The domain of convergence of series of generalized exponentials

A. F. Leont'ev

Sbornik. Mathematics, Tome 51 (1985) no. 1, pp. 1-8

Résumé

Let $f(z)$ be an entire function of exponential type and of completely regular growth, let $\gamma(t)$ be the Borel transform of $f(z)$, let $\overline D$ be the smallest closed convex set containing all the singular points of $\gamma(t)$, with $\overline D\ne\{0\}$, and let $\{\lambda_n\}$ be a sequence of complex numbers such that $$ \lim_{n\to\infty}\frac{\ln n}{\lambda_n}=0. $$ We ask for the domain of convergence of the series \begin{equation} \sum_{n=1}^\infty A_nf(\lambda_nz). \end{equation} Let $G$ be the open set in which (1) converges uniformly. It is proved that 1) if $0\not\in\partial\overline D$ then $G$ is convex, and 2) if $0\in\overline D$ and $0\in G$, then $G$ is also convex. Generally speaking, $G$ cannot be an arbitrary convex set. It is shown that $G$ can be an arbitrary convex set with $0\in\overline G$, if and only if the singular points of $\gamma(t)$ all lie on a line segment with one end at the origin. Bibliography: 2 titles.