The Laplace Transform of Generalized Functions

Abstract

In this paper we develop a theory of Laplace transforms for generalized functions. Some fundamental results in this field were given by Schwartz in (3) for the n-dimensional bilateral case from the point of view of topological vector spaces, and in (4) in a form amenable to operational use. Our presentation characterizes a one-dimensional theory of Laplace transforms with a half-plane of convergence (indicating an extension of the usual classical transform) and with the property that Laplace transforms are analytic functions satisfying the fundamental convolution-multiplication theorem. Section 1 is devoted to defining the Laplace transform of generalized functions and also to showing how the property of a half-plane of convergence is intrinsic to this definition.