## 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

Cesàro means

Zbl 0939.40002
Full Text:

### References:

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