Summary: 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 representations, with applications. We generalise a result of Batty and Blake, on abscissae 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 $f_1, f_2$ on two disjoint vertical strips, and extends analytically between the strips, when is $f_1=f_2$? 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.

44A10, 30E20, 31A10, 35K05

Laplace transform, Paley--Wiener theorem, heat equation, maximum principle, analytic continuation, Hardy spaces, Bergman spaces