On the expansions of analytic functions on convex locally closed sets in exponential series Текст научной статьи по специальности «Математика»

Владикавказский математический журнал 2011, Том 13, Выпуск 1, С. 44-58

YflK 517.9

ON THE EXPANSIONS OF ANALYTIC FUNCTIONS ON CONVEX LOCALLY CLOSED SETS IN EXPONENTIAL SERIES

S. N. Melikhov, S. Momm

In memory of G. P. Akilov

Let Q be a bounded, convex, locally closed subset of CN with nonempty interior. For N > 1 sufficient conditions are obtained that an operator of the representation of analytic functions on Q by exponential series has a continuous linear right inverse. For N = 1 the criterions for the existence of a continuous linear right inverse for the representation operator are proved.

Mathematics Subject Classification (2000): Primary 30B50; Secondary 32U35, 32A15, 47B37.. Key words: locally closed set, analytic functions, exponential series, continuous linear right inverse.

Introduction

In the late sixties Leont'ev (see [10]) proved that each analytic function f on a convex bounded domain Q c C can be expanded in an exponential series ^jgN Cj exp(Aj■). This series converges absolutely to f in the Frechet space A(Q) of all functions analytic on Q, and its exponents Aj are zeroes of an entire function on C which does not depend on f e A(Q). A formula for the coefficients of a some expansion in such exponential series (with the help of a system orthogonal to (exp(Aj-))jeN) was obtained only for the analytic functions on the closure of Q. Later similar results for the analytic functions on convex bounded domain Q c CN were obtained by Leont'ev [9], Korobeinik, Le Khai Khoi [3] (if Q is a polydomain) and Sekerin [15] (if Q is a domain of which the support function is a logarithmic potential).

In [4, 5, 11] was investigated a problem of the determination of the coefficients of the expansions of all f e A(Q), where Q is a convex bounded domain in C, in following setting. Let K c C be a convex set and suppose that L is an entire function on C with zero set (Aj)jSN and with the indicator Hq + HK, where Hq and HK is the support function of Q resp. of K. By Ai(Q) we denote a Frechet space of all number sequence (cjjeN such that the series XjeNCjexp(Aj■) converges absolutely in A(Q). In [4, 5, 11] were established the necessary and sufficient conditions under which a sequence of the coefficients (cjjeN e A1 (Q) in a representation f = ^jgN Cj exp(Aj■) can be selected in such way that they depend continuously and linearly on f e A(Q). In other words, in [4, 5, 11] was solved the problem of the existence of continuous linear right inverse for the representation operator R : A1 (Q) ^ A(Q), c n Cj exp(Aj■). Note that in [4, 5] a formula for continuous linear right inverse

for R (if it exists) was not obtained.

2011 Melikhov S. N., Momm S.

In the present article we consider the following situation. Let Q C CN be a bounded convex set with nonempty interior. We assume that Q is locally closed, i. e. Q has a fundamental sequence of compact convex subsets Qn, n G N. By A(Q) we denote the space of all analytic functions on Q with the topologie of proj^n A(Qn), where A(Qn) is endowed with natural (LF)-topologie. We put e\(z) := exp(^^=1 Amzm), A,z G CN. For an infinite set M C Nn, for a sequence (A(k))(k)eM C CnN with |A(k)| ^ to as |(k)| ^ to we define a locally convex space A1(Q) of all number sequence (c(k))(k)eM such that the series ^(k)eM c(k)e\k) converges absolutely in A(Q). The representation operator c ^ ^(k)eM c(k)eaw maps continuously and linearly A1 (Q) into A(Q). We solve the problem of the existence of a continuous linear right inverse for R.

In this paper for N ^ 1 we assume that (A(k))(k)eM is a subset of zero set of an entire function L on CN with "planar zeroes" and with indicator Hq + Hk, where Hq and Hk are the support functions of Q resp. of some convex compact set K C CN. By [15] such function L exists if and only if the support function of clQ + K is so-called logarithmic potential (for N = 1 a function L exists for each Q and each K). In contrast to [4, 5] here we do not use the structure theory of locally convex spaces. As in [11], we reduce the problem of existence a continuous linear right inverse for the representation operator to one of an extension of input function L to an entire function L on C2N satisfying some upper bounds. With the help of L we construct a continuous linear left inverse for the transposed map to R. Using <9-technique, we obtain that the existence of such extension L is equivalent to two conditions, namely, to the existence of two families of plurisubharmonic functions, first of which is associated only with Q and second is associated with K and Q. The evaluation of first condition was realized in [21]. For the evaluation of second condition we adapt as in [21] the theory of the boundary behavior of the pluricomplex Green functions of a convex domain and of a convex compact set in CN which was developed in [23, 25].

For N = 1 we obtain more complete results. In the first place we prove the criterions for the existence of a continuous linear right inverse for R without additional suppositions on Q and K. Secondly, with the help of a function L as above we give a formula for a continuous linear right inverse for R.

1. Preliminaries

1.1. Notations. If B C CN, by clB and int B we will denote the closure and the interior of B, respectively. By intrB, drB we denote the relative interior and the relative boundary of B with respect to a certain larger set. For notations from convex analysis, we refer to Schneider [26].

1.2. Definitions and Remarks. A convex set Q C CN admitting a countable fundamental system (Qn)nen of compact subsets of Q is called locally closed. Let Q C CN be a locally closed convex set. We will write u := Q n drQ, where drQ denotes the relative boundary of Q in its affine hull. By [21, Lemma 1.2] u is open in drQ. We may assume that the sets Qn are convex and that Qn C Qn+1 for all n G N. A convex set Q C CN will be called strictly convex at dru if the intersection of Q with each supporting hyperplane to cl Q C CN is compact. If int Q = 0, Q is strictly convex at dru if and only if each line segment of u is relatively compact in u.

