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9. Лебедев, А.В. Степенные хвосты и кластеры в линейных рекуррентных случайных последовательностях [Текст] // Труды VI Колмогоровеких чтений. - Ярославль, 2008. - С. 132-136.
Е.И. Смирнов
ХАУСДОРФОВЫ СПЕКТРЫ И ПУЧКИ ЛОКАЛЬНО ВЫПУКЛЫХ
ПРОСТРАНСТВ
В настоящей статье рассматриваются обобщения подготовительной теоремы Вейерштрасса и глобальной теоремы Вейерштрасса о делении для ростков голоморфных функций в точке n-мерного комплексного пространства. Автор формулирует глобальную теорему о делении в терминах существования и непрерывности линейного оператора.
Ключевые слова: глобальная теорема Вейерштрасса о делении, теорема о замкнутом графике, ростки голоморфных функций, Н-пространства.
E.I. Smirnov
HAUSDORFF SPECTRA AND SHEAVES OF LOCALLY
CONVEX SPACES
In the present article generalisation of the preparatory theorem by Vejershtrass and the global theorem by Vejershtrass about division for sprouts of holomorphic functions in a point of n-dimensional complex space are considered. The author formulates the global theorem about division in terms of existence and a continuity of the linear operator.
Keywords: The global theorem by Vejershtrass about division, the theorem of the closed schedule, sprouts of holomorphic functions, H-space.
Let {Su, puv} be a presheaf of abelian groups over a topological space V, Па nonempty partially ordered set and F an admissible class for П (we may assume without loss of generality that П = |F|). Let us denote by H(S) a covariant functor from Ord П to OrdU, where U is a base of open sets in V, and by H(S) a contravariant functor from Ord U to the category of abelian groups so that an abelian group Su is defined for each U EU and a homomorphism puv : Su ^ Sv is defined for each pair U С V, Then H = H(S) о H(S) is a contravariant functor of the Hausdorff spectrum X (S) = {SUs, F, Pus,us} which we will call the Hausdorff spectrum associated with the presheaf {Su, Puv}■ Let X be the H-limit of the Hausdorff spectrum X (S) in the category of abelian groups and let
= nu U.
f es selF |
Proposition 1. Let S be the sheaf of germs of holomorphic functions on an open set V C Cn, associated with the presheaf [Su, puv}, and let X (S) = [Sus, F,Pua,us} be the associated true Hausdorjf spectrum. Then the H-limit of the Hausdorff spectrum X(S) is isomorphic to the vector space of sections r(A, S) of the sheaf S over the set A.
Proof. By the conditions relating to [Su, puv}, we may put Su = r(U, S) (U E U). Further, let
X = imPus,uBr(Us, S) ,
f
so that
*=u n ^).
f es t eF
If x E X, there exists F E F such that x E ^(VF) (T E F), that is to say, there exists a selection
C(T) = (fT) se|F|
such that 4>(fsT) = x for each T E F. For any U E Uz (z E V) the homomorphism pzu : r(^, S) ^ Sz generates for f E r(U, S) the set of points
Pu(f)= U Pzu(f) CS ,
ze u
therefore let us put
Px = u Pus (f'T);
seT
it is clear that p^ generates the section fT on the open set UT = UseT Us, since the correspondence
z E Ut-A pT nSz CS
is single-valued and continuous. Moreover, if puv : pv (g) m- ptj (f), then ptj(f) C pv(g), so let us put
px = U U pus*us (pus us)) ,
F*yF s* E IF* I se T
where necessarily
Let us put
Pus*us (Pus (fj)) = Pus*us (Pus (fT)) (T,T' E F).
U* = fl UPi , where Upi c u Us;
? se|F|
in this connection we have in particular,
Pus (f'T) n pus (f'T') D pus*us (pus (f'T)). It is also clear that for each £ the correspondence
z E Upe ^ pi n Sz
is single-valued and continuous. Although, in general, it is not guaranteed that UPx = 0, we will show nevertheless that UPx D A under the conditions of the proposition, specifically because the Я-limit of the Hausdorff spectrum X(S) is true. Let the selection £(T) = (fj)seiF| (T E F) generating the element x E X be fixed. Then because the Hausdorff spectrum X(S) is true we may assume that fjl = fj2 (s E Ti П T2) and, consequently, there exists £ = (fs)se|F| E n^F ^f such that
x E ^((fs)iFi) and fs = pus,ub (fs) (s,s' E |F|).
