Fp(n)p Fo(=0)960 1639 y Fr(h)p Fu(\010)1011 1620 y Fo(\()p Fp(n)p Fo(\))1062 1639 y Fs(;)8 b(\022)1105 1620 y Fq(\012)p Fp(n)1156 1639 y Fr(i)k Fs(:)551 b Fu(\(28\))59 1781 y(F)l(or)15 b Fs(\022)f Fr(2)f(N)p 262 1788 V 253 1788 a Fp(C)298 1781 y Fu(suc)o(h)i(that)g(2)522 1764 y Fp(q)541 1781 y Fr(j)p Fs(\022)q Fr(j)589 1764 y Fo(2)589 1792 y Fp(p)621 1781 y Fs(<)e Fu(1)i(w)o(e)g(ha)o(v)o(e)g(b)o(y)g (de\014nition)i(\(F)l(orm)o(ula)d(\(2\)\))649 1883 y Fr(j)-8 b(j)g(j)o Fs(S)699 1890 y Fp(\026)723 1883 y Fu(\010)p Fr(j)f(j)h(j)778 1898 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)q(;)p Fq(\000)p Fo(1)947 1883 y Fu(=)13 b Fr(k)p Fu(\010)p Fr(k)1073 1896 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)q(;\026)1229 1883 y Fs(:)59 1985 y Fu(By)i(Cauc)o(h)o(y{Sc)o(h)o (w)o(arz)f(inequalit)o(y)400 2115 y Fr(j)p Fs(S)441 2122 y Fp(\026)464 2115 y Fu(\010\()p Fs(\022)q Fu(\))p Fr(j)41 b(\024)701 2062 y Fq(1)689 2075 y Fj(X)686 2165 y Fp(n)p Fo(=0)760 2115 y Fr(j)p Fu(\010)806 2096 y Fo(\()p Fp(n)p Fo(\))856 2115 y Fr(j)869 2122 y Fq(\000)p Fp(p)916 2115 y Fr(j)p Fs(\022)q Fr(j)964 2096 y Fp(n)964 2126 y(p)609 2271 y Fr(\024)686 2199 y Fj( )734 2218 y Fq(1)722 2231 y Fj(X)719 2321 y Fp(n)p Fo(=0)793 2271 y Fu(2)816 2253 y Fq(\000)p Fp(nq)883 2271 y Fr(j)p Fu(\010)929 2253 y Fo(\()p Fp(n)p Fo(\))980 2271 y Fr(j)993 2253 y Fo(2)993 2283 y Fq(\000)p Fp(p)1040 2199 y Fj(!)1072 2211 y Fo(1)p Fp(=)p Fo(2)1135 2199 y Fj( )1184 2218 y Fq(1)1171 2231 y Fj(X)1168 2321 y Fp(n)p Fo(=0)1242 2271 y Fu(2)1265 2253 y Fp(nq)1305 2271 y Fr(j)p Fs(\022)q Fr(j)1353 2253 y Fo(2)p Fp(n)1353 2283 y(p)1394 2199 y Fj(!)1427 2211 y Fo(1)p Fp(=)p Fo(2)609 2403 y Fu(=)h Fr(k)p Fu(\010)p Fr(k)764 2416 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)q(;\026)904 2356 y Fj(\020)929 2403 y Fu(1)10 b Fr(\000)g Fu(2)1030 2384 y Fp(q)1049 2403 y Fr(j)p Fs(\022)q Fr(j)1097 2384 y Fo(2)1097 2414 y Fp(p)1117 2356 y Fj(\021)1141 2367 y Fq(\000)p Fo(1)p Fp(=)p Fo(2)1247 2403 y Fs(:)59 2527 y Fu(Th)o(us)i(the)g(series)g(\(28\))f(con)o(v)o(erges)g(uniformly)i (on)f(an)o(y)g(closed)h(ball)1220 2480 y Fj(n)1252 2527 y Fs(\022)i Fr(2)d(H)1368 2534 y Fp(p;)p 1402 2534 V -3 w(C)1423 2527 y Fr(j)20 b(j)p Fs(\022)q Fr(j)1504 2511 y Fo(2)1504 2539 y Fp(p)1536 2527 y Fr(\024)13 b Fs(r)o(;)22 b(r)13 b(<)g Fu(2)1744 2511 y Fq(\000)p Fp(q)1790 2480 y Fj(o)1818 2527 y Fu(.)59 2592 y(Hence)j Fs(S)222 2599 y Fp(\026)245 2592 y Fu(\010)d Fr(2)g Fu(Hol)403 2599 y Fo(0)423 2592 y Fu(\()p Fr(N)p 484 2599 V 475 2599 a Fp(C)505 2592 y Fu(\))h(and)543 2698 y(n)568 2705 y Fp(p;l;)p Fq(1)654 2698 y Fu(\()p Fs(S)700 2705 y Fp(\026)723 2698 y Fu(\010\))e Fr(\024)h(k)p Fu(\010)p Fr(k)912 2712 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)q(;\026)1053 2698 y Fu(\(1)c Fr(\000)i Fu(2)1172 2679 y Fp(q)q Fq(\000)p Fo(2)p Fp(l)1246 2698 y Fu(\))1264 2679 y Fq(\000)p Fo(1)p Fp(=)p Fo(2)p eop %%Page: 32 32 32 31 bop 59 50 a Fn(8)45 b(CHARA)o(CTERIZA)l(TION)17 b(THEOREMS)888 b Fu(32)59 199 y(if)17 b(2)p Fs(l)d(>)h(q)r Fu(.)23 b(This)17 b(pro)o(v)o(es)f(that)f Fs(S)635 206 y Fp(\026)675 199 y Fu(is)i(a)f(con)o(tin)o(uous)g(mapping)h(from)f(\() p Fr(N)7 b Fu(\))1366 183 y Fq(\000)p Fo(1)1366 210 y Fp(\026)1428 199 y Fu(to)16 b(Hol)1554 206 y Fo(0)1574 199 y Fu(\()p Fr(N)p 1635 206 2 18 v 1626 206 a Fp(C)1656 199 y Fu(\).)23 b(In)17 b(the)59 256 y(language)e(of)g(section)h(2.2)e (this)i(reads)673 358 y(ind)h(lim)691 387 y Fp(p;q)q Fq(2)p Fp(I)-9 b(N)835 358 y Fu(E)866 339 y Fq(\000)p Fo(1)866 369 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)980 358 y Fr(\032)13 b Fu(Hol)1097 365 y Fo(0)1117 358 y Fu(\()p Fr(N)p 1178 365 V 1169 365 a Fp(C)1199 358 y Fu(\))59 479 y(top)q(ologically)l(.)130 561 y(Con)o(v)o(ersely)l(,)h(let)h Fs(F)k Fr(2)13 b Fu(Hol)593 568 y Fo(0)613 561 y Fu(\()p Fr(N)p 674 568 V 665 568 a Fp(C)694 561 y Fu(\))h(b)q(e)h(giv)o(en,)g (i.e.,)f(there)g(exist)h Fs(p;)8 b(l)k Fr(2)h Fs(I)-12 b(N)19 b Fu(suc)o(h)14 b(that)g(n)1626 568 y Fp(p;l;)p Fq(1)1712 561 y Fu(\()p Fs(F)6 b Fu(\))12 b Fs(<)59 617 y Fr(1)p Fu(.)20 b(The)c(\014rst)f(step)g(is)h(to)e(sho)o(w)h(that)f (there)h(exists)h Fs(p)998 601 y Fq(0)1009 617 y Fs(;)8 b(q)14 b Fr(2)f Fs(I)-12 b(N)20 b Fu(suc)o(h)c(that)651 719 y Fr(j)-8 b(j)g(j)p Fs(F)6 b Fr(j)-7 b(j)f(j)732 733 y Fq(\000)p Fp(p)777 723 y Fh(0)789 733 y Fp(;)p Fq(\000)p Fp(q)q(;)p Fq(\000)p Fo(1)912 719 y Fs(<)13 b Fu(n)985 726 y Fp(p;l;)p Fq(1)1071 719 y Fu(\()p Fs(F)6 b Fu(\))k Fr(\001)g Fs(C)18 b(;)59 821 y Fu(for)d(su\016cien)o(tly)h (large)f Fs(C)h(>)d Fu(0.)19 b(This)d(implies)h(immediately)672 924 y(Hol)741 931 y Fo(0)761 924 y Fu(\()p Fr(N)p 822 931 V 813 931 a Fp(C)843 924 y Fu(\))12 b Fr(\032)k Fu(ind)g(lim)942 953 y Fp(p;q)q Fq(2)p Fp(I)-9 b(N)1085 924 y Fu(E)1116 904 y Fq(\000)p Fo(1)1116 935 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)59 1045 y Fu(top)q(ologically)l(,)18 b(whic)o(h)f(is)f(the)g (missing)h(part)f(in)h(the)f(pro)q(of)g(of)f(the)h(second)h(statemen)o (t)e(in)i(Theorem)59 1102 y(8.)130 1158 y(By)e(assumption)g(the)h(T)l (a)o(ylor)e(expansion)706 1288 y Fs(F)6 b Fu(\()p Fs(\022)q Fu(\))13 b(=)875 1235 y Fq(1)863 1248 y Fj(X)860 1338 y Fp(n)p Fo(=0)947 1257 y Fu(1)p 939 1278 40 2 v 939 1319 a Fs(n)p Fu(!)1025 1272 y Fj(d)984 1288 y Fu(d)1009 1275 y Fp(n)1032 1288 y Fs(F)6 b Fu(\(0\)\()p Fs(\022)q Fu(\))59 1441 y(con)o(v)o(erges)15 b(uniformly)h(on)f(an)o(y)g(closed)h (ball)841 1394 y Fj(n)874 1441 y Fs(\022)e Fr(2)f(H)990 1448 y Fp(p;)p 1024 1448 2 18 v -3 w(C)1045 1441 y Fr(j)20 b(j)p Fs(\022)q Fr(j)1126 1424 y Fo(2)1126 1452 y Fp(p)1158 1441 y Fr(\024)13 b Fs(r)o(;)22 b(r)13 b(<)g Fu(2)1366 1424 y Fq(\000)p Fp(l)1406 1394 y Fj(o)1449 1441 y Fu(and)750 1551 y Fr(j)p Fs(F)6 b Fu(\()p Fs(\022)q Fu(\))p Fr(j)13 b(\024)g Fu(n)955 1558 y Fp(p;l;)p Fq(1)1041 1551 y Fu(\()p Fs(F)6 b Fu(\))15 b Fs(:)59 1653 y Fu(Pro)q(ceeding)h(analogously)g(to) e(Lemma)i(6,)e(an)h(application)i(of)e(Cauc)o(h)o(y's)f(inequalit)o(y)j (giv)o(es)579 1741 y(1)p 570 1761 40 2 v 570 1803 a Fs(n)p Fu(!)656 1755 y Fj(d)615 1772 y Fu(d)640 1759 y Fp(n)664 1772 y Fs(F)6 b Fu(\(0\))o(\()p Fs(\022)q Fu(\))42 b Fr(\024)g Fu(2)957 1753 y Fp(l)969 1772 y Fr(j)p Fs(\022)q Fr(j)1017 1753 y Fp(n)1017 1783 y(p)1082 1772 y Fu(sup)1048 1814 y Fq(j)p Fp(\022)q Fq(j)1085 1818 y Fg(p)1103 1814 y Fq(\024)p Fo(2)1148 1805 y Fh(\000)p Fg(l)1192 1772 y Fr(j)p Fs(F)6 b Fu(\()p Fs(\022)q Fu(\))p Fr(j)857 1886 y(\024)42 b Fu(n)959 1893 y Fp(p;l;)p Fq(1)1045 1886 y Fu(\()p Fs(F)6 b Fu(\))25 b Fr(\001)9 b Fu(2)1186 1867 y Fp(nl)1231 1886 y Fr(\001)g(j)p Fs(\022)q Fr(j)1301 1867 y Fp(n)1301 1897 y(p)59 1988 y Fu(The)15 b(p)q(olarization)i(iden) o(tit)o(y)f(giv)o(es)431 2057 y Fj(\014)431 2082 y(\014)431 2106 y(\014)431 2131 y(\014)458 2087 y Fu(1)p 450 2107 V 450 2149 a Fs(n)p Fu(!)494 2118 y(d)519 2099 y Fp(n)543 2118 y Fs(F)6 b Fu(\(0\)\()p Fs(\022)676 2125 y Fo(1)696 2118 y Fs(;)i(:)g(:)g(:)t(;)g(\022)818 2125 y Fp(n)841 2118 y Fu(\))859 2057 y Fj(\014)859 2082 y(\014)859 2106 y(\014)859 2131 y(\014)886 2118 y Fr(\024)13 b Fu(n)959 2125 y Fp(p;l;)p Fq(1)1044 2118 y Fu(\()p Fs(F)6 b Fu(\))26 b Fr(\001)9 b Fs(e)1184 2099 y Fp(n)1218 2118 y Fr(\001)h Fu(2)1264 2099 y Fp(nl)1325 2065 y(n)1310 2077 y Fj(Y)1305 2169 y Fp(j)r Fo(=1)1374 2118 y Fr(j)p Fs(\022)1408 2125 y Fp(j)1427 2118 y Fr(j)1440 2125 y Fp(p)59 2277 y Fu(Then)19 b(b)o(y)f(k)o(ernel)i(theorem)e(\(Theorem)g(2\))g(there)g(exist)h(k)o (ernels)g(\010)1255 2260 y Fo(\()p Fp(n)p Fo(\))1324 2277 y Fr(2)f(H)1415 2252 y Fo(^)1410 2258 y Fq(\012)r Fp(n)1410 2291 y Fq(\000)p Fp(p)1455 2282 y Fh(0)1467 2291 y Fp(;)p 1483 2291 2 18 v -3 w(C)1522 2277 y Fu(for)g Fs(p)1618 2260 y Fq(0)1648 2277 y Fs(>)g(p)g Fu(with)59 2302 y Fj(\015)59 2327 y(\015)82 2338 y Fs(i)98 2346 y Fp(p)116 2336 y Fh(0)127 2346 y Fp(;p)156 2302 y Fj(\015)156 2327 y(\015)179 2354 y Fp(H)s(S)249 2338 y Fs(<)13 b Fr(1)j Fu(suc)o(h)f(that)714 2430 y Fs(F)6 b Fu(\()p Fs(\022)q Fu(\))13 b(=)884 2377 y Fq(1)872 2389 y Fj(X)868 2480 y Fp(n)p Fo(=0)935 2430 y Fr(h)p Fu(\010)986 2411 y Fo(\()p Fp(n)p Fo(\))1036 2430 y Fs(;)8 b(\022)1079 2411 y Fq(\012)p Fp(n)1130 2430 y Fr(i)15 b Fs(:)59 2547 y Fu(Moreo)o(v)o(er)f(w)o(e)h(ha)o(v)o(e)g(the)g(follo)o(wing)h(norm)f (estimate)570 2607 y Fj(\014)570 2631 y(\014)570 2656 y(\014)584 2655 y Fu(\010)617 2636 y Fo(\()p Fp(n)p Fo(\))668 2607 y Fj(\014)668 2631 y(\014)668 2656 y(\014)681 2683 y Fq(\000)p Fp(p)726 2674 y Fh(0)752 2655 y Fr(\024)e Fu(n)825 2662 y Fp(p;l;)p Fq(1)911 2655 y Fu(\()p Fs(F)6 b Fu(\))1005 2608 y Fj(\020)1030 2655 y Fu(2)1053 2636 y Fp(l)1065 2655 y Fs(e)1094 2619 y Fj(\015)1094 2644 y(\015)1117 2655 y Fs(i)1133 2663 y Fp(p)1151 2653 y Fh(0)1162 2663 y Fp(;p)1192 2619 y Fj(\015)1192 2644 y(\015)1215 2671 y Fp(H)s(S)1272 2608 y Fj(\021)1296 2620 y Fp(n)p eop %%Page: 33 33 33 32 bop 59 50 a Fn(8)45 b(CHARA)o(CTERIZA)l(TION)17 b(THEOREMS)888 b Fu(33)59 199 y(Th)o(us)410 315 y Fr(j)-8 b(j)g(j)p Fs(F)6 b Fr(j)-8 b(j)g(j)491 294 y Fo(2)491 329 y Fq(\000)p Fp(p)536 319 y Fh(0)547 329 y Fp(;)p Fq(\000)p Fp(q)q(;)p Fq(\000)p Fo(1)700 315 y Fu(=)792 263 y Fq(1)780 275 y Fj(X)777 366 y Fp(n)p Fo(=0)851 315 y Fu(2)874 297 y Fq(\000)p Fp(nq)949 267 y Fj(\014)949 292 y(\014)949 317 y(\014)963 315 y Fu(\010)996 297 y Fo(\()p Fp(n)p Fo(\))1047 267 y Fj(\014)1047 292 y(\014)1047 317 y(\014)1060 280 y Fo(2)1060 344 y Fq(\000)p Fp(p)1105 334 y Fh(0)700 458 y Fr(\024)42 b Fu(n)802 439 y Fo(2)802 469 y Fp(p;l;)p Fq(1)888 458 y Fu(\()p Fs(F)6 b Fu(\))982 405 y Fq(1)969 417 y Fj(X)966 508 y Fp(n)p Fo(=0)1040 411 y Fj(\020)1065 458 y Fu(2)1088 439 y Fo(2)p Fp(l)p Fq(\000)p Fp(q)1163 458 y Fs(e)1184 439 y Fo(2)1211 421 y Fj(\015)1211 446 y(\015)1234 458 y Fs(i)1250 