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International Journal of Mathematics and Mathematical Sciences
Volume 27 (2001), Issue 7, Pages 399-406
http://dx.doi.org/10.1155/S0161171201006688

Necessary and sufficient conditions under which convergence follows from summability by weighted means

1University of Szeged, Aradi vértanúk tere 1, Szeged 6720, Hungary
2Universität Ulm, Abt. Math. III, Ulm D-89069, Germany

Received 7 February 2001

Copyright © 2001 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove necessary and sufficient Tauberian conditions for sequences summable by weighted mean methods. The main results of this paper apply to all weighted mean methods and unify the results known in the literature for particular methods. Among others, the conditions in our theorems are easy consequences of the slowly decreasing condition for real numbers, or slowly oscillating condition for complex numbers. Therefore, practically all classical (one-sided as well as two-sided) Tauberian conditions for weighted mean methods are corollaries of our two main theorems.