By [13, Lemma 3] Q C CN is strictly convex at dru if and only if Q has a fundamental system of convex neigborhoods.

1.3. Convention. For the sequel, we fix a bounded, convex and locally closed set Q C CN with 0 in its nonempty interior and with a fundamental system of compact convex

subsets Qn C Qn+1, n G N. By (wn)raeN we shall denote some fundamental system of compact subsets of u = Q n dr Q.

K will always denote a compact convex set in CN.

1.4. Notations. For each convex set D C CN we denote by Hd the support function of D, i. e. HD(z) := supwgD Re(z, w), z G CN. Here (z, w) := ZjWj. We put Hn := Hqu, n G N.

Let e^(z) := exp(A, z), A, z G CN. For a locally convex space E by E we denote the strong dual space of E.

1.5. Function spaces. We set |z| := (z,z)1/2, z G CN; U(t,R):= {z G CN : —z| < R}, t eCN, R> 0;U:= U(0,1). For all n,m G N let £?ra>m := A°° (Qra + if/) denote the Banach space of all bounded holomorphic functions on Qn + ^U, equipped with the sup-norm. We consider the spaces A(Qn) = (JmgN Era,m of all functions holomorphic in some neighborhood of Qn, n G N, and endow them with there natural inductive limit topology. By A(Q) we denote the vector space of all functions which are holomorphic on some neighborhood of Q. We have A(Q) = HneN A(Qn), and we endow this vector space with the topology of A(Q) := proj^n A(Qn). This topology does not depend on the choice of the fundamental system of compact sets (Qn)ragM. If Q is open, A(Q) is a Frechet space of all holomorphic functions on Q.

For all n, m G N let

A„>m :={ f G A(CN): ||f ||ra>m := sup |f (z)| exp ( — H„(z) — |z|/m) < to)

I ze€N J

Aq := indproj

and

—m

1.6. Duality. The (LF)-space A(Q) := ind„^ A(Qn)'6 and Aq are isomorphic by the Laplace transformation

F : A(Q) ^ Aq, F(p)(z):= ^(e^), z G CN.

In addition (LF)-topology of A(Q)' equals the strong topology.

The assertion has been proved in [21, Lemma 1.10] (see Remark after 1.10, too) If we identify the dual space of A(Q) with Aq by means of the bilinear form (■, ■), then (eA, f) = f (A) for all A G CN and all f G Aq.

1.7. Sequence spaces. Representation operator. Let M C Nn be an infinite set and (A(k))(fe)SM C CN be a sequence with |A(k)| ^ to as |(k)| ^ to. For all n,m G N we introduce the Banach spaces

An,m(Q) := ■{ c = (c(k))(fe)eM C C : |c(fc)| exp (H„(A(fc)) + |A(fc) |/m) < to^

^ (fc)SM ^

(Q) := i c = (c(k))(fe)eM C C : sup |c(fc)| exp ( - H„(A(fc)) - |A(fc)|/m) < to I (fc)eM

K

and put

Ai(Qn) := indA„,m(Q), Ai(Q) := projAi(Q„), K^(Q) := indproj Kn,m(Q).

—n

—m

We note that the series ^(k)eM c(k)e\{k) converges absolutely in A(Q) if and only if c G A1 (Q) (see [2, Ch.I, §§ 1, 9]).

The operator R(c) := ^(k)eM c(k)eA(k) maps continuously and linearly A1 (Q) in A(Q). We call R the representation operator. By Korobeinik [2], if R : A1 (Q) ^ A(Q) is surjectiv, (e^(k) )(k)eM is called an absolutely representing system in A(Q).

Let e(k) := (^(k),(m) )(m)sM, (k) G M, where £(k),(m) is the Kronecker delta.

1.8. Duality. (i) The transformation p ^ (p(e(k)))(k)eM is an isomorphism of (LF)-space A1(Q)/ := indn^ A1(Qn)b onto K^(Q). The duality between A1(Q) and K^(Q) is defined by the bilinear form (c, d) := ^(k)eM c(k)d(k).

(ii) A transposed map R/ : Aq ^ K^(Q) to R : A1(Q) ^ A(Q) is the restriction operator / ^ (/ (A(k)))(k)eM•

(iii) R has a continuous linear right inverse if and only if R/ has a continuous linear left inverse.

< The assertions (i) and (ii) were in [13, Lemma 6] proved.

(iii): This can be proved in the same way as (i) ^ (ii) in [21, Lemma 1.12]. (We note that we can not assume in advance the surjectivity of R.) >

1.9. Notations. Let S := {z G CN : |z| = 1}. For a convex set D C CN, 7 C D and A C S we define

SY(D) := {a G S : Re(w, a) = HD(a) for some w G 7}

and

Fa(D) := {w G D : Re(w,a) = HD(a) for some a G A}. We will write S7 := S7(Q), A := Sfa(k)(K), So := S\SW.

Definition 1.10. (a) Given an open subset B C S and a compact convex set K C CN. K is called smooth in the directions of the boundary of B if for each compact set k C B the compact set k := Sfk(k)(K) is still contained in B.

Note that the condition is fulfilled if dK is of class C1.

(b) A convex compact set K C CN is called not degenerate in the directions of B C S, if K is not contained all in the supporting hyperplane {z G CN Re(z, a) = HK(a)} of K for each a G B.

Note that the condition is fulfilled if int K = 0.

Remark 1.11. (a) Under the hypotheses of the Definition 1.10 (a) the following holds: Let S1 C S be an open neighborhood of S\B (with respect to S). For k := S\S1, the set k is a compact subset of B. Hence if S2 C S\k is compact, we have S2 n k = 0 and thus S2 C S1. (Otherwise it would follow that k n S2 = 0.)

