TAUBERIAN-THEOREMS FOR THE LAPLACE-STIELTJES TRANSFORM

Let alpha: [0, infinity)-->C be a function of locally bounded variation, with alpha-(0) = 0, whose Laplace-Stieltjes transform g(z) = integral-0-infinity e(-zt) dalpha-(t) is absolutely convergent for Re z > 0. Let E be the singular set of g in iR, and suppose that O is-not-a-member-of E. Various estimates for lim sup(t-->infinity) \alpha-(t) - g(O)\ are obtained. In particular, alpha-(t)-->g(O) as t-->infinity if [GRAPHICS] This result contains Tauberian theorems for Laplace transforms, power series, and Dirichlet series.