465 y Fp(p)1268 456 y Fh(0)1279 465 y Fp(;p)1309 421 y Fj(\015)1309 446 y(\015)1332 435 y Fo(2)1332 474 y Fp(H)s(S)1389 411 y Fj(\021)1413 422 y Fp(n)700 586 y Fu(=)42 b(n)802 567 y Fo(2)802 597 y Fp(p;l;)p Fq(1)888 586 y Fu(\()p Fs(F)6 b Fu(\))967 539 y Fj(\020)991 586 y Fu(1)k Fr(\000)g Fu(2)1092 567 y Fo(2)p Fp(l)p Fq(\000)p Fp(q)1167 586 y Fs(e)1188 567 y Fo(2)1215 549 y Fj(\015)1215 574 y(\015)1239 586 y Fs(i)1255 593 y Fp(p)1273 584 y Fh(0)1283 593 y Fp(;p)1313 549 y Fj(\015)1313 574 y(\015)1336 562 y Fo(2)1336 602 y Fp(H)s(S)1393 539 y Fj(\021)1418 550 y Fq(\000)p Fo(1)59 701 y Fu(if)16 b Fs(q)e Fr(2)f Fs(I)-12 b(N)20 b Fu(is)c(suc)o(h)f(that)g Fs(\032)d Fu(:=)h(2)610 684 y Fo(2)p Fp(l)p Fq(\000)p Fp(q)684 701 y Fs(e)705 684 y Fo(2)733 665 y Fj(\015)733 690 y(\015)756 701 y Fs(i)772 709 y Fp(p)790 699 y Fh(0)801 709 y Fp(;p)830 665 y Fj(\015)830 690 y(\015)853 678 y Fo(2)853 717 y Fp(H)s(S)938 701 y Fs(<)g Fu(1.)20 b(So)15 b(w)o(e)g(ha)o(v)o(e)f(in)i(fact)583 806 y Fr(j)-8 b(j)g(j)p Fs(F)6 b Fr(j)-7 b(j)f(j)664 819 y Fq(\000)p Fp(p)709 810 y Fh(0)721 819 y Fp(;)p Fq(\000)p Fp(q)q(;)p Fq(\000)p Fo(1)844 806 y Fr(\024)13 b Fu(n)917 813 y Fp(p;l;)p Fq(1)1003 806 y Fu(\()p Fs(F)6 b Fu(\)\(1)k Fr(\000)g Fs(\032)p Fu(\))1212 787 y Fq(\000)p Fo(1)p Fp(=)p Fo(2)1294 806 y Fs(:)59 908 y Fu(No)o(w)15 b(the)g(rest)g(is)g(simple.)22 b(De\014ne)15 b(\010)e Fr(2)g Fu(\()p Fr(N)7 b Fu(\))844 892 y Fq(\000)p Fo(1)844 920 y Fp(\026)906 908 y Fu(b)o(y)772 1042 y(\010)12 b(=)881 989 y Fq(1)869 1002 y Fj(X)865 1092 y Fp(n)p Fo(=0)939 1042 y Fs(Q)975 1023 y Fp(\026)975 1053 y(n)999 1042 y Fu(\(\010)1050 1023 y Fo(\()p Fp(n)p Fo(\))1100 1042 y Fu(\))59 1173 y(then)k Fs(S)191 1180 y Fp(\026)214 1173 y Fu(\010)c(=)h Fs(F)22 b Fu(and)679 1229 y Fr(k)p Fu(\010)p Fr(k)758 1243 y Fq(\000)p Fp(p)803 1233 y Fh(0)814 1243 y Fp(;)p Fq(\000)p Fp(q)q(;\026)914 1229 y Fu(=)13 b Fr(j)-8 b(j)g(j)p Fs(F)6 b Fr(j)-8 b(j)g(j)1043 1243 y Fq(\000)p Fp(p)1088 1233 y Fh(0)1099 1243 y Fp(;)p Fq(\000)p Fp(q)q(;)p Fq(\000)p Fo(1)59 1310 y Fu(This)18 b(pro)o(v)o(es)f(the)h(existence)g(of)f(a)h(con)o(tin)o(uous)g(in)o(v)o (erse)f(of)h(the)f Fs(S)1224 1317 y Fp(\026)1247 1310 y Fu({transform.)26 b(Uniqueness)19 b(of)e(\010)59 1367 y(follo)o(ws)e(from)g(the)g(fact)g(that)f Fs(\026)p Fu(-exp)q(onen)o (tials)j(are)e(total)g(in)h(an)o(y)f(\()p Fr(H)1266 1374 y Fp(p)1286 1367 y Fu(\))1304 1350 y Fo(1)1304 1378 y Fp(q)1323 1367 y Fu(.)461 b Ff(2)130 1470 y Fu(W)l(e)15 b(can)g(extract)f(some)g(useful)i(estimates)f(from)f(the)h(ab)q(o)o(v)o (e)g(pro)q(of)f(whic)o(h)i(describ)q(e)g(the)f(degree)59 1526 y(of)g(singularit)o(y)h(of)f(a)g(distribution.)59 1628 y Fk(Cor)o(ollar)m(y)j(35)23 b Fm(L)n(et)d Fs(F)27 b Fr(2)20 b Fu(Hol)655 1635 y Fo(0)674 1628 y Fu(\()p Fr(N)p 735 1635 2 18 v 726 1635 a Fp(C)756 1628 y Fu(\))g Fm(b)n(e)g(holomorphic)i(for)f(al)r(l)f Fs(\022)i Fr(2)f(N)p 1400 1635 V 1391 1635 a Fp(C)1442 1628 y Fm(with)g Fr(j)p Fs(\022)q Fr(j)1593 1635 y Fp(p)1633 1628 y Fr(\024)f Fu(2)1711 1611 y Fq(\000)p Fp(l)1752 1628 y Fm(.)33 b(If)59 1690 y Fs(p)82 1673 y Fq(0)114 1690 y Fs(>)20 b(p)h Fm(with)315 1653 y Fj(\015)315 1678 y(\015)339 1690 y Fs(i)355 1697 y Fp(p)373 1688 y Fh(0)383 1697 y Fp(;p)413 1653 y Fj(\015)413 1678 y(\015)436 1706 y Fp(H)s(S)513 1690 y Fs(<)g Fr(1)f Fm(and)h Fs(q)h Fr(2)e Fs(I)-12 b(N)26 b Fm(is)20 b(such)g(that)h Fs(\032)f Fu(:=)g(2)1282 1673 y Fo(2)p Fp(l)p Fq(\000)p Fp(q)1357 1690 y Fs(e)1378 1673 y Fo(2)1405 1653 y Fj(\015)1405 1678 y(\015)1428 1690 y Fs(i)1444 1697 y Fp(p)1462 1688 y Fh(0)1473 1697 y Fp(;p)1503 1653 y Fj(\015)1503 1678 y(\015)1526 1666 y Fo(2)1526 1706 y Fp(H)s(S)1603 1690 y Fs(<)g Fu(1)p Fm(.)33 b(Then)59 1750 y Fu(\010)13 b Fr(2)f Fu(\()p Fr(H)203 1758 y Fq(\000)p Fp(p)248 1749 y Fh(0)262 1750 y Fu(\))280 1731 y Fq(\000)p Fo(1)280 1761 y Fq(\000)p Fp(q)343 1750 y Fm(and)594 1816 y Fr(k)p Fu(\010)p Fr(k)672 1829 y Fq(\000)p Fp(p)717 1820 y Fh(0)729 1829 y Fp(;q)q(;\026)801 1816 y Fr(\024)h Fu(n)874 1823 y Fp(p;l;)p Fq(1)960 1816 y Fu(\()p Fs(F)6 b Fu(\))k Fr(\001)g Fu(\(1)f Fr(\000)i Fs(\032)p Fu(\))1202 1797 y Fq(\000)p Fo(1)p Fp(=)p Fo(2)1283 1816 y Fs(:)130 1924 y Fu(F)l(or)h(a)h(more)f(detailed)j(discussion)g(of)d(the)h(degree)h (of)e(singularit)o(y)i(the)f(spaces)h(\()p Fr(N)7 b Fu(\))1568 1908 y Fq(\000)p Fp(\014)1618 1924 y Fs(;)20 b(\014)15 b Fr(2)e Fu([0)p Fs(;)8 b Fu(1\))59 1981 y(are)19 b(useful.)33 b(In)20 b(the)f(follo)o(wing)h(theorem)f(w)o(e)f(will)j(c)o (haracterize)f(these)f(spaces)g(b)o(y)h(means)f(of)f Fs(S)1792 1988 y Fp(\026)1816 1981 y Fu(-)59 2037 y(transform.)59 2138 y Fk(Theorem)g(36)23 b Fm(The)17 b Fs(S)476 2145 y Fp(\026)499 2138 y Fm(-tr)n(ansform)h(is)f(a)h(top)n(olo)n(gic)n(al)f (isomorphism)i(fr)n(om)f Fu(\()p Fr(N)7 b Fu(\))1497 2122 y Fq(\000)p Fp(\014)1497 2150 y(\026)1547 2138 y Fm(,)18 b Fs(\014)g Fr(2)d Fu([0)p Fs(;)8 b Fu(1\))16 b Fm(on)59 2209 y Fr(E)87 2185 y Fo(2)p Fp(=)p Fo(\(1)p Fq(\000)p Fp(\014)r Fo(\))83 2214 y(max)219 2209 y Fu(\()p Fr(N)p 280 2216 V 271 2216 a Fp(C)300 2209 y Fu(\))p Fm(.)59 2310 y Fk(Remark.)73 b Fu(The)16 b(pro)q(of)e(will)j(also)e (complete)h(the)g(pro)q(of)e(of)h(Theorem)g(8.)59 2367 y Fk(Pr)o(oof.)77 b Fu(Let)17 b(\010)f Fr(2)f Fu(\()p Fr(H)526 2374 y Fq(\000)p Fp(p)573 2367 y Fu(\))591 2345 y Fq(\000)p Fp(\014)591 2378 y Fq(\000)p Fp(q)q(;\026)685 2367 y Fu(with)i(the)g(canonical)h(represen)o(tation)f(\010)e(=)1469 2335 y Fj(P)1513 2348 y Fq(1)1513 2378 y Fp(n)p Fo(=0)1589 2367 y Fs(Q)1625 2350 y Fp(\026)1625 2378 y(n)1648 2367 y Fu(\(\010)1699 2350 y Fo(\()p Fp(n)p Fo(\))1750 2367 y Fu(\))h(b)q(e)59 2423 y(giv)o(en.)k(The)c Fs(S)317 2430 y Fp(\026)340 2423 y Fu(-transform)e(of)h(\010)g(is)g(giv)o(en)h (b)o(y)698 2553 y Fs(S)726 2560 y Fp(\026)749 2553 y Fu(\010\()p Fs(\022)q Fu(\))c(=)916 2500 y Fq(1)903 2512 y Fj(X)900 2603 y Fp(n)p Fo(=0)967 2553 y Fr(h)p Fu(\010)1018 2534 y Fo(\()p Fp(n)p Fo(\))1068 2553 y Fs(;)c(\022)1111 2534 y Fq(\012)p Fp(n)1162 2553 y Fr(i)p Fs(:)59 2683 y Fu(Hence)537 2758 y Fr(j)-8 b(j)g(j)p Fs(S)588 2765 y Fp(\026)611 2758 y Fu(\010)p Fr(j)g(j)g(j)667 2738 y Fo(2)667 2774 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)q(;)p Fq(\000)p Fp(\014)840 2758 y Fu(=)903 2706 y Fq(1)891 2718 y Fj(X)888 2809 y Fp(n)p Fo(=0)954 2758 y Fu(\()p Fs(n)p Fu(!\))1030 2740 y Fo(1)p Fq(\000)p Fp(\014)1106 2758 y Fu(2)1129 2740 y Fq(\000)p Fp(nq)1196 2758 y Fr(j)p Fu(\010)1242 2740 y Fo(\()p Fp(n)p Fo(\))1293 2758 y Fr(j)1306 2740 y Fo(2)1306 2770 y Fq(\000)p Fp(p)p eop %%Page: 34 34 34 33 bop 59 50 a Fn(8)45 b(CHARA)o(CTERIZA)l(TION)17 b(THEOREMS)888 b Fu(34)59 199 y(is)16 b(\014nite.)21 b(W)l(e)15 b(will)i(sho)o(w)d(that)h(there)g(exist)h Fs(l)d Fr(2)g Fs(I)-12 b(N)20 b Fu(and)15 b Fs(C)h(<)c Fu(0)j(suc)o(h)h(that)537 301 y(n)562 311 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(l;)p Fo(2)p Fp(=)p Fo(\(1)p Fq(\000)p Fp(\014)r Fo(\))797 301 y Fu(\()p Fs(S)843 308 y Fp(\026)866 301 y Fu(\010\))c Fr(\024)h Fs(C)18 b Fr(j)-8 b(j)g(j)o Fs(S)1078 308 y Fp(\026)1102 301 y Fu(\010)p Fr(j)f(j)i(j)1157 317 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)q(;)p Fq(\000)p Fp(\014)1340 301 y Fs(:)130 403 y Fu(W)l(e)15 b(can)g(estimate)g(as)g (follo)o(ws)264 525 y Fs(S)292 532 y Fp(\026)315 525 y Fu(\010\()p Fs(\022)q Fu(\))41 b Fr(\024)540 472 y Fq(1)528 484 y Fj(X)524 575 y Fp(n)p Fo(=0)598 476 y Fj(\014)598 501 y(\014)598 526 y(\014)612 525 y Fu(\010)645 506 y Fo(\()p Fp(n)p Fo(\))696 476 y Fj(\014)696 501 y(\014)696 526 y(\014)710 553 y Fq(\000)p Fp(p)765 525 y Fr(j)o Fs(\022)q Fr(j)812 504 y Fp(n)812 538 y(p)447 681 y Fr(\024)524 609 y Fj( )573 628 y Fq(1)560 641 y Fj(X)557 731 y Fp(n)p Fo(=0)624 681 y Fu(\()p Fs(n)p Fu(!\))700 662 y Fo(1)p Fq(\000)p Fp(\014)768 681 y Fu(2)791 662 y Fq(\000)p Fp(nq)858 681 y Fr(j)p Fu(\010)904 662 y Fo(\()p Fp(n)p Fo(\))955 681 y Fr(j)968 662 y Fo(2)968 692 y Fq(\000)p Fp(p)1014 609 y Fj(!)1047 620 y Fo(1)p Fp(=)p Fo(2)1110 609 y Fj( )1158 628 y Fq(1)1146 641 y Fj(X)1143 731 y Fp(n)p Fo(=0)1282 650 y Fu(1)p 1222 670 144 2 v 1222 712 a(\()p Fs(n)p Fu(!\))1298 699 y Fo(1)p Fq(\000)p Fp(\014)1371 681 y Fu(2)1394 662 y Fp(nq)1441 681 y Fr(j)p Fs(\022)q Fr(j)1489 660 y Fo(2)p Fp(n)1489 694 y(p)1530 609 y Fj(!)1563 620 y Fo(1)p Fp(=)p Fo(2)447 837 y Fu(=)h Fr(j)-8 b(j)g(j)p Fs(S)575 844 y Fp(\026)598 837 y Fu(\010)p Fr(j)g(j)g(j)654 853 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)q(;)p Fq(\000)p Fp(\014)822 765 y Fj( )870 784 y Fq(1)858 797 y Fj(X)854 887 y Fp(n)p Fo(=0)928 837 y Fs(\032)952 818 y Fp(n\014)1007 837 y Fr(\001)1095 806 y Fu(1)p 1035 827 V 1035 868 a(\()p Fs(n)p Fu(!