Sufficient Conditions for the Representability as a Laplace Transform of a Function

Abstract

When asking what class of functions can be represented by a power series, $\sum\limits_{v = 0}^\infty {{a_v}{z^v}}$, we obtain a simple answer: all those functions which are analytic on a circular disk centred at the origin. As an analogy to the above answer, one might expect that the class of functions which can be represented as L-transforms is composed of those functions which are analytic in right half-planes. However, merely a subset of all functions which are analytic in right half-planes constitutes, in fact, the class of functions which can be represented as L-transforms. Our aim is to delineate this subset. While investigating this problem, we shall have to distinguish between functions that can be represented as L-transforms of functions and those which can be represented as L-transforms of distributions. For the subset of those functions which can be represented as L-transforms of functions there exists no simple criterion in terms of the theory of analytic functions. Only sufficient conditions are known which describe merely a portion of the sought subset of representable functions. By contrast, the subset of functions which may be represented as L-transforms of distributions is characterized by a necessary and sufficient condition which is taken from the theory of functions. In this Chapter 28, we deal with the representability as L-transforms of functions.