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Tauberian conditions, under which statistical convergence follows from statistical summability \((C,1)\). (English) Zbl 1021.40002

Let a real or complex sequence \((x_k)\) be given. We say that it converges statistically to some limit \(L\) if for all \(\varepsilon >0\) \[ {1\over n+1} |\bigl\{k\leq n:|x_k-L|\geq \varepsilon \bigr\} |\to 0\quad \text{as}\quad n\to\infty. \] Let \(\sigma_n= \sum^n_{k=0} x_k/ (n+1)\) denote the \((C,1)\)-transform of the sequence. Now, consider a sequence \((x_k)\) such that \((\sigma_n)\) is statistically convergent. The main results in this paper give necessary and sufficient one- and two-sided ‘statistical’ oscillation conditions which imply statistical convergence of the sequence. The results immediately show that ‘statistical’ slow oscillation or ‘statistical’ slow decrease are Tauberian conditions from statistical \((C,1)\)-convergence to statistical convergence. These Tauberian conditions hold under ordinary slow oscillation or slow decrease, which in turn are Tauberian conditions from statistical convergence to ordinary convergence; see J. A. Fridy and M. K. Khan [Proc. Am. Math. Soc. 128, 2347-2355 (2000; Zbl 0939.40002)].

MSC:

40E05 Tauberian theorems
40G05 Cesàro, Euler, Nörlund and Hausdorff methods

Keywords:

Cesàro means

Citations:

Zbl 0939.40002
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References:

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