(b) Let K have 0 as an interior point. K is smooth in the directions of the boundary of B if and only if the convex set intK0 U u/ is strictly convex at $ru/, where u/ := $K0 n r(B), K0 := {z G CN | Hk(z) < 1} and r(B) := {tb 11 > 0}.

2. Conditions of existence of a continuous linear right inverse for the representation operator

2.1. Notations, Definitions and Remarks. (a) Let / be an entire functions of exponential type on CN. By hf we denote the (radial) indicator of /, i. e.

hj(z) := limsup(limsuplog|/(rz')|/r) for all z G C

N

(b) An entire function / of exponential type on C is called function of completelly regular growth (by Levin-Pfliiger), if there is a set of circles U(^j,rj), j G N, with | ^ to as j —> to, such that lim^oo S^ ^ii rj = 0 and outside of Li the following

asymptotic equality holds:

log |/(^)| = h*f(z)+ o(\z\) as

^ to.

By Krasichkov-Ternovskii [6], in Definition (b) we can choose the exclusive circles U(^j,rj) so that they are mutually disjoint.

(c) By Gruman [18] an entire function / of exponential type on CN is called function of completelly regular growth, if for almost all a G S the function /(az) of one complex variable has completelly regular growth on C.

(d) There are other definitions of the functions of completelly regular growth of Azarin [1] and of Lelon, Gruman [8, Ch.IV, 4.1]. By Papush [14], if / is an entire function on CN with "planar" zeroes, i. e. the zero set {z G CN : /(z) = 0} of / is the union of the hyperplanes {z G CN : (z,ak) = ck}, ak G S, ck G C, k G N, all these definitions (for /) are equivalent. From this and from [7, 22] it follows that an entire function / on CN with "planar" zeroes has completely regular growth on CN if and only if / is slowly decreasing on CN.

We recall the some definitions and results from Sekerin [15].

2.2. A special entire function. A structure of the exponents A(k). (a) Below we shall exploit an entire function L on CN of order 1, which satisfies the following conditions:

(i) The zero set V(L) of L is a sequence of pairwise distinct hyperplanes Pk := {z G CN : (ak, z) = ck}, k G N, where |ak| = 1 and ck = 0. If for k1 < k2 < ... < kN the intersection Pki n Pk2 n ... n PkN is not empty, then it consits of a single point A(k), where (k) denotes multiindex (k1,k2,...,kN). Further M is the set of the such multiindexes (k). Moreover, L(k) (A(k)) = 0, where L(k)(z) := L(z)/l(k) (z) and l(k) (z) := n5=1 ((akj ,z) - ckj), (k) G M.

(ii) L is a function of completely regular growth with indicator Hq + HK.

(iii) |L(k)(A(k))| =exp(HQ(A(k)) + Hk(A(k)) + o(|Afc)|)) as |(k)| ^ to. We write lk (z) := (ak,z) - ck, z G CN, k G N.

(b) (i) By [15, Theorem 1], for each / G Aintq+k the Lagrange interpolation formula holds:

/(A)= £ XeCN, (1)

(k)iM L(k)(A(k))

where the series converges uniformly on compact sets of CN. From (1) it follows that (A(k))(k)eM is the uniqueness set for A^tq+k, i. e. from / G A(CN), h*f (z) < Hq(z) + Hk(z) for all z G CN\{0} it follows that / = 0.

(ii) There is a function a(z) = o(|z|) as |z| ^ to such that |L(k)(z)| ^ exp(HQ(z) + Hk(z) + a(z)) for all z G CN and all (k) G M.

(c) A plurisubharmonic function u on CN will be called a logarithmic potential if there exists a Borel measure ^ ^ 0 on [0, to) x Sn such that for every R G (0, to) there is a pluriharmonic function ur on U(0, R) with

u(z) = J log11 — (z, w)| d^(i) + uR(z) for all z G U(0, R).

[0,r]xsn

By [15] for a bounded convex domain D with 0 G D the support function HD is a logarithmic potential for example if D is a polydomain, a ball, an ellipsoid, a polyhedra with

symmetric faces, and in the case of C2, if D = D1 + iD2, where D1 and D2 are any centrally symmetric convex domains in R2; if D is symmetric with respect to 0 and clD is a Steiner compact set (see Matheron [19, §4.5]).

For each bounded convex domain D C C with 0 G int D the function Hd is a logarithmic potential.

(d) By [15, Theorem 5], there exists a function L satisfying the conditions (i)-(iii) in 2.2 (a) if and only if Hq + Hk is a logarithmic potential. Hq + Hk is a logarithmic potential if Hq and Hk are the logarithmic potentials.

(e) Let Hq+k = Hq + HK be a logarithmic potential. By [15] the representation operator R : A1 (intQ + K) ^ A (int Q + K) is surjective. By [13, Theorem 14] R : A^Q) ^ A(Q) is surjective, if Q is strictly convex at dru, K is smooth in the directions of drSw and not degenerate in the directions of Sw.

Theorem 2.3. Let Q be strictly convex at dru and L be an entire function on CN satisfying the conditions 2.2 (a). Then (II) ^ (III) ^ (I):

(I) The representation operator R : A1 (A) ^ A(Q) has a continuous linear right inverse.

(II) There is a positively homogeneous of order 1 plurisubharmonic function P on C2N such that P(z, z) ^ Hq(z) + Hk(z) and (Vn) (3n') (Vs) (3 s') with

P(z,^) < Hn(z) + |z|/s + Hk+ Hq(^) — H„Gu) — M/s' (Vz,^ G CN).