It is clear that pi = (Jse|F| pUs,Us(fs). Now let z E A Then z E Upi for any £(F) (F E F) and, moreover,
pi(z) = pi П Sz = PzUs (fs) for Z E Us (s E IF |).
Let us show that pi(z) = pi(z) for any £,£'. In fact, let £ = (/s)|F£' = (f's>)|F'| and x = ф(£), x' = '). Sinee £ ~ £', there exists F* E F, where F* ^ F' and F* ^ F', such that for each T* E F* we can find T E ^d T' E F' such that
Штт* : т* ^ T , ШТ'Т* : T* ^ T' and Pus*us (fs) = Pus*us, (fi>) , where s* E T*. However, z E (Js*e|F*| Us*, and so it remains to choose s* E |F*|, such that
z E Us* and pzus (fs) = Pzus, (fi>) (s* ^ s,s* ^ s').
Thus z E UPx. Furthermore, let us put x(z) = pi(z)|A, so that x(z) is a section of S on Д x(z) E Г(ДS). In this way we have constructed a morphism H : X ^ Г(ДS). Given fA = H(x), /а = H(y), let us prove that x = y. In fact, at each point z E A there exists an open Ml В(z, e) of the local homeomorphism n : S ^ V at the point fA(z). Let us put U = UzeA В(z,e/2) and determine the section fz E Г(В(z,t/2), S) passing through the point s = fA(z) E S such that
fz| a = ¡a
(we note that e = e(z)). Let
Bi:i = B(zi, ti/2) П B(zj, tj/2), Bi:i П A = 0 , zo E B^ П A
for some zi,zj E A Then fzi (z0) = fzj (z0), and, consequently, there is an open ball B0 С В(z0, t0/2) of the local homeomorphism at the point s0 = fzi(z0) such that B0 С B^ and fzi |Bo = fzj |Bo- However, because of the isomorphism Г(В^, S) ^ SBij the holomorphie functions /z^d fzj coincide on the connected open set Bij [1, Theorem A6], The last observation means that fzi | Bij = fzj | Bij. Now suppose that
Bij П A = 0 , but B'ij(ei, tj) П A = 0 , z' E В'^ П A.
Clearly, we will obtain by similar reasoning f'z. | B' = f'z. . But we have f'z. |B(Ziy4/2) = fZi and f'z.Bzjъ/2) = fzp so that fzi^^ = fzj^ (in the case where B^ = 0). Now there remains the third possibility for B^ = 0, when B'j П A = 0. In this case the sections
fz^ fzj on B^ do not necessarily coincide, therefore let us put M = (J Bij, where the bar denotes closure in Cn and ^te union is taken over all B^ of this third type. It is clear that M П A = 0, since in the contrary сase for z* E M П A ^^^re would exist B* of the third type
such that z* G which is impossible by construction. Let us put U(¡a) = U\M, so that U(ja) D ^^d U(ja) is an open subset of Cra, Then there exists f G T(U(/a), S) such that f I a = fA and, moreover,
f IU (fA)nB(z,t/2) = fz IU (fA)nB(z,£/2) (z G A);
also the section on U(fa) of f with the property f Ia = ¡a is uniquely determined (nevertheless, the corresponding holomorphic function on A, is extended holomorphieallv to U(ja) in a manner which, in general, is not unique).
Now if = x, ^(rj) = y, it follows from the fact that the familv of open sets IU, e|F| us}feF is fundamental for A that there exists F* G F such that U(fA) D Uf* Us*, and moreover by construction
P^uf* us* = Pvy |uf* us* .
