I know that if $f(t) \in L^1(\mathbb{R}) \cap C^k(\mathbb{R})$ then we must have $\hat{f}(s)=o(s^{-k})$ for the Fourier transform. Is there some sort of converse to this statement?
Let $f \in L^1(\mathbb{R})$ be such that $\hat{f}=o(s^{-k})$, What can we conclude about the regularity of $f$?
I'm willing to replace $o(s^{-k})$ with $o(s^{-k-\epsilon})$ for some small $\epsilon > 0$ if that makes a big difference.