Content-Type: multipart/mixed; boundary="-------------0203261812530" This is a multi-part message in MIME format. ---------------0203261812530 Content-Type: text/plain; name="02-152.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-152.keywords" M. G. Krein's continuous analogs of orthogonal polynomials on the circle; matrix canonical system of differential equations; density of the absolutely continuous component of the spectral measure; convergence of the adjoint polynomials; Szeg\"{o}--Kolmogorov--Krein condition ---------------0203261812530 Content-Type: application/postscript; name="teplyaev.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="teplyaev.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: teplyaev.dvi %%Pages: 18 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -f teplyaev.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.03.26:1559 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 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(\(1.2\),)h(but)g(the)f(con)m(v)m(erse)j(is)d(not)h(true)150 3014 y(in)30 b(general.)42 b(In)31 b(Section)f(4)g(w)m(e)i(pro)m(v)m(e) g(that)e(there)h(are)g(situations)e(when)j(\(1.1\))e(and)h(\(1.2\))f (hold,)150 3131 y(but)40 b(\005\()p Fr(\025)p Ft(\))g(has)h(to)f(b)s(e) g(de\014ned)i(as)e(a)g(limit)d(of)i(a)h(con)m(v)m(ergen)m(t)i (subsequence.)70 b(W)-8 b(e)41 b(sho)m(w)g(that)150 3247 y(this)j(situation)f(is)h(not)h(\\pathological",)f(but)h(can)g(o)s (ccur)f(ev)m(en)j(if)c(the)i(sp)s(ectral)g(measure)g Fr(\034)150 3363 y Ft(is)39 b(absolutely)h(con)m(tin)m(uous)g(with)g(p) s(ositiv)m(e)f(con)m(tin)m(uous)i(densit)m(y)f(\(Theorem)h(1\).)65 b(In)40 b(another)150 3479 y(example)32 b(\(Theorem)h(2\),)f(this)g (happ)s(ens)i(ev)m(en)g(though)799 3711 y Fm(j)p Ft(\005\()p Fr(\025)p Ft(\))p Fm(j)1061 3670 y Fq(2)1127 3711 y Ft(=)48 b(lim)1231 3771 y Fn(r)r Fl(!1)1422 3711 y Fm(j)p Fr(p)1499 3670 y Fl(\003)1538 3711 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))p Fm(j)1784 3670 y Fq(2)1851 3711 y Ft(=)28 b(2Im)o Fr(\025)2194 3575 y Fk(Z)2293 3601 y Fl(1)2249 3801 y Fq(0)2384 3711 y Fm(j)p Fr(p)p Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))p Fm(j)2707 3670 y Fq(2)2746 3711 y Fr(dr)30 b(<)d Fm(1)p Fr(;)150 3944 y Ft(and)33 b(the)g(co)s(e\016cien)m(ts)h(are)f (in)f Fr(L)1345 3908 y Fn(p)1418 3944 y Ft(for)g(an)m(y)h Fr(p)c(>)e Ft(2.)44 b(Moreo)m(v)m(er,)35 b(the)e(function)f(\005\()p Fr(\025)p Ft(\))h(can)g(not)g(b)s(e)150 4060 y(de\014ned)39 b(uniquely)-8 b(,)39 b(but)f(only)g(up)g(to)f(a)g(constan)m(t)i(factor) e(of)h(absolute)f(v)-5 b(alue)37 b(one)h(\(up)g(to)g(left)150 4177 y(m)m(ultiplication)33 b(b)m(y)39 b(a)e(unitary)g(matrix)f(in)h (the)h(case)h(of)e(Sakhno)m(vic)m(h's)i(theorem\).)59 b(Section)37 b(6)150 4293 y(con)m(tains)30 b(sev)m(eral)g(results)g (related)f(to)g(the)h(con)m(v)m(ergence)j(of)c(the)h(limit)c(\(1.3\))j (in)g(the)h(case)g(of)g(the)150 4409 y(Sakhno)m(vic)m(h's)k(di\013eren) m(tial)d(equations.)363 4525 y(Note)39 b(that)f(results)h(of)g(Section) f(6)h(apply)f(to)h(Krein's)f(system,)k(since)d(it)f(is)g(a)h (particular)150 4641 y(case)33 b(of)f(Sakhno)m(vic)m(h's)i(system.)44 b(Tw)m(o)33 b(of)f(the)h(three)g(results)f(there)h(are)f(new)i(ev)m(en) f(for)f(Krein's)150 4758 y(system.)48 b(A)m(t)34 b(the)g(same)f(time)f (results)i(of)g(Section)f(4)g(are)h(stated)g(for)f(Krein's)h(system,)h (but)f(are)150 4874 y(applicable)d(for)h(Sakhno)m(vic)m(h's)i(system)g (as)e(w)m(ell.)363 4990 y(The)39 b(fundamen)m(tal)f(pap)s(er)h([K])f (presen)m(ts)j(a)e(n)m(um)m(b)s(er)g(of)f(imp)s(ortan)m(t)f(results,)j (though)f(it)150 5106 y(do)s(es)24 b(not)g(con)m(tain)g(pro)s(ofs)f (due)i(to)e(the)i(t)m(yp)s(e)g(of)e(the)h(journal)f(it)g(w)m(as)i (published)f(in.)40 b(Later)23 b(pro)s(ofs)p eop %%Page: 4 4 4 3 bop 150 6 a Fu(4)1500 b(A.)25 b(TEPL)-6 b(Y)g(AEV)150 329 y Ft(of)33 b(Krein's)h(results)g(w)m(ere)h(giv)m(en)f(indep)s (enden)m(tly)h(b)m(y)g(the)f(author)f(in)g(1990)g(\([T1],)i(partly)e (pub-)150 446 y(lished)f(in)f([T2]\))i(and)f(L.)h(A.)f(Sakhno)m(vic)m (h)i(in)d(1998)g(\([S2{S4]\).)44 b(The)33 b(main)d(sub)5 b(ject)34 b(of)e([T1,T2])150 562 y(w)m(as)40 b(to)f(pro)m(v)m(e)i(that) e(the)g(sp)s(ectral)g(measure)h Fr(\034)51 b Ft(is)38 b(absolutely)h(con)m(tin)m(uous)h(with)f(probabilit)m(y)150 678 y(one)33 b(if)e(the)i(co)s(e\016cien)m(t)g Fr(a)p Ft(\()p Fr(r)s Ft(\))g(is)f(a)g(random)g(function)g(satisfying)g (certain)g(conditions.)363 794 y(In)46 b([T1,T2])h(the)f(author)g (noted)h(and)f(recti\014ed)g(an)g(inaccuracy)h(in)e(the)i(statemen)m(t) f(of)150 911 y(Krein's)36 b(theorem,)h(and)f(ga)m(v)m(e)i(a)e(pro)s(of) f(of)h(the)g(corrected)i(main)c(theorem)i(\(see)h(Section)g(3)e(for)150 1027 y(more)d(details\).)42 b(Theorems)34 b(1)e(and)h(2)f(in)g(Section) g(4)h(pro)m(v)m(e,)h(in)d(particular,)g(that)i(a)f(part)g(of)h(the)150 1143 y(statemen)m(t)g(of)f(Krein's)h(theorem)f(in)g([K])g(needs)i(to)f (b)s(e)f(revised.)363 1259 y(In)22 b([S1{S5])f(L.)h(A.)g(Sakhno)m(vic)m (h)h(de\014ned)g(and)f(studied)g(matrix)e(v)-5 b(alued)22 b(con)m(tin)m(uous)g(analogs)150 1376 y(of)41 b(orthogonal)f(p)s (olynomials)f(on)j(the)g(unit)f(circle,)i(and)f(pro)m(v)m(ed)h(matrix)e (generalizations)e(of)150 1492 y(Krein's)49 b(results.)92 b(Unfortunately)-8 b(,)53 b(these)d(w)m(orks)g(con)m(tain)f(the)g(same) f(kind)h(of)f(inaccuracy)150 1608 y(as)30 b([K].)f(In)h(Section)f(5)g (w)m(e)i(presen)m(t)g(the)f(corrected)h(statemen)m(t,)f(and)g(the)g (corrected)g(part)g(of)f(the)150 1724 y(pro)s(of.)363 1840 y(W)-8 b(e)30 b(emphasize)f(that)h(the)g(inaccuracy)g(in)f(the)h (statemen)m(t)g(of)g(Krein's)f(and)h(Sakhno)m(vic)m(h's)150 1957 y(theorems)37 b(is)g(not)g(signi\014can)m(t,)g(and)g(do)s(es)h (not)f(undermine)f(kno)m(wn)j(implications)33 b(from)j(these)150 2073 y(imp)s(ortan)m(t)j(results.)70 b(F)-8 b(or)40 b(instance,)k(if)c (\(1.1\))h(and)g(\(1.2\))g(hold,)h(then)g(there)g(is)f(the)g(function) 150 2189 y(\005\()p Fr(\025)p Ft(\))32 b(whic)m(h)i(is)e(analytic)f (and)i(has)f(no)h(zeros)g(for)f(Im)p Fr(\025)27 b(>)h Ft(0,)k(and)1565 2406 y Fr(\034)1618 2365 y Fl(0)1642 2406 y Ft(\()p Fr(\025)p Ft(\))c(=)2096 2339 y(1)p 1916 2383 409 4 v 1916 2475 a(2)p Fr(\031)t Fm(j)p Ft(\005\()p Fr(\025)p Ft(\))p Fm(j)2286 2446 y Fq(2)150 2635 y Ft(for)49 b(Leb)s(esgue)h(almost)e(all)f Fr(\025)56 b Fm(2)g Fp(R)k Ft(\(there)50 b(is)f(an)g(analogous)f(matrix)g(v)m(ersion)h(pro)m(v)m (ed)i(b)m(y)150 2751 y(L.)24 b(A.)h(Sakhno)m(vic)m(h)h(in)d([S4]\).)41 b(This)25 b(result)f(remains)f(unc)m(hanged)j(ev)m(en)g(if)e(the)g (limit)41 b(lim)3252 2811 y Fn(r)r Fl(!1)3444 2751 y Fr(p)3493 2715 y Fl(\003)3532 2751 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))150 2867 y(div)m(erges,)34 b(and)e(the)h(non)m (uniqueness)i(of)d(\005\()p Fr(\025)p Ft(\))h(men)m(tioned)f(ab)s(o)m (v)m(e)h(tak)m(es)h(place.)150 3065 y Fi(A)-5 b(cknow)5 b(le)-5 b(dgments.)47 b Ft(The)30 b(author)f(is)f(deeply)i(grateful)d (to)i(I.)g(A.)g(Ibragimo)m(v)f(for)g(his)h(in)m(v)-5 b(aluable)150 3181 y(guidance)36 b(during)g(an)g(early)g(part)g(of)f (this)h(w)m(ork,)i(whic)m(h)f(w)m(as)g(completed)f(in)f(St.