On the problem of mean periodic extension

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 $\mathbb{R}^n$.