(III) There are the plurisubharmonic functions ut, vt, t G S, on CN such that ut(t) ^ 0, vt(t) ^ 0 and (V n) (3 n') (V s) (3 s') with

(a) ut(z) ^ Hra/(z) — Hra(t) + |z|/s — 1/s' and

(b) vt< Hk+ Hq(^) — H„(^) — Hk(t) — HQ(t) + Hra/(t) — H/s' + 1/s for all z, ^ G CN and all t G S.

< (II) ^ (III). We may choose

ut(z) := P(z,t) — HQ(t) — Hk(t), vt(^) := P(t,^) — HQ(t) — Hk(t)

for all z, ^ G CN and t G S. (III) ^ (II). We put

Po(z,^):= (sup (ut(z) + vt+ Hq(z) + Hk(^)))*, z,^ G CN V teS y

where f * denotes the regularization of a function f. P0 is the plurisubharmonic function on C2N with

P(z, z) ^ Hq(z) + Hk(z) (V z G S). By (III) we have: (V m) (3 n') (V s) (3 r) with

ut(z) ^ Hn/(z) — Hm(t) + |z|/s — 1/r for all z G CN and all t G S

and (V n) (3 m) (V r) (3 s') with

vt(^) + HQ(t) + Hk(t) < Hk+ Hq(^) — Hra(^) + Hm(t) — M/s + 1/r

for all ^ G CN and all t G S. By adding the last inequalities, we obtain that (V n) (3 n') (V s) (3 s') with

ut(z) + vt(^) + HQ(t) + Hk(t) < Hra/(z) + Hk+ Hq(^) — Hra(^) + |z|/s — H/s'

for all z,^ G CN and t G S. From this it follows that P0 satisfies the upper bounds in (II). As P we may choose P(z,^) := (limsupt^+p P(tz,t^)/t)*, z,^ G CN.

(III) ^ (I). By (the proof of) [16, Theorem 4.4.3] (see [8, Theorem 7.1], too) there is a L G A(C2N) with L(z, z) = L(z) and (Vn) (3 n/) (Vs) (3 s/) (3 C): (Vz,^ G CN)

|L(z,^)| < Cexp (Hn(z) + |z|/s + Hk+ Hq(^) — Hn(^) — |^|/s/). (2)

We define

< M \ ST^ L(k)(z)L(z,A(k)) N

«1(c)(2) := > —7T—c £ Koo(Q), z £ C .

(k)iM L(k) (A(k))

(3)

From (2) it follows that the series in (3) converges absolutely in A2q+k. (By [21, Remark 1.5] 2Q + K is locally closed and (2Qn + K)neN is a fundamental system of compact subsets of 2Q + K.) Hence k1 maps Kp(Q) in A2q+k continuously (and linearly). Since, by (2), for all / G Aq and z G CN the function L(z, ■)/ belongs to Aintq+k, by 2.2 (b) for all z G CN

Kl(Bf(f))(z)= \[k)) = i{z,z)f{z)=L{z)f{z).

(k)eM (k)( (k))

From here it follows that k1 oR/ is the operator of multiplication by L. By [21, Proposition 2.7] there is a continuous linear left inverse k2 : A2q+k — Aq for k1 oR/. The operator k := k1 ok2 is a continuous linear left inverse for R/.

Now we shall evaluate the abstract condition (III) (b) of Theorem 2.3. The condition (III) (a) was evalueted in [21, Proposition 3.6]. >

We recall some definitions from [23] and [25].

Definition 2.4. If D C CN is bounded, convex and c > 0, let v0D c be the largest plurisubharmonic function on CN bounded by HD and with v°-D c(z) ^ clog |z| + O(1) as |z| — 0. A function C°D : S — [0, to] is defined by

{z G CN : v%D,c(z) = Hd(z)} = {Aa : a G S, 1/C0d(a) < A < to}.

If 0 G int D and if C > 0, let v pD c be the largest plurisubharmonic function on CN bounded by HD and with v pD(z) ^ C log |z|+O(1) as |z| — to. A function Cp0: S — [0, to] is defined by

{z G CN : v^d,c(z) = Hd(z)} = {Aa : a G S, 0 < A < 1/CgD(a)} .

Instead C°D and CpD we shall write briefly CD resp. Cpf.

Proposition 2.5. Let Q be strictly convex at the dru and suppose that 0 G int K. For N > 1 assume that K is smooth in the directions of dr Sw. The following are equivalent:

(i) There are plurisubharmonic functions vt (t G S) on CN with vt(t) ^ 0 such that: (V n) (3 n/) (V s) (3 s/) with

vt < Hk + Hq — Hn — | ■ |/s/ — Hk(t) — HQ(t) + H„/(t) + 1/s (Vt G S).

(ii) 1 /CK is bounded on some neighborhood of S0 and Cp is bounded on each compact subset of S^.

< (i) ^ (ii). Choose n' according to (i) for n = 1. On So we have Hn/ < Hq. Thus there are a neigborhood S of So and some e > 0 with Hn/ + e ^ Hq on S We put

v := (sup(vt + Hk(i)))*. tes

This function is plurisubharmonic on CN with v ^ HK on S and satisfies: (V n) (3 n') (V s) (3 s') such that

v < Hk + | ■ |/n + max{-HQ(i) + Hra/} + 1/s. Since Hn/ ^ Hq, this gives v ^ Hk on

CN. The bounds for n = 1 give v(0) ^ —e. From [25, 2.14] it follows that 1/C0 is bounded on S Let к С S^. We define

v := (sup(vt + Hk(i)))*.

This function is plurisubharmonic on CN with v ^ HK on к and satisfies: (V n) (3 n') (V s) (3 s') such that v ^ HK + HQ — Hn — | ■ |/s' + 1/s ^ HK + HQ — Hn + 1/s. This shows that v < Hk .