\))1111 855 y Fo(1)p Fq(\000)p Fp(\014)1184 837 y Fu(2)1207 818 y Fp(nq)1247 837 y Fs(\032)1271 818 y Fq(\000)p Fp(n\014)1350 837 y Fr(j)p Fs(\022)q Fr(j)1398 816 y Fo(2)p Fp(n)1398 851 y(p)1447 837 y Fr(\001)1460 765 y Fj(!)1492 777 y Fo(1)p Fp(=)p Fo(2)1555 837 y Fs(;)59 975 y Fu(where)18 b(w)o(e)f(ha)o(v)o(e)g(in)o(tro)q(duced)i(a)e(parameter)g Fs(\032)f Fr(2)g Fu(\(0)p Fs(;)8 b Fu(1\).)25 b(An)18 b(application)h(of)e(H\177)-23 b(older's)18 b(inequalit)o(y)59 1032 y(for)d(the)g(conjugate)g(indices)569 1014 y Fo(1)p 567 1021 22 2 v 567 1048 a Fp(\014)609 1032 y Fu(and)726 1014 y Fo(1)p 702 1021 67 2 v 702 1048 a(1)p Fq(\000)p Fp(\014)789 1032 y Fu(giv)o(es)225 1197 y Fr(j)p Fs(S)266 1204 y Fp(\026)289 1197 y Fu(\010\()p Fs(\022)q Fu(\))p Fr(j)41 b(\024)h(j)-8 b(j)g(j)o Fs(S)561 1204 y Fp(\026)585 1197 y Fu(\010)p Fr(j)f(j)i(j)640 1213 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)q(;)p Fq(\000)p Fp(\014)808 1125 y Fj( )856 1144 y Fq(1)844 1157 y Fj(X)841 1247 y Fp(n)p Fo(=0)915 1197 y Fs(\032)939 1179 y Fp(n)962 1125 y Fj(!)995 1137 y Fp(\014)r(=)p Fo(2)1064 1197 y Fr(\001)1086 1125 y Fj( )1135 1144 y Fq(1)1123 1157 y Fj(X)1119 1247 y Fp(n)p Fo(=0)1207 1167 y Fu(1)p 1198 1187 40 2 v 1198 1228 a Fs(n)p Fu(!)1251 1150 y Fj(\020)1276 1197 y Fu(2)1299 1179 y Fp(q)1317 1197 y Fs(\032)1341 1179 y Fq(\000)p Fp(\014)1392 1197 y Fr(j)p Fs(\022)q Fr(j)1440 1179 y Fo(2)1440 1209 y Fp(p)1460 1150 y Fj(\021)1509 1148 y Fg(n)p 1489 1154 58 2 v 1489 1175 a Fi(1)p Fh(\000)p Fg(\014)1554 1125 y Fj(!)1592 1121 y Fi(1)p Fh(\000)p Fg(\014)p 1592 1129 V 1613 1150 a Fi(2)434 1331 y Fu(=)42 b Fr(j)-8 b(j)g(j)o Fs(S)561 1338 y Fp(\026)585 1331 y Fu(\010)p Fr(j)f(j)i(j)640 1347 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)q(;)p Fq(\000)p Fp(\014)808 1331 y Fu(\(1)9 b Fr(\000)i Fs(\032)p Fu(\))945 1310 y Fq(\000)p Fp(\014)r(=)p Fo(2)1039 1331 y Fu(exp)1116 1272 y Fj(\022)1152 1311 y Fo(1)p Fq(\000)p Fp(\014)p 1152 1321 67 2 v 1176 1347 a Fo(2)1231 1331 y Fu(2)1280 1293 y Fg(q)p 1258 1301 58 2 v 1258 1322 a Fi(1)p Fh(\000)p Fg(\014)1331 1331 y Fs(\032)1355 1309 y Fq(\000)1406 1293 y Fg(\014)p 1387 1301 V 1387 1322 a Fi(1)p Fh(\000)p Fg(\014)1459 1331 y Fr(j)p Fs(\022)q Fr(j)1534 1284 y Fi(2)p 1512 1290 V 1512 1311 a(1)p Fh(\000)p Fg(\014)1507 1337 y Fp(p)1577 1272 y Fj(\023)59 1451 y Fu(If)16 b Fs(l)d Fr(2)g Fs(I)-12 b(N)20 b Fu(is)15 b(suc)o(h)h(that)750 1513 y(2)773 1490 y Fp(l)p Fq(\000)838 1474 y Fg(q)p 816 1483 V 816 1503 a Fi(1)p Fh(\000)p Fg(\014)894 1513 y Fs(>)947 1492 y Fo(1)p Fq(\000)p Fp(\014)p 947 1502 67 2 v 971 1528 a Fo(2)1019 1513 y Fs(\032)1043 1490 y Fq(\000)1094 1474 y Fg(\014)p 1074 1483 58 2 v 1074 1503 a Fi(1)p Fh(\000)p Fg(\014)59 1596 y Fu(w)o(e)f(ha)o(v)o(e)349 1698 y(n)374 1708 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(l;)p Fo(2)p Fp(=)p Fo(\(1)p Fq(\000)p Fp(\014)r Fo(\))609 1698 y Fu(\()p Fs(S)655 1705 y Fp(\026)678 1698 y Fu(\010\))41 b(=)67 b(sup)847 1737 y Fp(\022)q Fq(2H)918 1743 y Fg(p;)p 948 1743 2 14 v -3 w(C)974 1698 y Fr(j)o Fs(S)1014 1705 y Fp(\026)1038 1698 y Fu(\010\()p Fs(\022)q Fu(\))p Fr(j)14 b Fu(exp)1233 1651 y Fj(\020)1258 1698 y Fr(\000)p Fu(2)1316 1680 y Fp(l)1329 1698 y Fr(j)p Fs(\022)q Fr(j)1377 1680 y Fo(2)p Fp(=)p Fo(\(1)p Fq(\000)p Fp(\014)r Fo(\))1377 1710 y Fp(p)1508 1651 y Fj(\021)770 1814 y Fr(\024)42 b Fu(\()o(1)10 b Fr(\000)h Fs(\032)p Fu(\))984 1793 y Fq(\000)p Fp(\014)r(=)p Fo(2)1078 1814 y Fr(j)-8 b(j)g(j)o Fs(S)1128 1821 y Fp(\026)1152 1814 y Fu(\010)p Fr(j)f(j)h(j)1207 1830 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)q(;)p Fq(\000)p Fp(\014)59 1935 y Fu(This)21 b(sho)o(ws)g(that)f Fs(S)437 1942 y Fp(\026)481 1935 y Fu(is)h(con)o(tin)o(uous)g(from)f(\()p Fr(N)7 b Fu(\))958 1918 y Fq(\000)p Fp(\014)958 1946 y(\026)1029 1935 y Fu(to)20 b Fr(E)1118 1911 y Fo(2)p Fp(=)p Fo(\(1)p Fq(\000)p Fp(\014)r Fo(\))1114 1940 y(max)1250 1935 y Fu(\()p Fr(N)p 1311 1942 2 18 v 1302 1942 a Fp(C)1332 1935 y Fu(\))p Fs(:)g Fu(Or)h(in)g(the)g(language)g(of)59 1991 y(Theorem)15 b(8)638 2048 y(ind)h(lim)656 2077 y Fp(p;q)q Fq(2)p Fp(I)-9 b(N)800 2048 y Fu(E)831 2025 y Fq(\000)p Fp(\014)831 2059 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)945 2048 y Fr(\032)13 b(E)1021 2029 y Fo(2)p Fp(=)p Fo(\(1)p Fq(\000)p Fp(\014)r Fo(\))1017 2059 y(max)1152 2048 y Fu(\()p Fr(N)p 1213 2055 V 1204 2055 a Fp(C)1234 2048 y Fu(\))59 2151 y(top)q(ologically)l(.)130 2232 y(The)k(pro)q(of)g(of)g(the)h(in)o(v)o(erse)f(direction)i(is)f(closely) g(related)g(to)f(the)g(pro)q(of)g(of)g(Lemma)h(7.)26 b(So)17 b(w)o(e)59 2288 y(will)g(b)q(e)f(more)f(sk)o(etc)o(h)o(y)f(in)i (the)f(follo)o(wing.)59 2345 y(Let)g Fs(F)20 b Fr(2)13 b(E)260 2328 y Fp(k)256 2356 y Fo(max)323 2345 y Fu(\()p Fr(N)p 384 2352 V 375 2352 a Fp(C)405 2345 y Fu(\))p Fs(;)22 b(k)14 b Fu(=)573 2327 y Fo(2)p 549 2334 67 2 v 549 2360 a(1)p Fq(\000)p Fp(\014)620 2345 y Fu(.)20 b(Hence)c(there)g(exist)f Fs(p;)8 b(l)k Fr(2)h Fs(I)-12 b(N)1175 2352 y Fo(0)1210 2345 y Fu(suc)o(h)15 b(that)502 2459 y Fr(j)p Fs(F)6 b Fu(\()p Fs(\022)q Fu(\))p Fr(j)12 b(\024)h Fu(n)706 2466 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(l;k)831 2459 y Fu(\()p Fs(F)6 b Fu(\))i(exp\(2)1020 2440 y Fp(l)1032 2459 y Fr(j)p Fs(\022)q Fr(j)1080 2440 y Fp(k)1080 2470 y(p)1102 2459 y Fu(\))14 b Fs(;)99 b(\022)14 b Fr(2)f(N)p 1367 2466 2 18 v 1358 2466 a Fp(C)59 2561 y Fu(F)l(rom)j(this)i(w)o(e)f(ha)o(v)o(e)g(completely)h(analogous)f(to) f(the)i(pro)q(of)e(of)h(Lemma)g(7)g(b)o(y)g(Cauc)o(h)o(y)g(inequalit)o (y)59 2617 y(and)e(k)o(ernel)h(theorem)f(the)h(represen)o(tation)728 2747 y Fs(F)6 b Fu(\()p Fs(\022)q Fu(\))13 b(=)898 2694 y Fq(1)886 2707 y Fj(X)882 2797 y Fp(n)p Fo(=0)949 2747 y Fr(h)p Fu(\010)1000 2728 y Fo(\()p Fp(n)p Fo(\))1050 2747 y Fs(;)8 b(\022)1093 2728 y Fq(\012)p Fp(n)1144 2747 y Fr(i)p eop %%Page: 35 35 35 34 bop 59 50 a Fn(9)45 b(THE)15 b(WICK)h(PR)o(ODUCT)1165 b Fu(35)59 199 y(and)15 b(the)h(b)q(ound)393 253 y Fj(\014)393 278 y(\014)393 302 y(\014)407 301 y Fu(\010)440 283 y Fo(\()p Fp(n)p Fo(\))491 253 y Fj(\014)491 278 y(\014)491 302 y(\014)504 329 y Fq(\000)p Fp(p)549 320 y Fh(0)576 301 y Fr(\024)d Fu(n)649 308 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(l;k)773 301 y Fu(\()p Fs(F)6 b Fu(\))13 b(\()p Fs(n)p Fu(!\))933 283 y Fq(\000)p Fo(1)p Fp(=k)1024 254 y Fj(n)1051 301 y Fu(\()p Fs(k)q Fu(2)1117 283 y Fp(l)1130 301 y Fu(\))1148 283 y Fo(1)p Fp(=k)1204 301 y Fs(e)1233 265 y Fj(\015)1233 290 y(\015)1256 301 y Fs(i)1272 309 y Fp(p)1290 300 y Fh(0)1301 309 y Fp(;p)1330 265 y Fj(\015)1330 290 y(\015)1353 318 y Fp(H)s(S)1410 254 y Fj(o)1438 266 y Fp(n)1484 301 y Fs(;)59 421 y Fu(where)i Fs(p)213 404 y Fq(0)238 421 y Fs(>)e(p)i Fu(is)g(suc)o(h)h(that)e Fs(i)586 428 y Fp(p)604 419 y Fh(0)615 428 y Fp(;p)657 421 y Fu(:)e Fr(H)720 428 y Fp(p)738 419 y Fh(0)765 421 y Fs(,)-8 b Fr(!)13 b(H)866 428 y Fp(p)901 421 y Fu(is)j(Hilb)q(ert{Sc) o(hmidt.)22 b(Using)15 b(this)h(w)o(e)f(ha)o(v)o(e)219 556 y Fr(j)-8 b(j)g(j)o Fs(F)6 b Fr(j)-7 b(j)f(j)300 535 y Fo(2)300 569 y Fq(\000)p Fp(p)345 560 y Fh(0)356 569 y Fp(;)p Fq(\000)p Fp(q)q(;)p Fq(\000)p Fp(\014)513 556 y Fu(=)605 503 y Fq(1)593 515 y Fj(X)589 606 y Fp(n)p Fo(=0)656 556 y Fu(\()p Fs(n)p Fu(!\))732 537 y Fo(1)p Fq(\000)p Fp(\014)800 556 y Fu(2)823 537 y Fq(\000)p Fp(q)q(n)898 507 y Fj(\014)898 532 y(\014)898 557 y(\014)912 556 y Fu(\010)945 537 y Fo(\()p Fp(n)p Fo(\))996 507 y Fj(\014)996 532 y(\014)996 557 y(\014)1009 520 y Fo(2)1009 584 y Fq(\000)p Fp(p)1054 575 y Fh(0)513 698 y Fr(\024)41 b Fu(n)614 679 y Fo(2)614 709 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(l;k)739 698 y Fu(\()p Fs(F)6 b Fu(\))833 645 y Fq(1)821 658 y Fj(X)818 748 y Fp(n)p Fo(=0)884 698 y Fu(\()p Fs(n)p Fu(!\))960 679 y Fo(1)p Fq(\000)p Fp(\014)r Fq(\000)p Fo(2)p Fp(=k)1110 698 y Fu(2)1133 679 y Fq(\000)p Fp(q)q(n)1208 651 y Fj(n)1236 698 y Fu(\()p Fs(k)q Fu(2)1302 679 y Fp(l)1314 698 y Fu(\))1332 679 y Fo(1)p Fp(=k)1389 698 y Fs(e)1418 662 y Fj(\015)1417 687 y(\015)1441 698 y Fs(i)1457 706 y Fp(p)1475 696 y Fh(0)1485 706 y Fp(;p)1515 662 y Fj(\015)1515 687 y(\015)1538 714 y Fp(H)s(S)1595 651 y Fj(o)1622 663 y Fo(2)p Fp(n)513 840 y Fr(\024)41 b Fu(n)614 822 y Fo(2)614 852 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(l;k)739 840 y Fu(\()p Fs(F)6 b Fu(\))833 788 y Fq(1)821 800 y Fj(X)818 891 y Fp(n)p Fo(=0)892 840 y Fs(\032)916 822 y Fp(n)59 989 y Fu(where)18 b(w)o(e)f(ha)o(v)o(e)g(set)h Fs(\032)e Fu(:=)g(2)570 973 y Fq(\000)p Fp(q)q Fo(+2)p Fp(l=k)709 989 y Fs(k)734 973 y Fo(2)p Fp(=k)791 989 y Fs(e)812 973 y Fo(2)840 953 y Fj(\015)840 978 y(\015)863 989 y Fs(i)879 997 y Fp(p)897 987 y Fh(0)907 997 y Fp(;p)937 953 y Fj(\015)937 978 y(\015)960 966 y Fo(2)960 1005 y Fp(H)s(S)1034 989 y Fu(.)27 b(If)18 b Fs(q)h Fr(2)e Fs(I)-12 b(N)22 b Fu(is)c(c)o(hosen)g(large)f(enough)h(suc)o(h)59 1045 y(that)d Fs(\032)d(<)h Fu(1)i(the)g(sum)g(on)g(the)g(righ)o(t)g (hand)h(side)g(is)g(con)o(v)o(ergen)o(t)e(and)i(w)o(e)f(ha)o(v)o(e)487 1148 y Fr(j)-8 b(j)g(j)o Fs(F)6 b Fr(j)-7 b(j)f(j)568 1161 y Fq(\000)p Fp(p)613 1152 y Fh(0)624 1161 y Fp(;)p Fq(\000)p Fp(q)q(;)p Fq(\000)p Fp(\014)752 1148 y Fr(\024)13 b Fu(n)825 1157 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(l;)p Fo(2)p Fp(=)p Fo(\(1)p Fq(\000)p Fp(\014)r Fo(\))1060 1148 y Fu(\()p Fs(F)6 b Fu(\))k Fr(\001)f Fu(\(1)h Fr(\000)g Fs(\032)p Fu(\))1301 1129 y Fq(\000)p Fo(1)p Fp(=)p Fo(2)1398 1148 y Fs(:)339 b Fu(\(29\))59 1250 y(That)15 b(means)637 1306 y Fr(E)665 1287 y Fo(2)p Fp(=)p Fo(\(1)p Fq(\000)p Fp(\014)r Fo(\))661 1317 y(max)796 1306 y Fu(\()p Fr(N)p 857 1313 2 18 v 848 1313 a Fp(C)878 1306 y Fu(\))d Fr(\032)k Fu(ind)g(lim)968 1336 y Fp(p;q)q Fq(2)p Fp(I)-9 b(N)1074 1341 y Fi(0)1120 1306 y Fu(E)1151 1284 y Fq(\000)p Fp(\014)1151 1317 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)59 1409 y Fu(top)q(ologically)l (.)130 1466 y(If)15 b(w)o(e)g(set)766 1541 y(\010)d(:=)887 1488 y Fq(1)875 1501 y Fj(X)872 1591 y Fp(n)p Fo(=0)946 1541 y Fs(Q)982 1522 y Fp(\026)982 1552 y(n)1005 1541 y Fu(\(\010)1056 1522 y Fo(\()p Fp(n)p Fo(\))1107 1541 y Fu(\))59 1672 y(then)k Fs(S)191 1679 y Fp(\026)214 1672 y Fu(\010)c(=)h Fs(F)22 b Fu(and)15 b(\010)e Fr(2)g Fu(\()p Fr(H)591 1679 y Fq(\000)p Fp(p)636 1670 y Fh(0)649 1672 y Fu(\))667 1650 y Fq(\000)p Fp(\014)667 1683 y Fq(\000)p Fp(q)733 1672 y Fu(since)726 1757 y Fq(1)714 1770 y Fj(X)711 1860 y Fp(n)p Fo(=0)777 1810 y Fu(\()p Fs(n)p Fu(!