Does there exist two functions $f,g \in L^1(\mathbb R)$, with $f,g \neq 0$, such that $\operatorname{supp}(f)\subset[0,\infty[ $ and $ \operatorname{supp}(g) \subset[0,\infty[ $ and $f\ast g =0$ ?
I've done similar exercises where taking the Fourier transform gives a quick solution but I don't know how to make it work for that one.