One-sided Tauberian theorems for Dirichlet series methods of summability

We extend recently established two-sided or O-Tauberian results concerning the summability method D-lambda,D-a based on the Dirichlet series Sigma a(n)e(-lambda)n(x) to one-sided Tauberian results. More precisely, we formulate one-sided Tauberian conditions, under which D-lambda,D-a-summability implies convergence. Our theorems contain various known results on power series methods of summability and, in the so-called high index case we even obtain a new result for such methods. Our method of proof uses asymptotic properties of the Dirichlet series subject to the assumption that an and an can be interpolated by smooth functions. In addition we develop refined Vijayaraghavan-type results which enable us to infer the boundedness of sequences from the boundedness of their D-lambda,D-a-means and the one-sided Tauberian conditions.