Now choose n with к С S^n, i. e. with Hq = Hn on к. Choose n' ^ n according to (i). Choose s' for s = 1. Then there is a neighborhood v of к in S such that

Hq — Hn — | - |/s' < —| ■ |/(2s') on Г(к)

and thus

v < Hk — | ■ |/(2s') + 1 on Г(к).

In order to reach our claim that Cp is bounded on к, we need an estimate like the previous one on all CN (not only on the particular cone). For this purpose we are going to modify v. First note that, if N = 1, it follows from what we have already proved that dK has to be of class C1 (see [20, 2.10, 2.14]). For N

> 1 we use our special hypothesis. For this reason we may assume that we have constructed v for the set к instead of к. Define

L(z) := sup Re(w,z), z £ CN. weFK

The positively homogeneous function L satisfies L ^ Hk on

CN, and L = Hk

on к. If

L(a) = HK(a), there is w £ with Re(w, a) = HK(a), hence a £ Sfk. Thus L < H on S outside the compact set к. We replace v by v := v/2 + L/2 and obtain v ^ HK on CN, v = HK on к and v < HK outside a neighborhood of the origin. By [23, 2.1] this shows that Cp is bounded on к.

(ii) ^ (i). By the hypothesis, 1/CK is bounded on some neighborhood S? of So. Hence there is c > 0 such that the plurisubharmonic function v0-K c equals HK on S Let n £ N. Since Hn < Hq on S0, there is a compact neighborhood Sn of S0 with Hn < Hq on Sn. We may assume Sn С Sn-1 С ... С S1 С S Since Cp is bounded on S\Sn, there is Cn > 0 with vP := v^k,Cn = Hk on S\S„+2.

Again for N = 1 it follows from (ii) that dK is of class C1. For N > 1 we apply the extra hypothesis to obtain (as in the first part of the proof) a positively homogeneous function Ln bounded by H on CN, which equals H on к = Sn+1, and such that Ln < H outside the compact set Sn+1 С Sn (see Remark 1.11 (a)). Then the plurisubharmonic function vn := v°K,c/2 + Ln/2 satisfy v„ ^ Hk on CN, v„ = Hk on S„+b v„ ^ (Hk + L„)/2 < Hk on S\Sn.

Fix n e N. Since ^ (HK + Ln)/2 < HK + Hq - Hn on S, and since (0) < 0, there is n with

< < Hk + Hq - H„ - D/2 - 1/n on CN,

where

D := HK + HQ - Hn - (HK + Ln)/2 = (HK - Ln)/2 + HQ - Hn.

Choose n' with Hq - Hn ^ 1/n on Sn+i. Then for each s there is s' with D/2 ^ | ■ |/s' on CN such that

vn < Hk + Hq - Hn - | ■ |/s' - HQ(t) + H„/(t) + 1/s (Vt e Sn+i).

For the functions vf we get: Choose n' (in addition) so large that Hq = Hn' on S\Sn+2. For each s we choose s' (in addition) so large that vf ^ HK — | ■ |/s' + 1/s (see Definition 2.4). This gives

vf < Hk + Hq — Hn — | ■ |/s' — Hq(î) + Hra/(t) + 1/s (Vt G S\Sn+2).

Note that v0 ^ ... ^ vJ ^ vJ+1 and that vf ^ ... ^ vf ^ v^. That is why for each l G N the following holds: (V n) (3 n') (V s) (3 s') with

v0 < Hk + Hq — Hn — | ■ |/s' — HQ(t) + H„/ (t) + 1/s (V t G Sn+i),

and (V n) (3 n') (V s) (3 s') with

vf < Hk + Hq — Hn — | ■ |/s' — HQ(t) + H„z(t) + 1/s (Vt G S\S„+2)

By the construction, lim^r v0 =: v^ exists and defines a plurisubharmonic function with v^ = Hk on S0.

For t G S\S2 define vt:= vf. For t G Si+i\Si+2 we put vt := v0/2 + vf°/2. For t G So we define ït := v . Obviously ït(t) = HK(t) for all t G S.

Let t G S1+i\S1+2. For n ^ l and n', s and s' as above we get

it < (Hk + Hq — Hn — | ■ |/s' — HQ(t) + Hn'(t) + 1/s)/2 + Hk/2.

By the strict convexity of Q at dru (see [21], the proof of Proposition 3.6), there is n'' such that (HQ + Hn')/2 ^ Hn« and thus (HQ — Hn')/2 ^ HQ — Hn». This gives

it < Hk + Hq — Hn — | ■ |/(2s') — HQ(t) + Hn»(t) + 1/(2s).

For n ^ l and n', s and s' as above we get

it < Hk/2 + (Hk + Hq — Hn — | ■ |/s' — HQ(t) + Hn'(t) + 1/s)/2.

0

As above we get the desired estimate.

For t e So = P|ieN Si, we see as in the first part of the previous arguing that = v, satisfies these estimates for all n (^ l = to).

For t e S\S2, as in the second part of the arguing just done, we see that these estimates hold for all n (^ l = 1).

Finally we put vt := - HK(t), t e S and are done. >

Remark 2.6. Let Q be strictly convex at the drw. By [21, Proposition 3.6] the following are equivalent:

(i) There are plurisubharmonic functions ut (t G S) on CN with ut(t) ^ 0 such that: (Vn) (3 n/) (V s) (3 s/) with

ut(z) ^ Hn(z) — Hn(t) + |z|/s — 1/s/ (Vz G CN, t G S).

(ii) CQP is bounded on some neighborhood of S0 and 1/CQ is bounded on each compact subset of Sw.

