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Zbl 0588.40001
Fridy, J.A.
On statistical convergence. (English)
[J] Analysis 5, 301-313 (1985). ISSN 0174-4747

A sequence $\{x\sb k\}\sp{\infty}\sb{k=1}$ is said to be statistically convergent to L provided that the density of the set $\{$ $k\in N:$ $\vert x\sb K-L\vert \ge \epsilon \}$ is 0 for each $\epsilon >0$ (the density of the set $M\subset N$ is the number $\lim\sb{n\to \infty}M(n)/n$, where M(n) denotes the number of elements of M not exceeding n). The author gives an equivalent condition of Cauchy type for the statistical convergence. This convergence can be regarded as a regular summability method. This method cannot be included by any matrix summability method. Two Tauberian conditions are given for the statistical convergence. One of them is the following: $\Delta x\sb k=O(1/k)$.
[T.Šalát]

MSC 2000:: *40A05 Convergence of series and sequences
40D05 General summability theorems
40C05 Matrix methods in summability
40E05 Tauberian theorems, general

Keywords: sequence; statistically convergent; statistical convergence; regular summability method

Cited in: Zbl 1185.40005 Zbl 1175.26062 Zbl 1147.54015 Zbl 1174.60361 Zbl 1032.40001 Zbl 0835.43006

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