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Equivalence results for Cesàro submethods. (English) Zbl 0949.40015

In this paper the author defines Cesàro submethods \(C_\lambda\) as follows. Let \(E\) be the range of a strictly increasing sequence of positive integers, say \[ E=\bigl\{\lambda(n)\bigr\}^\infty_{n=1}\quad\text{and}\quad(c_\lambda x)_n= {1\over\lambda(n)} \sum^{\lambda(n)}_{k=1} x_k \] where \(\{x_k\}\) is a sequence of real or complex numbers and \(n=1,2,3,\dots,C_\lambda\) being a subsequence of the \(C_1\) method and regular for any \(\lambda\). D. H. Armitage and I. J. Maddox [A new type of Cesàro mean, Analysis 9, No. 1-2, 195-204 (1989; Zbl 0693.40009)] proved inclusion and Tauberian results for \(C_\lambda\) methods. The author here has expanded the previous results by examining further inclusion properties of the \(C_\lambda\) method for bounded sequences and its relationship to statistical convergence. His theorems are used to prove a “condensation test” for statistical convergence.
Reviewer: I.L.Sukla (Orissa)

MSC:

40G05 Cesàro, Euler, Nörlund and Hausdorff methods
40D25 Inclusion and equivalence theorems in summability theory
40C05 Matrix methods for summability
40D20 Summability and bounded fields of methods

Citations:

Zbl 0693.40009
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