Theorem 2.7. Let Q be strictly convex at the dru and suppose that 0 G int K and L is a function as in 2.2 (a). For N > 1 assume that K is smooth in the directions of dSw. If CQ3 and 1/CK are bounded on some neighborhood of S0, 1/CQ and Cp are bounded on each compact subset of Sw then the representation operator R : A1 (Q) — A(Q) has a continuous linear right inverse.

< The assertion hold by Theorem 2.3, Proposition 2.5 and Remark 2.6. >

The equivalent conditions of Theorem 2.7 are fulfilled if dQ and dK are of Holder class C1iA for some A > 0. They are not fulfilled if Q or K is a polyedra, and for N = 1 if dQ or dK has a corner [24].

3. The case of one complex variable

In this section we consider the case N = 1 for which the results of the previous sections can be improved.

Convention 3.1. Further L is an entire function on C satisfying following conditions:

(i) The zero set of L is a sequence of pairwise distinct simple zeros Ak , k G N, such that |Ak | ^ |Ak+1| for each k G N.

(ii) L is a function of completely regular growth with indicator Hq + HK.

(iii) The asymptotic equality holds:

|L/(Ak)| = exp(HQ(Ak) + Hk(Ak) + o(|Ak|)) as k — to.

Such function L exists (see for example [10]).

Leont'ev (see [10]) introduced an interpolating function, which is defined with the help of an entire function of one complex variable. Leont'ev's interpolating function is a functional from A(clQ + K)/\A(Q)/ for every K (if Q = clQ). With the help of an entire function of two complex variables we give the analogous definition of an interpolating functional from A(Q)/.

Definition 3.2. Let L be an entire function on C2 such that L(-,^) G Aq for each ^ G C. Q-interpolating functional we shall call a functional

t

(z,^,/):= F-1(L(-,^))^y /(t — £) exp(z£) (%) , z,^ G C, / G A(Q),

0

where the integral is taken along the interval [0, t].

We show certain properties of

Lemma 3.3. (a) ) G Aq for all ^ G C and / G A(Q).

(b) For all z, ^ G C the equality ^¿(z, z, ) = z) holds where a function l G A(C2) is such that L(u, z) — L(z, z) = l(^, z)(^ — z).

(c) z, ■) G A(Q)/ for all z, ^ G C.

< (a): We fix ^ e C, f e A(Q) and a domain G with Q c G and f e A(G). We choose a contour C in G which contains in its interior the conjugate diagram of If 7is

Borel conjugate of we have:

^ /) = ¿r J 7(i, Ai) I y /(i - £)exp(z£K di, z G C.

c \o /

Since the function (t,^) ^ 7(t,^) ^/0 f (t — £)exp(z£) d£J is continuous by t e C and entire by z, the function ^¿(z,^,f) is entire (with respect to z). From direct upper bounds for ^¿(z,^,f)| it follows that ) e Aq.

(b): Obvious.

(c): Since the map f ^ /0 f (t — £)exp(z£) d£, t e Q, is continuous and linear in A(Q) and F-1 (Q(^)) is a continuous and linear on A(Q), the functional Qq(z,^, •) is continuous and linear on A(Q), too. >

Lemma 3.4. We assume that a function L, as in 3.2, satisfies in addition the following conditions: L(z,z) = L(z) for each z e C and (Vn) (3 n') (Vs) (3 s') (3 C) with

|L(z,^)| < Cexp (Hra/(z) + Hk+ Hq(^) — + |z|/s — H/s') (Vz,^ e C).

Then n(f) := (^¿(Ak, Ak, f )/L'(Ak))kgN, f e A(Q), is continuous linear operator from A(Q) into A1(Q).

< We define Lk(z) := L(z, Ak)/(L'(Ak)(z — Ak)), k e N. By using upper bounds for |L|, 3.1 (iii) and 3.3(b), we obtain, that Lk is entire function on C and (Vn) (3 n') (Vs) (3 s') (3 Ci,C2) such that for all z e U(Ak, (1 + |Ak|)-2)

|Lfc(z)| < Ci exp (Hn/(z) + Hk(Afc) + HQ(Afc) — H„(Afc) + |z|/s — |Ak|)(s' — 1) +2log(1 + |Afc|) — log|L'(Afc)|) < C2 exp (Hn'(z) — Hn(Ak) + |z|/s — |Ak|/s') (Vk e N).

Applying the maximum principle we get that (Vn) (3 n') (Vs) (3 s') (3 C3) with

|Lfc(z)| < C3 exp (Hn'(z) — Hn(Afc) + |z|/s — |Afc|/s') (Vz e C, k e N).

From this it follows that the series ^kgN ckLk converges absolutely in Aq for each c = (ck)keN e K^(Q) and k : c ^ ^keN ckLk is continuous linear operator from K^(Q) into Aq. We shall find its adjoint operator k' : A(Q) ^ A1 (Q):

(c, k' (eM)) = (K(C) , f) = cfc Lfc

fcSN

Lfc= £ cfc^¿(Afc,Ak,eM)/L'(Ak) (V^ e C, c e Ai(Q)).

fcSN fceN

Hence K'(eM) = (^¿(Ak, Ak, eM)/L'(Ak))fcgN, ^ e C. Let CN be a space of all number sequence with its natural topologie. The maps k' : A(Q) ^ CN and n : A(Q) ^ CN are continuous and linear. Since the set {eM : ^ e C} is total in A(Q), we have n = k' on A(Q) and n is continuous and linear from A(Q) into A1(Q). >

Theorem 3.5. (I) Let 0 e intrK. The following assertions are equivalent:

(i) The representation operator R : A1(Q) ^ A(Q) has a continuous linear right inverse.

(ii) There is an entire function L on C2 such that L(z,z) = L(z) and (Vn) (3 n') (Vs) (3 s') (3 C) with

|L(z,u)| < Cexp (H„/(z) + Hk(u) + Hq(u) - H„(u) + |z|/s - |u|/s') (Vz,u e C).

