Tauberian conditions under which the original convergence of double sequences follows from the statistical convergence of their weighted means

In this paper, we introduce a new type of slow oscillation and slow decrease conditions. We prove that these or their variants are Tauberian conditions from smn st →s to smn →s. We also prove that they are Tauberian conditions from t11 mn st →s to smn→s, where t11 mn are the weighted means of the double sequence {smn}∞m,n=0. Our results not only generalize well-known results, but also solve the conjecture of Móricz posed in [F. Móricz, Tauberian theorems for double sequences that are statistically summable (C, 1, 1), J. Math. Anal. Appl. 286 (2003) 340–350]. © 2006 Elsevier Inc. All rights reserved