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Zbl 1219.40004
Mursaleen, Mohammad; Çakan, Celal; Mohiuddine, Syed Abdul; Savaş, Ekrem
Generalized statistical convergence and statistical core of double sequences.
(English)
[J] Acta Math. Sin., Engl. Ser. 26, No. 11, 2131-2144 (2010). ISSN 1439-8516; ISSN 1439-7617/e

$\lambda$-statistical convergence was introduced in [{\it Mursaleen}, Math. Slovaca, 50, No. 1, 111--115 (2000; Zbl 0953.40002)] for single sequences as follows: Let $\lambda=(\lambda_n)$ be a non-decreasing sequence of positive numbers tending to $\infty$ such that $$\lambda_{n+1}\leq \lambda_n+1, \qquad \lambda_1=0.$$ A double sequence $x=x_{jk}$ is said to be $(\lambda, \mu)$-statistically convergent to $l$ if $\delta_{\lambda\mu}(E)=0$, where $E=\{j\in J_m, k\in I_n:|x_{jk}-l|\geq\varepsilon\}$, i.e., if for every $\varepsilon>0$, $$(P)\lim_{m,n} \frac{1}{\lambda_m\mu_n} \big|\{j\in J_m,\ k\in I_n:|x_{jk}-l|\geq\varepsilon\}\big|=0.$$ In this case the authors write $(st_{\lambda,\mu})\lim_{j,k}x_{j,k}=l$ and they denote the set of all $(\lambda, \mu)$-statistically convergent double sequences by $S_{\lambda,\mu}$. In this paper, they extended the notion of $\lambda$-statistical convergence to the $(\lambda, \mu)$-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 $S_{\lambda,\mu}$.
[Yilmaz Erdem (Aydin)]
MSC 2000:
*40A35
40B05 Multiple sequences and series
40C05 Matrix methods in summability

Keywords: double sequence; statistical convergence; matrix transformation; core

Citations: Zbl 0953.40002

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