On the problem of mean periodic extension

V. V. Volchkov ; Vit. V. Volchkov

Problemy analiza, Tome 9 (2020) no. 2, pp. 138-151

Résumé

This paper is devoted to a study of the following version of the mean periodic extension problem: (i) Suppose that $T\in\mathcal{E}'(\mathbb{R}^n)$, $n\geq 2$, and $E$ is a non-empty subset of $\mathbb{R}^n$. Let $f\in C(E)$. What conditions guarantee that there is an $F\in C(\mathbb{R}^n)$ coinciding with $f$ on $E$, such that $F\ast T=0$ in $\mathbb{R}^n$? (ii) If such an extension $F$ exists, then estimate the growth of $F$ at infinity. In this paper, we present a solution of this problem for a broad class of distributions $T$ in the case when $E$ is a segment in $\mathbb{R}^n$.

Détail

Keywords: mean periodicity, continuous extension, spherical transform.
Mots-clés : convolution equation