On the problem of mean periodic extension

V. V. Volchkov, Vit. V. Volchkov

Donetsk National University, 24 Universitetskaya str., Donetsk 283001, Russia

Abstract: 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 $\mathbb{R}^n$.

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

UDC: 517.444

MSC: 44A35, 45E10, 46F10

Received: 14.03.2020
Revised: 23.05.2020
Accepted: 23.05.2020

Language: English

DOI: 10.15393/j3.art.2020.8630

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