I was going through various posts about differrentiability of convolutions. What I would like to ask is:
Suppose $f \in C^{1}(\mathbb{R})$. Then what conditions on the function $g$ would ensure that their convolution is smooth? I think that $g \in L^{1}$ and $g$ having compact support are sufficient? Am I right?
Also, can I relax the condition of compact support and put some other conditions, say $g \in C^{1}(\mathbb{R})$?
I would also appreciate if you could provide some references where I could find about these things.