-P)m (etersburg,)150 3297 y(former)26 b(Leningrad,)i(State)f(Univ)m(ersit)m (y)h(and)f(Steklo)m(v)h(Mathematical)d(Institute)i(\(POMI\).)h(The)150 3414 y(author)g(also)f(thanks)i(S.)g(A.)f(Deniso)m(v,)h(M.)g(I.)f (Gordin,)g(M.)h(L.)f(Lapidus,)h(L.)f(A.)g(Sakhno)m(vic)m(h)i(and)150 3530 y(M.)j(Z.)f(Solom)m(y)m(ak)g(for)g(helpful)g(discussions)h(during) f(the)h(preparation)f(of)g(this)g(pap)s(er.)874 3761 y(2.)49 b Fs(Or)-7 b(thogonal)39 b(pol)-7 b(ynomials)39 b(on)f(the)f(cir)n(cle.)363 3935 y Ft(If)g Fm(f)p Ft(\010)585 3950 y Fn(n)632 3935 y Ft(\()p Fr(z)t Ft(\))p Fm(g)807 3899 y Fl(1)807 3960 y Fn(n)p Fq(=0)983 3935 y Ft(are)g(monic)f(p)s (olynomials)f(of)i(degree)i Fr(n)e Ft(orthogonal)f(on)i(the)g(unit)f (circle)150 4051 y(with)27 b(resp)s(ect)i(to)f(a)f(probabilit)m(y)f (Borel)h(measure)h Fr(\034)40 b Ft(then)28 b(there)h(exists)f(a)g (sequence)i(of)d(complex)150 4167 y(n)m(um)m(b)s(ers)33 b Fm(f)p Fr(a)644 4182 y Fn(n)691 4167 y Fm(g)741 4131 y Fl(1)741 4192 y Fn(n)p Fq(=0)911 4167 y Ft(suc)m(h)h(that)1348 4328 y(\010)1418 4343 y Fn(n)p Fq(+1)1556 4328 y Ft(\()p Fr(z)t Ft(\))28 b(=)g Fr(z)t Ft(\010)1932 4343 y Fn(n)1980 4328 y Ft(\()p Fr(z)t Ft(\))23 b Fm(\000)g Ft(\026)-50 b Fr(a)2278 4343 y Fn(n)2325 4328 y Ft(\010)2395 4287 y Fl(\003)2395 4352 y Fn(n)2443 4328 y Ft(\()p Fr(z)t Ft(\))1348 4479 y(\010)1418 4438 y Fl(\003)1418 4503 y Fn(n)p Fq(+1)1556 4479 y Ft(\()p Fr(z)t Ft(\))28 b(=)g(\010)1883 4438 y Fl(\003)1883 4503 y Fn(n)1930 4479 y Ft(\()p Fr(z)t Ft(\))23 b Fm(\000)f Fr(a)2228 4494 y Fn(n)2276 4479 y Fr(z)t Ft(\010)2395 4494 y Fn(n)2443 4479 y Ft(\()p Fr(z)t Ft(\))3549 4410 y(\(2.1\))150 4653 y(where)34 b Fm(f)p Ft(\010)552 4617 y Fl(\003)552 4678 y Fn(n)599 4653 y Ft(\()p Fr(z)t Ft(\))p Fm(g)774 4617 y Fl(1)774 4678 y Fn(n)p Fq(=0)944 4653 y Ft(is)e(an)h(auxiliary)d(sequence)35 b(of)e(p)s(olynomials)c(and)1545 4822 y(\010)1615 4837 y Fq(0)1655 4822 y Ft(\()p Fr(z)t Ft(\))g(=)e(\010)1982 4781 y Fl(\003)1982 4846 y Fq(0)2022 4822 y Ft(\()p Fr(z)t Ft(\))h(=)g(1)p Fr(:)150 4990 y Ft(The)33 b(p)s(olynomials)c(\010)962 4954 y Fl(\003)962 5015 y Fn(n)1010 4990 y Ft(\()p Fr(z)t Ft(\))j(are)h(adjoin)m(t)e(to)h(the)g(orthogonal)f(p)s(olynomials)e (\010)3046 5005 y Fn(n)3093 4990 y Ft(\()p Fr(z)t Ft(\))k(in)f(the)g (sense)150 5106 y(that)g(\010)431 5070 y Fl(\003)431 5131 y Fn(n)479 5106 y Ft(\()p Fr(z)t Ft(\))c(=)g(1)22 b(+)g Fm(\001)17 b(\001)g(\001)j Ft(+)k(\026)-51 b Fr(c)1183 5121 y Fn(j)1219 5106 y Fr(z)1268 5070 y Fn(n)p Fl(\000)p Fn(j)1426 5106 y Ft(+)22 b Fm(\001)17 b(\001)g(\001)47 b Ft(if)31 b(\010)1848 5070 y Fl(\003)1848 5131 y Fn(n)1896 5106 y Ft(\()p Fr(z)t Ft(\))d(=)g Fr(z)e Ft(+)d Fm(\001)17 b(\001)g(\001)j Ft(+)i Fr(c)2601 5121 y Fn(j)2637 5106 y Fr(z)2686 5070 y Fn(j)2746 5106 y Ft(+)g Fm(\001)17 b(\001)g(\001)e Ft(.)p eop %%Page: 5 5 5 4 bop 713 6 a Fu(ON)33 b(THE)g(THEOREMS)g(OF)h(M.)g(G.)f(KREIN)g(AND) f(L.)h(A.)g(SAKHNO)n(VICH.)523 b(5)363 329 y Ft(So)28 b(called)f(circular)h(\(re\015ection,)h(Sh)m(ur's\))h(parameters)f Fm(f)p Fr(a)2531 344 y Fn(n)2578 329 y Fm(g)2628 293 y Fl(1)2628 354 y Fn(n)p Fq(=0)2793 329 y Ft(satisfy)g Fm(j)p Fr(a)3173 344 y Fn(n)3220 329 y Fm(j)e Fh(6)h Ft(1.)42 b(More-)150 446 y(o)m(v)m(er,)1783 603 y Fm(j)p Fr(a)1862 618 y Fn(n)1909 603 y Fm(j)27 b Fr(<)h Ft(1)1432 b(\(2.2\))150 782 y(for)32 b(all)f Fr(n)h Ft(if)g(and)g(only)g(if)g (the)h(measure)g(is)f(not)g(concen)m(trated)i(in)e(a)h(\014nite)f(n)m (um)m(b)s(er)h(of)f(atoms.)363 898 y(Con)m(v)m(ersely)-8 b(,)51 b(if)45 b(conditions)g(\(2.2\))h(are)g(satis\014ed,)k(then)d (there)g(exists)f(a)g(unique)h(Borel)150 1014 y(probabilit)m(y)35 b(measure)j Fr(\034)49 b Ft(on)37 b(the)h(unit)f(circle)g(suc)m(h)h (that)g(p)s(olynomials)c Fm(f)p Ft(\010)3021 1029 y Fn(n)3068 1014 y Ft(\()p Fr(z)t Ft(\))p Fm(g)3243 978 y Fl(1)3243 1039 y Fn(n)p Fq(=0)3381 1014 y Ft(,)39 b(de\014ned)150 1130 y(b)m(y)c(\(2.1\))o(,)e(are)g(orthogonal)d(with)j(resp)s(ect)g(to) g Fr(\034)11 b Ft(.)363 1246 y(The)33 b(theory)h(of)e(orthogonal)f(p)s (olynomials)f(on)j(the)g(circle)f(w)m(as)i(dev)m(elop)s(ed)g(b)m(y)g (G.)e(Szeg\177)-49 b(o,)150 1363 y(N.)28 b(I.)g(Akhiezer,)h(L.)f(Y)-8 b(a.)27 b(Geronim)m(us)g Fi(et)j(al)p Ft(.)42 b(\([A,G,Se]\).)g(The)29 b(follo)m(wing)c(theorem)i(is)g(a)g(com)m(bi-)150 1479 y(nation)k(of)g(results)i(of)e(G.)h(Szeg\177)-49 b(o,)32 b(A.)h(N.)f(Kolmogoro)m(v,)e(M.)i(G.)g(Krein)f(and)h(L.)g(Y)-8 b(a.)32 b(Geronim)m(us)150 1595 y(\(see)i([G]\).)150 1753 y FH(Theorem.)k Fi(The)30 b(line)-5 b(ar)31 b(sp)-5 b(an)30 b(of)h Fm(f)p Ft(\010)1575 1768 y Fn(n)1622 1753 y Ft(\()p Fr(z)t Ft(\))p Fm(g)1797 1716 y Fl(1)1797 1777 y Fn(n)p Fq(=0)1965 1753 y Fi(is)g(not)g(dense)f(in)h Fr(L)2681 1716 y Fq(2)2681 1777 y Fn(\034)2755 1753 y Fi(if)g(and)f(only)h(if)f(any)h(of)g(the)150 1869 y(fol)5 b(lowing)33 b(\014ve)i(e)-5 b(quivalent)34 b(statements)h(hold)312 2026 y Ft(\(I\))1436 2117 y Fk(Z)1536 2144 y Fq(2)p Fn(\031)1491 2343 y Fq(0)1634 2253 y Ft(log)17 b Fr(\034)1830 2212 y Fl(0)1854 2253 y Ft(\()p Fr(e)1937 2212 y Fo(i)i Fn(\022)2019 2253 y Ft(\))p Fr(d\022)30 b(>)e Fm(\0001)1085 b Ft(\(2.3\))465 2486 y Fi(wher)-5 b(e)26 b Fr(\034)785 2450 y Fl(0)836 2486 y Fi(is)i(the)f(density)g(of)g(the)h(absolutely)f(c)-5 b(ontinuous)27 b(c)-5 b(omp)g(onent)26 b(of)h Fr(\034)39 b Fi(with)27 b(r)-5 b(esp)g(e)g(ct)465 2602 y(to)35 b(the)g(L)-5 b(eb)g(esgue)34 b(me)-5 b(asur)g(e)34 b(on)h(the)g(unit)g(cir)-5 b(cle.)274 2719 y Ft(\(I)s(I\))42 b Fi(Ther)-5 b(e)34 b(exists)g(at)h(le)-5 b(ast)35 b(one)f Fr(z)40 b Fi(in)34 b(the)h(unit)h(disk)e Fr(D)c Ft(=)e Fm(f)p Fr(z)k Ft(:)c Fm(j)p Fr(z)t Fm(j)g Fr(<)f Ft(1)p Fm(g)35 b Fi(such)f(that)1609 2873 y Fl(1)1572 2903 y Fk(X)1577 3112 