Tte last assertion means that £ ~ ^ and, consequently, x = y. Moreover, the fact that IUf Us}feF '1S fundamental for A and the constructions carried out above allow us to conclude that 'H : X ^ S) is an isomorphism. The proposition is proved. □
The absence of sufficiency restrictions on |F| in Proposition 1 allows us to apply it to any nonempty A C Cn\ it is enough to take F = |F| and Us (s G F) a fundamental system of open sets containing A (in general, of uncountable cardinality). The investigation of topological properties of ^-limits in this case produces substantial difficulties, therefore the investigation of T(A, S) with no more than countable |F| is of interest. It is clear that, for example, any closed bounded set A C Cn will be of this type, and the space of sections S) is the inductive limit of the sequence of spaces T(US, S) (s G |F|). Furthermore, the corresponding Hausdorif spectrum {r(Us, S),F,Pus,us} will be true in this case. In general, the Hausdorif spectrum X(S) will be true if all open sets Us (s G |F|) are connected. We recall that each space T(US, S) can be given the separated locally convex topology of uniform convergence on the compact subsets of Us (s G |F|), under which it is a Freehet space; we will denote this topology by Ts.
Proposition 2. Let X(S) = {T(US, S), F,Pus,us} be a true countable Hausdorff spectrum and suppose that A = p|f (jF Us has a countable fundamental system of compact sets, is
O i-
connected and A= 0. Then the H-limit X = limpUg/UsT(US, S) is a separated H-space in the
f
topology r* and is continuously embedded in oa (oa is the algebra of holomorphic functions on A).
Proof. First of all, by Proposition 1 we have the isomorphism H : X ^ S);
O
because of the connectedness of A and the fact that A= 0 each holomorphic function on A, 0 G oa, is generated by some holomorphic function on the open set U(4>); moreover, any two holomorphic functions G Ou and 02 G Ov (U D A, V D A) which coineide on A must coincide on a connected component of the intersection U fl V (see [1, p. 104]), which also implies the isomorphism S) = oa- Since A has a countable fundamental system of compact subsets
Kn (n =1, 2,...), Ki C K2 C ...,
on putting
MU = max It(z)I (t gOa) ,
zeKn
we obtain a seminorm on О a (or о п Г( A, S), which is permissible according to the construction). Furthermore, on putting
n=1
for example, we obtain a quasinorm on Oa under which Oa becomes a separated locally convex space with a countable base of neighbourhoods of zero, therefore metrizable, but in general not complete; we will denote this space by (Oa, p).
We now show that on Oa the locally convex topologv r* of the #-limit of the Hausdorif spectrum X(S) is not weaker than p. In fact, let W = (0 G Oa : ||0||w < e} be some neighbourhood of zero in (Oa,p) and let F G F- Let us choose s0 G |F| such that Uso D KN - this choice turns out to be possible because of the compactness of KN and the condition A C |JF US ; also we can fed a compact set K^ C US0 such that K^ D KN - here the choice is possible because of the availability of a fundamental system (K^^Li in US0. Now it is clear that
Ho^(MF) c W,
where
£=( fs)F , MF = (ÇGVS0 : sup | fso (z)l < e} , H = fao |a = 0.
zeK°n
Since 4>(MF) is itself a neighbourhood of zero in the MVG X(F^d F G F was chosen arbitrarily, we have that
H(co MF)) c W
F
and is a neighbourhood of zero in the topology t*. This also shows that t* > p. The proposition is proved. □
The conditions of Proposition 2 are satisfied, for example, by A = A(0, r), the compact polvdisk in Cra, or by any domain V C Cn. It is not ^^^ralt to see that if A is a connected set and A = P|FUF Us, where F is an admissible class for Q, then without loss of generality we may assume that the Us (s G |F|) ^re connected open sets in Cn. In fact, let each Us have
A
of transformation of indices ( s G |F|). Let us denote by Us the opsn connected component of Us which contains Us HA (s s |F|) Now it is clear that A = P| ^ UF Us for the admissible class F in Q and moreover, if (UF Us}F were a fundamental system of neighbourhoods for A, then (UF Us}F would be the same. Now let us consider the question: For what classes of sets A c Cn do we have a representation
^ = niK •
f es seF
where F is an admissible class for Q (a countable set)?
