Expansion in series of exponential polynomials of mean-periodic functions with growth conditions

Ouerdiane, Habib; Ounaies, Myriam

Comptes Rendus. Mathématique, Tome 346 (2008) no. 9-10, pp. 509-514.

Soit θ une fonction de Young. Considérons l'espace ${\mathcal{F}}_{\theta }(\mathbb{C})$ de toutes les fonctions entières sur $\mathbb{C}$ à croissance θ-exponentielle. On s'intéresse dans cette Note aux solutions $f\in {\mathcal{F}}_{\theta }(\mathbb{C})$ de l'équation de convolution $T\star f=0$, appelées fonctions T-moyenne-périodiques, où T est dans le dual topologique de ${\mathcal{F}}_{\theta }(\mathbb{C})$. On montre que toute fonction moyenne-périodique admet un développement en série de polynômes exponentiels. De plus cette série est convergente pour la topologie de l'espace ${\mathcal{F}}_{\theta }(\mathbb{C})$.

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

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