Clarification: I have seen in quantum mechanics many examples of wavefunctions and their Fourier transforms. I understand that a square pulse has a Fourier transform which is nonzero on an infinite interval. I am curious to know whether there exists any function which is nonzero on only a finite interval, whose Fourier transform is also nonzero on only a finite interval. Is it impossible? I have not been able to find any theorem about it. If an example exists I would be very interested in seeing it because I think it would make a great wavefunction to think about.