y Fn(n)p Fq(=0)1732 2997 y Fm(j)p Ft(\010)1830 3012 y Fn(n)1877 2997 y Ft(\()p Fr(z)t Ft(\))p Fm(j)2030 2956 y Fq(2)2098 2997 y Fr(<)27 b Fm(1)p Fr(:)1221 b Ft(\(2.4\))236 3287 y(\(I)s(I)s(I\))42 b Fi(Ther)-5 b(e)34 b(exists)g(at)h(le)-5 b(ast)35 b(one)f Fr(z)f Fm(2)28 b Fr(D)37 b Fi(such)e(that)1528 3485 y Ft(lim)17 b(inf)1572 3545 y Fn(n)p Fl(!1)1815 3485 y Fm(j)p Ft(\010)1913 3444 y Fl(\003)1913 3510 y Fn(n)1960 3485 y Ft(\()p Fr(z)t Ft(\))p Fm(j)28 b Fr(<)g Fm(1)p Fr(:)239 3720 y Ft(\(IV\))42 b Fi(The)34 b(series)42 b Ft(\(2.4\))35 b Fi(c)-5 b(onver)g(ges)33 b(uniformly)i(on)f(c)-5 b(omp)g(act)34 b(subsets)h(of)g Fr(D)s Fi(.)274 3837 y Ft(\(V\))42 b Fi(Ther)-5 b(e)34 b(exists)g(a)h(function)g Ft(\005\()p Fr(z)t Ft(\))p Fi(,)g(analytic)g(in)f Fr(D)s Fi(,)h(such)f(that)i(the)e(limit)1563 4035 y Ft(\005\()p Fr(z)t Ft(\))29 b(=)51 b(lim)1893 4095 y Fn(n)p Fl(!1)2094 4035 y Ft(\010)2164 3994 y Fl(\003)2164 4060 y Fn(n)2211 4035 y Ft(\()p Fr(z)t Ft(\))1213 b(\(2.5\))465 4265 y Fi(is)34 b(uniformly)h(c)-5 b(onver)g(gent)34 b(on)g(c)-5 b(omp)g(act)34 b(subsets)h(of)g Fr(D)s Fi(.)150 4422 y(Mor)-5 b(e)g(over,)34 b(the)h(statements)g(\(I)16 b({V\))35 b(ar)-5 b(e)35 b(e)-5 b(quivalent)34 b(to)h(the)g(c)-5 b(ondition)1681 4576 y Fl(1)1644 4606 y Fk(X)1650 4815 y Fn(n)p Fq(=0)1805 4700 y Fm(j)p Fr(a)1884 4715 y Fn(n)1931 4700 y Fm(j)1959 4659 y Fq(2)2025 4700 y Fr(<)28 b Fm(1)p Fr(:)363 4990 y Ft(Note)39 b(that)g(in)g(\(I\))g(the)h(in)m(tegral)d (is)i(alw)m(a)m(ys)h(less)g(than)f(+)p Fm(1)p Ft(,)i(and)e(that)g (there)h(is)f(no)g(re-)150 5106 y(strictions)32 b(on)g(the)h(singular)e (part)i(of)f Fr(\034)11 b Ft(.)p eop %%Page: 6 6 6 5 bop 150 6 a Fu(6)1500 b(A.)25 b(TEPL)-6 b(Y)g(AEV)1454 329 y Ft(3.)49 b Fs(Krein's)37 b(theorem.)363 504 y Ft(In)25 b([K])h(M.)g(G.)f(Krein)f(studied)i(the)g(follo)m(wing)d(canonical)h (system)i(of)f(ordinary)g(di\013eren)m(tial)150 620 y(equations)1156 730 y Fr(d)p 1133 774 98 4 v 1133 866 a(dr)1274 797 y(p)17 b Ft(\()p Fr(r)m(;)g(\025)p Ft(\))105 b(=)49 b Fg(i)12 b Fr(\025)17 b(p)p Ft(\()p Fr(r)m(;)g(\025)p Ft(\))k Fm(\000)p 2297 711 174 4 v 23 w Fr(a)p Ft(\()p Fr(r)s Ft(\))16 b Fr(p)2536 756 y Fl(\003)2576 797 y Ft(\()p Fr(r)m(;)h(\025)p Ft(\))1156 972 y Fr(d)p 1133 1017 98 4 v 1133 1108 a(dr)1257 1040 y(p)1306 998 y Fl(\003)1345 1040 y Ft(\()p Fr(r)m(;)g(\025)p Ft(\))100 b(=)38 b Fm(\000)23 b Fr(a)p Ft(\()p Fr(r)s Ft(\))16 b Fr(p)p Ft(\()p Fr(r)m(;)h(\025)p Ft(\))3549 909 y(\(3.1\))150 1224 y(where)36 b Fr(a)p Ft(\()p Fm(\001)p Ft(\))e(is)g(a)h(measurable)f(lo)s(cally)e(in)m (tegrable)h(function)i(on)f([0)p Fr(;)17 b Fm(1)p Ft(\),)34 b(and)h(the)g(initial)c(con-)150 1341 y(ditions)g(are)1487 1470 y Fr(p)p Ft(\(0)p Fr(;)17 b(\025)p Ft(\))27 b(=)g Fr(p)1941 1429 y Fl(\003)1981 1470 y Ft(\(0)p Fr(;)17 b(\025)p Ft(\))27 b(=)g(1)p Fr(:)150 1619 y Ft(There)i(is)f(a)g(unique) g(Borel)g(measure)g Fr(\034)39 b Ft(on)28 b Fp(R)5 b Ft(,)35 b(whic)m(h)29 b(called)e(the)h(sp)s(ectral)g(measure,)i(suc)m (h)f(that)1511 1701 y Fk(Z)1567 1926 y Fj(R)1754 1769 y Ft(1)p 1646 1814 266 4 v 1646 1905 a(1)21 b(+)h Fr(\025)1871 1876 y Fq(2)1921 1836 y Fr(d\034)11 b Ft(\()p Fr(\025)p Ft(\))27 b Fr(<)h Fm(1)1160 b Ft(\(3.2\))150 2067 y(and)33 b(the)g(map)e Ff(U)d Ft(:)g Fr(L)944 2031 y Fq(2)944 2097 y([0)p Fn(;)p Fl(1)p Fq(\))1149 2067 y Fm(!)f Fr(L)1342 2031 y Fq(2)1342 2092 y Fn(\034)1418 2067 y Ft(de\014ned)34 b(b)m(y)1385 2313 y Ff(U)p Fr(f)11 b Ft(\()p Fr(\025)p Ft(\))27 b(=)1779 2178 y Fk(Z)1878 2204 y Fl(1)1834 2403 y Fq(0)1970 2313 y Fr(f)11 b Ft(\()p Fr(r)s Ft(\))p Fr(p)p Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))p Fr(dr)1035 b Ft(\(3.3\))150 2540 y(is)30 b(an)h(isometry)-8 b(.)42 b(Con)m(v)m(ersely)-8 b(,)33 b(if)d(the)h(condition)f(\(3.2\))g(is)g(satis\014ed)i(for)e(a)h (Borel)f(measure)h Fr(\034)42 b Ft(on)150 2656 y Fp(R)5 b Ft(,)39 b(then)33 b(there)h(exists)f(a)f(unique)i(measurable)e(lo)s (cally)e(in)m(tegrable)i(function)g Fr(a)p Ft(\()p Fm(\001)p Ft(\))g(on)h Fp(R)3465 2671 y Fq(+)3562 2656 y Ft(suc)m(h)150 2773 y(that)f Ff(U)h Ft(de\014ned)h(b)m(y)h(\(3.1\))d(and)h(\(3.3\))f (is)g(an)g(isometry)g Ff(U)c Ft(:)g Fr(L)2446 2736 y Fq(2)2446 2802 y([0)p Fn(;)p Fl(1)p Fq(\))2650 2773 y Fm(!)g Fr(L)2844 2736 y Fq(2)2844 2797 y Fn(\034)2887 2773 y Ft(.)363 2902 y(A)42 b(simple)f(example)h(is)g(the)g(situation)f (when)j Fr(a)p Ft(\()p Fr(r)s Ft(\))g Fm(\021)h Ft(0)d(and)g Ff(U)g Ft(is)g(the)h(usual)f(F)-8 b(ourier)150 3018 y(transform.)70 b(In)42 b(this)f(case)h Fr(\034)53 b Ft(is)41 b(the)h(Leb)s(esgue)h (measure)f(normalized)e(b)m(y)i(2)p Fr(\031)t Ft(.)71 b(F)-8 b(or)40 b(a)i(more)150 3134 y(detailed)32 b(study)h(see)h ([AR,R,D2{5,DKu].)363 3251 y(The)26 b(follo)m(wing)d(imp)s(ortan)m(t)g (result)j(w)m(as)g(stated)g(b)m(y)h(M.)f(G.)f(Krein)f(in)h([K])h (without)f(a)g(pro)s(of.)150 3394 y FH(Theorem.)41 b Fi(The)34 b(isometry)g Ff(U)h Fi(is)f(not)g(onto)h(if)f(and)g(only)g (if)g(any)h(of)f(the)g(fol)5 b(lowing)33 b(\014ve)h(e)-5 b(quiv-)150 3510 y(alent)35 b(statements)f(hold)312 3653 y Ft(\(I\))1494 3706 y Fk(Z)1549 3931 y Fj(R)1628 3774 y Ft(log)17 b Fr(\034)1824 3738 y Fl(0)1847 3774 y Ft(\()p Fr(\025)p Ft(\))p 1628 3818 353 4 v 1671 3910 a(1)22 b(+)g Fr(\025)1897 3881 y Fq(2)1990 3841 y Fr(d\025)27 b(>)h Fm(\0001)1143 b Ft(\(3.4\))465 4046 y Fi(wher)-5 b(e)26 b Fr(\034)785 4010 y Fl(0)836 4046 y Fi(is)i(the)f(density)g(of) g(the)h(absolutely)f(c)-5 b(ontinuous)27 b(c)-5 b(omp)g(onent)26 b(of)h Fr(\034)39 b Fi(with)27 b(r)-5 b(esp)g(e)g(ct)465 4162 y(to)35 b(the)g(L)-5 b(eb)g(esgue)34 b(me)-5 b(asur)g(e)34 b(on)h Fp(R)5 b Fi(.)274 4278 y Ft(\(I)s(I\))42 b Fi(Ther)-5 b(e)34 b(exists)g(at)h(le)-5 b(ast)35 b(one)f Fr(\025)h Fi(in)g(the)g(domain)e Fr(H)i Ft(=)28 b Fm(f)p Fr(\025)f Ft(:)h Fi(Im)o Fr(\025)g(>)f Ft(0)p Fm(g)35 b Fi(such)g(that)1496 4368 y Fk(Z)1596 4394 y Fl(1)1551 4593 y Fq(0)1687 4503 y Fm(j)p Fr(p)p Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))p Fm(j)2010 4462 y Fq(2)2049 4503 y Fr(dr)30 b(<)d Fm(1)p Fr(:)1145 b Ft(\(3.5\))236 4730 y(\(I)s(I)s(I\))42 b Fi(Ther)-5 b(e)34 b(exists)g(at)h(le)-5 b(ast)35 b(one)f Fr(\025)28 b Fm(2)g Fr(H)42 b Fi(such)35 b(that)1497 4900 y Ft(lim)17 b(inf)1545 4960 y Fn(r)r Fl(!1)1784 4900 y Fm(j)p Fr(p)1861 4859 y Fl(\003)1900 4900 y Ft(\()p Fr(r)m(;)g(\025)p Ft(\))p Fm(j)27 b Fr(<)g Fm(1)p Fr(:)1146 b Ft(\(3.6\))239 5106 y(\(IV\))42 b Fi(The)34 b(inte)-5 