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 Probl. Anal. Issues Anal., 2020, Volume 9(27), Issue 2, Pages 138–151 (Mi pa301)

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.

DOI: https://doi.org/10.15393/j3.art.2020.8630

Full text: PDF file (457 kB)
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Bibliographic databases:

UDC: 517.444
MSC: 44A35, 45E10, 46F10
Revised: 23.05.2020
Accepted:23.05.2020
Language:

Citation: V. V. Volchkov, Vit. V. Volchkov, “On the problem of mean periodic extension”, Probl. Anal. Issues Anal., 9(27):2 (2020), 138–151

Citation in format AMSBIB
\Bibitem{VolVol20}
\by V.~V.~Volchkov, Vit.~V.~Volchkov
\paper On the problem of mean periodic extension
\jour Probl. Anal. Issues Anal.
\yr 2020
\vol 9(27)
\issue 2
\pages 138--151
\mathnet{http://mi.mathnet.ru/pa301}
\crossref{https://doi.org/10.15393/j3.art.2020.8630}
\elib{https://elibrary.ru/item.asp?id=45452312}