F-Statistical convergence of order α and strong Cesàro summability of order α with respect to a modulus

Vinod K Bhardwaj,Shweta Dhawan

doi:10.1186/s13660-015-0850-x

Vinod K Bhardwaj, Shweta Dhawan

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https://doi.org/10.1186/s13660-015-0850-x

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Abstract

In this paper, following a very recent and new approach of Aizpuru et al. (Quaest. Math. 37:525-530, 2014), we further generalize a concept of α-density to that of f α $f_{\alpha}$ -density, where f is an unbounded modulus and 0 < α ≤ 1 $0 < \alpha\leq1$ . As a consequence, we obtain a new nonmatrix convergence method, namely f-statistical convergence of order α or S α f $S_{\alpha}^{f}$ -convergence, which is intermediate between the ordinary convergence and the statistical convergence of order α. We also introduce a new concept of strong Cesaro summability of order α with respect to a modulus function f, and finally we investigate the relationship between the set S α f $S_{\alpha}^{f}$ of all f-statistically convergent sequences of order α and the set w α f $w_{\alpha}^{f}$ of all strongly Cesaro summable sequences of order α with respect to f.

Highlights

Introduction and historical backgroundThe credit of introducing the idea of statistical convergence, which is a generalization of the usual notion of convergence, goes to Zygmund [ ]
Later on statistical convergence was further investigated from the sequence space point of view and linked with summability theory by Connor [ ], Çinar et al [ ], Et et al [ ], Savas [ ], Sengül and Et [ ], and many others
In the year, Aizpuru et al [ ] defined a new concept of density with the help of an unbounded modulus function and, as a consequence, they obtained a new concept of nonmatrix convergence which is intermediate between the ordinary convergence and the statistical convergence, and agrees with the statistical convergence when the modulus function is the identity mapping

Summary

Introduction

Introduction and historical backgroundThe credit of introducing the idea of statistical convergence, which is a generalization of the usual notion of convergence, goes to Zygmund [ ]. We first extend the notion of α-density to that of fα-density in the same way as natural density was extended to f -density by Aizpuru et al [ ] and introduce a new and more general nonmatrix summability method, namely f -statistical convergence of order α, where f is an unbounded modulus function and α is a real number such that < α ≤ .

Results

Conclusion