I am not sure how to prove this property of Fourier transforms:

Two functions that have compact support and are continuous $ f, g \in C_c(\mathbb R)$ and therefore the Fourier transform of their convolution is infintely differentiable, $$ \widehat {f*g} \in C^\infty(\mathbb R).$$

I tried writing the whole integral down - but I think that I am missing a certain property of functions with compact support. Any tip/idea on how to approach this problem is welcome.

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    $\begingroup$ See the answer here for the basic result you're missing. $\endgroup$ – David C. Ullrich Jan 28 '18 at 20:37

By the Paley-Wiener theorem, the Fourier transform of a continuous and compact-supported function is much more than $C^\infty$: it is an entire function with finite order. The space of such functions is closed with respect to $\cdot$ and $\widehat{f*g}=\widehat{f}\cdot \widehat{g}$.


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