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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 ε>0

1 n+1|kn:| x k -L|ε|0asn·

Let σ n = k=0 n x k /(n+1) denote the (C,1)-transform of the sequence. Now, consider a sequence (x k ) such that (σ 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)].

40E05Tauberian theorems, general
40G05Cesàro, Euler, Nörlund and Hausdorff methods
Cesàro means