(iii) Q is strictly convex at drw, the interior of K is not empty, C° and 1/C0 are bounded on some neighborhood of S0, 1/CQ and C, are bounded on each compact subset of Sw.

(II) (iv) If L is a function as in (ii), the operator

n(f) ^ (fiL(Afc, Afc, f )/L'(Afc))fceN, f e A(Q),

is a continuous linear right inverse for R.

(v) If n : A(Q) ^ A1(Q) is a continuous linear right inverse for R, then there is a unique function L as in (ii) such that n(f) = (^¿(Ak, Ak, f)/L'(Ak))keN, f e A(Q).

< (iv) (and (ii) ^ (i)): Let L be a function as in (ii). Then

EL(-> Afc) cfc—-——--—-

kL'(Ak)(■ - Ak)

(fe)eN v

maps continuously (and linearly) K, (Q) into Aq. Since for each f e Aq the function f-L(z, ■) belongs to AintQ+K, taking into account the Lagrange interpolation formula (1), we obtain:

= L(z, z)f (z) = L(z)f (z) (Vz e C, f e Aq).

This implies that k = n' is a left inverse for R'. By the proof of Lemma 3.4 k is the adjoint to n for each function L as in (ii). Hence n is a right inverse for R.

(i) ^ (ii): Let n be a continuous linear right inverse for R. Then k:= n' : K,(A) ^ Aq is a left inverse for R'. We put fk := K(e(k)), where e(k) := (¿k,n)ngM, k e N. By Grothendieck's factorization theorem, for each n there is n' such that k maps continuously proj^mKn,m(Q) in proj^mAn/Hence the following holds: (Vn) (3 n') (Vs) (3 r) (3 C) with

|fk(z)| < Cexp (H„/(z) - H„(A(fc)) + |z|/s - |A(fc)|/r) (Vz e C, k e N). (4)

For f e Aq let

WH/x) := E iixt* " ^ G C'

fcSN U k

By 2.2 (b) (ii) and (4) the series converges absolutely in Aq and converges uniformly (by u) on compact sets of C. Fix z e C. Then Tz(uf)(u)) = (f)(u) for all f e Aq and u e C. By [12, Lemma 1.7] there is a function az e A(C) such that Tz(f)(u) = az(u)f (u) for all U e C, f e Aq. The function L(z,u) := (u), z,u e C, satisfies the conditions in (ii) (see the proof of (i) ^ (ii) in [12, Theorem 1.8] too). (iii) ^ (i) holds by Theorem 2.7.

(i) ^ (iii): Since the operator R has a continuous linear righ inverse, R : Ai(Q) ^ A(Q) is surjectiv. By [13, Theorem 8] the set Q is strictly convex at örw.

Since (i) is equivalent to (ii) there is a function L which satisfies the conditions in (ii). Let P be the (radial) indicator of L, i. e.

P{z,ß) :=flimsup z>/iGC.

V t )

Then P is a plurisubharmonic function on C2 satisfying the conditions in (II) of Theorem 2.3. Hence, by Theorem 2.3, there are subharmonic functions vt (t £ S) as in (III) (b). We put gt:= |t|vt/|t|(^/|t|), t £ C, t = 0. Then gt are subharmonic functions on C such that gt (t) ^ 0 and (V n) (3 n') (V s) (3 s') with

gtM < Hk+ Hq(^) - H„(^) - Hk(t) - HQ(t) + Hra/(t) - M/s' + |t|/s

for all t £ C, t = 0. If Sw = 0, the set Q is open. Hence the following holds: (Vn) (3 n') with

gtM < Hk- Hk(t) + M/s' -|t|/s (V£ C, t = 0).

Then, by [12, Proposition 1.17], an angle with the corner at 0 doesn't exist in which the support function Hk of K is harmonic. Hence int K = 0. If Sw = 0, there is an open (with respect to S) subset A of S such that Hn = Hq on A for large n. Let r(A) := |ra : r > 0}. Then for each s there is s' with

gtM < Hk- Hk(t) + |t|/s - M/s' (V£ C, t = 0).

As in [12, Proposition 1.17] from the maximum principle for harmonic functions it follows that the interior of K is not empty.

By Theorem 2.3 , Proposition 2.5 and Remark 2.6 CQ and 1/C0 are bounded on some neighborhood of So, 1/CQ and are bounded on each compact subset of Sw.

(v): By the proof of (i) ^ (ii) there is an entire function L satisfying the conditions in (ii) and such that n'(e(fc)) = ¿/ff)'(A-Afc) for each ^ £ N. Hence n'(c) = EfcSNcfcL'(\{k)(k-^k) for each c £ K^(Q) and n(f) = (^¿(Ak, Ak, f )/L'(Ak))kgN for all f £ A(Q) (see the proof of Lemma 3.4). We shall show uniqueness of such function L. Let L1, L2 be two such functions. Then L1(z,Ak) = L2(z,Ak) for all k £ N, z £ C. Since {Ak : k £ N} is the uniqueness set for AintQ+K (see 2.2 (b)) and Li(z, ■), L2(z, ■) £ AintQ+K, we get Li(z, ■) = L2(z, ■) for each z £ C and, consequently, L1 = L2 on C2.

Acknowledgement. The first named author thanks for the support by the Deutscher Akademischer Austauschdienst during his stay at the University of Düsseldorf in autumn 2005.

References

1. Азарин В. С. О субгармонических функциях вполне регулярного роста в многомерном промтран-стве // Докл. АН СССР.—1961.—Т. 146, № 4.—С. 743-746.

2. Коробейник Ю. Ф. Представляющие системы // Успехи мат. наук.—1981.—Т. 36, вып. 1.—С. 73126.

