International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 65, Pages 3499-3511
doi:10.1155/S0161171204403329
Summability of double sequences by weighted mean methods and
Tauberian conditions for convergence in Pringsheim's sense
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.
Abstract
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.