Let A be any nonempty bounded connected subset of Cn and B = B (z0, r) an open ball such that A C B. By Proposition 3.2 for the s-set B\A we have the representation
B\A = U '
BeicteS
n
where the Lt are open subse ts of В (z0, r) (t e IK-I, а шип table set) an d К is some family of subsets В С П ; moreover, for each В e К the intersections (P|g form a fundamental system of compact subsets of B\A. Since Cra is a finite-dimensional space, by Proposition 3.10 we will obtain the representation
B\A = и n,
век teS
where the Lt С B(z0,r + e) are compact sets (t e Now
A = B\ UP Lt = d(B\ Q Lt) = П U (В\Т<),
к _§ к _§ к _§
and Gt = B\Lt is ад open set in Cra (t e We will show that (Уg Gt}ic form a
fundamental system of open connected neighbourhoods of A In feet, \iW D A,W С В is an open set, then without loss of generality we may assume that W С В(z0,r — 8) for some 8 > 0, so that P = (B(zq, r — 8) U dB)\W is a compact b^^^ of В(z0, r). Therefore there exists a compact subset Lt = П go Lt such th at P С P| go Lt and, consequently,
B\P D U(B\Lt) or W U [z : r — 8< |z — Zq| < r} D ^ Gt D ^ Gt.
Bo Bo Bo
However, Ьесаше o^the connectedness of Gt and the ordering of B0 e K, we obtain the inclusion W D Ugo Gt, which was to be established. Thus we obtain the following
Proposition 3. Every connected bounded subset А С Cn has a representation
A = OU , (6)
' s
f eF seF
where F is an admissible class for the cowritable set Q and the Us are connected open subsets (domains) in Cn.
In particular, for such a set A the Hausdorff spectrum
X(S) = {T(US, S), F,pus,us}
is true (it suffices to apply the uniqueness theorem for holomorphic functions). In the representation (6) it is natural to require that if Us fl Us' = 0 E |F|) then it is a
connected set. Only such sets A will be considered further.
In what follows the space Oa of germs of holomorphic functions on A will be provided with the topology p (in general not separated) of uniform convergence on the compact subsets of A and with the locally convex topology of the #-limit. As has already been noted above (Proposition 1), for a connected bounded subset A C Cn we have the linear isomorphism
X = T(A, S) = Oa .
We also note that if the set A has an interior point then Oa coincides with the space of holomorphic functions on A (up to isomorphism).
Weierstrass's Global Division Theorem
Weierstrass's preparation theorem and the division theorem for germs of holomorphie functions at a point w E Cn allow us to establish a series of properties of local rings nOw and modules over these rings (Noetherian, Oka's Lemma on the exactness of homomorphisms of „^-modules, etc. [1]). The proofs have a number of algebraic characteristics, therefore consideration of a global variant of the theorems is significantly different and uses topological results of linear analysis (see, for example, [1]). A more careful analysis makes it possible to formulate a global division theorem in terms of the existence and continuity of a linear operator acting on locally convex spaces so that the local and global variants of Weierstrass's theorem turn out to be in fact special cases of a more general theorem. In this section we obtain a stronger form of Theorems II.B.3 and II.D.l in [1] for the case of #-spaces, C-1 denotes C x ■ - x C x C x - • x ^d nm : Cn ^ C™-1 is the projeetion of Cn onto C™-1;
m- 1 n-m
at the same time nm : Cn ^ Cm and Cra = C™-1 x Cm so that nm is the projection of Cra onto Cm. For notational convenience in what follows the germ of a holomorphie function is denoted by capital Roman letters F, G, H, ... .
We will say that the germ H E Oa (A C Cra) is a w-local Weierstrass polynomial in zm (1 < m < n) of degree k (k > 0) if there exists w E A and function h E H which is holomorphie on an open neighbourhood U D A and has representation on U
h(z) = (zm - wm)k + a\(z')(zm - wm)k 1 +-----+ ak(z')
z' = (Z1,Z2, . . .,Zm- 1 ■) Zm+1 ■>...■> Zn) j
(7)
where the aj(z') ^re holomorphie functions on nm(U), aj(w') = 0, and w = w' x wm (j = 1, 2,..., k). It is clear that the holomorphic function h is regular of order k 111 Zm at the point w E A.
