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

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 -density, where f is an unbounded modulus and 0 < α ≤ 1. As a consequence, we

On (f, I) - Lacunary statistical convergence of Order α of sequences of sets

“…Schoenberg [32] established some basic properties of statistical convergence and studied the concept as a summability method. Later on it was further investigated from the sequence space point of view and linked with summability theory by Altınok et al ( [2], [3]), Bhardwaj and Dhawan [4], Caserta et al [5], Ç ınar et al [9], Connor [12], Ç akallı et al ( [6], [7], [8]), Ç olak et al ( [10], [11], [23]), Et et al ( [13], [14], [15], [16]), Fridy [19], Gadjiev and Orhan [21], Işık and Akbaş [22], Salat [29], Savaş and Et [31], Şengül [34], Taylan [38], and many others. Nuray and Rhoades [28] extended the notion to statistical convergence of sequences of sets and gave some basic theorems.…”

Density by moduli and Wijsman lacunary statistical convergence of sequences of sets

“…This idea of replacing natural density with density by moduli has motivated us to look for some new generalizations of statistical convergence and consequently we have introduced and studied the concepts of f -statistical convergence of order α [34] and f -lacunary statistical convergence [35]. Using the notion of density by moduli Bhardwaj et al [36] have also introduced and studied the concept of f -statistical boundedness which is a generalization of statistical boundedness [38] and intermediate between the usual boundedness and the statistical boundedness.…”

f-statistical convergence of order β for sequences of fuzzy numbers

“…After then, Bhardwaj [7] introduced f −statistical convergence of order α and strong Cesaro summability of order α with respect to a modulus function f for real sequences. The purpose of this paper is to generalize the study of Bhardwaj [7] and Ç olak [11] applying to sequences of fuzzy numbers so as to fill up the existing gaps in the summability theory of fuzzy numbers. For a detailed account of many more interesting investigations concerning statistical convergence of order α and β, one may refer to ([12], [14], [15], [16], [2], [3], [5]) This paper organizes as follows: In section 2, we give the basic notions which will be used throughout the paper.…”