Continuous mean periodic extension of functions from an interval

We study the following version of the mean periodic extension problem.
(i) Suppose that $T\in\mathscr{E}'(\mathbb{R}^n)$, $n\ge2$, and $E$ is a nonempty closed subset of $\mathbb{R}^n$. What conditions guarantee that, for a function $f\in C(E)$, 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 $\mathbb{R}^n$.