b(gr)g(al)45 b Ft(\(3.5\))34 b Fi(c)-5 b(onver)g(ges)34 b(uniformly)g(on)h(c)-5 b(omp)g(act)34 b(subsets)h(of)f Fr(H)8 b Fi(.)p eop %%Page: 7 7 7 6 bop 713 6 a Fu(ON)33 b(THE)g(THEOREMS)g(OF)h(M.)g(G.)f(KREIN)g(AND) f(L.)h(A.)g(SAKHNO)n(VICH.)523 b(7)274 329 y Ft(\(V\))42 b Fi(Ther)-5 b(e)32 b(exists)h(a)g(function)g Ft(\005\()p Fr(\025)p Ft(\))p Fi(,)h(analytic)f(in)g Fr(H)8 b Fi(,)33 b(such)g(that)h(on)f(any)g(c)-5 b(omp)g(act)33 b(subset)465 446 y(of)h Fr(H)43 b Fi(ther)-5 b(e)34 b(is)h(a)g(uniform)f(limit)1503 609 y Ft(\005\()p Fr(\025)p Ft(\))28 b(=)52 b(lim)1840 668 y Fn(n)p Fl(!1)2041 609 y Fr(p)2090 567 y Fl(\003)2129 609 y Ft(\()p Fr(r)2211 624 y Fn(n)2258 609 y Fr(;)17 b(\025)p Ft(\))1152 b(\(3.7\))465 802 y Fi(for)34 b(a)h(se)-5 b(quenc)g(e)34 b Fr(r)1145 817 y Fn(n)1220 802 y Fm(!)27 b(1)p Fi(.)363 942 y Ft(P)m(arts)39 b(\(I)s(I)s(I,V\))g(of)g(this)f (theorem)h(w)m(ere)h(not)f(stated)h(in)e([K])h(correctly)-8 b(.)62 b(Namely)-8 b(,)40 b(it)e(w)m(as)150 1058 y(written)32 b(as)h(if)f(\(I,I)s(I,IV\))h(w)m(ere)h(equiv)-5 b(alen)m(t)32 b(to:)209 1198 y(\(I)s(I)s(I'\))42 b(There)33 b(exists)h(at)e(least)g (one)h Fr(\025)28 b Fm(2)g Fr(H)40 b Ft(suc)m(h)34 b(that)e(sup)2420 1221 y Fn(r)r Fe(>)p Fq(0)2565 1198 y Fm(j)p Fr(p)2642 1162 y Fl(\003)2681 1198 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))p Fm(j)27 b Fr(<)g Fm(1)p Ft(.)247 1315 y(\(V'\))42 b(The)36 b(limit)c(\005\()p Fr(\025)p Ft(\))27 b(=)h(lim)1376 1330 y Fn(r)r Fl(!1)1572 1315 y Fr(p)1621 1279 y Fl(\003)1660 1315 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))35 b(con)m(v)m(erges)j (uniformly)33 b(on)i(compact)g(subsets)i(of)465 1431 y Fr(H)8 b Ft(.)150 1571 y(In)28 b(Section)f(4)g(w)m(e)i(presen)m(t)g (t)m(w)m(o)f(coun)m(terexamples.)43 b(The)29 b(corrected)f(statemen)m (t)g(app)s(eared)g(\014rst)150 1687 y(in)k([T1,T2].)363 1803 y(In)39 b(Section)h(4)f(w)m(e)h(pro)m(v)m(e)h(that,)g(in)d (general,)j(\005\()p Fr(\025)p Ft(\))e(can)h(not)f(b)s(e)h(de\014ned)h (uniquely)-8 b(,)41 b(but)150 1920 y(only)26 b(up)h(to)g(a)f(factor)h (of)f(absolute)g(v)-5 b(alue)27 b(one.)41 b(In)28 b(Section)e(6)h(w)m (e)g(sho)m(w)h(that)f(if)f Fr(a)p Ft(\()p Fr(r)s Ft(\))h Fm(2)h Fr(L)3453 1883 y Fq(2)3493 1920 y Ft([0)p Fr(;)17 b Fm(1)p Ft(\))150 2036 y(then)30 b(\(I)16 b({V\))31 b(as)f(w)m(ell)f(as)h(\(I)s(I)s(I',V'\))g(hold,)g(but)g(the)g(result)g (is)f(sharp)i(in)e(the)h(sense)i(of)d(Theorem)h(2)150 2152 y(and)j(Remark)f(6.1.)363 2268 y(Note)39 b(that)g(in)g(\(I\))g (the)h(in)m(tegral)d(is)i(alw)m(a)m(ys)h(less)g(than)f(+)p Fm(1)p Ft(,)i(and)e(that)g(there)h(is)f(no)g(re-)150 2384 y(strictions)32 b(on)g(the)h(singular)e(part)i(of)f Fr(\034)11 b Ft(.)1037 2599 y(4.)49 b Fs(Tw)n(o)38 b(resul)-7 b(ts)37 b(on)h(nonconver)n(gence.)150 2773 y FH(Theorem)33 b(1.)38 b Fi(Ther)-5 b(e)31 b(exists)g(a)g(r)-5 b(e)g(al-value)g(d)47 b Fr(C)1939 2737 y Fl(1)2045 2773 y Fi(function)31 b Fr(a)p Ft(\()p Fr(r)s Ft(\))g Fi(such)g(that)h(the)f(sp)-5 b(e)g(ctr)g(al)31 b(me)-5 b(a-)150 2890 y(sur)g(e)34 b Fr(\034)45 b Fi(is)34 b(absolutely)g(c)-5 b(ontinuous)33 b(with)h(p)-5 b(ositive)33 b(c)-5 b(ontinuous)34 b(density,)g (statements)f(\(I)16 b({V\))34 b(of)150 3006 y(Kr)-5 b(ein)-10 b('s)34 b(the)-5 b(or)g(em)35 b(hold,)f(but)1222 3169 y Ft(lim)17 b(inf)1269 3228 y Fn(r)r Fl(!1)1509 3169 y Fm(j)p Fr(p)1586 3128 y Fl(\003)1625 3169 y Ft(\()p Fr(r)m(;)g(\025)p Ft(\))p Fm(j)27 b Fr(<)g Ft(lim)17 b(sup)2063 3248 y Fn(r)r Fl(!1)2317 3169 y Fm(j)p Fr(p)2394 3128 y Fl(\003)2433 3169 y Ft(\()p Fr(r)m(;)g(\025)p Ft(\))p Fm(j)870 b Ft(\(4.1\))150 3387 y Fi(for)41 b(any)h Fr(\025)d Fm(2)i Fr(H)8 b Fi(.)63 b(In)41 b(addition,)h(the)g Ft(lim)17 b(sup)39 b Fi(in)i(\(4.1\))g(c)-5 b(an)41 b(b)-5 b(e)41 b(either)g(\014nite)g(or)h(identic)-5 b(al)5 b(ly)150 3504 y Ft(+)p Fm(1)34 b Fi(on)h Fr(H)8 b Fi(.)363 3643 y Ft(First,)31 b(w)m(e)j(need)g(the)f(follo)m(wing)c(lemma.)150 3783 y FH(Lemma)g(4.1.)37 b Fi(Supp)-5 b(ose)27 b Fr(b)p Ft(\()p Fr(r)s Ft(\))i Fi(b)-5 b(e)28 b(a)g(r)-5 b(e)g(al)29 b(nonzer)-5 b(o)27 b(c)-5 b(ontinuous)28 b(function)g(with)h(supp)-5 b(ort)28 b(in)h Ft([0)p Fr(;)17 b Ft(2])150 3899 y Fi(such)38 b(that)h Fr(b)p Ft(\()p Fr(r)s Ft(\))g Fi(is)f(nonne)-5 b(gative)38 b(for)g Fr(r)f Fm(2)e Ft([0)p Fr(;)17 b Ft(1])p Fi(,)39 b(and)f Fr(b)p Ft(\()p Fr(r)s Ft(\))d(=)f Fm(\000)p Fr(b)p Ft(\()p Fr(r)29 b Fm(\000)c Ft(2\))39 b Fi(for)f Fr(r)f Fm(2)e Ft([1)p Fr(;)17 b Ft(2])p Fi(.)56 b(L)-5 b(et)150 4015 y Fr(p)199 4030 y Fn(")236 4015 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))p Fi(,)34 b Fr(p)567 3979 y Fl(\003)567 4040 y Fn(")607 4015 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))34 b Fi(b)-5 b(e)35 b(the)g(solutions)f(of)h(\(3.1\))f (with)1334 4192 y Fr(a)p Ft(\()p Fr(r)s Ft(\))28 b(=)f Fr(a)1690 4207 y Fn(")1727 4192 y Ft(\()p Fr(r)s Ft(\))g(=)h Fm(\000)2068 4145 y Fq(log)13 b Fl(j)f Fq(log)g Fn(")p Fl(j)p 2068 4169 290 4 v 2197 4227 a Fn(")2368 4192 y Fr(b)p Ft(\()2457 4153 y Fn(r)p 2457 4169 34 4 v 2458 4227 a(")2501 4192 y Ft(\))p Fr(;)150 4356 y Ft(0)27 b Fr(<)h(")f(<)h Ft(1)p Fi(,)34 b(and)h(initial)f(c)-5 b(onditions)34 b Fr(p)1611 4371 y Fn(")1648 4356 y Ft(\(0)p Fr(;)17 b(\025)p Ft(\))26 b(=)i Fr(c)p Fi(,)35 b Fr(p)2160 4320 y Fl(\003)2160 4381 y Fn(")2199 4356 y Ft(\(0)p Fr(;)17 b(\025)p Ft(\))27 b(=)h Fr(c)2598 4320 y Fl(\003)2637 4356 y Fi(.)363 4473 y(If)34 b Fr(\025)h Fi(is)f(any)h(c)-5 b(omplex)34 b(numb)-5 b(er,)34 b(then)595 4635 y Fr(p)644 4650 y Fn(")681 4635 y Ft(\(2)p Fr(";)17 b(\025)p Ft(\))26 b(=)i Fr(c)22 b Ft(+)g Fr(o)p Ft(\()1330 4559 y Fm(p)p 1413 4559 46 4 v 76 x Fr(")o Ft(\))1496 4650 y Fn(")p Fl(!)p Fq(0)1838 4635 y Fi(and)199 b Fr(p)2241 4594 y Fl(\003)2241 4660 y Fn(")2280 4635 y Ft(\(2)p Fr(";)17 b(\025)p Ft(\))27 b(=)g Fr(c)2724 4594 y Fl(\003)2786 4635 y Ft(+)22 b Fr(o)p Ft(\()2969 4559 y Fm(p)p 3052 4559 V 76 x Fr(")o Ft(\))3135 4650 y Fn(")p Fl(!)p Fq(0)3278 4635 y Fr(;)244 b Ft(\(4.2\))150 4798 y Fi(wher)-5 b(e)37 b(the)g(limits)h(ar)-5 b(e)37 b(uniform)g(for)g Fr(\025)p Fi(,)h Fr(c)p Fi(,)g Fr(c)1845 4762 y Fl(\003)1922 4798 y Fi(in)g(any)f(c)-5 b(omp)g(act)37 b(subset)h(of)f Fp(C)20 b Fi(.)59 b(If)37 b(in)g(addition)150 4915 y Fr(c)28 b Fm(6)p Ft(=)f Fm(\000)p Fr(c)442 4878 y Fl(\003)482 4915 y Fi(,)35 b(then)1237 5037 y Ft(lim)1236 5096 y Fn(")p Fl(!)p Fq(0)1391 5037 y Fm(j)p Fr(p)1468 5052 y Fn(")1504 5037 y Ft(\()p Fr(";)17 b(\025)p Ft(\))p