Continuous Mean Periodic Extension of Functions from an Interval

Abstract

We study the following version of the mean periodic extension problem. (i) Suppose that $T \in \mathcal{E}'({{\mathbb{R}}^{n}})$, n ≥ 2, and E is a nonempty closed subset of ${{\mathbb{R}}^{n}}$. What conditions guarantee that, for a function f ∈ C(E), there is a function $F \in C({{\mathbb{R}}^{n}})$ coinciding with f on E such that $F * T = 0$ in ${{\mathbb{R}}^{n}}$? (ii) If such an extension F exists, then estimate the growth of F at infinity. We present a solution of this problem for a broad class of distributions T in the case when E is an interval in ${{\mathbb{R}}^{n}}$.