@reruns The open problem is a characterization of the form $$f\in \mathcal{F}(L^1(\mathbb{R}^d))$$ if and only if ___________________. Yes, $\widehat{f}\in C_0$ for $f\in L^1$ but not all $f\in C_0$ are in the image of the FT when the domain is $L^1$ funtions.