Tauberian Theorems in the Statistical Sense for the Weighted Means of Double Sequences

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

“…We write s mn → s or lim m,n→∞ s mn = s, if s mn → s as min(m, n) → ∞. Following [2] and [9], we say that {s mn } ∞ m,n=0 is statistically convergent to s, in symbols, s mn st → s, if the following equality holds for each > 0: where | · | denotes the cardinality of a given set. We write s mn → s (N, p, q; 1, 1) if t 11 mn → s. We also write s mn st → s (N, p, q; 1, 1) for the case that t 11 mn st → s.…”

“…[1,4,8]). A parallel theory to [4] for the statistical convergence was recently established by the authors (see [2]). In [9, p. 344], Móricz considered the (C, 1, 1) summability, that is, the case p j = q k = 1.…”

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]). …”

On M λ m ,

“…In this case, we write st 2 − limx � L. By st 2 , we denote the set of all statistically convergent double sequences. Later, a lot of works have been done on the statistical convergence of double sequences (see [27][28][29][30][31][32][33]). e following definition which is required for our study is given by Natarajan.…”