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 Mat. Sb. (N.S.), 1988, Volume 136(178), Number 2(6), Pages 260–273 (Mi msb1740)

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 $P_m=ż:\langle a_m,z\rangle=c_m\},$ $m\in\mathbf N,$ $|a_m|=1$. Suppose the following conditions hold}:
a) {\it there exist constants $c,$ $r_0,$ $d_0,$ $\gamma\in(0,1),$ such that the estimate
$$|\ln|L(z)|-H(z)|\leqslant c|\ln d||z|^{1-\gamma}$$
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.

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English version:
Mathematics of the USSR-Sbornik, 1989, 64:1, 263–276

Bibliographic databases:

UDC: 517.537
MSC: Primary 32A15; Secondary 32A30, 30A50

Citation: A. B. Sekerin, “On sufficient sets in spaces of entire functions of several variables”, Mat. Sb. (N.S.), 136(178):2(6) (1988), 260–273; Math. USSR-Sb., 64:1 (1989), 263–276

Citation in format AMSBIB
\Bibitem{Sek88} \by A.~B.~Sekerin \paper On sufficient sets in spaces of entire functions of several variables \jour Mat. Sb. (N.S.) \yr 1988 \vol 136(178) \issue 2(6) \pages 260--273 \mathnet{http://mi.mathnet.ru/msb1740} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=954928} \zmath{https://zbmath.org/?q=an:0668.32004|0651.32002} \transl \jour Math. USSR-Sb. \yr 1989 \vol 64 \issue 1 \pages 263--276 \crossref{https://doi.org/10.1070/SM1989v064n01ABEH003306}