International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 65, Pages 3499-3511

Summability of double sequences by weighted mean methods and Tauberian conditions for convergence in Pringsheim's sense

Ferenc Móricz1 and U. Stadtmüller2

1Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged 6720, Hungary
2Abteilung Zahlentheorie und Wahrscheinlichkeitstheorie, Fakultät für Mathematik und Wirtschaftswissenschaften, Universität Ulm, Ulm 89069, Germany

Received 16 March 2004

Copyright © 2004 Ferenc Móricz and U. Stadtmüller. 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.


After a brief summary of Tauberian conditions for ordinary sequences of numbers, we consider summability of double sequences of real or complex numbers by weighted mean methods which are not necessarily products of related weighted mean methods in one variable. Our goal is to obtain Tauberian conditions under which convergence of a double sequence follows from its summability, where convergence is understood in Pringsheim's sense. In the case of double sequences of real numbers, we present necessary and sufficient Tauberian conditions, which are so-called one-sided conditions. Corollaries allow these Tauberian conditions to be replaced by Schmidt-type slow decrease conditions. For double sequences of complex numbers, we present necessary and sufficient so-called two-sided Tauberian conditions. In particular, these conditions are satisfied if the summable double sequence is slowly oscillating.