Representation of mean-periodic functions in series of exponential polynomials

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.