On sufficient sets in spaces of entire functions of several variables
A. B. Sekerin
Abstract:
The main result is
Theorem 1. {\it Let
$D$ be a bounded convex domain in
$\mathbf C^n,$ $n\geqslant2,$ with
$0\in D$. Let $H(z)=\max_{\lambda\in\overline D}\mathbf{Re}\langle\lambda,z\rangle$. Let
$L(z)$ be an entire function of exponential type whose zero set
$S$ is the union of planes
$P_m=\{z:\langle a_m,z\rangle=c_m\},$ $m\in\mathbf N,$ $|a_m|=1$. Suppose the following conditions hold}:
a) {\it there exist constants
$c,$ $r_0,$ $d_0,$ $\gamma\in(0,1),$ such that the estimate
$$
\left|\ln|L(z)|-H(z)\right|\leqslant c\left|\ln d\right||z|^{1-\gamma}
$$
holds if the point
$z\in\mathbf C^n,$ satisfies
$|z|\geqslant r_0,$ $\inf_{w\in S}|z-w|=d(z,S)\geqslant d>0,$ $d<d_0$};
b) {\it for every
$m$ the restriction of the entire function
$(\langle a_m,z\rangle-c_m)^{-1}L(z)$ to the plane
$P_m$ is not identically zero};
c) {\it there exist constants
$c$ and
$N$ such that for
$m\ne k$ either
$d(P_m,P_k)\geqslant c|c_m|^{-N}|c_k|^{-N}$ or $1-|\langle a_m,\overline a_k\rangle|\geqslant c|c_m|^{-N}|c_k|^{-N}$.
Then every analytic function
$f(z)$ in the domain
$D$ can be represented by a series
$$
f(z)=\sum_{m=1}^\infty\int_{P_m}\exp\langle\lambda,z\rangle\,d\mu_m(\lambda)
$$
converging in the topology of
$H(D)$.}
Bibliography: 11 titles.
UDC:
517.537
MSC: Primary
32A15; Secondary
32A30,
30A50 Received: 27.06.1987
Fulltext:
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