3. Коробейник Ю. Ф., Ле Хай Хой. Представляющие системы экспонент в полицилиндрических областях // Мат. сб.—1983.—Т. 122(164), № 4.—С. 458-474.

4. Коробейник Ю. Ф., Мелихов С. Н. Линейный непрерывный правый обратный для оператора представления и конформные отображения // Докл. АН.—1992.—Т. 323, № 5.—С. 826-829.

5. Коробейник Ю. Ф., Мелихов С. Н. Линейный непрерывный правый обратный для оператора представления и приложения к операторам свертки // Сиб. мат. журн.—1993.—Т. 34, № 1.—С. 7084.

6. Красичков-Терновский И. Ф. Одна геометрическая лемма, полезная в теории целых функций, и теоремы типа Левинсона // Мат. заметки.—1978.—Т. 24, вып. 4.—С. 531-546.

7. Кривошеев А. С. Критерий разрешимости неоднородных уравнений свертки в выпуклых областях пространства Cn // Изв. АН СССР. Сер. мат.—1990.—Т. 54, № 3.—С. 480-450.

8. Лелон П., Груман Л. Целые функции многих комплексных переменных.—М.: Мир, 1989.—350 с.

9. Леонтьев А. В. О представлении аналитических функций рядами экспонент в полицилиндрической области // Мат. сб.—1972.—Т. 89, № 4.—С. 364-383.

10. Леонтьев А. Ф. Ряды экспонент.—М.: Наука, 1976.—536 с.

11. Мелихов С. Н. Продолжение целых функций вполне регулярного роста и правый обратный для оператора представления аналитических функций рядами квазиполиномов // Мат. сб.—2000.— Т. 191, № 7.—С. 105-128.

12. Мелихов С. Н. О левом обратном к оператору сужения на весовых пространствах целых функций // Алгебра и анализ.—2002.—Т. 14, вып. 1.—С. 99-133.

13. Мелихов С. Н., Момм З. О свойстве внутрь-продолжаемости представляющих систем экспонент на выпуклых локально замкнутых множествах // Владикавк. мат. журн.—2008.—Т. 10, вып. 2.— С. 36-45.

14. Папуш Д. Е. О росте целых функций с «плоскими» нулями // Теория функций, функцион. анализ и их прил.—1987.—Вып. 48.—С. 117-125.

15. Секерин А. Б. О представлении аналитических функций многих переменных рядами экспонент // Изв. РАН. Сер. мат.—1992.—Т. 56, № 3.—С. 538-565.

16. Хермандер Л. Введение в теорию функций нескольких комплексных переменных.—М.: Мир, 1968.—280 с.

17. Эдвардс Р. Функциональный анализ. Теория и приложения.—М.: Мир, 1969.—1071 с.

18. Gruman L. Entire functions of several variables and their asymptotic growth // Ark. Mat.—1971.— Vol. 9.—P. 141-163.

19. Matheron G. Random sets and integral geometry.— New York: Wiley, 1975.—xxiii+261 p.

20. Melikhov S. N., Momm S. Solution operators for convolution equations on the germs of analytic functions on compact convex sets on CN // Studia Math.—1995.—Vol. 117.—P. 79-99.

21. Melikhov S. N., Momm S. Analytic solutions of convolution equations on convex sets with obstacle in the boundary // Math. Scand.—2000.—Vol. 86.—P. 293-319.

22. Momm S. Division problem in spaces of entire functions of finite order // Functional Analysis.—1993.— P. 435-457.

23. Momm S. The boundary behavior of extremal plurisubharmonic functions // Acta Math.—1994.— Vol. 172.—P. 51-75.

24. Momm S. Extremal plurisubharmonic functions for convex bodies in CN // Complex analysis, harmonic analysis and appl. / Deville R. (Bordeaux, France, 1995).—Harlow: Longman, Bitman.—1996.— Vol. 347.—P. 87-103.—(Res. Notes Math. Ser.)

25. Momm S. An extremal plurisubharmonic functions associated to a convex pluricomplex Green function with pole at infinity // J. Reine Angew. Math.—1996.—Vol. 471.—P.139-163.

26. Schneider R. Convex bodies: The Brunn-Minkowski theory.—Cambridge: Cambridge University Press, 1993.—490 p.

Received February 1, 2010.

Melikhov Sergej Nikolaevich

Southern Mathematical Institute

Vladikavkaz Science Center of the RAS,

chief of complex analysis laboratory

RUSSIA, 362027, Vladikavkaz, Markus street, 22;

Southern Federal University

Department of Mathematics, Mechanics and Computer Science RUSSIA, 344090 Rostov on Don, 344090, Mil'chakova street, 8 a E-mail: [email protected]

Momm Siegfried Mathematisches Institut Heinrich-Heine Universitat Düsseldorf GERMANY, 40225, Düsseldorf, Universitütsstrasse, 1 E-mail: [email protected]

О РАЗЛОЖЕНИИ В РЯДЫ ЭКСПОНЕНТ ФУНКЦИЙ, АНАЛИТИЧЕСКИХ НА ВЫПУКЛЫХ ЛОКАЛЬНО ЗАМКНУТЫХ МНОЖЕСТВАХ

Мелихов С. Н., Момм З.

Пусть Q — ограниченное, выпуклое, локально замкнутое подмножество С^ с непустой внутренностью. Для N > 1 получены достаточные условия того, что оператор представления рядами экспонент функций, аналитических на Q, имеет линейный непрерывный правый обратный. Для N =1 доказаны критерии существования линейного непрерывного правого обратного к оператору представления.

Ключевые слова: локально замкнутое множество, аналитические функции, ряды экспонент, линейный непрерывный правый обратный.