\))853 1792 y Fo(1)p Fq(\000)p Fp(\014)921 1810 y Fu(2)944 1792 y Fq(\000)p Fp(q)q(n)1012 1810 y Fr(j)p Fu(\010)1058 1792 y Fo(\()p Fp(n)p Fo(\))1108 1810 y Fr(j)1121 1792 y Fo(2)1121 1822 y Fq(\000)p Fp(p)1166 1812 y Fh(0)59 1946 y Fu(is)j(\014nite.)21 b(Hence)670 2002 y Fs(S)698 2009 y Fp(\026)734 2002 y Fu(:)12 b(\()p Fr(N)7 b Fu(\))839 1984 y Fq(\000)p Fp(\014)839 2014 y(\026)902 2002 y Fr(!)14 b(E)989 1984 y Fo(2)p Fp(=)p Fo(\(1)p Fq(\000)p Fp(\014)r Fo(\))985 2014 y(max)1120 2002 y Fu(\()p Fr(N)p 1181 2009 V 1172 2009 a Fp(C)1202 2002 y Fu(\))59 2086 y(is)21 b(one)g(to)f(one.)37 b(The)21 b(con)o(tin)o(uit)o(y)h(of)e(the)h(in)o(v)o(erse)g(mapping)h(follo)o (ws)e(from)g(the)h(norm)g(estimate)59 2142 y(\(29\).)1628 b Ff(2)59 2286 y Ft(9)67 b(The)22 b(Wic)n(k)i(pro)r(duct)59 2387 y Fu(In)d(Gaussian)f(Analysis)h(it)f(has)g(b)q(een)i(sho)o(wn)d (that)h(\()p Fr(N)7 b Fu(\))1083 2371 y Fq(\000)p Fo(1)1083 2398 y Fp(\015)1101 2404 y Fh(H)1151 2387 y Fu(\(and)20 b(other)f(distribution)j(spaces\))e(is)59 2443 y(closed)j(under)g(so)f (called)i(Wic)o(k)e(m)o(ultiplication)j(\(see)d([KLS94])g(and)g ([BeS95,)f(\037k95,)h(V)l(a95])f(for)59 2500 y(applications\).)g(This) 16 b(concept)g(has)f(a)g(natural)g(generalization)h(to)f(the)g(presen)o (t)g(setting.)59 2606 y Fk(Definition)h(37)40 b Fm(L)n(et)15 b Fu(\010)p Fs(;)8 b Fu(\011)k Fr(2)17 b Fu(\()p Fr(N)7 b Fu(\))712 2590 y Fq(\000)p Fo(1)712 2617 y Fp(\026)758 2606 y Fm(.)21 b(Then)15 b(we)i(de\014ne)e(the)i(Wick)f(pr)n(o)n(duct)h Fu(\010)10 b Fr(\005)g Fu(\011)p Fm(by)700 2708 y Fs(S)728 2715 y Fp(\026)751 2708 y Fu(\(\010)f Fr(\005)h Fu(\011\))j(=)g Fs(S)986 2715 y Fp(\026)1009 2708 y Fu(\010)d Fr(\001)g Fs(S)1103 2715 y Fp(\026)1126 2708 y Fu(\011)17 b Fs(:)p eop %%Page: 36 36 36 35 bop 59 50 a Fn(9)45 b(THE)15 b(WICK)h(PR)o(ODUCT)1165 b Fu(36)130 199 y(This)17 b(is)g(w)o(ell)g(de\014ned)h(b)q(ecause)f (Hol)772 206 y Fo(0)791 199 y Fu(\()p Fr(N)p 852 206 2 18 v 843 206 a Fp(C)873 199 y Fu(\))f(is)h(an)f(algebra)h(and)f(th)o (us)h(b)o(y)f(the)h(c)o(haracterization)59 256 y(Theorem)e(34)g(there)g (exists)h(an)f(elemen)o(t)h(\010)9 b Fr(\005)h Fu(\011)j Fr(2)g Fu(\()p Fr(N)7 b Fu(\))1031 239 y Fq(\000)p Fo(1)1031 267 y Fp(\026)1093 256 y Fu(suc)o(h)15 b(that)g Fs(S)1322 263 y Fp(\026)1345 256 y Fu(\(\010)10 b Fr(\005)g Fu(\011\))i(=)h Fs(S)1580 263 y Fp(\026)1603 256 y Fu(\010)d Fr(\001)g Fs(S)1697 263 y Fp(\026)1720 256 y Fu(\011.)130 312 y(By)15 b(de\014nition)i(w)o(e)e(ha)o(v)o(e)505 414 y Fs(Q)541 396 y Fp(\026)541 426 y(n)565 414 y Fu(\(\010)616 396 y Fo(\()p Fp(n)p Fo(\))666 414 y Fu(\))10 b Fr(\005)g Fs(Q)763 396 y Fp(\026)763 426 y(m)796 414 y Fu(\(\011)849 396 y Fo(\()p Fp(m)p Fo(\))910 414 y Fu(\))i(=)h Fs(Q)1024 392 y Fp(\026)1024 425 y(n)p Fo(+)p Fp(m)1106 414 y Fu(\(\010)1157 396 y Fo(\()p Fp(n)p Fo(\))1214 407 y Fu(^)1208 414 y Fr(\012)p Fu(\011)1278 396 y Fo(\()p Fp(m)p Fo(\))1339 414 y Fu(\))i(,)59 527 y(\010)92 510 y Fo(\()p Fp(n)p Fo(\))155 527 y Fr(2)e Fu(\()p Fr(N)265 501 y Fo(^)260 507 y Fq(\012)p Fp(n)p 259 540 V 250 540 a(C)311 527 y Fu(\))329 510 y Fq(0)355 527 y Fu(and)h(\011)477 510 y Fo(\()p Fp(m)p Fo(\))551 527 y Fr(2)f Fu(\()p Fr(N)661 501 y Fo(^)656 507 y Fq(\012)p Fp(n)p 655 540 V 646 540 a(C)707 527 y Fu(\))725 510 y Fq(0)736 527 y Fu(.)20 b(So)14 b(in)h(terms)f(of)p 1075 527 3 25 v 17 w Fs(Q)p Fu({decomp)q(ositions)i(\010)c(=)1532 494 y Fj(P)1576 508 y Fq(1)1576 538 y Fp(n)p Fo(=0)1652 527 y Fs(Q)1688 510 y Fp(\026)1688 538 y(n)1712 527 y Fu(\(\010)1763 510 y Fo(\()p Fp(n)p Fo(\))1813 527 y Fu(\))59 583 y(and)j(\011)e(=)243 551 y Fj(P)287 564 y Fq(1)287 594 y Fp(n)p Fo(=0)363 583 y Fs(Q)399 567 y Fp(\026)399 594 y(n)423 583 y Fu(\(\011)476 567 y Fo(\()p Fp(n)p Fo(\))527 583 y Fu(\))i(the)g(Wic)o(k)g(pro)q (duct)h(is)g(giv)o(en)f(b)o(y)734 716 y(\010)10 b Fr(\005)g Fu(\011)j(=)921 663 y Fq(1)909 675 y Fj(X)906 766 y Fp(n)p Fo(=0)980 716 y Fs(Q)1016 723 y Fp(n)1039 716 y Fu(\(\004)1087 697 y Fo(\()p Fp(n)p Fo(\))1138 716 y Fu(\))59 851 y(where)713 927 y(\004)743 908 y Fo(\()p Fp(n)p Fo(\))807 927 y Fu(=)876 874 y Fp(n)857 886 y Fj(X)855 979 y Fp(k)q Fo(=0)927 927 y Fu(\010)960 908 y Fo(\()p Fp(k)q Fo(\))1015 920 y Fu(^)1008 927 y Fr(\012)q Fu(\011)1079 908 y Fo(\()p Fp(n)p Fq(\000)p Fp(k)q Fo(\))59 1045 y Fu(This)j(allo)o(ws)f(for)g (concrete)g(norm)g(estimates.)59 1152 y Fk(Pr)o(oposition)h(38)23 b Fm(The)17 b(Wick)h(pr)n(o)n(duct)g(is)e(c)n(ontinuous)h(on)g Fu(\()p Fr(N)7 b Fu(\))1224 1135 y Fq(\000)p Fo(1)1224 1163 y Fp(\026)1271 1152 y Fm(.)23 b(In)17 b(p)n(articular)h(the)f(fol) r(lowing)59 1214 y(estimate)f(holds)g(for)g Fu(\010)c Fr(2)h Fu(\()p Fr(H)572 1221 y Fq(\000)p Fp(p)617 1226 y Fi(1)637 1214 y Fu(\))655 1195 y Fq(\000)p Fo(1)655 1225 y Fq(\000)p Fp(q)698 1230 y Fi(1)715 1225 y Fp(;\026)764 1214 y Fs(;)23 b Fu(\011)13 b Fr(2)g Fu(\()p Fr(H)947 1221 y Fq(\000)p Fp(q)990 1226 y Fi(2)1009 1214 y Fu(\))1027 1195 y Fq(\000)p Fo(1)1027 1225 y Fq(\000)p Fp(q)1070 1230 y Fi(2)1105 1214 y Fm(and)j Fs(p)d Fu(=)g(max)o(\()p Fs(p)1402 1221 y Fo(1)1422 1214 y Fs(;)8 b(p)1466 1221 y Fo(2)1484 1214 y Fu(\))p Fs(;)23 b(q)15 b Fu(=)e Fs(q)1641 1221 y Fo(1)1670 1214 y Fu(+)c Fs(q)1734 1221 y Fo(2)1763 1214 y Fu(+)h(1)502 1316 y Fr(k)p Fu(\010)g Fr(\005)f Fu(\011)p Fr(k)658 1330 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)q(;\026)804 1316 y Fu(=)k Fr(k)o Fu(\010)p Fr(k)930 1330 y Fq(\000)p Fp(p)975 1335 y Fi(1)992 1330 y Fp(;)p Fq(\000)p Fp(q)1045 1335 y Fi(1)1063 1330 y Fp(;\026)1103 1316 y Fr(k)p Fu(\011)p Fr(k)1184 1330 y Fq(\000)p Fp(p)1229 1335 y Fi(2)1247 1330 y Fp(;)p Fq(\000)p Fp(q)1300 1335 y Fi(2)1317 1330 y Fp(;\026)1374 1316 y Fm(.)59 1422 y Fk(Pr)o(oof.)72 b Fu(W)l(e)16 b(can)f(estimate)g(as)g(follo)o(ws)303 1544 y Fr(k)p Fu(\010)10 b Fr(\005)g Fu(\011)p Fr(k)460 1523 y Fo(2)460 1557 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)q(;\026)634 1544 y Fu(=)726 1491 y Fq(1)714 1503 y Fj(X)711 1594 y Fp(n)p Fo(=0)785 1544 y Fu(2)808 1525 y Fq(\000)p Fp(nq)883 1495 y Fj(\014)883 1520 y(\014)883 1545 y(\014)897 1544 y Fu(\004)927 1525 y Fo(\()p Fp(n)p Fo(\))978 1495 y Fj(\014)978 1520 y(\014)978 1545 y(\014)992 1508 y Fo(2)992 1572 y Fq(\000)p Fp(p)634 1696 y Fu(=)726 1644 y Fq(1)714 1656 y Fj(X)711 1747 y Fp(n)p Fo(=0)785 1696 y Fu(2)808 1678 y Fq(\000)p Fp(nq)883 1624 y Fj( )937 1644 y Fp(n)918 1656 y Fj(X)916 1748 y Fp(k)q Fo(=0)988 1648 y Fj(\014)988 1673 y(\014)988 1698 y(\014)1002 1696 y Fu(\010)1035 1678 y Fo(\()p Fp(k)q Fo(\))1083 1648 y Fj(\014)1083 1673 y(\014)1083 1698 y(\014)1097 1724 y Fq(\000)p Fp(p)1152 1648 y Fj(\014)1152 1673 y(\014)1152 1698 y(\014)1166 1696 y Fu(\011)1201 1678 y Fo(\()p Fp(n)p Fq(\000)p Fp(k)q Fo(\))1299 1648 y Fj(\014)1299 1673 y(\014)1299 1698 y(\014)1313 1724 y Fq(\000)p Fp(p)1360 1624 y Fj(!)1393 1636 y Fo(2)634 1840 y Fr(\024)726 1787 y Fq(1)714 1800 y Fj(X)711 1890 y Fp(n)p Fo(=0)785 1840 y Fu(2)808 1822 y Fq(\000)p Fp(nq)883 1840 y Fu(\()p Fs(n)g Fu(+)h(1\))1053 1787 y Fp(n)1034 1800 y Fj(X)1031 1892 y Fp(k)q Fo(=0)1103 1792 y Fj(\014)1103 1816 y(\014)1103 1841 y(\014)1117 1840 y Fu(\010)1150 1822 y Fo(\()p Fp(k)q Fo(\))1199 1792 y Fj(\014)1199 1816 y(\014)1199 1841 y(\014)1213 1805 y Fo(2)1213 1868 y Fq(\000)p Fp(p)1268 1792 y Fj(\014)1268 1816 y(\014)1268 1841 y(\014)1281 1840 y Fu(\011)1316 1822 y Fo(\()p Fp(n)p Fq(\000)p Fp(k)q Fo(\))1414 1792 y Fj(\014)1414 1816 y(\014)1414 1841 y(\014)1428 1805 y Fo(2)1428 1868 y Fq(\000)p Fp(p)634 1984 y Fr(\024)726 1931 y Fq(1)714 1944 y Fj(X)711 2034 y Fp(n)p Fo(=0)806 1931 y Fp(n)787 1944 y Fj(X)785 2036 y Fp(k)q Fo(=0)857 1984 y Fu(2)880 1965 y Fq(\000)p Fp(nq)944 1970 y Fi(1)971 1935 y Fj(\014)971 1960 y(\014)971 1985 y(\014)985 1984 y Fu(\010)1018 1965 y Fo(\()p Fp(n)p Fo(\))1069 1935 y Fj(\014)1069 1960 y(\014)1069 1985 y(\014)1082 1949 y Fo(2)1082 2012 y Fq(\000)p Fp(p)1137 1984 y Fu(2)1160 1965 y Fq(\000)p Fp(nq)1224 1970 y Fi(2)1252 1935 y Fj(\014)1252 1960 y(\014)1252 1985 y(\014)1265 1984 y Fu(\011)1300 1965 y Fo(\()p Fp(n)p Fq(\000)p Fp(k)q Fo(\))1398 1935 y Fj(\014)1398 1960 y(\014)1398 1985 y(\014)1412 1949 y Fo(2)1412 2012 y Fq(\000)p Fp(p)634 2130 y Fr(\024)711 2058 y Fj( )759 2078 y Fq(1)747 2090 y Fj(X)744 2181 y Fp(n)p Fo(=0)818 2130 y Fu(2)841 2112 y Fq(\000)p Fp(nq)905 2117 y Fi(1)932 2082 y Fj(\014)932 2107 y(\014)932 2132 y(\014)946 2130 y Fu(\010)979 2112 y Fo(\()p Fp(k)q Fo(\))1027 2082 y Fj(\014)1027 2107 y(\014)1027 2132 y(\014)1041 2095 y Fo(2)1041 2159 y Fq(\000)p Fp(p)1086 2164 y Fi(1)1106 2058 y Fj(!)c( )1195 2078 y Fq(1)1182 2090 y Fj(X)1179 2181 y Fp(n)p Fo(=0)1253 2130 y Fu(2)1276 2112 y Fq(\000)p Fp(nq)1340 2117 y Fi(2)1367 2082 y Fj(\014)1367 2107 y(\014)1367 2132 y(\014)1381 2130 y Fu(\011)1416 2112 y Fo(\()p Fp(n)p Fo(\))1468 2082 y Fj(\014)1468 2107 y(\014)1468 2132 y(\014)1481 2095 y Fo(2)1481 2159 y Fq(\000)p Fp(p)1526 2164 y Fi(2)1546 2058 y Fj(!)634 2244 y Fu(=)42 b Fr(k)o Fu(\010)p Fr(k)789 2223 y Fo(2)789 2257 y Fq(\000)p Fp(p)834 2262 y Fi(1)852 2257 y Fp(;)p Fq(\000)p Fp(q)905 2262 y Fi(1)922 2257 y Fp(;\026)963 2244 y Fr(k)o Fu(\011)p Fr(k)1043 2223 y Fo(2)1043 2257 y Fq(\000)p Fp(p)1088 2262 y Fi(2)1106 2257 y Fp(;)p Fq(\000)p Fp(q)1159 2262 y Fi(2)1176 2257 y Fp(;\026)1232 2244 y Fu(.)1797 2346 y Ff(2)130 2452 y Fu(Similar)17 b(to)d(the)i(Gaussian)g(case)f(the)h(sp)q(ecial)h(prop)q(erties)f(of)f (the)h(space)g(\()p Fr(N)7 b Fu(\))1487 2436 y Fq(\000)p Fo(1)1487 2463 y Fp(\026)1549 2452 y Fu(allo)o(w)15 b(the)h(de\014-)59 2509 y(nition)h(of)e Fm(Wick)i(analytic)f(functions)g Fu(under)g(v)o(ery)f(general)h(assumptions.)21 b(This)16 b(has)g(pro)o(v)o(en)f(to)g(b)q(e)59 2565 y(of)j(some)g(relev)m(ance)i (to)e(solv)o(e)g(equations)h(e.g.,)f(of)g(the)g(t)o(yp)q(e)h(\010)12 b Fr(\005)g Fs(X)21 b Fu(=)d(\011)h(for)f Fs(X)j Fr(2)d Fu(\()p Fr(N)7 b Fu(\))1658 2549 y Fq(\000)p Fo(1)1658 2576 y Fp(\026)1723 2565 y Fu(.)29 b(See)59 2621 y([KLS94])15 b(for)f(the)i(Gaussian)f(case.)59 2728 y Fk(Theorem)j(39)23 b Fm(L)n(et)11 b Fs(F)20 b Fu(:)p 518 2728 V 16 w Fs(C)15 b Fr(!)p 629 2728 V 18 w Fs(C)g Fm(b)n(e)d(analytic)g(in)g(a)h(neighb)n (orho)n(o)n(d)f(of)h(the)g(p)n(oint)f Fs(z)1493 2735 y Fo(0)1526 2728 y Fu(=)h Fs(I)-12 b(E)1618 2735 y Fp(\026)1641 2728 y Fu(\(\010\))12 b Fs(;)20 b Fu(\010)12 b Fr(2)59 2784 y Fu(\()p Fr(N)7 b Fu(\))139 2768 y Fq(\000)p Fo(1)139 2795 y Fp(\026)186 2784 y Fm(.)20 b(Then)c Fs(F)374 2768 y Fq(\005)394 2784 y Fu(\(\010\))g Fm(de\014ne)n(d)f(by)h Fs(S)719 2791 y Fp(\026)742 2784 y Fu(\()p Fs(F)795 2768 y Fq(\005)815 2784 y Fu(\(\010\)\))c(=)h Fs(F)6 b Fu(\()p Fs(S)1043 2791 y Fp(\026)1066 2784 y Fu(\010\))16 b Fm(exists)f(in)h Fu(\()p Fr(N)7 b Fu(\))1393 2768 y Fq(\000)p Fo(1)1456 2784 y Fm(.)p eop %%Page: 37 37 37 36 bop 59 50 a Fn(10)45 b(POSITIVE)16 b(DISTRIBUTIONS)1031 b Fu(37)59 199 y Fk(Pr)o(oof.)96 b Fu(By)23 b(the)g(c)o (haracterization)g(Theorem)g(34)f Fs(S)1113 206 y Fp(\026)1136 199 y Fu(\010)j Fr(2)h Fu(Hol)1319 206 y Fo(0)1339 199 y Fu(\()p Fr(N)p 1400 206 2 18 v 1391 206 a Fp(C)1421 199 y Fu(\).)42 b(Then)23 b Fs(F)6 b Fu(\()p Fs(S)1701 206 y Fp(\026)1725 199 y Fu(\010\))24 b Fr(2)59 256 y Fu(Hol)128 263 y Fo(0)148 256 y Fu(\()p Fr(N)p 209 263 V 200 263 a Fp(C)230 256 y Fu(\))16 b(since)h(the)f(comp)q(osition)h (of)e(t)o(w)o(o)g(analytic)i(functions)g(is)f(also)g(analytic.)23 b(Again)17 b(b)o(y)f(c)o(ha-)59 312 y(racterization)f(Theorem)h(w)o(e)e (\014nd)i Fs(F)709 296 y Fq(\005)730 312 y Fu(\(\010\))11 b Fr(2)i Fu(\()p Fr(N)7 b Fu(\))933 296 y Fq(\000)p Fo(1)933 323 y Fp(\026)980 312 y Fs(:)804 b Ff(2)59 418 y Fk(Remark.)73 b Fu(If)15 b Fs(F)6 b Fu(\()p Fs(z)r Fu(\))13 b(=)526 386 y Fj(P)570 399 y Fq(1)570 430 y Fp(n)p Fo(=0)646 418 y Fs(a)670 425 y Fp(k)691 418 y Fu(\()p Fs(z)f Fr(\000)f Fs(z)809 425 y Fo(0)828 418 y Fu(\))846 402 y Fp(n)885 418 y Fu(then)k(the)g Fm(Wick)i(series)1311 386 y Fj(P)1355 399 y Fq(1)1355 430 y Fp(n)p Fo(=0)1431 418 y Fs(a)1455 425 y Fp(k)1476 418 y Fu(\(\010)10 b Fr(\000)g Fs(z)1603 425 y Fo(0)1623 418 y Fu(\))1641 402 y Fq(\005)p Fp(n)1697 418 y Fu(\(where)59 475 y(\011)94 458 y Fq(\005)p Fp(n)148 475 y Fu(=)j(\011)e Fr(\005)e Fs(:)f(:)g(:)h Fr(\005)g Fu(\011)16 b(n-times)g(con)o(v)o(erges)e(in)i(\()p Fr(N)7 b Fu(\))919 458 y Fq(\000)p Fo(1)981 475 y Fu(and)15 b Fs(F)1104 458 y Fq(\005)1125 475 y Fu(\(\010\))d(=)1254 443 y Fj(P)1297 456 y Fq(1)1297 486 y Fp(n)p Fo(=0)1374 475 y Fs(a)1398 482 y Fp(k)1419 475 y Fu(\(\010)d Fr(\000)i Fs(z)1546 482 y Fo(0)1566 475 y Fu(\))1584 458 y Fq(\005)p Fp(n)1640 475 y Fu(holds.)59 581 y Fk(Example)j(9)g Fu(The)e(ab)q(o)o (v)o(e)f(men)o(tioned)h(equation)h(\010)s Fr(\005)s Fs(X)i Fu(=)e(\011)f(can)g(b)q(e)h(solv)o(ed)f(if)g Fs(I)-12 b(E)1489 588 y Fp(\026)1512 581 y Fu(\(\010\))12 b(=)h Fs(S)1669 588 y Fp(\026)1692 581 y Fu(\010\(0\))e Fr(6)p Fu(=)59 638 y(0.)33 b(That)19 b(implies)i(\()p Fs(S)450 645 y Fp(\026)473 638 y Fu(\010\))524 621 y Fq(\000)p Fo(1)591 638 y Fr(2)f Fu(Hol)710 645 y Fo(0)730 638 y Fu(\()p Fr(N)p 791 645 V 782 645 a Fp(C)812 638 y Fu(\).)32 b(Th)o(us)20 b(\010)1028 621 y Fq(\005)p Fo(\()p Fq(\000)p Fo(1\))1140 638 y Fu(=)g Fs(S)1226 621 y Fq(\000)p Fo(1)1223 649 y Fp(\026)1280 603 y Fj(\000)1299 638 y Fu(\()p Fs(S)1345 645 y Fp(\026)1368 638 y Fu(\010\))1419 621 y Fq(\000)p Fo(1)1466 603 y Fj(\001)1505 638 y Fr(2)g Fu(\()p Fr(N)7 b Fu(\))1635 621 y Fq(\000)p Fo(1)1635 649 y Fp(\026)1681 638 y Fu(.)33 b(Then)59 701 y Fs(X)24 b Fu(=)d(\010)210 684 y Fq(\005)p Fo(\()p Fq(\000)p Fo(1\))316 701 y Fr(\005)13 b Fu(\011)20 b(is)h(the)f(solution)h(in)g(\()p Fr(N)7 b Fu(\))856 684 y Fq(\000)p Fo(1)856 712 y Fp(\026)902 701 y Fu(.)35 b(F)l(or)19 b(more)h(instructiv)o(e)h(examples)g(w)o(e)f (refer)g(the)59 757 y(reader)15 b(to)g([KLS94].)59 901 y Ft(10)67 b(P)n(ositiv)n(e)23 b(distributions)59 1002 y Fu(In)e(this)f(section)h(w)o(e)f(will)i(c)o(haracterize)e(the)g(p)q (ositiv)o(e)h(distributions)h(in)f(\()p Fr(N)7 b Fu(\))1455 986 y Fq(\000)p Fo(1)1455 1013 y Fp(\026)1502 1002 y Fu(.)34 b(W)l(e)20 b(will)i(pro)o(v)o(e)59 1058 y(that)c(the)g(p)q (ositiv)o(e)h(distributions)h(can)f(b)q(e)g(represen)o(ted)g(b)o(y)f (measures)g(in)i Fr(M)1455 1065 y Fp(a)1475 1058 y Fu(\()p Fr(N)1537 1042 y Fq(0)1548 1058 y Fu(\).)29 b(In)19 b(the)g(case)59 1115 y(of)g(the)h(Gaussian)f(Hida)h(distribution)i(space)d(\()p Fr(S)s Fu(\))956 1098 y Fq(0)987 1115 y Fu(similar)h(statemen)o(ts)f (can)g(b)q(e)i(found)e(in)i(w)o(orks)59 1171 y(of)h(Kondratiev)i ([Ko80a)n(,)h(b])d(and)h(Y)l(ok)o(oi)g([Y)l(ok90)o(,)f(Y)l(ok93)o(],)i (see)f(also)g([P)o(o87)n(])f(and)h([Lee91].)42 b(In)59 1228 y(the)21 b(Gaussian)g(setting)g(also)g(the)g(p)q(ositiv)o(e)h (distributions)g(in)g(\()p Fr(N)7 b Fu(\))1264 1211 y Fq(\000)p Fo(1)1331 1228 y Fu(ha)o(v)o(e)21 b(b)q(een)h(discussed,)h (see)59 1284 y([KoSW95)o(].)130 1341 y(Since)14 b(\()p Fr(N)7 b Fu(\))326 1324 y Fo(1)358 1341 y Fu(=)13 b Fr(E)434 1324 y Fo(1)430 1353 y(min)491 1341 y Fu(\()p Fr(N)553 1324 y Fq(0)564 1341 y Fu(\))f(w)o(e)h(sa)o(y)f(that)g Fs(')h Fr(2)g Fu(\()p Fr(N)7 b Fu(\))997 1324 y Fo(1)1023 1341 y Fu(is)14 b(p)q(ositiv)o(e)g(\()p Fs(')e Fr(\025)h Fu(0\))f(if)h(and)g(only)h(if)f Fs(')p Fu(\()p Fs(x)p Fu(\))f Fr(\025)h Fu(0)59 1397 y(for)i(all)h Fs(x)c Fr(2)h(N)317 1381 y Fq(0)329 1397 y Fu(.)59 1503 y Fk(Definition)j(40)23 b Fm(A)o(n)18 b(element)f Fu(\010)f Fr(2)h Fu(\()p Fr(N)7 b Fu(\))812 1487 y Fq(\000)p Fo(1)812 1515 y Fp(\026)877 1503 y Fm(is)18 b(p)n(ositive)g(if)g(for)g(any)h(p)n(ositive)e Fs(')g Fr(2)f Fu(\()p Fr(N)7 b Fu(\))1642 1487 y Fo(1)1673 1503 y Fm(we)18 b(have)59 1566 y Fr(h)-8 b(h)p Fu(\010)p Fs(;)8 b(')p Fr(i)-9 b(i)197 1579 y Fp(\026)233 1566 y Fr(\025)13 b Fu(0)j Fm(.)21 b(The)16 b(c)n(one)g(of)g(p)n(ositive)g (elements)f(in)h Fu(\()p Fr(N)7 b Fu(\))1093 1549 y Fq(\000)p Fo(1)1093 1577 y Fp(\026)1155 1566 y Fm(is)16 b(denote)n(d)g(by)h Fu(\()p Fr(N)7 b Fu(\))1510 1547 y Fq(\000)p Fo(1)1510 1577 y Fp(\026;)p Fo(+)1569 1566 y Fm(.)59 1680 y Fk(Theorem)18 b(41)23 b Fm(L)n(et)18 b Fu(\010)f Fr(2)h Fu(\()p Fr(N)7 b Fu(\))613 1661 y Fq(\000)p Fo(1)613 1691 y Fp(\026;)p Fo(+)691 1680 y Fm(.)29 b(Then)18 b(ther)n(e)h(exists)f(a)h(unique)g (me)n(asur)n(e)g Fs(\027)i Fr(2)c(M)1613 1687 y Fp(a)1634 1680 y Fu(\()p Fr(N)1696 1664 y Fq(0)1707 1680 y Fu(\))i Fm(such)59 1737 y(that)e Fr(8)p Fs(')c Fr(2)g Fu(\()p Fr(N)7 b Fu(\))343 1720 y Fo(1)675 1810 y Fr(h)-8 b(h)o Fu(\010)p Fs(;)8 b(')p Fr(i)-9 b(i)813 1823 y Fp(\026)849 1810 y Fu(=)897 1752 y Fj(Z)920 1846 y Fq(N)954 1837 y Fh(0)975 1810 y Fs(')p Fu(\()p Fs(x)p Fu(\))15 b(d)p Fs(\027)s Fu(\()p Fs(x)p Fu(\))h Fs(:)527 b Fu(\(30\))59 1918 y Fm(Vic)n(e)14 b(versa,)g(any)g(\(p)n(ositive\))f(me)n(asur)n(e)i Fs(\027)h Fr(2)d(M)887 1925 y Fp(a)907 1918 y Fu(\()p Fr(N)969 1902 y Fq(0)980 1918 y Fu(\))h Fm(de\014nes)f(a)i(p)n(ositive) f(distribution)g Fu(\010)f Fr(2)g Fu(\()p Fr(N)7 b Fu(\))1771 1899 y Fq(\000)p Fo(1)1771 1929 y Fp(\026;)p Fo(+)59 1975 y Fm(by)16 b(\(30\).)59 2081 y Fk(Remarks.)59 2150 y Fu(1.)31 b(F)l(or)18 b(a)h(giv)o(en)g(measure)g Fs(\027)j Fu(the)d(distribution)i(\010)d(ma)o(y)h(b)q(e)g(view)o(ed)h(as)e(the)h (generalized)i(Radon-)59 2206 y(Nik)o(o)q(dym)c(deriv)m(ativ)o(e)472 2188 y Fo(d)p Fp(\027)p 471 2195 41 2 v 471 2222 a Fo(d)p Fp(\026)534 2206 y Fu(of)f Fs(\027)k Fu(with)d(resp)q(ect)g(to)g Fs(\026)p Fu(.)24 b(In)18 b(fact)e(if)h Fs(\027)j Fu(is)e(absolutely)f (con)o(tin)o(uous)g(with)59 2263 y(resp)q(ect)f(to)e Fs(\026)i 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b(in)o(tegrable)g(with)59 2639 y(resp)q(ect)g(to)e Fs(\027)s Fu(.)p 59 2678 709 2 v 110 2705 a Fh(\003)128 2721 y Fc(Since)g(the)f(algebra)h(of)e(exp)q (onen)o(tial)j(functions)f(is)g(not)e(con)o(tained)j(in)e(\()p Fb(N)6 b Fc(\))1232 2705 y Fi(1)1232 2728 y Fg(\026)1265 2721 y Fc(w)o(e)12 b(cannot)i(use)e(Minlos')j(theorem)e(to)59 2767 y(construct)h(the)f(measure.)18 b(This)13 b(w)o(as)g(the)g(metho)q (d)h(used)g(in)f(Y)m(ok)o(oi's)h(w)o(ork)f([Y)m(ok90].)p eop %%Page: 38 38 38 37 bop 59 50 a Fn(10)45 b(POSITIVE)16 b(DISTRIBUTIONS)1031 b Fu(38)130 199 y(Since)16 b Fr(P)t Fu(\()p Fr(N)346 183 y Fq(0)357 199 y Fu(\))c Fr(\032)h Fu(\()p Fr(N)7 b Fu(\))515 178 y Fo(1)549 199 y Fu(w)o(e)15 b(ma)o(y)g(de\014ne)h (momen)o(ts)f(of)f(a)h(p)q(ositiv)o(e)h(distribution)h(\010)c Fr(2)g Fu(\()p Fr(N)7 b Fu(\))1714 183 y Fq(\000)p Fo(1)1714 210 y Fp(\026)1775 199 y Fu(b)o(y)276 328 y(M)318 335 y Fp(n)341 328 y Fu(\()p Fs(\030)379 335 y Fo(1)399 328 y Fs(;)h(:::;)g(\030)500 335 y Fp(n)520 328 y Fu(\))13 b(=)599 256 y Fj(*)-23 b(*)643 328 y Fu(\010)p Fs(;)731 275 y Fp(n)715 288 y Fj(Y)711 379 y Fp(j)r Fo(=1)780 328 y Fr(h)o(\001)p Fs(;)8 b(\030)851 335 y Fp(j)868 328 y Fr(i)886 256 y Fj(+)-22 b(+)930 392 y Fp(\026)976 328 y Fs(;)68 b(n)13 b Fr(2)g Fs(I)-12 b(N)5 b(;)52 b(\030)1275 335 y Fp(j)1306 328 y Fr(2)13 b(N)7 b Fs(;)15 b Fu(1)d Fr(\024)h Fs(j)i Fr(\024)e Fs(n)789 470 y Fu(M)831 477 y Fo(0)863 470 y Fu(=)g Fr(h)-8 b(h)p Fu(\010)p Fs(;)22 b Fu(1)-15 b(1)o Fr(i)-7 b(i)22 b Fu(.)59 547 y(W)l(e)14 b(w)o(an)o(t)f(to)g(get)g(estimates)g(on)h(the)g(momen)o(ts.)19 b(Since)c(\010)d Fr(2)h Fu(\()p Fr(H)1183 554 y Fq(\000)p Fp(p)1231 547 y Fu(\))1249 528 y Fq(\000)p Fo(1)1249 558 y Fq(\000)p Fp(q)q(;\026)1339 547 y Fu(for)g(some)h Fs(p;)8 b(q)14 b(>)f Fu(0)g(w)o(e)g(ma)o(y)59 604 y(estimate)i(as)g (follo)o(ws)351 654 y Fj(\014)351 679 y(\014)351 704 y(\014)351 729 y(\014)373 643 y(*)-23 b(*)417 715 y Fu(\010)p Fs(;)471 643 y Fj(*)503 715 y Fs(x)529 696 y Fq(\012)p Fp(n)580 715 y Fs(;)620 675 y Fp(n)608 683 y Fj(N)600 754 y Fp(j)r Fo(=1)669 715 y Fs(\030)689 722 y Fp(j)707 643 y Fj(+)q(+)g(+)785 768 y Fp(\026)816 654 y Fj(\014)816 679 y(\014)816 704 y(\014)816 729 y(\014)842 715 y Fr(\024)13 b(k)p Fu(\010)p Fr(k)968 729 y Fq(\000)p Fp(p;)p Fq(\000)p Fp(q)q(;\026)1109 642 y Fj(\015)1109 667 y(\015)1109 691 y(\015)1109 