Representation of mean-periodic functions in series of exponential polynomials

TL;DR

This paper studies mean-periodic functions within a space of entire functions of exponential type, demonstrating that they can be explicitly represented as convergent series of exponential polynomials.

Contribution

It provides an explicit series representation for mean-periodic functions in the context of entire functions with exponential growth.

Findings

Mean-periodic functions can be expressed as convergent series of exponential polynomials.

The paper characterizes solutions to convolution equations in the space of entire functions with exponential growth.

Abstract

Let $\theta$ be a Young function and consider the space $\mathcal{F}_{\theta}(\C)$ of all entire functions with $\theta$-exponential growth. In this paper, we are interested in the solutions $f\in \mathcal{F}_{\theta}(\C)$ of the convolution equation $T\star f=0$, called mean-periodic functions, where $T$ is in the topological dual of $\mathcal{F}_{\theta}(\C)$. We show that each mean-periodic function can be represented in an explicit way as a convergent series of exponential polynomials.