On the representation of analytic functions by series of exponentials in a~polycylindrical domain

A. F. Leont'ev

Sbornik. Mathematics, Tome 29 (1976) no. 3, pp. 327-344

Résumé

We prove the following Theorem. {\it Let $D_p$ $(1\leqslant p\leqslant m)$ be a finite convex domain in the plane of the complex variable $z_p$, let $K_p(\varphi)$ be the support function of the domain $D_p$, and let $h_p(\varphi)=K_p(-\varphi)$. Then there exists a sequence of exponents $\{\lambda^{(p)}_k\}_{k=1}^\infty$ $($where the $\lambda^{(p)}_k$ $(k=1,2,\dots)$ are the zeros of an entire function $L_p(\lambda)$ of completely regular growth with indicator function $h_p(\varphi))$ such that any function $f(z_1,\dots,z_m)$ analytic in the domain $D=D_1\times\dots\times D_m$ can be represented in $D$ by the series $$ f(z_1,\dots,z_m)=\sum^\infty_{k_1,\dots,k_m=1}a_{k_1,\dots,k_m}\exp\bigl\{\lambda^{(1)}_{k_1}z_1+\dots+ \lambda^{(m)}_{k_m}z_m\bigr\}, $$ which is absolutely convergent in $D$ and uniformly convergent inside $D$.} For the case $m=1$ the theorem has been proved earlier (RZhMat., 1970, 10B132). Bibliography: 5 titles.