On sufficient sets in spaces of entire functions of several variables

A. B. Sekerin

Sbornik. Mathematics, Tome 64 (1989) no. 1, pp. 263-276

Résumé

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 $|z|\geqslant r_0,$ $\inf_{w\in S}|z-w|=d(z,S)\geqslant d>0,$ $d$}; 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.