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The Laplace Transform of Generalized Functions

Published online by Cambridge University Press:  20 November 2018

John Benedetto
John Benedetto*
Affiliation:
University of Torontoand New York University
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 357 - 374
Copyright
Copyright © Canadian Mathematical Society 1966

References

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6. Widder, D. V., Necessary and sufficient conditions for the representation of a function as a Laplace integral, Trans. Amer. Math. Soc., 33, 851–892.Google Scholar