We know that the range oh Fourier transform $\mathcal{F}{L^1}$ is dense in $C_0$.
I would like to know if we can have something more precise :
let $f \in C_0$, is there exist a sequence $\phi_n$ such that for all $n$, $f \star \phi_n \in \mathcal{F}(L^1)$ and $f \star \phi_n \rightarrow f$ in $C_0$ ? ( here $f \star \phi_n $ is the convolution)
This previous fact is true when $f \in C_0\cap L^1$ because we can take something like $\phi_n(x) = n\phi(nx)$ where $\phi(x) = e^{-\pi x^2}$. And then it is easy to show that $f \star \phi_n \in \mathcal{F}L^1$ and $f \star \phi_n \rightarrow f$ in $C_0$ (since $\phi_n$ is an approximate identity).But I guess in this case we don't know if $f \star \phi_n \in \mathcal{F}L^1$ when $f\in C_0$.