Representation of mean-periodic functions in series of exponential polynomials

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.