Theorem 1. (Weierstrass's global division theorem.) Let A C Cn be a nonempty connected bounded set such, that /nm(A) is closed and let H E Oa be a w-local Weierstrass polynomial in zm of degree k (k > 0) with representation hu E H such, that
[z E n^1 o nm(A) n U : hu(z) = 0} C A. Then there exists a continuous linear operator L : Oa ^ Oa x Oa , where
L(F) = (G,P), F = GH + P, k-1
P = Y, Pj(z')zjn , Pj E OA . j=0
First of all we recall [1, Chapter 2, §5] that Oa has the topologv p of uniform convergence on the compact subsets, which in general is neither separated nor complete, and Oa x Oa has the usual product topology. In the course of the proof of Theorem 1 Oa will also be given another stronger locally convex topology, again in general not separated, under which it is an H-space. Therefore we first present a lemma for Theorem 1.
Lemma 1. Let A : X ^ Y be a closed linear operator, where X is an H-space under the locally convex topology r* and (Y, a) is an H-space (in general X, Y are not separated spaces). Then A is continuous.
Proof. Let М, N be the respective nonseparated parts of X, Y and X/M, Y/N the separated quotient spaces with quotient maps £ : X ^ X/M and ц : Y ^ Y/N. Then the quotient topology * on X/M is in general weaker than the topology *)*, the limit of the corresponding Hausdorif spectrum (see, for example, [4]); let r/a be the quotient topology on Y/N. Then the diagram
X/M .....■> Y/N
(8)
A
X -> Y
is commutative and the induced mapping A* exists because of the elosedness of the operator A In fact, the closedness of A implies that N = Q {U + AV}, where U, AV are bases of
и ей ,v ev
neighbourhoods of zero for the topologies a, At Respectively. But AM С AV for any V G V and 0 G U, therefore AM С U + AV (V U, V) and, consequentlv, AM С N. Moreover, the induced mapping A* is clearly linear; we will show that A* is a closed operator. For this we have to show that
0= П + A*^v} •
и ей ,v ev
Since rjA = A£, this is equivalent to the relation 0 = P|M v r/{U + AV} ; let us suppose that a G Пи,v 'n{U + AV}. Then r]-1a П (U + AV) = 0 (V U,V). But r]-la = у + N and because of the absolute convexity of U + AV and Theorem 1.3 of [2] we obtain r/-1a С U + AV (V U, V), This implies that r/-1a С ^consequently a = ^d A* is a closed operator.
Thus by the Closed Graph Theorem for the Я-space (Y/N,r/a) and complete MVGs the closed operator A is continuous from (X/M, (£т*)*) to (Y/N,"qa). The existence of the Hausdorif spectrum for (Y/N,rja) follows from Proposition 4.10 and [4].
Now we will establish the continuity of the operator A : X ^ У, Let W be a closed absolutely convex neighbourhood of zero in Y and (VF) a base of absolutely convex neighbourhoods of zero in the TVG X(F) (F G F), where
X = un •
f eF seF
If it is shown that A : X(F) ^ Y is continuous, then by the definition of the topology т* and the local convexity of (Y, a) this will imply that A : X ^ У is continuous. Therefore let F G F be fixed. Then (£VJF) is a base of neighbourhoods of zero for the TVG (X/M)( F) (see Proposition 4.10) and r/W is a neighbourhood of zero in (Y/N,^a). By the commutativitv of Diagram (8) A*£V,F = 'qAVF (V n G N) and by the continuity of A there exists N G N such that A*£V-F С ^W or rjAV-F С rjW.Rence, AV-F С W + N, to since W is a closed set and N С W, then W + N С W and the continuity of A : X(F) ^ У is established. This means that A : X ^ У is continuous and the lemma is proved.
Lemma 2. Let L : (Oa,p) ^ (0A,p) be a closed linear operator. Then L : (0A,p*) ^ (0A,p*) is continuous (A is a nonempty connected bounded subset of Cn).
Proof. We recall that the locally convex topology p* of the #-limit of a Hausdorif spectrum on the space of germs of holomorphie functions on A is not weaker than the locally convex topology p of uniform convergence on the compact subsets of A. Therefore the operator L : (0A,p*) ^ (Oa,p*) is closed. Moreover, by Proposition 3.10 the set A has a representation
^=nuu- ■
F eF seF
where F is an admissible class for the countable set Q and Us is a domain in Cn; moreover, each Us (s E |F|) has a countable fundamental system of compact subsets (K,S1 with
KS C K% C____We will show that each space (Oa)(F) (F E F) is complete and so (Oa,p*)
is an H-space. We recall that
x = U PI ^(ys)
f es seF
and k : X ^ r(A, S) = Oa is an isomorphism. The TVG (Oa)(F) is an isomorphic image of the restriction of the complete TVG of countable character S(F) (notation of 3.2) to X. Therefore it is enough to establish the elosedness of k-1(Oa)(F) in S(F). The arguments are carried out more easily for the germs of holomorphie functions on A.