Fm(j)27 b Ft(=)i(lim)1885 5096 y Fn(")p Fl(!)p Fq(0)2040 5037 y Fm(j)p Fr(p)2117 4996 y Fl(\003)2117 5061 y Fn(")2156 5037 y Ft(\()p Fr(";)17 b(\025)p Ft(\))p Fm(j)27 b Ft(=)g Fm(1)p Fr(:)885 b Ft(\(4.3\))p eop %%Page: 8 8 8 7 bop 150 6 a Fu(8)1500 b(A.)25 b(TEPL)-6 b(Y)g(AEV)150 329 y Fi(Pr)h(o)g(of.)41 b Ft(First,)32 b(w)m(e)h(consider)g (di\013eren)m(tial)e(equations)1445 476 y Fr(d)p 1422 521 98 4 v 1422 612 a(dr)1546 544 y(q)1589 559 y Fn(")1626 544 y Ft(\()p Fr(r)s Ft(\))105 b(=)39 b Fm(\000)23 b Fr(a)2120 559 y Fn(")2157 544 y Ft(\()p Fr(r)s Ft(\))16 b Fr(q)2343 502 y Fl(\003)2339 568 y Fn(")2382 544 y Ft(\()p Fr(r)s Ft(\))1445 719 y Fr(d)p 1422 763 V 1422 854 a(dr)1546 786 y(q)1593 745 y Fl(\003)1589 811 y Fn(")1632 786 y Ft(\()p Fr(r)s Ft(\))99 b(=)39 b Fm(\000)23 b Fr(a)2120 801 y Fn(")2157 786 y Ft(\()p Fr(r)s Ft(\))16 b Fr(q)2339 801 y Fn(")2376 786 y Ft(\()p Fr(r)s Ft(\))3549 656 y(\(4.4\))150 987 y(with)32 b(initial)d(conditions)j Fr(q)1164 1002 y Fn(")1201 987 y Ft(\(0\))27 b(=)g Fr(c)p Ft(,)33 b Fr(q)1605 951 y Fl(\003)1601 1012 y Fn(")1644 987 y Ft(\(0\))28 b(=)f Fr(c)1942 951 y Fl(\003)1982 987 y Ft(.)43 b(Then)34 b(w)m(e)f(ha)m(v)m(e)1009 1208 y Fr(q)1052 1223 y Fn(")1089 1208 y Ft(\()p Fr(r)s Ft(\))22 b Fm(\000)g Fr(q)1380 1167 y Fl(\003)1376 1233 y Fn(")1420 1208 y Ft(\()p Fr(r)s Ft(\))27 b(=)16 b(\()p Fr(c)22 b Fm(\000)h Fr(c)1906 1167 y Fl(\003)1945 1208 y Ft(\))17 b(exp)2166 1098 y Fk(n)2249 1073 y(Z)2348 1099 y Fn(r)2304 1298 y Fq(0)2403 1208 y 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Fq(0)1371 1932 y Ft(=)1485 1864 y(1)p 1485 1909 49 4 v 1485 2000 a(2)1544 1932 y(\()p Fr(c)g Ft(+)g Fr(c)1786 1891 y Fl(\003)1825 1932 y Ft(\))17 b(exp)2045 1821 y Fk(n)2112 1932 y Ft(log)f Fm(j)h Ft(log)f Fr(")p Fm(j)22 b(\001)2586 1796 y Fk(Z)2686 1822 y Fq(1)2642 2022 y(0)2742 1932 y Fr(b)p Ft(\()p Fr(r)s Ft(\))p Fr(dr)3004 1821 y Fk(o)3092 1932 y Ft(+)g Fr(o)p Ft(\(1\))3362 1947 y Fn(")p Fl(!)p Fq(0)3504 1932 y Fr(:)150 2149 y Ft(Th)m(us)34 b(our)f(aim)d(is)i(to)h(sho)m(w)g(that) g(for)f(0)27 b Fh(6)h Fr(r)j Fh(6)d Ft(2)p Fr(")k Ft(w)m(e)h(ha)m(v)m (e)450 2315 y Fm(j)p Fr(p)527 2330 y Fn(")563 2315 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))22 b Fm(\000)h Fr(q)946 2330 y Fn(")983 2315 y Ft(\()p Fr(r)s Ft(\))p Fm(j)k Ft(=)g Fr(o)p Ft(\()1349 2239 y Fm(p)p 1432 2239 46 4 v 76 x Fr(")p Ft(\))1516 2330 y Fn(")p Fl(!)p Fq(0)1853 2315 y Ft(and)196 b Fm(j)p Fr(p)2283 2274 y Fl(\003)2283 2340 y Fn(")2322 2315 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))22 b Fm(\000)g Fr(q)2708 2274 y Fl(\003)2704 2340 y Fn(")2747 2315 y Ft(\()p Fr(r)s Ft(\))p Fm(j)27 b Ft(=)h Fr(o)p Ft(\()3114 2239 y Fm(p)p 3197 2239 V 76 x Fr(")o Ft(\))3280 2330 y Fn(")p Fl(!)p Fq(0)3423 2315 y Fr(:)363 2481 y Ft(T)-8 b(o)47 b(sho)m(w)i(this,)j(w)m(e)c(use)h(Gron)m(w)m (all's)e(lemma:)71 b Fi(if)49 b Fr(\013)q Ft(\()p Fr(r)s Ft(\))f Fi(is)h(a)f(nonne)-5 b(gative)48 b(inte)-5 b(gr)g(able)150 2598 y(function)41 b(such)h(that)g(for)f(some)g(c)-5 b(onstants)41 b Fr(c)1870 2613 y Fq(1)1910 2598 y Fr(;)17 b(c)1996 2613 y Fq(2)2075 2598 y Fh(>)41 b Ft(0)g Fi(we)g(have)g Fr(\013)q Ft(\()p Fr(r)s Ft(\))f Fh(6)g Fr(c)3050 2613 y Fq(1)3106 2517 y Fk(R)3173 2544 y Fn(r)3153 2632 y Fq(0)3227 2598 y Fr(\013)q Ft(\()p Fr(s)p Ft(\))p Fr(ds)27 b Ft(+)g Fr(c)3681 2613 y Fq(2)3720 2598 y Fi(,)150 2715 y(then)35 b Fr(\013)q Ft(\()p Fr(r)s Ft(\))27 b Fh(6)h Fr(c)727 2730 y Fq(2)766 2715 y Fr(e)811 2679 y Fn(c)842 2688 y Fd(1)877 2679 y Fn(r)914 2715 y Fi(.)363 2831 y Ft(First,)34 b(w)m(e)i(can)f(use)g(this)g(lemma)d(to)j(estimate)e Fr(\013)q Ft(\()p Fr(r)s Ft(\))e(=)g Fm(j)p Fr(p)2551 2846 y Fn(")2587 2831 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))p Fm(j)23 b Ft(+)h Fm(j)p Fr(p)3033 2795 y Fl(\003)3033 2856 y Fn(")3072 2831 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))p Fm(j)p Ft(.)49 b(By)35 b(\(3.1\))150 2948 y(and)e(the)g (de\014nition)e(of)h Fr(p)1101 2963 y Fn(")1138 2948 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))32 b(and)h Fr(p)1627 2912 y Fl(\003)1627 2972 y Fn(")1666 2948 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))33 b(w)m(e)g(ha)m(v)m(e)1156 3117 y Fm(j)p Fr(p)1233 3132 y Fn(")1270 3117 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))p Fm(j)k Ft(+)h Fm(j)p Fr(p)1712 3076 y Fl(\003)1712 3142 y Fn(")1751 3117 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))p Fm(j)27 b Fh(6)h Ft(\()p Fm(j)p Fr(c)p Fm(j)22 b Ft(+)g Fm(j)p Fr(c)2455 3076 y Fl(\003)2494 3117 y Fm(j)p Ft(\))p Fr(e)2605 3076 y Fn(M)2673 3084 y Fc(")2706 3076 y Fn(r)3549 3117 y Ft(\(4.6\))150 3284 y(where)1330 3409 y Fr(M)1424 3424 y Fn(")1488 3409 y Ft(=)28 b Fm(j)p Fr(\025)p Fm(j)21 b Ft(+)1834 3362 y Fq(log)13 b Fl(j)f Fq(log)h Fn(")p Fl(j)p 1834 3386 290 4 v 1963 3443 a Fn(")2166 3409 y Ft(max)2150 3470 y Fq(0)p Fe(6)p Fn(s)p Fe(6)p Fq(1)2380 3409 y Fr(b)p Ft(\()p Fr(s)p Ft(\))p Fr(:)150 3603 y Ft(Then)45 b(w)m(e)g(use)f(Gron)m(w)m (all's)f(lemma)f(once)i(more)f(to)h(estimate)f Fr(\013)q Ft(\()p Fr(r)s Ft(\))i(=)i Fm(j)p Fr(p)3023 3618 y Fn(")3059 3603 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))30 b Fm(\000)g Fr(q)3457 3618 y Fn(")3494 3603 y Ft(\()p Fr(r)s Ft(\))p Fm(j)f Ft(+)150 3719 y Fm(j)p Fr(p)227 3683 y Fl(\003)227 3744 y Fn(")266 3719 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))31 b Fm(\000)h Fr(q)671 3683 y Fl(\003)667 3744 y Fn(")710 3719 y Ft(\()p Fr(r)s Ft(\))p Fm(j)p Ft(.)84 b(Using)45 b(\(3.1\))h(and)g(\(4.5\))g(w)m(e)h(obtain)e Fr(\013)q Ft(\()p Fr(r)s Ft(\))50 b Fh(6)h Fr(c)2836 3734 y Fq(1)2892 3639 y Fk(R)2959 3665 y Fn(r)2939 3754 y Fq(0)3013 3719 y Fr(\013)q Ft(\()p Fr(s)p Ft(\))p Fr(ds)31 b Ft(+)g Fr(c)3475 3734 y Fq(2)3560 3719 y Ft(with)150 3837 y Fr(c)192 3852 y Fq(1)276 3837 y Ft(=)44 b Fr(M)490 3852 y Fn(")569 3837 y Ft(and)e Fr(c)810 3852 y Fq(2)894 3837 y Fh(>)i Fr(r)s Fm(j)p Fr(\025)17 b(p)1213 3852 y Fn(")1249 3837 y Ft(\()p Fr(s;)g(\025)p Ft(\))p Fm(j)41 b Ft(for)h(an)m(y)h(0)h Fh(6)g Fr(s)h Fh(6)f Fr(r)s Ft(.)72 b(By)43 b(estimate)e(\(4.6\))h(w)m (e)h(can)g(use)150 3953 y Fr(c)192 3968 y Fq(2)259 3953 y Ft(=)28 b(2)p Fr(")p Fm(j)p Fr(\025)p Fm(j)p Ft(\()p Fm(j)p Fr(c)p Fm(j)20 b Ft(+)i Fm(j)p Fr(c)895 3917 y Fl(\003)934 3953 y Fm(j)p Ft(\))p Fr(e)1045 3917 y Fq(2)p Fn(M)1148 3925 y Fc(")1181 3917 y Fn(")1250 3953 y Ft(when)34 b(0)27 b Fh(6)i Fr(r)h Fh(6)e Ft(2)p Fr(")p