716 y(\015)1109 741 y(\015)1132 643 y(*)1165 715 y Fs(x)1191 696 y Fq(\012)p Fp(n)1242 715 y Fs(;)1282 675 y Fp(n)1270 683 y Fj(N)1262 754 y Fp(j)r Fo(=1)1331 715 y Fs(\030)1351 722 y Fp(j)1369 643 y Fj(+)1403 642 y(\015)1403 667 y(\015)1403 691 y(\015)1403 716 y(\015)1403 741 y(\015)1426 768 y Fp(p;q)q(;\026)1526 715 y Fu(.)59 847 y(T)l(o)i(pro)q(ceed)h(w)o(e)f(use)g(the)h(prop)q(ert)o(y)f(\(12\)) f(and)h(the)g(estimate)g(\(7\))g(to)f(obtain)248 904 y Fj(\015)248 929 y(\015)248 954 y(\015)248 979 y(\015)248 1004 y(\015)271 906 y(*)304 978 y Fs(x)330 959 y Fq(\012)p Fp(n)381 978 y Fs(;)421 937 y Fp(n)409 945 y Fj(N)401 1016 y Fp(j)r Fo(=1)470 978 y Fs(\030)490 985 y Fp(j)508 906 y Fj(+)542 904 y(\015)542 929 y(\015)542 954 y(\015)542 979 y(\015)542 1004 y(\015)565 917 y Fo(2)565 1031 y Fp(p;q)q(;\026)684 978 y Fu(=)782 925 y Fp(n)763 937 y Fj(X)761 1029 y Fp(k)q Fo(=0)833 906 y Fj( )866 947 y Fs(n)867 1009 y(k)893 906 y Fj(!)926 917 y Fo(2)953 904 y Fj(\015)953 929 y(\015)953 954 y(\015)953 979 y(\015)953 1004 y(\015)976 906 y(*)1009 978 y Fs(P)1044 955 y Fp(\026)1038 992 y(k)1075 971 y Fu(^)1068 978 y Fr(\012)q Fu(M)1146 955 y Fp(\026)1146 992 y(n)p Fq(\000)p Fp(k)1215 978 y Fs(;)1256 937 y Fp(n)1243 945 y Fj(N)1236 1016 y Fp(j)r Fo(=1)1305 978 y Fs(\030)1325 985 y Fp(j)1343 906 y Fj(+)1376 904 y(\015)1376 929 y(\015)1376 954 y(\015)1376 979 y(\015)1376 1004 y(\015)1399 917 y Fo(2)1399 1031 y Fp(p;q)q(;\026)684 1135 y Fr(\024)782 1083 y Fp(n)763 1095 y Fj(X)761 1187 y Fp(k)q Fo(=0)833 1063 y Fj( )866 1105 y Fs(n)867 1167 y(k)893 1063 y Fj(!)926 1075 y Fo(2)945 1135 y Fu(\()p Fs(k)q Fu(!\))1019 1117 y Fo(2)1046 1135 y Fu(2)1069 1117 y Fp(k)q(q)1114 1135 y Fr(j)p Fu(M)1169 1113 y Fp(\026)1169 1150 y(n)p Fq(\000)p Fp(k)1239 1135 y Fr(j)1252 1117 y Fo(2)1252 1147 y Fq(\000)p Fp(p)1326 1083 y(n)1310 1095 y Fj(Y)1306 1186 y Fp(j)r Fo(=1)1375 1135 y Fr(j)p Fs(\030)1408 1142 y Fp(j)1426 1135 y Fr(j)1439 1117 y Fo(2)1439 1147 y Fp(p)684 1295 y Fu(=)781 1255 y Fp(n)772 1263 y Fj(Q)761 1334 y Fp(j)r Fo(=1)830 1295 y Fr(j)p Fs(\030)863 1302 y Fp(j)880 1295 y Fr(j)893 1277 y Fo(2)893 1307 y Fp(p)942 1242 y(n)923 1255 y Fj(X)920 1347 y Fp(k)q Fo(=0)992 1223 y Fj( )1025 1265 y Fs(n)1026 1326 y(k)1052 1223 y Fj(!)1085 1235 y Fo(2)1105 1295 y Fu(\()p Fs(k)q Fu(!\))1179 1277 y Fo(2)1205 1295 y Fu(\(\()p Fs(n)c Fr(\000)h Fs(k)q Fu(\)!\))1397 1274 y Fo(2)1424 1295 y Fs(K)1466 1277 y Fo(2\()p Fp(n)p Fq(\000)p Fp(k)q Fo(\))1581 1295 y Fu(2)1604 1277 y Fp(k)q(q)684 1439 y Fr(\024)781 1399 y Fp(n)772 1407 y Fj(Q)761 1477 y Fp(j)r Fo(=1)830 1439 y Fr(j)p Fs(\030)863 1446 y Fp(j)880 1439 y Fr(j)893 1420 y Fo(2)893 1450 y Fp(p)925 1439 y Fu(\()p Fs(n)p Fu(!\))1001 1420 y Fo(2)1028 1439 y Fu(2)1051 1420 y Fp(nq)1120 1386 y(n)1101 1399 y Fj(X)1099 1491 y Fp(k)q Fo(=0)1171 1439 y Fu(2)1194 1420 y Fq(\000)p Fo(\()p Fp(n)p Fq(\000)p Fp(k)q Fo(\))p Fp(q)1335 1439 y Fs(K)1377 1420 y Fo(2\()p Fp(n)p Fq(\000)p Fp(k)q Fo(\))684 1583 y Fr(\024)781 1543 y Fp(n)772 1551 y Fj(Q)761 1621 y Fp(j)r Fo(=1)830 1583 y Fr(j)p Fs(\030)863 1590 y Fp(j)880 1583 y Fr(j)893 1564 y Fo(2)893 1594 y Fp(p)925 1583 y Fu(\()p Fs(n)p Fu(!\))1001 1564 y Fo(2)1028 1583 y Fu(2)1051 1564 y Fp(nq)1113 1530 y Fq(1)1101 1543 y Fj(X)1099 1635 y Fp(k)q Fo(=0)1171 1583 y Fu(2)1194 1564 y Fq(\000)p Fp(k)q(q)1259 1583 y Fs(K)1301 1564 y Fo(2)p Fp(k)59 1711 y Fu(whic)o(h)16 b(is)g(\014nite)g(for)f Fs(p;)8 b(q)16 b Fu(large)f(enough.)20 b(Here)c Fs(K)i Fu(is)d(determined)i(b)o (y)e(equation)h(\(7\).)59 1767 y(Then)g(w)o(e)f(arriv)o(e)g(at)614 1794 y Fj(\014)614 1819 y(\014)614 1844 y(\014)628 1843 y Fu(M)670 1850 y Fp(n)693 1843 y Fu(\()p Fs(\030)731 1850 y Fo(1)751 1843 y Fs(;)8 b(:::\030)831 1850 y Fp(n)852 1843 y Fu(\))870 1794 y Fj(\014)870 1819 y(\014)870 1844 y(\014)896 1843 y Fr(\024)13 b Fs(K)18 b(C)1037 1824 y Fp(n)1076 1843 y Fs(n)p Fu(!)1143 1790 y Fp(n)1127 1802 y Fj(Y)1123 1894 y Fp(j)r Fo(=1)1192 1843 y Fr(j)p Fs(\030)1225 1850 y Fp(j)1243 1843 y Fr(j)1255 1858 y Fp(p)1750 1843 y Fu(\(31\))59 1961 y(for)d(some)g Fs(K)q(;)8 b(C)13 b(>)g Fu(0.)130 2017 y(Due)i(to)g(the)g(k)o(ernel)h(theorem)f(2) g(w)o(e)f(then)i(ha)o(v)o(e)f(the)g(represen)o(tation)578 2113 y(M)620 2120 y Fp(n)644 2113 y Fu(\()p Fs(\030)682 2120 y Fo(1)701 2113 y Fs(;)8 b(:::\030)781 2120 y Fp(n)802 2113 y Fu(\))13 b(=)881 2066 y Fj(D)906 2113 y Fu(M)948 2094 y Fo(\()p Fp(n)p Fo(\))999 2113 y Fs(;)8 b(\030)1040 2120 y Fo(1)1068 2113 y Fr(\012)j Fs(:::)e Fr(\012)h Fs(\030)1227 2120 y Fp(n)1251 2066 y Fj(E)1299 2113 y Fu(,)59 2225 y(where)18 b(M)235 2209 y Fo(\()p Fp(n)p Fo(\))301 2225 y Fr(2)f Fu(\()p Fr(N)414 2203 y Fo(^)409 2209 y Fq(\012)q Fp(n)460 2225 y Fu(\))478 2209 y Fq(0)489 2225 y Fu(.)26 b(The)18 b(sequence)814 2178 y Fj(n)842 2225 y Fu(M)884 2209 y Fo(\()p Fp(n)p Fo(\))935 2225 y Fs(;)k(n)13 b Fr(2)g Fs(I)-12 b(N)1103 2232 y Fo(0)1122 2178 y Fj(o)1167 2225 y Fu(has)17 b(the)h(follo)o(wing)g(prop)q(ert)o (y)f(of)f(p)q(o-)59 2304 y(sitivit)o(y:)29 b(for)19 b(an)o(y)f (\014nite)j(sequence)f(of)f(smo)q(oth)g(k)o(ernels)1091 2257 y Fj(n)1118 2304 y Fs(g)1142 2287 y Fo(\()p Fp(n)p Fo(\))1193 2304 y Fs(;)8 b(n)k Fr(2)h Fs(I)-12 b(N)1345 2257 y Fj(o)1392 2304 y Fu(\(i.e.,)19 b Fs(g)1524 2287 y Fo(\()p Fp(n)p Fo(\))1594 2304 y Fr(2)g(N)1692 2282 y Fo(^)1687 2288 y Fq(\012)1715 2287 y Fp(n)1757 2304 y Fu(and)59 2375 y Fs(g)83 2359 y Fo(\()p Fp(n)p Fo(\))146 2375 y Fu(=)13 b(0)i Fr(8)20 b Fs(n)13 b Fr(\025)g Fs(n)392 2382 y Fo(0)427 2375 y Fu(for)i(some)g Fs(n)638 2382 y Fo(0)670 2375 y Fr(2)e Fs(I)-12 b(N)5 b Fu(\))15 b(the)g(follo)o (wing)h(inequalit)o(y)h(is)f(v)m(alid)663 2443 y Fp(n)684 2448 y Fi(0)652 2456 y Fj(X)660 2549 y Fp(k)q(;j)720 2450 y Fj(D)745 2497 y Fu(M)787 2478 y Fo(\()p Fp(k)q Fo(+)p Fp(j)r Fo(\))892 2497 y Fs(;)8 b(g)937 2478 y Fo(\()p Fp(k)q Fo(\))994 2497 y Fr(\012)p 1040 2454 70 2 v 11 w Fs(g)1064 2484 y Fo(\()p Fp(j)r Fo(\))1109 2450 y Fj(E)1147 2497 y Fr(\025)13 b Fu(0)i(.)504 b(\(32\))59 2635 y(This)16 b(follo)o(ws)f(from)g(the)g(fact)f(that)h(the)g(left)h (hand)f(side)i(can)e(b)q(e)h(written)f(as)1409 2600 y Fj(\012)-8 b(\012)1440 2635 y Fu(\010)p Fs(;)8 b Fr(j)p Fs(')p Fr(j)1550 2618 y Fo(2)1568 2600 y Fj(\013)-8 b(\013)1615 2635 y Fu(with)602 2758 y Fs(')p Fu(\()p Fs(x)p Fu(\))12 b(=)767 2705 y Fp(n)788 2710 y Fi(0)757 2718 y Fj(X)754 2809 y Fp(n)p Fo(=0)828 2711 y Fj(D)853 2758 y Fs(x)879 2740 y Fq(\012)p Fp(n)930 2758 y Fs(;)c(g)975 2740 y Fo(\()p Fp(n)p Fo(\))1024 2711 y Fj(E)1057 2758 y Fs(;)53 b(x)13 b Fr(2)f(N)1248 2740 y Fq(0)1275 2758 y Fu(,)p eop %%Page: 39 39 39 38 bop 59 50 a Fn(11)45 b(CHANGE)15 b(OF)g(MEASURE)1108 b Fu(39)59 199 y(whic)o(h)19 b(is)f(a)f(smo)q(oth)g(p)q(olynomial.)29 b(F)l(ollo)o(wing)18 b([BS71,)f(BeKo88])g(inequalities)j(\(31\))d(and)h (\(32\))e(are)59 256 y(su\016cien)o(t)j(to)e(ensure)h(the)g(existence)h (of)f(a)g(uniquely)i(de\014ned)f(measure)f Fs(\027)j Fu(on)d(\()p Fr(N)1525 239 y Fq(0)1536 256 y Fs(;)8 b Fr(C)1581 263 y Fp(\033)1604 256 y Fu(\()p Fr(N)1666 239 y Fq(0)1677 256 y Fu(\)\),)17 b(suc)o(h)59 312 y(that)e(for)f(an)o (y)h Fs(')d Fr(2)h(P)t Fu(\()p Fr(N)496 296 y Fq(0)507 312 y Fu(\))i(w)o(e)g(ha)o(v)o(e)672 416 y Fr(h)-8 b(h)p Fu(\010)p Fs(;)8 b(')p Fr(i)-9 b(i)810 430 y Fp(\026)846 416 y Fu(=)894 359 y Fj(Z)917 453 y Fq(N)951 444 y Fh(0)972 416 y Fs(')p Fu(\()p Fs(x)p Fu(\))14 b(d)p Fs(\027)s Fu(\()p Fs(x)p Fu(\))h(.)130 523 y(F)l(rom)c(estimate)h(\(31\))g(w)o(e) g(kno)o(w)f(that)h Fs(\027)k Fr(2)d(M)930 530 y Fp(a)950 523 y Fu(\()p Fr(N)1012 507 y Fq(0)1024 523 y Fu(\).)18 b(Then)13 b(Lemma)g(9)f(sho)o(ws)f(that)h(there)g(exists)59 580 y Fs(")i(>)f Fu(0)8 b Fs(;)14 b(p)f Fr(2)h Fs(I)-12 b(N)20 b Fu(suc)o(h)c(that)f(exp)q(\()p Fs(")p Fr(j)p Fs(x)p Fr(j)708 587 y Fq(\000)p Fp(p)754 580 y Fu(\))g(is)h Fs(\027)s Fr(\000)p Fu(in)o(tegrable.)23 b(Corollary)15 b(25)g(then)h(implies)i(that)d(eac)o(h)59 636 y Fs(')d Fr(2)h Fu(\()p Fr(N)7 b Fu(\))224 620 y Fo(1)259 636 y Fu(is)15 b Fs(\027)s Fu(-in)o(tegrable.)130 733 y(Con)o(v)o(ersely)22 b(let)g Fs(\027)28 b Fr(2)d(M)599 740 y Fp(a)619 733 y Fu(\()p Fr(N)681 717 y Fq(0)692 733 y Fu(\))d(b)q(e)h(giv)o(en.)41 b(Then)23 b(the)f(same)g(argumen)o(t)g(sho)o(ws)f(that)h(eac)o(h)59 789 y Fs(')12 b Fr(2)h Fu(\()p Fr(N)7 b Fu(\))224 773 y Fo(1)259 789 y Fu(is)15 b Fs(\027)s Fu(-in)o(tegrable)i(and)e(from)g (Corollary)g(25)f(w)o(e)h(kno)o(w)g(that)440 836 y Fj(\014)440 861 y(\014)440 886 y(\014)440 911 y(\014)454 840 y(Z)477 934 y Fq(N)511 925 y Fh(0)532 897 y Fs(')p Fu(\()p Fs(x)p Fu(\)d)p Fs(\027)s Fu(\()p Fs(x)p Fu(\))736 836 y Fj(\014)736 861 y(\014)736 886 y(\014)736 911 y(\014)761 897 y Fr(\024)21 b Fs(C)d Fr(k)o Fs(')p Fr(k)943 911 y Fp(p;q)q(;\026)1028 840 y Fj(Z)1051 934 y Fq(N)1085 925 y Fh(0)1106 897 y Fu(exp\()p Fs(")p Fr(j)p Fs(x)p Fr(j)1266 904 y Fq(\000)p Fp(p)1312 897 y Fu(\))8 b(d)p Fs(\027)s Fu(\()p Fs(x)p Fu(\))59 1006 y(for)16 b(some)h Fs(p;)8 b(q)16 b Fr(2)g Fs(I)-12 b(N)12 b(;)j(C)j(>)e Fu(0.)24 b(Th)o(us)17 b(the)g(con)o(tin)o (uit)o(y)g(of)f Fs(')f Fr(7!)1191 971 y Fj(R)1211 1019 y Fq(N)1245 1010 y Fh(0)1265 1006 y Fs(')g Fu(d)p Fs(\027)21 b Fu(is)c(established,)i(sho)o(wing)59 1062 y(that)c(\010)g(de\014ned)h (b)o(y)f(equation)h(\(30\))e(is)h(in)i(\()p Fr(N)7 b Fu(\))886 1043 y Fq(\000)p Fo(1)886 1073 y Fp(\026;)p Fo(+)945 1062 y Fu(.)824 b Ff(2)59 1203 y Ft(11)67 b(Change)21 b(of)h(measure)59 1304 y Fu(Supp)q(ose)17 b(w)o(e)f(are)g(giv)o(en)h(t) o(w)o(o)e(measures)h Fs(\026;)11 b Fu(^)-26 b Fs(\026)14 b Fr(2)h(M)975 1311 y Fp(a)995 1304 y Fu(\()p Fr(N)1057 1288 y Fq(0)1068 1304 y Fu(\))h(b)q(oth)h(satisfying)f(Assumption)h(2.) 