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

Abstract

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 s m n → s t s to s m n → s . We also prove that they are Tauberian conditions from t m n 11 → s t s to s m n → s , where t m n 11 are the weighted means of the double sequence { s m n } 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].