Let F E F, UF D A, UF = (JseF Us (F is no more than countable and is totally linearly ordered for s). Further, let (Gn) be a sequence of germs of holomorphie functions on A which is fundamental in (Oa)(F). Since (Oa)(F) is a quotient group (up to isomorphism) of the complete MVG (Y\F 0Us )(F), where 0Us is the Freehet space with the topology of uniform convergence on the compact sets (K,it follows from Proposition 4.10 that there exists a subsequence gnk E nF ®us (k = 1, 2,...) such th at gnk converges in (Y[F 0Us )(F) to some element g E nF = Gnk (k = 1, 2,...). The last condition implies in
particular that gnk = ( fnk)seF, where /pfclUs = fnk ( s < p, p E F), p = p(k), (k = 1, 2,...). Put p0 = infk p(k), p0 E F. Then, clearly, fn lUs = fnk (s < p0, k = 1, 2,...); we will denote by fk = fn the holomorphie functions on the open connected set UPo (k = 1, 2,...). Since gnk = g and g = (gs)seF, in particular, fk converges to gpo in OUp0 and
moreover g — gn E (s E F, nkl = nkl (s),l = 1, 2,...). The last observation means that for s > po tte holomorphie function gPo — fn has a unique extension to the set Us ( s E F). However, eaA element gnk is equivalent to elements ak E nFfc 0U^.e. ^gnk = 4>ak and moreover ak E P|seFk Vsk (k = 1, 2,...). Furthermore, we may assume without loss of
generality that Fi -< F2 <____Thus the holomorphie function fnk has a unique extension
to the set UFk D A (I = 1, 2,...) and, consequently, the holomorphie functio n gpo has a unique extensron to the set Us fl UFk (s > p0,s E F). Since Us R UFk = (JqeFk (Us H Uq) and Us fUq cUs f Uq' (q < q'), the n Us f UFki is a connected op en set (1 = 1, 2,... ,ki = ki (s)),
and since the set (s E F : s > p0] can be enumerated, let its points be s^ s2,____
Thus on each nonempty open connected set Us f UFk a holomorphie function gpos is defined such that gPo^ = gPo ( s > p0 ,s E F). But since each nonempty intersection (Usi fUFk,) f (Usj fUFk,) is connected by construction and has nonempty intersection with i7Po, then a holomorphie function ¿/is defined on the ope n set \J°=1(USi f UFk,) such that gluSi nU^ = gPo si (i = 1, 2,...). The n gluF * generates an element of HseF * VF* such that = ^qIuf ^^rnequentlv, ^g = G E O^d limn^^ Gn = G in the TVG (Oa)(F). Thus the space (Oa)(F) is complete and (Oa,p*) is an ^^^^ce (UF* C U^ 1(Usi f UFk,), F* E F).
Continuity of the operator A now follows from the Closed Graph Theorem, Lemma 1 and the elosedness of the operator A : (Oa,P*) ^ (Oa,P*)■ The lemma is proved.
