  The Gaver-Stehfest algorithm for numerical inversion of Laplace transform was
developed in the late 1960s. Due to its simplicity and good performance it is
becoming increasingly more popular in such diverse areas as Geophysics,
Operations Research and Economics, Financial and Actuarial Mathematics,
Computational Physics and Chemistry. Despite the large number of applications
and numerical studies, this method has never been rigorously investigated. In
particular, it is not known whether the Gaver-Stehfest approximations converge
and what is the rate of convergence. In this paper we answer the first of these
two questions: We prove that the Gaver-Stehfest approximations converge for
functions of bounded variation and functions satisfying an analogue of Dini
criterion.