Ft(.)43 b(Hence)34 b(w)m(e)g(ha)m(v)m(e)446 4123 y Fm(j)p Fr(p)523 4138 y Fn(")559 4123 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))22 b Fm(\000)h Fr(q)942 4138 y Fn(")979 4123 y Ft(\()p Fr(r)s Ft(\))p Fm(j)e Ft(+)h Fm(j)p Fr(p)1326 4082 y Fl(\003)1326 4147 y Fn(")1365 4123 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))22 b Fm(\000)h Fr(q)1752 4082 y Fl(\003)1748 4147 y Fn(")1791 4123 y Ft(\()p Fr(r)s Ft(\))p Fm(j)k Fh(6)h Ft(2)p Fr(")p Fm(j)p Fr(\025)p Fm(j)p Ft(\()p Fm(j)p Fr(c)p Fm(j)20 b Ft(+)i Fm(j)p Fr(c)2606 4082 y Fl(\003)2645 4123 y Fm(j)p Ft(\))p Fr(e)2756 4082 y Fq(4)p Fn(M)2859 4090 y Fc(")2892 4082 y Fn(")2957 4123 y Ft(=)27 b Fr(o)p Ft(\()3145 4046 y Fm(p)p 3228 4046 46 4 v 77 x Fr(")p Ft(\))3312 4138 y Fn(")p Fl(!)p Fq(0)150 4289 y Ft(for)32 b(0)27 b Fh(6)i Fr(r)h Fh(6)e Ft(2)p Fr(")p Ft(,)k(whic)m(h)h (completes)g(the)g(pro)s(of.)1711 b Fh(\003)150 4466 y Fi(Pr)-5 b(o)g(of)35 b(of)f(The)-5 b(or)g(em)34 b(1.)41 b Ft(In)31 b(this)f(pro)s(of)f Fr(n)f Fm(!)f(1)j Ft(means)g(that)g(the) h(limit)26 b(is)k(tak)m(en)h(o)m(v)m(er)h(p)s(ositiv)m(e)150 4583 y(in)m(tegers,)h(and)g Fr(r)d Fm(!)d(1)32 b Ft(means)h(that)f(the) h(limit)d(is)i(tak)m(en)h(o)m(v)m(er)h(p)s(ositiv)m(e)e(reals.)363 4699 y(W)-8 b(e)33 b(\014x)h(a)f(function)g Fr(b)p Ft(\()p Fr(r)s Ft(\))h(whic)m(h)g(satis\014es)g(the)f(conditions)g(of)g(Lemma)f (4.1.)45 b(Let)34 b Fr(a)p Ft(\()p Fr(r)s Ft(\))f(b)s(e)150 4815 y(de\014ned)h(b)m(y)887 5015 y Fr(a)p Ft(\()p Fr(r)s Ft(\))28 b(=)1229 4891 y Fl(1)1192 4920 y Fk(X)1198 5130 y Fn(n)p Fq(=1)1336 5015 y Ft(\(2)1423 4974 y Fn(n)1487 5015 y Ft(log)16 b Fr(n)p Ft(\))p Fr(b)p Ft(\(2)1853 4974 y Fn(n)1900 5015 y Fr(r)25 b Fm(\000)e Fr(n)p Ft(2)2176 4974 y Fn(n)2223 5015 y Ft(\))k(=)2429 4891 y Fl(1)2392 4920 y Fk(X)2397 5130 y Fn(n)p Fq(=1)2553 5015 y Fr(a)2604 5030 y Fn(")2637 5038 y Fc(n)2683 5015 y Ft(\()p Fr(r)e Fm(\000)d Fr(n)p Ft(\))p Fr(;)p eop %%Page: 9 9 9 8 bop 713 6 a Fu(ON)33 b(THE)g(THEOREMS)g(OF)h(M.)g(G.)f(KREIN)g(AND) f(L.)h(A.)g(SAKHNO)n(VICH.)523 b(9)150 329 y Ft(where)39 b Fr(a)488 344 y Fn(")525 329 y Ft(\()p Fm(\001)p Ft(\))e(is)g (de\014ned)h(as)g(in)f(Lemma)f(4.1,)i(and)g Fr(")2137 344 y Fn(n)2219 329 y Ft(=)e(2)2380 293 y Fl(\000)p Fn(n)2482 329 y Ft(.)58 b(This)37 b(sum)h(is)f(a)53 b Fr(C)3288 293 y Fl(1)3400 329 y Ft(function)150 446 y(since)34 b(for)f(an)m(y)h Fr(r)i Ft(the)d(sum)h(con)m(tains)f(at)g(most)g(one)g (nonzero)h(term.)45 b(Then)35 b(b)m(y)f(Lemma)e(4.1)h(w)m(e)150 562 y(ha)m(v)m(e)1040 716 y Fm(j)p Fr(p)1117 675 y Fl(\003)1156 716 y Ft(\()p Fr(n;)17 b(\025)p Ft(\))22 b Fm(\000)h Fr(p)1562 675 y Fl(\003)1601 716 y Ft(\()p Fr(n)f Ft(+)g(2)1866 675 y Fq(1)p Fl(\000)p Fn(n)2003 716 y Fr(;)17 b(\025)p Ft(\))p Fm(j)27 b Ft(=)h Fr(o)p Ft(\(2)2435 675 y Fl(\000)p Fn(n=)p Fq(2)2607 716 y Ft(\))2645 731 y Fn(n)p Fl(!1)2833 716 y Fr(:)150 890 y Ft(Note)41 b(that)g Fr(p)663 854 y Fl(\003)703 890 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))41 b(do)s(es)g(not)h(c)m(hange)g(when)g Fr(r)i Ft(is)d(in)f(an)h(in)m (terv)-5 b(al)40 b([)p Fr(n)29 b Ft(+)f(2)3056 854 y Fq(1)p Fl(\000)p Fn(n)3193 890 y Fr(;)17 b(n)28 b Ft(+)f(1])42 b(since)150 1006 y Fr(a)p Ft(\()p Fr(r)s Ft(\))27 b(=)h(0)k(on)h(suc)m (h)h(in)m(terv)-5 b(als,)32 b(and)g(so)1109 1209 y Fm(j)p Fr(p)1186 1167 y Fl(\003)1225 1209 y Ft(\()p Fr(n;)17 b(\025)p Ft(\))22 b Fm(\000)g Fr(p)1630 1167 y Fl(\003)1670 1209 y Ft(\()p Fr(n)g Ft(+)g(1)p Fr(;)17 b(\025)p Ft(\))p Fm(j)27 b Ft(=)g Fr(o)p Ft(\(2)2366 1167 y Fl(\000)p Fn(n=)p Fq(2)2538 1209 y Ft(\))2576 1224 y Fn(n)p Fl(!1)2764 1209 y Fr(:)758 b Ft(\(4.7\))150 1403 y(Hence)53 b(the)g(limit)48 b(lim)1035 1418 y Fn(n)p Fl(!1)1239 1403 y Fr(p)1288 1367 y Fl(\003)1328 1403 y Ft(\()p Fr(n;)17 b(\025)p Ft(\))51 b(exists)i(and)f(is)g(\014nite)f(for)h(an)m(y)h Fr(\025)60 b Fm(2)h Fp(C)20 b Ft(.)108 b(Note)52 b(that)150 1519 y(lim)285 1534 y Fn(n)p Fl(!1)490 1519 y Fr(p)539 1483 y Fl(\003)579 1519 y Ft(\()p Fr(n;)17 b(\025)p Ft(\))27 b Fm(6)p Ft(=)h(0)k(for)g(Im)o Fr(\025)c Fh(>)g Ft(0)k(since,)h(b)m(y)h (\(3.1\),)967 1700 y Fr(d)p 944 1745 98 4 v 944 1836 a(dr)1067 1687 y Fk(\000)1113 1767 y Fm(j)p Fr(p)1190 1726 y Fl(\003)1229 1767 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))p Fm(j)1475 1726 y Fq(2)1536 1767 y Fm(\000)23 b(j)p Fr(p)p Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))p Fm(j)1959 1726 y Fq(2)1997 1687 y Fk(\001)2071 1767 y Ft(=)27 b(2Im)o Fr(\025)p Fm(j)p Fr(p)p Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))p Fm(j)2719 1726 y Fq(2)2785 1767 y Fh(>)29 b Ft(0)p Fr(:)582 b Ft(\(4.8\))150 2008 y(By)28 b(the)g(same)g(argumen)m(t,)g (for)g(an)m(y)g(Im)p Fr(\025)f(>)h Ft(0)f(w)m(e)i(ha)m(v)m(e)g(lim)2360 2023 y Fn(n)p Fl(!1)2564 2008 y Fr(p)p Ft(\()p Fr(n;)17 b(\025)p Ft(\))28 b(=)f(0)h(since)3317 1968 y Fn(d)p 3300 1985 71 4 v 3300 2042 a(dr)3380 2008 y Fr(p)p Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))27 b(=)150 2129 y Fg(i)12 b Fr(\025)44 b Ft(on)g(an)m(y)i(in)m(terv)-5 b(al)43 b([)p Fr(n)30 b Ft(+)g(2)1276 2093 y Fq(1)p Fl(\000)p Fn(n)1413 2129 y Fr(;)17 b(n)30 b Ft(+)h(1].)79 b(Hence)45 b Fr(p)p Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))48 b Fm(6)p Ft(=)g Fr(p)2623 2093 y Fl(\003)2662 2129 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))44 b(for)g(an)m(y)h Fr(r)51 b(>)d Ft(0)c(and)150 2246 y(Im)o Fr(\025)28 b(>)g Ft(0.)43 b(Lemma)31 b(4.1)h(implies)e (that)950 2440 y(lim)925 2500 y Fn(n)p Fl(!1)1126 2440 y Fm(j)p Fr(p)p Ft(\()p Fr(n)22 b Ft(+)g(2)1468 2399 y 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Fl(\003)1926 4946 y Ft(\()p Fr(n;)17 b(\025)p Ft(\))p Fm(j)2189 4917 y Fq(2)2265 4878 y Ft(=)2558 4810 y(1)p 2379 4855 409 4 v 2379 4946 a(2)p Fr(\031)t Fm(j)p Ft(\005\()p Fr(\025)p Ft(\))p Fm(j)2749 4917 y Fq(2)150 5106 y Ft(is)32 b(a)g(p)s(ositiv)m(e)g(con)m(tin)m(uous)i(function)e(on)g Fp(R)5 b Ft(,)39 b(whic)m(h)33 b(completes)f(the)h(pro)s(of.)698 b Fh(\003)p eop %%Page: 10 10 10 9 bop 150 6 a Fu(10)1462 b(A.)25 b(TEPL)-6 b(Y)g(AEV)150 341 y FH(Theorem)44 b(2.)h Fi(Ther)-5 b(e)40 b(exists)h(a)57 b Fr(C)1507 305 y Fl(1)1622 341 y Fi(function)41 b Fr(a)p Ft(\()p Fr(r)s Ft(\))f Fi(such)h(that)g(\(I)16 b({V\))41 b(of)f(Kr)-5 b(ein)-10 b('s)40 b(the)-5 b(or)g(em)150 458 y(hold,)30 b Fr(a)p Ft(\()p Fr(r)s Ft(\))e Fm(2)g Fr(L)747 422 y Fn(p)787 458 y Ft([0)p Fr(;)17 b Fm(1)p Ft(\))29 b Fi(for)h(any)h