ON THE PROBLEM OF MEAN PERIODIC EXTENSION

Abstract

This paper is devoted to a study of the following version of the mean periodic extension problem: (i) Suppose that 𝑇 ∈ ℰ′ (R^𝑛), 𝑛 ≥ 2, and 𝐸 is a non-empty subset of R^𝑛. Let 𝑓 ∈ 𝐶(𝐸). What conditions guarantee that there is an 𝐹 ∈ 𝐶(R^𝑛) coinciding with 𝑓 on 𝐸, such that 𝐹 *𝑇 = 0 in 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 R^n.

Keywords

• convolution equation
• mean periodicity
• continuous extension
• spherical transform