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\(\lambda \)-statistical convergence. (English) Zbl 0953.40002
This paper deals with a generalization of the Cesàro mean summability of the sequences \[ [C,1]:=\{x=(x_n): \text{ there is } L \in \mathbb{R}\text{ such that } \lim _{n \to \infty }{1 \over n} \sum _{k=1}^n {}x_k - L{}=0 \} \] to \[ [V,\lambda ]:= \{x=(x_n): \text{ there is } L \in \mathbb{R} \text{ such that } \lim_{n \to \infty }{1 \over \lambda _n} \sum _{k \in I_n} {}x_k - L{}=0{}\} \] 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.

MSC:
40A05 Convergence and divergence of series and sequences
40C05 Matrix methods for summability
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