Proof of Theorem 1. Let H e О a be a w-local Weierstrass polynomial in zm of degr ее к and let h e H be a holomorphic function on the open connected set U D A which satisfies the conditions of the theorem and the relation (7) Furthermore, let F e О a be an arbitrary germ, let f e F and suppose that f is a holomorphic function on the domain V С Ui (it may be assumed without loss of generality that Ui С U). Let us fix a point a' e nm(A) and a closed (according to the condition) cross-section ra> (А) С ra>(U) and choose a closed pieeewise-smooth Jordan contour Га/ which encloses ra> (A) and lies in ra> (V) and has length l(Га'). Since the function h(z) is continuous on the open neighbourhood of the compact set
Q = [z e V : zm e Га/ , nm(z) = a'} ,
there exists an open ball B(0,8) such that for zm e Га/ and z' e nm[(a',zm) + В(0,5)] we have the inequality
\h(z) - h(a',zm)l < inf \h(a!,zm)| (9)
ra'
and the inclusion
(a',zm) + B(0,8) С V (zm e Га/). Moreover, by the compactness of Q we can choose a polvdisk Д'(0, S'a,) С Cn-i such that
[a' + Д'(0,8fa,)] x Г^ С U [(a',zm) + В(0,5)]. (10)
zm&n,
In fact, we cover the compact set Q with the open balls (a!, zm) + B(0,5) (zm e r^), in each of which we choose a polvdisk (a1, zm) + A(0,8a>) (zm e r^, 8a> = (8'a, ,8m)) with these taken together also covering Q. Put Rai = a1 + A'(0, S'a,), Then
U [(a1, zm) + A(0,5a/)] = [ U (zm + A(0,5m))] x Ra, d ra, x Ra,,
zm&ai ra,
from which (10) follows.
The inclusion (10) allows us to conclude in particular that (9) and the inclusion Ta/ C rz>(V) are valid for z' e Ra' ■ Now for fte indicated z' e Ra' the fonction hz> = h(z',zm) as a holomorphic function of one variable zm has exactly k zeros inside the contour Ta/ by Rouche's Theorem for the domain rz> (V) fl ra> (V); in particular, hz> = ^n ra/ and outside this contour in the domain rz>(V) (and even rz>(U)).
We will denote by the domain bounded by r„/ and put
V = U (Va, x Ra,) .
a' enm(A)
It is clear that V is an open connected set s uch that A cPc V C U\.
Further, for each open set Va> x Ra> (a' G ■nm(A)) we define a holomorphic function (see
2ni Jr , h(z', () ( — z„
and a holomorphie function ра> (z) = f (z) — ga> (z)h(z). Therefore
~h(z',() — h(z',zm)
Pa — 2niJr h(z',( )
С - ^
d(
where
k-1
P< —
Pa' (z) = Y Pa'j (Z')(Zm - Wm)3 , 3=0
1 f dc (J — 0,h...,k- 1)
2Wra, h{zC)
and the holomorphie functions h* (j = 0,1,... ,k — 1) are defined from (7) by consideration of the expression
h(z',() — h(z',zm) C zm
The uniqueness of the functions ga> and pa' is established similarly to [1, p. 93] by using Rouehe's Theorem.
If (£>a' X Ra') n {Va'' x Ra'') = 0, then for z G (V^ x Ra') n {pal' x Ra'') we have zm G Va> H Va" , Because the contours ra and ra^ are homotopie this implies that the following identity holds:
f (* X ) d(
f (* X ) d(
Jra, h(z', C) C - z™ Jra„ h{z', () ( - zm'
Thus ga/(z) = ga''(z) and, consequently, a holomorphic function g(z) can be defined on the domain V such that glna/xva, = ga (a1 G nm(A)). In the same way a holomorphie function p(z) can be defined such that pIr ,xp , = pa (a1 G nm(A)) and
k-1
P(Z) — Pi (Z')(Zm - Wm)
3=0
so that we have the unique representation
f (z)— g(z)h(z)+ p(z) (z eV)
(11)
Thus a linear operator L : Oa ^ Oa x Oa is defined by the relation L(F) = (G,P), F = GH + P, f G F, g G G, h G H, p G P. The operator L has components Li : F ^ G and L2 : F ^ P, whose continuity in the respective topologies will also imply that of L. Let us therefore investigate the continuity of the operators Li and L2.
It follows clearly from the relation (11) that Li and L2 are closed linear operators from (Oa,p) into (Oa,p). Thus by Lemma 2 the operator Li : (Oa,p*) ^ (Oa,p*) is continuous (i = 1, 2), as also is the operator
L :(Oa,p*) ^ (Oa,p*) x (Oa,p*) .
