It's not hard to prove that both a function and its Fourier transform can't be supported compactly at the same time.
I was wondering if there is an example of a function from $C^{\infty}$ which is compactly supported and its Fourier transform can be actually calculated analytically. The only function I can imagine is a bump function $\psi(x)$ or something like this $f(x) * \psi(x)$ but I don't seem to be able to evaluate the transform of any of them.