Laplace Transforms and Generalized Laguerre Polynomials

Abstract

Various sets of necessary and sufficient conditions are known in order that a function ƒ(s), analytic for Re s > 0, be represented as the Laplace transform of a function in L _p(0,∞), 1 < p ⩽ ∞ . Most of these theories are based on the properties of some inversion operator for the transformation— see, for example, (7, chap. 7). However in the case p = 2 a number of representation theorems of a much simpler type are available.