Representation of analytic functions by exponential series in half-plane with given growth majorant

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. In the theory of exponential series one of fundamental results is the following general result by A.F. Leontiev: 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. 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 vicinity of zero. In order to do this, we employ the methods of estimating based on the Legendre transform. We prove a statement which generalizes the corresponding result by A.M. Gaisin on expanding analytic in half-plane functions into exponential series taking into consideration the growth order.