A Note on holomorphic functions and the fourier-laplace transform

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 H^p(ℂ₊).