Fr(p)c(>)h Ft(2)p Fi(,)j(and)e Ft(lim)2018 473 y Fn(r)r Fl(!1)2213 458 y Fr(a)p Ft(\()p Fr(r)s Ft(\))f(=)f(0)p Fi(.)43 b(In)30 b(addition,)g(for)g(a)g (function)150 574 y Ft(\005\()p 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b(the)h(follo)m(wing)d(estimates)i(w)m(e)i(assume)f(that)f Fr(\025)c Fm(2)h Fp(C)60 b Ft(is)33 b(\014xed.)47 b(W)-8 b(e)34 b(write)f(\\)p Fr(const)p Ft(")g(for)150 5106 y(a)d(constan)m(t,)h(di\013eren)m(t)f(in)f(di\013eren)m(t)h (inequalities,)f(whic)m(h)h(dep)s(ends)i(on)e Fr(\025)p Ft(,)g(but)g(is)g(indep)s(enden)m(t)p eop %%Page: 11 11 11 10 bop 713 6 a Fu(ON)33 b(THE)g(THEOREMS)g(OF)h(M.)g(G.)f(KREIN)g (AND)f(L.)h(A.)g(SAKHNO)n(VICH.)484 b(11)150 329 y Ft(of)37 b Fr(")p Ft(,)h Fr(r)i Ft(and)d Fr(b)h Ft(pro)m(vided)g(0)d Fr(<)h(";)17 b(r)m(;)g(b)35 b(<)h Ft(1.)57 b(First,)37 b(w)m(e)i(use)f(Gron)m(w)m(all's)e(lemma)f(and)j(\(3.1\))e(to)150 449 y(sho)m(w)e(that)e Fm(j)p Fr(p)680 464 y Fn(b;\030)s(;")819 449 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))p Fm(j)22 b Ft(+)g Fm(j)p Fr(p)1262 412 y Fl(\003)1262 475 y Fn(b;\030)s(;")1401 449 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))p Fm(j)27 b Fh(6)i Ft(1)22 b(+)g Fr(r)s Ft(\()p Fm(j)p Fr(\025)p Fm(j)f Ft(+)h Fr(b)p Ft(\))p Fr(e)2390 412 y Fq(\()p 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b(is)f(easy)i(to)f(see)h(that)f(estimates)g(lik)m(e)f(\(4.15\))g(and)h (\(4.16\))f(can)i(b)s(e)f(established)150 4758 y(uniformly)g(in)i Fr(\025)g Ft(in)f(a)h(compact)g(subsets)j(of)d Fp(C)20 b Ft(.)69 b(Also)39 b Fm(j)p Fr(p)p Ft(\()p Fr(r)2434 4773 y Fn(n)2481 4758 y Fr(;)17 b(\025)p Ft(\))p Fm(j)38 b(!)h Ft(0)g(uniformly)e(in)h Fr(\025)i Ft(in)e(a)150 4874 y(compact)d(subsets)i(of)e Fr(H)42 b Ft(as)35 b Fr(r)1286 4889 y Fn(n)p Fl(\000)p Fq(1)1459 4874 y Ft(is)f(\014xed)j (and)e(\()p Fr(r)2072 4889 y Fn(n)2143 4874 y Fm(\000)24 b Fr(r)2288 4889 y Fn(n)p Fl(\000)p Fq(1)2425 4874 y Ft(\))32 b Fm(!)g(1)p Ft(.)50 b(Th)m(us)37 b(for)e(an)m(y)g(compact)150 4990 y(subset)i Fr(A)f Ft(of)g(of)f Fr(H)43 b Ft(there)37 b(is)e(a)h(sequence)i Fr(r)1807 4954 y Fn(A)1804 5015 y(n)1900 4990 y Ft(suc)m(h)f(that)e(\(4.18\),)h(\(4.19\))f(and)h (\(4.20\))f(hold)g(for)150 5106 y Fr(r)194 5121 y Fn(n)269 5106 y Ft(=)27 b Fr(r)419 5070 y Fn(A)416 5131 y(n)476 5106 y Ft(,)32 b(and)h(also)f(for)g Fr(r)1114 5121 y Fn(n)1193 5106 y Ft(that)h(is)f(an)m(y)h(subsequence)j(of)c Fr(r)2396 5070 y Fn(A)2393 5131 y(n)2453 5106 y Ft(.)p eop %%Page: 13 13 13 12 bop 713 6 a Fu(ON)33 b(THE)g(THEOREMS)g(OF)h(M.)g(G.)f(KREIN)g (AND)f(L.)h(A.)g(SAKHNO)n(VICH.)484 b(13)363 329 y Ft(W)-8 b(e)28 b(can)f(represen)m(t)j Fr(H)35 b Ft(as)28 b(an)f(increasing)g (union)g(of)g(compact)g(subsets)j Fr(H)3098 344 y Fn(k)3141 329 y Ft(.)41 b(Without)27 b(loss)150 455 y(of)34 b(generalit)m(y)f(w)m (e)j(can)e(assume)h(that)f Fr(r)1639 402 y Fn(H)1697 414 y Fc(k)q Fd(+1)1636 466 y Fn(n)1851 455 y Ft(is)f(a)i(subsequence)i (of)d Fr(r)2746 419 y Fn(H)2804 431 y Fc(k)2743 480 y Fn(n)2881 455 y Ft(for)f(eac)m(h)j Fr(k)s Ft(.)48 b(Then)36 b(w)m(e)150 572 y(de\014ne)e Fr(r)476 587 y Fn(n)555 572 y Ft(b)m(y)g(the)f(\\diagonal)d(pro)s(cess")j Fr(r)1730 587 y Fn(n)1805 572 y Ft(=)27 b Fr(r)1955 536 y Fn(H)2013 544 y Fc(n)1952 596 y Fn(n)2060 572 y Ft(.)1586 b Fh(\003)150 777 y FH(Conjecture)34 b(4.3.)39 b Ft(W)-8 b(e)30 b(conjecture)h(that)e (if)g Fr(a)p Ft(\()p Fr(r)s Ft(\))g(is)g(a)g Fi(r)-5 b(e)g(al-value)g(d)39 b Ft(function,)29 b(and)h(conditions)150 893 y(\(I)8 b({V\))31 b(of)f(Krein's)g(theorem)g(hold,)g(then)h(\005\() p Fr(\025)p Ft(\))g(is)f(unique)g(in)g(the)h(follo)m(wing)d(sense:)44 b(if)29 b Fr(t)3448 908 y Fn(n)3523 893 y Fm(!)e(1)150 1009 y Ft(and)32 b(lim)474 1024 y Fn(n)p Fl(!1)679 1009 y Fr(p)p Ft(\()p Fr(t)801 1024 y Fn(n)848 1009 y Fr(;)17 b(\025)p Ft(\))27 b(=)h(0,)k(then)g(the)g(limit)c(lim)1982 1024 y Fn(n)p Fl(!1)2186 1009 y Fr(p)2235 973 y Fl(\003)2275 1009 y Ft(\()p Fr(t)2348 1024 y Fn(n)2395 1009 y Fr(;)17 b(\025)p Ft(\))27 b(=)h(\005\()p Fr(\025)p Ft(\))j(con)m(v)m(erges)j (uniformly)150 1125 y(on)39 b(compact)g(subsets)i(of)d Fr(H)8 b Ft(.)63 b(If)39 b(true,)i(this)e(conjecture)h(implies)d(that)h (the)i(original)c(form)h(of)150 1241 y(Krein's)c(theorem)f(holds)g(if)g Fr(a)p Ft(\()p Fr(r)s Ft(\))g(is)g(real)g(and)g(uniformly)f(in)m (tegrable)g(\(see)j(Theorem)f(3\).)150 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2056 y Fn(r)3051 2021 y Fm(j)p Fr(a)p Ft(\()p Fr(s)p Ft(\))p Fm(j)p Fr(ds)c(<)h Fm(1)p Ft(.)46 b(If)150 2139 y(true,)36 b(this)e(conjecture)i(also)d (implies)f(the)j(uniqueness)i(of)d(\005\()p Fr(\025)p Ft(\).)50 b(Note)34 b(that)h(in)e(the)j(situation)150 2255 y(of)i(Theorem)g(2)g(the)h(limit)c(in)i(a)m(v)m(erage)j(of)e Fr(p)1825 2219 y Fl(\003)1864 2255 y Ft(\()p Fr(t)1937 2270 y Fn(n)1984 2255 y Fr(;)17 b(\025)p Ft(\))38 b(do)s(es)h(not)f (exists)h(if)e Fr(r)2980 2270 y Fn(n)p Fq(+1)3143 2255 y Fm(\000)27 b Fr(r)3291 2270 y Fn(n)3376 2255 y Ft(are)38 b(large)150 2371 y(enough.)44 b(These)34 b(t)m(w)m(o)g(conjectures)g (ma)m(y)e(b)s(e)h(related)f(to)g(the)h(results)g(of)f([D2].)1310 2659 y(5.)49 b Fs(Sakhno)n(vich's)37 b(theorem.)363 2834 y Ft(In)h([S1{S5])f(L.)h(A.)g(Sakhno)m(vic)m(h)h(in)m(tro)s(duced)f (and)g(studied)g(matrix)e(analogs)h(of)g(Krein's)150 2950 y(system.)44 b(He)33 b(considered)h(a)e(system)i(of)e(canonical)f (di\013eren)m(tial)g(equations)1127 3119 y Fr(d)p 1103 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Fr(P)3180 4075 y Fq(2)3219 4060 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\).)64 b(Since)150 4177 y(Krein's)39 b(system)i(is)e(a)g(particular)f(case)i(of)f(Sakhno)m(vic)m(h's)j (system,)g(the)e(coun)m(terexamples)g(of)150 4293 y(Section)29 b(4)h(applies)e(to)i(this)f(situation)f(also.)41 b(It)30 b(is)f(easy)i(to)e(construct)i(a)e(\\true")g(matrix-v)-5 b(alued)150 4409 y(coun)m(terexample)33 b(along)e(the)i(lines)f(of)g (Section)h(4.)363 4525 y(In)28 b(Section)g(4)g(w)m(e)i(pro)m(v)m(e)f (that,)g(in)f(general,)h(\005\()p Fr(\025)p Ft(\))f(can)g(not)h(b)s(e)f (de\014ned)i(uniquely)-8 b(.)42 b(In)29 b(Sec-)150 4641 y(tion)i(6)g(w)m(e)i(sho)m(w)g(that)f(if)e Fr(A)1186 4656 y Fq(2)1226 4641 y Ft(\()p Fr(r)s Ft(\))d Fm(2)h Fr(L)1536 4605 y Fq(2)1576 4641 y Ft([0)p Fr(;)17 b Fm(1)p Ft(\),)31 b(then)i(the)f(\014nite)f(limit)e(\005\()p Fr(\025)p Ft(\))e(=)h(lim)3234 4656 y Fn(r)r Fl(!1)3430 4641 y Fr(P)3493 4656 y Fq(2)3532 4641 