We now establish the continuity of the operator L : (Oa,p) ^ (Oa,p) x (Oa,p). First of all, let us fix an open set Vh constructed be the method indicated above for the holomorphie
function h(z) on the domain U1; then by the compactness of A we choose a finite subcover Ui!i x > where by the construction it may be assumed without loss of generality that
on the distinguished boundary of the polvdisk R!\ the function h(z', () = 0 only for ( E Vf",
i i
(i = 1, 2,...,N), Therefo re h(z) = 0 on the distinguished boundary of the polvdomain Rf^, x , which is part of the boundary of the domain 1 [R1^ x V^ = U2 ■ If we now put
M = sup sup
h*(z>,()
Uf=! ХГ, ) h(Z,C> )
then M <
Now let F E Oa, L1(F) = G, L2(F) = P and choose f E F with domain of definition V C U2 ; construct the domain Vf C V such that Vf C V, while the functions p(z) and g(z) are defined on Vf (p E P, g E G) and the relation (11) holds. It is clear that Vf D A and the following diagrams are commutative:
L2 Li
F ..........> P F ..........> G
f -> P f -> 9
We will establish the continuity of the operator L2 : (0A,p) ^ (0A,p), the continuitv of L1 being obvious. Let a' E nm(A). Then
\p« (*)I < K E^-c1 bf. V )| < ^r £ to1 f0(ri') If V, ()l ■ K I < ^ ■ k ■ supp7 If (z',C)b l(rfa,).
Now we choose a sequence V1 = V D V2 D ... which is fundamental for A and compact
sets Vm = £>4, where £>4 C Vm such th at A = P| XX=1 £>4 and, moreover, the sequence (Vm) converges to A in the Hausdorff metric for all compact subsets of Cra, This means in particular that for f E Ov1 we have the relation
lim sup If (z)I < sup If (z)I.
In fact, let us assume the contrary, i.e. there exist t> 0 and a sequence ) such that
sup If (z)I + e < sup If (z)| (k E N).
a t>
Umk
From this we find a sequence (zmk) such th at zmk E Vmk and
sup If(z)I + e < If(zmk)I (k E N); a
but then we can find a subsequence (zmk ) such th at z* = lim zmk .The n z* E A and,
1 l^rx 1
consequently, we have the inequality
sup If(z)I + e <If(z*)I , a
which is impossible. Therefore
\\p !U = supA !p(z)! _
< supPm \p(z) \ = limm^ supPm \pa,(z)\
< ™ lim^ l(r%) ■ lirnm^ suppm \f (z)\
< KA ■ lA ■ supA \f (z)\ = KA ■ lA -\\F\U .
Thus the operator L2 : (0A,p) ^ (0A,p) is continuous, The theorem is proved. □
References
1. Gunning, R.C.R., Rossi, H. Analytic functions of several complex variables (Russian).
- Moscow, Mir (1969) (transl. from English edition: Prentice Hall 1965).
2. Schaefer, H.H. Topological vector spaces (Russian). - Moscow, Mir 1971 (transl. from English edition: New York, Maemillan 1966; New York, Heidelberg, Berlin, SpringerVerlag 1971).
3. Smirnov, E.I. Hausdorff spectra in functional analysis. - Springer-Verlag, London, 2002.
- 209 p.
4. Smirnov, E.I. On the Hausdorff limit of locally convex spaces (Russian). Editorial Board of the Sibirsk. Mat. Z. Novosibirsk 1986. - Dep. VINITI, 25.12. 86, 2507-B.
Ю.В. Бондаренко
СИЛЬНОЕ УСЛОВИЕ ШОКЕ ДЛЯ КОНУСОВ В ПРОСТРАНСТВЕ
ФУНКЦИЙ
В настоящей статье приведены некоторые теоремы о представлении конусов в пространстве функций на (0;?). Эти конструкции навеяны, с одной стороны , классической теоремой Каратеодори-Минковского о представлении элементов конуса через крайние точки , а с другой стороны, - конструкциями из работ, посвященных операторному представлению конусов убывающих и вогнутых функций в весовом пространстве.
Ключевые слова: конус в пространстве функций, крайние лучи, весовые пространства, конуса убывающих и вогнутых функций.
Ju.V.Bondarenko
STRONG CONDITION SHOKE FOR CONES IN SPACE OF FUNCTIONS
Some theorems about representation of cones in function spaces on (0;?) are considered. We use the classical Karatheodorv - Minkowski theorem about representation of cone elements by extremal points and operator representation of cones of monotone and concave functions in weight spaces.
Key words: cones in function spaces, extremal points, weight spaces, cones of monotone and concave functions.
