Let $\mathcal{E}'(\mathbb{R})$ be the space of all compactly supported distributions on $\mathbb{R}$.

For $f\in \mathcal{E}'(\mathbb{R})$, let $\widehat{f}$ denote the entire extension of the Fourier transform of $f$.

Question: If $f_n\stackrel{n\rightarrow\infty}{\longrightarrow}f$ in $\mathcal{E}'(\mathbb{R})$, then does $\widehat{f}_n$ converge to $\widehat{f}$ uniformly on compact subsets of $\mathbb{C}$ as $n\rightarrow\infty$?