On~the question of representing entire functions by exponential series

A. F. Leont'ev

Sbornik. Mathematics, Tome 33 (1977) no. 3, pp. 327-342

Résumé

It was established by the author (RZhMat., 1966, 4B 107) that any entire function $F(z)$ of finite order may be represented in the whole plane by a Dirichlet series $$ F(z)=\sum_{k=1}^\infty A_ke^{|\lambda_k|z}. $$ It is established that for suitable choice of the sequence $\{\lambda_k\}$ the expression $\sum_{k=1}^\infty|A_k|e^{|\lambda_k|r}$ has, for large $r$, the upper bounds 1) $\exp r^{\rho+\varepsilon}$ $\forall\,\varepsilon>0$, if $F(z)$ has order $\rho>1$; 2) $\exp(\sigma+\varepsilon)r^\rho$ $\forall\,\varepsilon>0$, if $F(z)$ has order $\rho>1$ and finite type $\sigma$. Bibliography: 7 titles.