23 b(Let)16 b(a)59 1361 y(distribution)315 1349 y(^)310 1361 y(\010)g Fr(2)i Fu(\()p Fr(N)7 b Fu(\))487 1342 y Fq(\000)p Fo(1)490 1375 y(^)-21 b Fp(\026)551 1361 y Fu(b)q(e)19 b(giv)o(en.)28 b(Since)19 b(the)f(test)f(function)i (space)f(\()p Fr(N)7 b Fu(\))1438 1344 y Fo(1)1475 1361 y Fu(is)19 b(in)o(v)m(arian)o(t)f(under)59 1417 y(c)o(hanges)d(of)g (measures)g(in)h(view)g(of)f(Theorem)g(27,)f(the)h(con)o(tin)o(uous)h (mapping)659 1505 y Fs(')c Fr(7!)h(h)-8 b(h)792 1493 y Fu(^)787 1505 y(\010)p Fs(;)8 b(')p Fr(i)-8 b(i)902 1512 y Fo(^)-21 b Fp(\026)936 1505 y Fs(;)98 b(')12 b Fr(2)h Fu(\()p Fr(N)7 b Fu(\))1212 1486 y Fo(1)59 1593 y Fu(can)19 b(also)h(b)q(e)f(represen)o(ted)h(as)f(a)g(distribution)h (\010)g Fr(2)f Fu(\()p Fr(N)7 b Fu(\))1088 1576 y Fq(\000)p Fo(1)1088 1604 y Fp(\026)1135 1593 y Fu(.)31 b(So)19 b(w)o(e)g(ha)o(v)o(e)g(the)g(implicit)j(relation)59 1658 y(\010)13 b Fr(2)f Fu(\()p Fr(N)7 b Fu(\))227 1641 y Fq(\000)p Fo(1)227 1669 y Fp(\026)287 1658 y Fr($)350 1646 y Fu(^)345 1658 y(\010)12 b Fr(2)h Fu(\()p Fr(N)7 b Fu(\))513 1638 y Fq(\000)p Fo(1)516 1672 y(^)-21 b Fp(\026)575 1658 y Fu(de\014ned)16 b(b)o(y)739 1756 y Fr(h)-8 b(h)772 1745 y Fu(^)767 1756 y(\010)p Fs(;)8 b(')p Fr(i)-8 b(i)882 1763 y Fo(^)-21 b Fp(\026)913 1756 y Fu(=)13 b Fr(h)-8 b(h)p Fu(\010)p Fs(;)8 b(')p Fr(i)-8 b(i)1101 1763 y Fp(\026)1138 1756 y Fs(:)59 1844 y Fu(This)17 b(section)g(will)h(pro)o(vide)g(form)o(ulae)e(whic)o(h)h(mak)o(e)f (this)h(relation)g(more)f(explicit)j(in)e(terms)f(of)h(re-)59 1901 y(decomp)q(osition)k(of)f(the)p 518 1901 3 25 v 23 w Fs(Q)p Fu(-series.)35 b(First)20 b(w)o(e)f(need)i(an)f(explicit)i (relation)f(of)f(the)g(corresp)q(onding)59 1957 y Fs(I)-12 b(P)7 b Fu(-systems.)59 2049 y Fk(Lemma)17 b(42)23 b Fm(L)n(et)16 b Fs(\026;)11 b Fu(^)-26 b Fs(\026)12 b Fr(2)h(M)565 2056 y Fp(a)586 2049 y Fu(\()p Fr(N)648 2032 y Fq(0)659 2049 y Fu(\))j Fm(then)491 2158 y Fs(P)526 2139 y Fp(\026)520 2169 y(n)550 2158 y Fu(\()p Fs(x)p Fu(\))c(=)724 2117 y Fj(X)672 2210 y Fp(k)q Fo(+)p Fp(l)p Fo(+)p Fp(m)p Fo(=)p Fp(n)896 2127 y Fs(n)p Fu(!)p 850 2147 133 2 v 850 2189 a Fs(k)q Fu(!)c Fs(l)q Fu(!)g Fs(m)p Fu(!)987 2158 y Fs(P)1025 2136 y Fo(^)-20 b Fp(\026)1016 2172 y(k)1046 2158 y Fu(\()p Fs(x)p Fu(\))1114 2151 y(^)1108 2158 y Fr(\012)p Fs(P)1178 2136 y Fp(\026)1172 2172 y(l)1201 2158 y Fu(\(0\))1266 2151 y(^)1260 2158 y Fr(\012)p Fu(M)1337 2139 y Fp(\026)1337 2169 y(m)1386 2158 y Fs(:)59 2288 y Fk(Pr)o(oof.)72 b Fu(Expanding)17 b(eac)o(h)e(factor)f(in)i(the)f (form)o(ula)645 2376 y Fs(e)666 2383 y Fp(\026)689 2376 y Fu(\()p Fs(\022)q(;)8 b(x)p Fu(\))k(=)h Fs(e)878 2383 y Fo(^)-21 b Fp(\026)899 2376 y Fu(\()p Fs(\022)q(;)8 b(x)p Fu(\))p Fs(l)1019 2357 y Fq(\000)p Fo(1)1018 2387 y Fp(\026)1064 2376 y Fu(\()p Fs(\022)q Fu(\))p Fs(l)1139 2383 y Fo(^)-21 b Fp(\026)1159 2376 y Fu(\()p Fs(\022)q Fu(\))15 b Fs(;)59 2464 y Fu(w)o(e)g(obtain)254 2517 y Fq(1)242 2530 y Fj(X)239 2620 y Fp(n)p Fo(=0)326 2539 y Fu(1)p 318 2560 40 2 v 318 2601 a Fs(n)p Fu(!)363 2570 y Fr(h)p Fs(P)416 2551 y Fp(\026)410 2581 y(n)439 2570 y Fu(\()p Fs(x)p Fu(\))p Fs(;)8 b(\022)544 2551 y Fq(\012)p Fp(n)594 2570 y Fr(i)13 b Fu(=)718 2517 y Fq(1)706 2530 y Fj(X)673 2622 y Fp(k)q(;l;m)p Fo(=0)866 2539 y Fu(1)p 811 2560 133 2 v 811 2601 a Fs(k)q Fu(!)8 b Fs(l)q Fu(!)g Fs(m)p Fu(!)949 2570 y Fr(h)p Fs(P)1002 2548 y Fp(\026)996 2584 y(k)1025 2570 y Fu(\()p Fs(x)p Fu(\))i Fr(\012)g Fs(P)1180 2548 y Fo(^)-20 b Fp(\026)1171 2584 y(l)1201 2570 y Fu(\(0\))9 b Fr(\012)h Fu(M)1356 2551 y Fp(\026)1356 2581 y(m)1389 2570 y Fs(;)e(\022)1432 2551 y Fq(\012)p Fo(\()p Fp(k)q Fo(+)p Fp(l)p Fo(+)p Fp(m)p Fo(\))1605 2570 y Fr(i)15 b Fs(:)59 2700 y Fu(A)g(comparison)h(of)e(co)q (e\016cien)o(ts)i(giv)o(es)g(the)f(ab)q(o)o(v)o(e)g(result.)724 b Ff(2)130 2796 y Fu(An)15 b(immediate)h(consequence)h(is)f(the)f(next) g(reordering)h(lemma.)p eop %%Page: 40 40 40 39 bop 59 50 a Fn(REFERENCES)1414 b Fu(40)59 199 y Fk(Lemma)17 b(43)23 b Fm(L)n(et)18 b Fs(')e Fr(2)h Fu(\()p Fr(N)7 b Fu(\))555 183 y Fo(1)592 199 y Fm(b)n(e)18 b(given.)26 b(Then)18 b Fs(')g Fm(has)g(r)n(epr)n(esentations)f(in)h Fs(I)-12 b(P)1464 181 y Fp(\026)1487 199 y Fm(-series)18 b(as)g(wel)r(l)g(as)59 256 y Fs(I)-12 b(P)106 238 y Fo(^)-21 b Fp(\026)126 256 y Fm(-series:)608 331 y Fs(')12 b Fu(=)714 278 y Fq(1)702 291 y Fj(X)698 381 y Fp(n)p Fo(=0)765 331 y Fr(h)p Fs(P)818 312 y Fp(\026)812 342 y(n)841 331 y Fs(;)c(')892 312 y Fo(\()p Fp(n)p Fo(\))942 331 y Fr(i)k Fu(=)1036 278 y Fq(1)1024 291 y Fj(X)1020 381 y Fp(n)p Fo(=0)1087 331 y Fr(h)p Fs(P)1143 312 y Fo(^)-21 b Fp(\026)1134 342 y(n)1163 331 y Fs(;)15 b Fu(^)-30 b Fs(')1214 312 y Fo(\()p Fp(n)p Fo(\))1264 331 y Fr(i)59 465 y Fm(wher)n(e)16 b Fs(')217 448 y Fo(\()p Fp(n)p Fo(\))268 465 y Fs(;)f Fu(^)-30 b Fs(')319 448 y Fo(\()p Fp(n)p Fo(\))385 465 y Fr(2)13 b(N)477 439 y Fo(^)472 445 y Fq(\012)q Fp(n)p 471 478 2 18 v 462 478 a(C)539 465 y Fm(for)k(al)r(l)f Fs(n)d Fr(2)g Fs(I)-12 b(N)810 472 y Fo(0)829 465 y Fm(,)17 b(and)f(the)g(fol)r(lowing)g(formula)h(holds:)373 599 y Fu(^)-30 b Fs(')396 581 y Fo(\()p Fp(n)p Fo(\))459 599 y Fu(=)538 547 y Fq(1)526 559 y Fj(X)507 651 y Fp(l;m)p Fo(=0)617 569 y Fu(\()p Fs(l)10 b Fu(+)g Fs(m)g Fu(+)h Fs(n)p Fu(\)!)p 617 589 241 2 v 670 631 a Fs(l)q Fu(!)d Fs(m)p Fu(!)g Fs(n)p Fu(!)870 552 y Fj(\020)895 599 y Fs(P)930 577 y Fp(\026)924 613 y(l)954 599 y Fu(\(0\))1019 592 y(^)1013 599 y Fr(\012)o Fu(M)1092 581 y Fo(^)-21 b Fp(\026)1089 611 y(m)1122 599 y Fs(;)8 b(')1173 581 y Fo(\()p Fp(l)p Fo(+)p Fp(m)p Fo(+)p Fp(n)p Fo(\))1320 552 y Fj(\021)1345 627 y Fq(H)1375 618 y Fh(\012)p Fi(\()p Fg(l)p Fi(+)p Fg(m)p Fi(\))1511 599 y Fs(:)226 b Fu(\(33\))59 747 y(No)o(w)15 b(w)o(e)g(ma)o(y)f(pro)o(v)o(e)h(the)g(announced)h (theorem.)59 854 y Fk(Theorem)i(44)23 b Fm(L)n(et)437 842 y Fu(^)432 854 y(\010)13 b(=)526 822 y Fj(P)569 835 y Fq(1)569 865 y Fp(n)p Fo(=0)638 854 y Fr(h)p Fs(Q)695 837 y Fo(^)-21 b Fp(\026)692 865 y(n)715 854 y Fs(;)740 842 y Fu(^)736 854 y(\010)769 837 y Fo(\()p Fp(n)p Fo(\))819 854 y Fr(i)12 b(2)h Fu(\()p Fr(N)7 b Fu(\))972 833 y Fq(\000)p Fo(1)974 867 y(^)-20 b Fp(\026)1019 854 y Fm(.)21 b(Then)15 b Fu(\010)e(=)1266 822 y Fj(P)1309 835 y Fq(1)1309 865 y Fp(n)p Fo(=0)1378 854 y Fr(h)p Fs(Q)1432 837 y Fp(\026)1432 865 y(n)1455 854 y Fs(;)8 b Fu(\010)1509 837 y Fo(\()p Fp(n)p Fo(\))1559 854 y Fr(i)16 b Fm(de\014ne)n(d)f(by) 596 966 y Fr(h)-8 b(h)p Fu(\010)p Fs(;)8 b(')p Fr(i)-8 b(i)736 973 y Fp(\026)770 966 y Fu(=)13 b Fr(h)-8 b(h)851 955 y Fu(^)846 966 y(\010)p Fs(;)8 b(')p Fr(i)-8 b(i)961 973 y Fo(^)-21 b Fp(\026)996 966 y Fs(;)100 b(')13 b Fr(2)g Fu(\()p Fr(N)7 b Fu(\))1275 948 y Fo(1)59 1069 y Fm(is)16 b(in)g Fu(\()p Fr(N)7 b Fu(\))244 1052 y Fq(\000)p Fo(1)244 1080 y Fp(\026)306 1069 y Fm(and)17 b(the)f(fol)r(lowing)g(r)n (elation)g(holds)566 1196 y Fu(\010)599 1177 y Fo(\()p Fp(n)p Fo(\))662 1196 y Fu(=)763 1156 y Fj(X)710 1248 y Fp(k)q Fo(+)p Fp(l)p Fo(+)p Fp(m)p Fo(=)p Fp(n)920 1165 y Fu(1)p 888 1186 88 2 v 888 1227 a Fs(l)q Fu(!)8 b Fs(m)p Fu(!)985 1185 y(^)980 1196 y(\010)1013 1177 y Fo(\()p Fp(k)q Fo(\))1068 1189 y Fu(^)1062 1196 y Fr(\012)p Fs(P)1132 1174 y Fp(\026)1126 1210 y(l)1156 1196 y Fu(\(0\))1221 1189 y(^)1215 1196 y Fr(\012)o Fu(M)1294 1177 y Fo(^)-21 b Fp(\026)1291 1207 y(m)59 1344 y Fk(Pr)o(oof.)72 b Fu(W)l(e)16 b(can)f(insert)h(form)o(ula)f(\(33\))f(in)i(the)f(form)o(ula)589 1423 y Fq(1)576 1436 y Fj(X)573 1526 y Fp(n)p Fo(=0)647 1476 y Fs(n)p Fu(!)8 b Fr(h)p Fu(\010)746 1457 y Fo(\()p Fp(n)p Fo(\))796 1476 y Fs(;)g(')847 1457 y Fo(\()p Fp(n)p Fo(\))897 1476 y Fr(i)k Fu(=)991 1423 y Fq(1)978 1436 y Fj(X)975 1526 y Fp(n)p Fo(=0)1049 1476 y Fs(n)p Fu(!)c Fr(h)1120 1465 y Fu(^)1115 1476 y(\010)1148 1457 y Fo(\()p Fp(n)p Fo(\))1198 1476 y Fs(;)15 b Fu(^)-30 b Fs(')1249 1457 y Fo(\()p Fp(n)p Fo(\))1299 1476 y Fr(i)59 1612 y Fu(and)15 b(compare)g(co)q(e\016cien)o(ts)h(again.)1104 b Ff(2)59 1774 y Fv(Ac)o(kno)o(wledgemen)o(ts.)59 b Fu(The)18 b(follo)o(wing)h(results)f(w)o(ere)f(presen)o(ted)i(at)e(the)h(w)o (orkshop)f(\\Mathe-)59 1831 y(matical)h(Approac)o(h)f(to)g (Fluctuations")h(organized)g(at)e(I)q(IAS)j(b)o(y)f(Professor)e(T.)h (Hida.)27 b(L.S.)17 b(w)o(ould)59 1887 y(lik)o(e)i(to)e(express)h(his)h (gratitude)f(to)f(I)q(IAS)i(not)f(only)g(for)g(the)g(opp)q(ortunit)o(y) g(to)f(do)h(so,)f(but)h(also)g(for)59 1944 y(its)j(academic)g (hospitalit)o(y)h(in)g(the)e(w)o(eeks)h(after)f(the)h(w)o(orkshop)f (whic)o(h)h(pro)o(vided)h(a)e(unique)j(op-)59 2000 y(p)q(ortunit)o(y)18 b(for)g(con)o(tin)o(ued)g(w)o(ork)f(and)i(collab)q(oration.)29 b(The)18 b(\014rst)g(author)f(w)o(as)g(supp)q(orted)i(b)o(y)f(the)59 2057 y(Europ)q(ean)13 b(program)e(`Human)h(capital)h(and)f(mobilit)o (y')h(and)g(b)o(y)f(the)g(`Deutsc)o(he)g(F)l(orsc)o(h)o(ungsgemein-)59 2113 y(sc)o(haft'.)20 b(W.W.)15 b(gratefully)h(ac)o(kno)o(wledges)f (\014nancial)j(supp)q(ort)d(of)h(a)f(sc)o(holarship)i(from)e (`Graduier-)59 2170 y(tenf\177)-23 b(orderung)20 b(des)h(Landes)g (Nordrhein{W)l(estfalen'.)37 b(The)20 b(last)h(author)f(\(J.A.Y\))f(w)o (as)h(partially)59 2226 y(supp)q(orted)c(b)o(y)f(STRIDE)h(and)f(the)g (National)h(Natural)f(Science)i(F)l(oundation)f(of)e(China.)59 2363 y Ft(References)59 2454 y Fw([AKS93])88 b(Alb)q(ev)o(erio,)10 b(S.,)g(Kondratiev,)g(Y)m(u.G.)g(and)g(Streit,)h(L.)f(\(1993\),)g Fa(How)i(to)g(gener)n(alize)f(White)i(Noise)299 2504 y(A)o(nalysis)g(to)h(Non-Gaussian)h(Sp)n(ac)n(es)p Fw(.)e(In:)g ('Dynamics)d(of)j(Complex)e(and)i(Irregular)g(Systems'.)299 2554 y(Eds.:)g(Ph.)h(Blanc)o(hard)g(et)g(al.,)e(W)m(orld)h(Scien)o (ti\014c.)59 2617 y([ADKS94])56 b(Alb)q(ev)o(erio,)13 b(S.,)g(Daletzky)m(,)g(Y.,)h(Kondratiev,)g(Y)m(u.)f(G.)g(and)h(Streit,) g(L.)g(\(1994\),)f Fa(Non-Gaussian)299 2667 y(in\014nite)i(dimensional) g(analysis,)g Fw(preprin)o(t,)f(to)g(app)q(ear)g(in)f(J.)h(F)m(unc.)g (Anal..)59 2730 y([BeS95])104 b(Ben)o(th,)12 b(F.)g(and)h(Streit,)f(L.) g(\(1995\),)f Fa(The)i(Bur)n(gers)g(Equation)i(with)d(a)i(Non-Gaussian) h(R)n(andom)299 2779 y(F)m(or)n(c)n(e.)f Fw(UMa)g(preprin)o(t.)p eop %%Page: 41 41 41 40 bop 59 50 a Fn(REFERENCES)1414 b Fu(41)59 199 y Fw([BeKo88])74 b(Berezansky)m(,)21 b(Y)m(u.)e(M.)g(and)h(Kondratiev,)g (Y)m(u.)f(G.)g(\(1988\),)g Fa(Sp)n(e)n(ctr)n(al)h(Metho)n(ds)h(in)f (In\014nite-)299 249 y(Dimensional)12 b(A)o(nalysis)p Fw(,)d(\(in)h(Russian\),)f(Nauk)o(o)o(v)n(a)g(Dumk)n(a,)f(Kiev.)h (English)h(translation,)f(1995,)299 299 y(Klu)o(w)o(er)14 b(Academic)f(Publishers,)h(Dordrec)o(h)o(t.)59 361 y([BeLy93])79 b(Berezansky)m(,)20 b(Y)m(u.M.)f(and)g(Lytvyno)o(v,)f(E.V.)h(\(1993\),) g Fa(Gener)n(alize)n(d)h(White)g(Noise)g(A)o(nalysis)299 411 y(c)n(onne)n(cte)n(d)15 b(with)f(p)n(ertub)n(e)n(d)h(\014eld)g(op)n (er)n(ators)p Fw(,)e(Dop)q(o)o(vidy)g(AN)h(Ukrain)o(y)m(,)e(No)i Fx(10)p Fw(.)59 473 y([BS71])122 b(Berezansky)m(,)16 b(Y)m(u.)e(M.)g(and)h(Shifrin,)f(S.N.)g(\(1971\),)g Fa(The)i(gener)n (alize)n(d)g(de)n(gr)n(e)n(e)f(symmetric)g(Mo-)299 522 y(ment)g(Pr)n(oblem,)e Fw(Ukrainian)f(Math.)i(J.)g(23)f(N3,)g(247-258.) 