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Zbl 0953.40002
Mursaleen
$\lambda$-statistical convergence.
(English)
[J] Math. Slovaca 50, No.1, 111-115 (2000). ISSN 0139-9918; ISSN 1337-2211/e

This paper deals with a generalization of the Cesàro mean summability of the sequences $$[C,1]:=\{x=(x_n): \text{ there is } L \in \bbfR\text{ such that } \lim _{n \to \infty }{1 \over n} \sum _{k=1}^n {}x_k - L{}=0 \}$$ to $$[V,\lambda ]:= \{x=(x_n): \text{ there is } L \in \bbfR \text{ such that } \lim_{n \to \infty }{1 \over \lambda _n} \sum _{k \in I_n} {}x_k - L{}=0{}\}$$ for some interval $I_n$. Comparisons of statistical convergence and $\lambda$-statistical convergence for a sequence $x=(x_n)$ defined by using limits $$\lim _{n \to \infty }{1 \over n} {}\{ k \leq n : {}x_n - L{} \geq \varepsilon \}=0 \quad \text{and}\quad\lim _{n \to \infty }{1 \over \lambda _n} {}\{k \in I_n : {}x_n - L{} \geq \varepsilon \}= 0$$ are given.
[Ondrej Kováčik (Žilina)]
MSC 2000:
*40A05 Convergence of series and sequences
40C05 Matrix methods in summability

Keywords: statistical convergence; summability of sequences

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