Representation of analytic functions in a closed polydomain by Dirichlet series (Q1117358)

In the paper, a generalization of the results of \textit{A. F. Leont'ev} [Izv. Akad. Nauk SSSR, Ser. Mat. 37, 577-592 (1973; Zbl 0275.30004)] on representations of analytic functions in closed convex domains in \({\mathbb{C}}\) by Dirichlet series to the case of several complex variables is given. There are given necessary and sufficient conditions in order that, for any analytic function f in a polydomain \(D=D_ 1\times...\times D_ n\subset {\mathbb{C}}^ n,\) where \(D_ i\) are convex domains in the complex plane, and having the \(C^ 2\)-extension to \(\bar D\) with respect to each variable, the Dirichlet series corresponding to f (depending on some auxiliary entire functions connected with D) be absolutely convergent in \(\bar D,\) uniformly convergent in \(\bar D,\) convergent to f in \(\bar D,\) respectively. The conditions are given in terms of the above entire functions connected with D.