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On statistical convergence. (English) Zbl 0588.40001
A sequence {x k } k=1 is said to be statistically convergent to L provided that the density of the set { kN: |x K -L|ϵ} is 0 for each ϵ>0 (the density of the set MN is the number lim n 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: Δx k =O(1/k).
Reviewer: T.Šalát

MSC:
40A05Convergence and divergence of series and sequences
40D05General summability theorems
40C05Matrix methods in summability
40E05Tauberian theorems, general