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))150 4758 y(exists,)33 b(and)g(so)g(\005\()p Fr(\025)p Ft(\))f(is)g(unique.)363 4874 y(Belo)m(w)39 b(w)m(e)h(giv)m(e)e(a)h(corrected)h(part)f(of)f(the)i(pro)s(of)e(of)g (Sakhno)m(vic)m(h's)j(theorem.)62 b(F)-8 b(ollo)m(w-)150 4990 y(ing)44 b(the)i(lines)e(of)h([S2{S4],)j(w)m(e)e(will)d(sho)m(w)j (that)f(statemen)m(ts)h(\(I)s(I)8 b({V\))46 b(are)f(equiv)-5 b(alen)m(t.)81 b(An)150 5106 y(alternativ)m(e)32 b(approac)m(h)h(can)g (b)s(e)f(found)h(in)f([T2].)p eop %%Page: 15 15 15 14 bop 713 6 a Fu(ON)33 b(THE)g(THEOREMS)g(OF)h(M.)g(G.)f(KREIN)g (AND)f(L.)h(A.)g(SAKHNO)n(VICH.)484 b(15)150 329 y Fi(A)35 b(pr)-5 b(o)g(of)35 b(of)f(a)h(p)-5 b(art)35 b(of)f(Sakhnovich's)f(the) -5 b(or)g(em.)41 b Ft(One)33 b(can)g(see)h(that)438 541 y Fr(P)515 500 y Fl(\003)501 566 y Fq(2)554 541 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))p Fr(P)835 556 y Fq(2)874 541 y Ft(\()p Fr(r)m(;)g(\025)p Ft(\))22 b Fm(\000)g Fr(P)1290 500 y Fl(\003)1276 566 y Fq(1)1329 541 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))p Fr(P)1610 556 y Fq(1)1649 541 y 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Fm(2)h Fr(L)1466 293 y Fq(2)1505 329 y Ft([0)p Fr(;)17 b Fm(1)p Ft(\))p Fi(.)48 b(Then)36 b(c)-5 b(onditions)35 b(\(I)8 b({V\))36 b(of)g(Sakhnovich's)e(the)-5 b(o-)465 446 y(r)g(em)34 b(hold)g(and,)g(mor)-5 b(e)g(over,)34 b(the)h(limits)1007 611 y Ft(lim)987 670 y Fn(r)r Fl(!1)1178 611 y Fr(P)1241 626 y Fq(2)1281 611 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))27 b(=)h(\005\()p Fr(\025)p Ft(\))104 b Fr(and)141 b Ft(lim)2222 670 y Fn(r)r Fl(!1)2413 611 y Fr(P)2476 626 y Fq(1)2515 611 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))28 b(=)f(0)636 b(\(6.3\))465 806 y Fi(c)-5 b(onver)g(ge)33 b(uniformly)i(on)f(c)-5 b(omp)g(act)34 b(subsets)h(of)g Fr(H)8 b Fi(.)150 947 y FH(Remark)47 b(6.1.)g Ft(This)41 b(result)h(is)f(sharp)h(in)f(the)h (sense)h(that)f(there)g(is)f(a)g(real)g Fr(C)3211 911 y Fl(1)3327 947 y Ft(co)s(e\016cien)m(t)150 1063 y Fr(A)223 1078 y Fq(2)263 1063 y Ft(\()p Fr(r)s Ft(\),)46 b(whic)m(h)f(is)f(in)g Fr(L)1052 1027 y Fn(p)1136 1063 y Ft(for)g(an)m(y)h Fr(p)i(>)h 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b(but)g(only)g(up)g(to)f(a)150 1761 y(constan)m(t)39 b(factor)f(ev)m(en)i(though)e(the)h(limit)c(lim)1946 1776 y Fn(r)r Fl(!1)2125 1761 y Fm(j)p Fr(P)2216 1776 y Fq(2)2255 1761 y Ft(\()p Fr(r)m(;)17 b(\025)p Ft(\))p Fm(j)37 b Ft(=)g Fm(j)p Ft(\005\()p Fr(\025)p Ft(\))p Fm(j)h Ft(exists)h(b)m(y)g(part)f(2)g(of)150 1877 y(Theorem)33 b(3.)363 1993 y(Note)g(that)g(in)g(statemen)m(ts)i(2)e(and)g(3)g(there) i(is)e(no)g(restriction)f(on)i(the)g(co)s(e\016cien)m(t)g Fr(A)3561 2008 y Fq(1)3600 1993 y Ft(\()p Fr(r)s Ft(\),)150 2109 y(except)g(for)e(the)h(usual)f(assumption)g(of)h(lo)s(cal)d(in)m (tegrabilit)m(y)-8 b(.)363 2250 y(It)47 b(w)m(as)i(pro)m(v)m(ed)g(in)e ([S2,S3])h(that)g(if)e Fr(A)1886 2265 y Fq(2)1926 2250 y Ft(\()p Fr(r)s Ft(\))53 b Fm(2)h Fr(L)2288 2214 y Fq(1)2328 2250 y Ft([0)p Fr(;)17 b Fm(1)p Ft(\),)50 b(then)f(conditions)d(\(I)8 b({V\))48 b(of)150 2366 y(Sakhno)m(vic)m(h's)33 b(theorem)e(hold,)g (and)h(the)f(limits)e(\(6.3\))h(con)m(v)m(erge)j(uniformly)c(on)j (compact)f(sub-)150 2482 y(sets)45 b(of)d Fr(H)37 b Fm([)30 b Fp(R)54 b Ft(and)44 b Fr(H)51 b Ft(resp)s(ectiv)m(ely)-8 b(.)76 b(This)44 b(fact)f(and)g(statemen)m(t)h(3)f(of)g(Theorem)h(3)f (w)m(ere)150 2599 y(form)m(ulated)31 b(in)i([K])f(for)h(Krein's)g (system.)45 b(Also,)32 b(for)h(Krein's)g(system)g(statemen)m(ts)h(2)f (and)g(3)f(of)150 2715 y(Theorem)h(3)f(are)h(related)f(to)g(the)h (results)g(of)f([D2].)150 2891 y Fi(Pr)-5 b(o)g(of)35 b(of)f(1.)41 b Ft(Let)33 b Fr(\025)879 2906 y Fq(0)946 2891 y Fm(2)28 b Fr(S)6 b Ft(.)44 b(Then)33 b(for)f(an)m(y)i Fr(\025)27 b Fm(2)h Fr(H)40 b Ft(w)m(e)34 b(ha)m(v)m(e)g(that)884 3105 y(lim)860 3165 y Fn(n)p Fl(!1)1060 3105 y Fr(P)1137 3064 y Fl(\003)1123 3129 y Fq(2)1176 3105 y Ft(\()p Fr(t)1249 3120 y Fn(n)1296 3105 y Fr(;)17 b(\025)1397 3120 y Fq(0)1436 3105 y Ft(\))p Fr(P)1537 3120 y Fq(2)1577 3105 y Ft(\()p Fr(t)1650 3120 y Fn(n)1697 3105 y Fr(;)g(\025)p Ft(\))27 b(=)1966 2969 y Fk(Z)2066 2995 y Fl(1)2022 3195 y Fq(0)2157 3105 y Fr(P)2234 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FG(preprin)n(t.)150 3313 y([D5])61 b(S.)22 b(A.)g(Deniso)n(v,)g Fv(On)h(the)h(c)l(o)l(existenc)l(e)g(of)h (absolutely)g(c)l(ontinuous)f(and)g(singular)h(c)l(ontinuous)e(c)l(omp) l(onents)362 3413 y(of)38 b(the)f(sp)l(e)l(ctr)l(al)g(me)l(asur)l(e)f (for)i(some)f(Sturm-Liouvil)t(le)h(op)l(er)l(ators)f(with)h(squar)l(e)f (summable)g(p)l(otential,)362 3512 y FG(preprin)n(t.)150 3612 y([DKu])42 b(S.)31 b(A.)h(Deniso)n(v)f(and)g(S.)g(Kupin,)66 b Fv(On)33 b(singular)g(sp)l(e)l(ctrum)g(of)h(Schr)l(o)l(dinger)g(op)l (er)l(ators)g(with)g(de)l(c)l(aying)362 3712 y(p)l(otential,)43 b FG(preprin)n(t.)150 3811 y([G])101 b(L.)39 b(Y)-7 b(a.)39 b(Geronim)n(us,)i Fv(Ortho)l(gonal)f(p)l(olynomials:)62 b(Estimates,)43 b(asymptotic)f(formulas,)i(and)c(series)h(of)362 3911 y(p)l(olynomials)h(ortho)l(gonal)e(on)g(the)f(unit)f(cir)l(cle)j (and)e(on)h(an)f(interval.)53 b FG(Authorized)38 b(translation)e(from) 362 4011 y(Russian.)27 b(Consultan)n(ts)g(Bureau,)g(New)h(Y)-7 b(ork)27 b(1961.)150 4110 y([K])101 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Fv(On)j(a)h(class)h(of)f(c)l(anonic)l(al)h(systems)f(on)f(half-axis,)45 b FG(In)n(tegral)27 b(Equations)h(Op)r(erator)362 4907 y(Theory)g Fb(31)f FG(\(1998\),)f(no.)i(1,)f(92{112.)150 5007 y([S3])78 b(L.)26 b(A.)h(Sakhno)n(vic)n(h,)e Fv(Sp)l(e)l(ctr)l(al) k(the)l(ory)g(of)g(c)l(anonic)l(al)h(di\013er)l(ential)g(systems.)f (Metho)l(d)h(of)f(op)l(er)l(ator)g(identi-)362 5106 y(ties.)42 b FG(Op)r(erator)26 b(Theory:)36 b(Adv)-5 b(ances)27 b(and)h(Applications,)f(107.)g(Birkh\177)-42 b(auser)25 b(V)-7 b(erlag,)27 b(Basel,)g(1999.)p eop %%Page: 18 18 18 17 bop 150 6 a Fu(18)1462 b(A.)25 b(TEPL)-6 b(Y)g(AEV)150 329 y FG([S4])78 b(L.)23 b(A.)h(Sakhno)n(vic)n(h,)e Fv(On)j(the)g(sp)l (e)l(ctr)l(al)h(the)l(ory)g(of)h(a)f(class)g(of)g(c)l(anonic)l(al)h (di\013er)l(ential)f(systems)44 b FG(\(Russian\),)362 429 y(F)-7 b(unktsional.)36 b(Anal.)h(i)g(Prilozhen.)e Fb(34)h FG(\(2000\),)i(50{62,)e(96;)k(translation)c(in)g(F)-7 b(unct.)38 b(Anal.)e(Appl.)i Fb(34)362 529 y FG(\(2000\),)27 b(119{128.)150 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