On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind

Analytic expressions are derived for the eigenfunctions and eigenvalues of the Laplace transform and similar dilationally invariant integral equations of the first kind. Some generalised concepts of information theory are introduced to show how the use of these eigenfunctions enables the maximum possible amount of information to be obtained when solving the inverse problem numerically. These concepts also explain how the amount of information available depends on the level of noise in the calculation and on the structure of the particular integral kernel. Some numerical examples which illustrate these points are presented.