On conditions for representability of entire functions by certain general series

A. F. Leont'ev ; Yu. N. Frolov

Izvestiya. Mathematics , Tome 13 (1979) no. 1, pp. 63-72

Résumé

Let $f(z)$ be an entire function of order $\rho$ and $L(\lambda)$ an entire function of order $\rho_1>\rho$ with simple zeros $\lambda_1,\dots,\lambda_k,\dots$ . A series $\sum_1^\infty\alpha_kf(\lambda_kz)$ is assigned (according to a specific rule) to an arbitrary entire function $F(z)$ of order $\nu\frac{\rho\rho_1}{\rho_1-\rho}$. Necessary and sufficient conditions on $L(\lambda)$ are found under which this series always converges to $F(z)$ in some topology. Bibliography: 5 titles.