Does the Fourier transform of a function with compact support itself have compact support?
I am reading many books on the Fourier transform. They all state that the Fourier transform is not a mapping from test function space ( functions of compact support) to the test function space. In one of the books I read, they explain that if is a test function, and F( ) is a test function, then is 0. Does anyone know how to start on proving this? Or can anyone give me a test function (by above, any non zero one will do!) that I can compute the FT of? I have tried, and end up with having to integrate something that even Mathematica can't do.
By one of the Paley–Wiener theorems, if a square-integrable function is compactly supported then its Fourier transform is the restriction to the real axis of an entire function. But if such a function has compact support then it must be identically zero, because a nonzero entire function can only have isolated zeros.