Exponential series in normed spaces of analytic functions

Abstract

There is a classical well-known theorem by A.F. Leontiev on representing functions analytic in a convex domain $D$ and continuous up to the boundary by series of form $\sum_{k=1}^\infty f_ke^{\lambda_kz}$ converging in the topology of the space $H(D)$, that is, uniformly on compact subsets in $D$. In the paper we prove the possibility of representing the functions in \begin{equation*} A_0(D)=\left \{f\in H(D)\bigcap C(\overline D):\ \|f \|:=\sup_{z\in \overline D}|f(z)|\right \} \end{equation*} by the exponential series converging in a stronger topology, namely, there exists an integer number $s>0$ such that 1) for each bounded convex domain $D$ there exists a system of exponentials $e^{\lambda_kz},$ ${k\in \mathbb{N}}$, such that each function $f\in H(D)\bigcap C^{(s)}(\overline D)$ is represented as a series over this system converging in the norm of the space $A_0(D)$; 2) for each bounded convex domain $D$ there exists a system of exponentials $e^{\lambda_kz},$ ${ k\in \mathbb{N}}$ such that each function $f\in A_0(D)$ is represented as a series over this system converging in the norm \begin{equation*} \|f\| = \sup_{z\in D}|f(z)|(d(z))^s, \end{equation*} where $d(z)$ is the distance from a point $z$ to the boundary of the domain $D$. The number $s$ is related with the existence of entire functions with a maximal possible asymptotic estimate. In particular cases, when $D$ is a polygon or a domina with a smooth boundary possessing a smooth curvature separated from zero, we can assume that $s=4$.