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Tauberian conditions under which statistical convergence follows from statistical summability by weighted means. (English) Zbl 1063.40007

Summary: The first named author has recently proved necessary and sufficient Tauberian conditions under which statistical convergence follows from statistical summability (C,1). The aim of the present paper is to generalize these results to a large class of summability methods (N ¯,p) by weighted means.

Let p=(p k :k=0,1,2,) be a sequence of nonnegative numbers such that p 0 >0 and

P n := k=0 n p k asn·

Let (x k ) be a sequence of real or complex numbers and set t n :=P n -1 k=0 n p k x k for n=0,1,2,. We present necessary and sufficient conditions under which the existence of the limit st-limx k =L follows that of st-limt n =L, where L is a finite number. If (x k ) is a sequence of real numbers, then these are one-sided Tauberian conditions. If (x k ) is a sequence of complex numbers, then these are two-sided Tauberian conditions.


MSC:
40E05Tauberian theorems, general
40G05Cesàro, Euler, Nörlund and Hausdorff methods