If $f\in L^{1}(\mathbb{R}^n)$, and $f$ has compact support, then how to show that $\hat{f}$ cannot satisfy $$ \hat{f}(x)=O(e^{-\epsilon|x|}) $$ for any $\epsilon>0$?

This is similar in the spirit of the Paley-Wiener-Schwartz theorem, which gives a characterization of a smooth function with compact support of a distribution with compact support in terms of the decay property of their Fourier transform. So similarly, I'm also interested to know if there is a characterization for $L^p$ function with compact support.