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 mn st → s to s mn → s. We also prove that they are Tauberian conditions from t 11 mn st → s to s mn → s, where t 11 mn are the weighted means of the double sequence {s mn } ∞ 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),

On Tauberian theorems for statistical weighted mean method of summability

“…The following theorem generalizes Theorem 3.4 given by Chen and Chang [4]. Let (p n ) satisfy the conditions (9), (12) and (13).…”

“…Later, Chen and Chang [4] generalized Móricz and Orhan's Tauberian theorem in the following theorem.…”

Tauberian theorems for the statistical convergence and the statistical (C,1,1) summability

“…Briefly, in the space of bounded sequences, the implications Pconvergence ⇒ statistical convergence ⇒ statistical (C, 1, 1) summable (14) are satisfied. Instead of using the conditions (10) and (11) as Tauberian conditions, Chen and Chang [6] recovered Pconvergence of a double sequence from its statistical (C, 1, 1) summability if it is slowly decreasing in senses (1, 0) and (0, 1), in addition, in the strong sense with respect to one of the types. Under weaker conditions, Móricz [16] obtained statistical convergence of a double sequence from its statistical (C, 1, 1) summability.…”

Some Tauberian conditions obtained through weighted generator sequences

“…Móricz and Orhan [10] proved the necessary and su cient Tauberian conditions under which (1.1) follows from (1.5). There are also some interesting studies related to Tauberian theorems in which statistical convergence is used (see [2,3,5]). …”

“…Case > 1. Since ( ) is statistically ( , )-summable to , we have st − lim (1) , = , and, by Lemma 2.1, we have st − lim(2) , = . It follows from the weighted Kronecker identity that st − lim(1) , (Δ ) So…”