ONE-SIDED TAUBERIAN THEOREMS
FOR DIRICHLET SERIES
METHODS OF SUMMABILITY

DAVID BORWEIN, WERNER KRATZ
AND ULRICH STADTMÜLLER

Abstract:

We extend recently established two-sided or $O$-Tauberian results concerning the summability method $D_{\lambda,a}$ based on the Dirichlet series
$\sum a_n e^{- \lambda_nx}$ to one-sided Tauberian results. More precisely, we formulate one-sided Tauberian conditions, under which $D_{\lambda,a}$-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 $a_n$ and $\lambda_n$ 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 $D_{\lambda,a}$-means and the one-sided Tauberian conditions.