Laplace transform representations and Paley-Wiener theorems for functions on vertical strips

Abstract

We consider the problem of representing an analytic function on a vertical strip by a bilateral Laplace transform. We give a Paley-Wiener theorem for weighted Bergman spaces on the existence of such representa- tions, with applications. We generalise a result of Batty an d Blake, on ab- scissae of convergence and boundedness of analytic functions on halfplanes, and also consider harmonic functions. We consider analytic continuations of Laplace transforms, and uniqueness questions: if an analytic function is the Laplace transform of functions f1, f2 on two disjoint vertical strips, and extends analytically between the strips, when is f1 = f2? We show that this is related to the uniqueness of the Cauchy problem for the heat equation with complex space variable, and give some applications, including a new proof of a Maximum Principle for harmonic functions.