Representation of mean-periodic functions in series of exponential polynomials

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