59 584 y([Bo76])124 b(Bourbaki,)10 b(N.)h(\(1976\),)f Fa(Elements)j(of)f(mathematics.)g(F)m(unctions)h(of)g(a)f(r)n(e)n(al)g (variable.)g Fw(Addison-)299 634 y(W)m(esley)m(.)59 696 y([Da91])121 b(Daletsky)m(,)15 b(Y)m(u.L.)g(\(1991\),)h Fa(A)h(biortho)n(gonal)g(analo)n(gy)h(of)f(the)g(Hermite)f(p)n (olynomials)i(and)g(the)299 746 y(inversion)f(of)g(the)h(F)m(ourier)e (tr)n(ansform)h(with)g(r)n(esp)n(e)n(ct)f(to)i(a)f(non)i(Gaussian)f(me) n(asur)n(e)p Fw(,)e(F)m(unct.)299 796 y(Anal.)c(Appl.)i Fx(25)p Fw(,)f(68-70.)59 858 y([Di81])130 b(Dineen,)11 b(S.)g(\(1981\),)g Fa(Complex)i(A)o(nalysis)f(in)h(L)n(o)n(c)n(al)r(ly) g(Convex)g(Sp)n(ac)n(es,)f Fw(Mathematical)e(Studies)299 908 y(57,)i(North)j(Holland,)d(Amsterdam.)59 970 y([GV68])110 b(Gel'fand,)9 b(I.)i(M.)g(and)g(Vilenkin,)g(N.Y)m(a.)f(\(1968\),)h Fa(Gener)n(alize)n(d)i(F)m(unctions)p Fw(,)e(V)m(ol.)f(IV,)h(Academic) 299 1020 y(Press,)k(New)f(Y)m(ork)f(and)h(London.)59 1081 y([Hi75])131 b(Hida,)16 b(T.)h(\(1975\),)g Fa(A)o(nalysis)h(of)g (Br)n(ownian)g(F)m(unctionals,)h Fw(Carleton)e(Math.)h(Lecture)h(Notes) 299 1131 y(No.)13 b(13,)g(Carleton.)59 1193 y([Hi80])131 b(Hida,)12 b(T.)i(\(1980\),)e Fa(Br)n(ownian)j(Motion)p Fw(.)f(Springer,)g(New)g(Y)m(ork.)59 1255 y([HKPS93])60 b(Hida,)18 b(T.,)h(Kuo,)g(H.H.,)f(P)o(ottho\013,)i(J.)f(and)g(Streit,)h (L.)f(\(1993\),)g Fa(White)h(Noise.)g(A)o(n)g(in\014nite)299 1305 y(dimensional)15 b(c)n(alculus)p Fw(.)e(Klu)o(w)o(er,)h(Dordrec)o (h)o(t.)59 1367 y([Ito88])122 b(Ito,)13 b(Y.)g(\(1988\),)g Fa(Gener)n(alize)n(d)i(Poisson)g(F)m(unctionals.)f Fw(Prob.)g(Th.)f (Rel.)g(Fields)h Fx(77)g Fw(1-28.)59 1429 y([IK88])127 b(Ito,)10 b(Y.)h(and)g(Kub)q(o,)f(I.)h(\(1988\),)f Fa(Calculus)i(on)h (Gaussian)g(and)g(Poisson)g(White)f(Noises.)f Fw(Nago)o(y)o(a)299 1479 y(Math.)i(J.)h Fx(111)f Fw(41-84.)59 1541 y([Ko78])121 b(Kondratiev,)14 b(Y)m(u.G.)f(\(1978\),)g Fa(Gener)n(alize)n(d)i (functions)h(in)g(pr)n(oblems)f(of)g(in\014nite)h(dimensional)299 1591 y(analysis)p Fw(.)d(Ph.D.)g(thesis,)h(Kiev)g(Univ)o(ersit)o(y)m(.) 59 1653 y([Ko80a])100 b(Kondratiev,)14 b(Y)m(u.G.)f(\(1980\),)g Fa(Sp)n(ac)n(es)j(of)g(entir)n(e)f(functions)h(of)f(an)h(in\014nite)g (numb)n(er)g(of)f(varia-)299 1703 y(bles,)f(c)n(onne)n(cte)n(d)h(with)f (the)h(rigging)f(of)h(a)g(F)m(o)n(ck)g(sp)n(ac)n(e.)g Fw(In:)e('Sp)q(ectral)h(Analysis)f(of)g(Di\013eren)o(tial)299 1752 y(Op)q(erators.')h(Math.)g(Inst.)h(Acad.)f(Sci.)g(Ukrainian)f (SSR,)h(p.)g(18-37.)e(English)i(translation:)f(Se-)299 1802 y(lecta)h(Math.)f(So)o(vietica)g Fx(10)h Fw(\(1991\),)f(165-180.) 59 1864 y([Ko80b])98 b(Kondratiev,)15 b(Y)m(u.G.)e(\(1980\),)h Fa(Nucle)n(ar)i(sp)n(ac)n(es)h(of)f(entir)n(e)f(functions)i(in)f(pr)n (oblems)g(of)g(in\014nite)299 1914 y(dimensional)f(analysis.)f Fw(So)o(viet)f(Math.)h(Dokl.)e Fx(22)p Fw(,)h(588-592.)59 1976 y([KLPSW94])22 b(Kondratiev,)13 b(Y)m(u.G.,)f(Leuk)o(ert,)i(P)m (.,)f(P)o(ottho\013,)h(J.,)f(Streit,)h(L.,)f(W)m(esterk)n(amp,)f(W.)h (\(1994\),)g Fa(Ge-)299 2026 y(ner)n(alize)n(d)i(F)m(unctionals)h(in)f (Gaussian)i(Sp)n(ac)n(es)f({)g(the)f(Char)n(acterization)g(The)n(or)n (em)g(R)n(evisite)n(d.)299 2076 y Fw(Man)o(uskripte)f(175/94,)e(Uni)h (Mannheim.)59 2138 y([KLS94])93 b(Kondratiev,)9 b(Y)m(u.G.,)e(Leuk)o (ert,)j(P)m(.,)e(Streit,)i(L.)f(\(1994\),)g Fa(Wick)i(Calculus)g(in)g (Gaussian)g(A)o(nalysis,)299 2187 y Fw(BiBoS)j(preprin)o(t)g(637,)f(to) g(app)q(ear)i(in)e(Acta)i(Applicandae)e(Mathematicae.)59 2249 y([KoSa78])77 b(Kondratiev,)10 b(Y)m(u.G.)f(and)h(Samoilenk)o(o,)e (Y)m(u.S.)h(\(1978\),)g Fa(Sp)n(ac)n(es)k(of)f(trial)e(and)j(gener)n (alize)n(d)e(func-)299 2299 y(tions)j(of)h(an)h(in\014nite)f(numb)n(er) g(of)g(variables)p Fw(,)e(Rep.)g(Math.)g(Ph)o(ys.)h Fx(14)p Fw(,)f(No.3,)g(325-350.)59 2361 y([KoS93])98 b(Kondratiev,)15 b(Y)m(u.G.)g(and)g(Streit,)h(L.)g(\(1993\),)f Fa(Sp)n(ac)n(es)i(of)g (White)g(Noise)g(distributions:)f(Con-)299 2411 y(structions,)e (Descriptions,)g(Applic)n(ations.)g Fw(I.)f(Rep.)h(Math.)f(Ph)o(ys.)h Fx(33)p Fw(,)f(341-366.)59 2473 y([KoSW95])55 b(Kondratiev,)14 b(Y)m(u.G.)g(,)h(Streit,)g(L.)g(and)g(W)m(esterk)n(amp,)e(W.\(1995\),)g Fa(A)j(Note)g(on)h(Positive)f(Dis-)299 2523 y(tributions)e(in)h (Gaussian)g(A)o(nalysis)p Fw(,)f(Ukrainian)e(Math.)i(J.)g Fx(47)f Fw(No.)g(5.)59 2585 y([KoTs91])75 b(Kondratiev,)15 b(Y)m(u.G.)e(and)i(Tsyk)n(alenk)o(o)g(T.V.)f(\(1991\),)g Fa(Dirichlet)i(Op)n(er)n(ators)f(and)i(Asso)n(ciate)n(d)299 2635 y(Di\013er)n(ential)d(Equations.)h Fw(Selecta)f(Math.)g(So)o (vietica)f Fx(10)p Fw(,)g(345-397.)59 2697 y([KMP65])76 b(Kristensen,)16 b(P)m(.,)f(Mejlb)q(o,)f(L.,)h(and)g(P)o(oulsen,)g (E.T.)g(\(1965\),)f Fa(T)m(emp)n(er)n(e)n(d)h(Distributions)h(in)h(In-) 299 2746 y(\014nitely)g(Many)i(Dimensions.)g(I.)e(Canonic)n(al)h(Field) g(Op)n(er)n(ators.)f Fw(Comm)n(un.)d(math.)h(Ph)o(ys.)i Fx(1)p Fw(,)299 2796 y(175{214.)p eop %%Page: 42 42 42 41 bop 59 50 a Fn(REFERENCES)1414 b Fu(42)59 199 y Fw([Kuo92])98 b(Kuo,)13 b(H.-H.)g(\(1992\),)g Fa(L)n(e)n(ctur)n(es)h (on)h(white)g(noise)g(analysis)p Fw(.)f(So)q(o)q(c)o(ho)o(w)f(J.)h (Math.)f Fx(18)p Fw(,)g(229-300.)59 262 y([Lee91])112 b(Lee,)19 b(Y.-J.)f(\(1991\),)f Fa(A)o(nalytic)i(V)m(ersion)g(of)g(T)m (est)g(F)m(unctionals,)h(F)m(ourier)e(T)m(r)n(ansform)g(and)i(a)299 312 y(Char)n(acterization)12 b(of)h(Me)n(asur)n(es)h(in)f(White)h (Noise)f(Calculus.)f Fw(J.)g(F)m(unct.)g(Anal.)g Fx(100)p Fw(,)f(359-380.)59 375 y([Lu70])125 b(Luk)n(acs,)13 b(E.)h(\(1970\),)e Fa(Char)n(acteristic)i(F)m(unctions)p Fw(,)g(2nd)g(edition,)e (Gri\016n,)h(London.)59 438 y([\037k95])120 b(\037ksendal,)18 b(B.)h(\(1995\),)f Fa(Sto)n(chastic)i(Partial)e(Di\013er)n(ential)h (Equations)h(and)h(Applic)n(ations)e(to)299 488 y(Hydr)n(o)n(dynamics.) 13 b Fw(In:)g(`Sto)q(c)o(hastic)f(Analysis)h(and)f(Applications)g(to)h (Ph)o(ysics')g(Eds.:)f(A.I.)g(Car-)299 537 y(doso)h(et)i(al;)d(Klu)o(w) o(er,)i(Dordrec)o(h)o(t.)59 600 y([Ou91])119 b(Ouerdiane,)13 b(H.)g(\(1991\),)g Fa(Applic)n(ation)i(des)f(m)o(\023)-20 b(etho)n(des)15 b(d'holomorphie)g(et)f(de)h(distributions)e(en)299 650 y(dimension)i(quelc)n(onque)h(\023)-21 b(a)15 b(l'analyse)g(sur)f (les)h(esp)n(ac)n(es)g(Gaussiens)p Fw(.)f(BiBoS)g(preprin)o(t)h(491.)59 713 y([P)o(o87])126 b(P)o(ottho\013,)13 b(J.)h(\(1987\),)e Fa(On)j(p)n(ositive)g(gener)n(alize)n(d)g(functionals.)g Fw(J.)e(F)m(unct.)h(Anal.)f Fx(74)p Fw(,)g(81-95.)59 776 y([PS91])123 b(P)o(ottho\013,)13 b(J.)h(and)f(Streit,)h(L.)f (\(1991\),)g Fa(A)i(char)n(acterization)f(of)h(Hida)f(distributions.)f Fw(J.)h(F)m(unct.)299 826 y(Anal.)e Fx(101)p Fw(,)h(212-229.)59 889 y([Sc)o(h71])111 b(Sc)o(haefer,)14 b(H.H.)f(\(1971\),)g Fa(T)m(op)n(olo)n(gic)n(al)h(V)m(e)n(ctor)g(Sp)n(ac)n(es,)h Fw(Springer,)f(New)h(Y)m(ork.)59 952 y([Sk74])129 b(Sk)o(oroho)q(d,)13 b(A.V.)g(\(1974\),)f Fa(Inte)n(gr)n(ation)j(in)g(Hilb)n(ert)e(Sp)n(ac)n (e,)j Fw(Springer,)e(Berlin.)59 1014 y([V)m(a95])125 b(V)-5 b(\027)-26 b(age,)11 b(G.)f(\(1995\),)h Fa(Sto)n(chastic)i (Di\013er)n(ential)g(Equations)g(and)h(Kondr)n(atiev)e(Sp)n(ac)n(es.)i Fw(Ph.D.)c(the-)299 1064 y(sis,)j(T)m(rondheim)f(Univ)o(ersit)o(y)m(.) 59 1127 y([V)o(GG75])78 b(V)m(ershik,)15 b(A.M.,)f(Gelfand,)g(I.M.)h (and)g(Graev,)g(M.I.)f(\(1975\),)h Fa(R)n(epr)n(esentations)h(of)h (di\013e)n(omor-)299 1177 y(phisms)d(gr)n(oups)p Fw(.)g(Russian)g (Math.)f(Surv)o(eys)i Fx(30)p Fw(,)e(No)g(6,)h(3-50.)59 1240 y([Y)m(ok90])103 b(Y)m(ok)o(oi,)10 b(Y.\(1990\),)g Fa(Positive)j(gener)n(alize)n(d)g(white)g(noise)h(functionals)p Fw(.)e(Hiroshima)e(Math.)h(J.)h Fx(20)p Fw(,)299 1290 y(137-157.)59 1353 y([Y)m(ok93])103 b(Y)m(ok)o(oi,)13 b(Y.)i(\(1993\),)f Fa(Simple)i(setting)g(for)f(white)h(noise)g(c)n (alculus)g(using)h(Bar)n(gmann)f(sp)n(ac)n(e)h(and)299 1402 y(Gauss)e(tr)n(ansform.)e Fw(Preprin)o(t.)588 1521 y Fu(Y)l(uri)k(G.)d(Kondratiev)976 1505 y Fo(1)p Fp(;)p Fo(2)588 1590 y Fu(Ludwig)j(Streit)862 1574 y Fo(1)p Fp(;)p Fo(3)588 1659 y Fu(W)l(erner)f(W)l(esterk)m(amp)992 1643 y Fo(1)588 1728 y Fu(Jia-an)g(Y)l(an)803 1711 y Fo(4)1104 1505 y(1)1124 1521 y Fm(BiBoS,)g(Universit\177)-23 b(at)15 b(Bielefeld,)1104 1578 y(D)h(33615)i(Bielefeld,)d(Germany)1104 1641 y Fo(2)1124 1658 y Fm(Institute)g(of)i(Mathematics,)1104 1714 y(Kiev,)f(Ukr)n(aine)1104 1778 y Fo(3)1124 1794 y Fm(CCM,)f(Universidade)g(da)i(Madeir)n(a,)1104 1851 y(P9000)g(F)m(unchal,)e(Portugal)1104 1914 y Fo(4)1124 1931 y Fm(Institute)g(of)i(Applie)n(d)e(Mathematics,)1104 1987 y(A)n(c)n(ademic)n(a)h(Sinic)n(a,)1104 2044 y(100080)i(Beijing,)d (China)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ENDBODY