Representation of Analytic Functions by Exponential Series in Half-Plane

Abstract

In this paper we study representations of analytic in the half-plane $\Pi_{0}=\{z=x+iy\colon\,x>0\}$ functions by the exponential series taking into consideration a given growth. A.F. Leontiev proved that for each bounded convex domain $D$ there exists a sequence $\{\lambda_{n}\}$ of complex numbers depending only on the given domain such that each function $F$ analytic in $D$ can be expanded into an exponential series $F(z)=\sum_{n=1}^{\infty}a_{n}e^{\lambda_{n}z}$ (the convergence of which is uniform on compact subsets of $D$). Later a similar results on expansions into exponential series, but taking into consideration the growth, was also obtained by A.F. Leontiev for the space of analytic functions of finite order in a convex polygon. He also showed that the series of absolute values $\sum_{n=1}^{\infty}\left|a_{n}e^{\lambda_{n}z}\right|$ admits the same upper bound as the initial function $F$. In 1982, this fact was extended to the half-plane $\Pi_{0}^{+}$ by A.M. Gaisin. However, the expansion in $\Pi_{0}^{+}$ contains an additional term — an entire function. In the present paper we study a similar case, when as a comparing function, some decreasing convex majorant serves and this majorant is unbounded in the neighborhood of zero. We have found out in which case the entire component from the expansion in the half-plane is bounded in the strip containing the imaginary axis.