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Generalized statistical convergence and statistical core of double sequences. (English) Zbl 1219.40004

λ-statistical convergence was introduced in [Mursaleen, Math. Slovaca, 50, No. 1, 111–115 (2000; Zbl 0953.40002)] for single sequences as follows:

Let λ=(λ n ) be a non-decreasing sequence of positive numbers tending to such that

λ n+1 λ n +1,λ 1 =0·

A double sequence x=x jk is said to be (λ,μ)-statistically convergent to l if δ λμ (E)=0, where E={jJ m ,kI n :|x jk -l|ε}, i.e., if for every ε>0,

(P)lim m,n 1 λ m μ n |{jJ m ,kI n :|x jk -l|ε}|=0·

In this case the authors write (st λ,μ )lim j,k x j,k =l and they denote the set of all (λ,μ)-statistically convergent double sequences by S λ,μ .

In this paper, they extended the notion of λ-statistical convergence to the (λ,μ)-statistical convergence for double sequences x=(x k ). They also determine some matrix transformations and establish some core theorems related to their new space of double sequences S λ,μ .

40A35Ideal and statistical convergence
40B05Multiple sequences and series
40C05Matrix methods in summability
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