We revisit the classical problem of when a given function, which is analytic in the upper half plane C+, can be written as the Fourier transform of a function or distribution with support on a half axis (−∞,b], b∈R. We derive slight improvements of the classical Paley-Wiener-Schwartz Theorem, as well as softer conditions for verifying membership in classical function spaces such as Hp(C+).
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