It is well known that the Fourier transform is well defined on the class of tempered distributions on the real line. It is easily seen that the Fourier transform of a tempered distribution, whose support lies on the positive semiaxis, defines a holomorphic function in the upper halfplane of the complex plane. My question is as follows:
Given a holomorphic function on the upper halfplane, how to decide whether it is the Fourier transform of a tempered distribution supported on the positive semiaxis?
I am sure the answer is well known, but I wasn't able to find it.