Generalized Deferred Statistical Convergence

Abstract

The idea of statistical convergence was given by Zygmund [45] in the first edition of his monograph published in Warsaw in 1935 and then was considered as a summability method by Schoenberg [38] in 1959 despite the fact that it made its initial appearance in a short note by Fast [16] and Steinhaus [42] in 1951. Along with the theory of summability it has played an important role in Fourier analysis, Ergodic theory, Number theory, Measure theory, Trigonometric series, Turnpike theory and Banach spaces. Salat [35] also showed that the set of bounded statistical convergent real valued sequences is a closed subspace of bounded sequences in 1980. After that Fridy [17] introduced the concept of statistically Cauchiness of sequences and proved that it is equivalent to statistical convergence. More importantly he proved that there is no any matrix summability method which involves statistical convergence in the same paper in 1985. Another magnificent development was presented to the literature by Connor [8] in 1988. For the first time in the literature his work confirmed the direct link between statistical convergence and strong $s-$Cesàro summability by unveiling that the notions are equivalent for bounded sequences, where $0<s<\infty$. Beside he showed that the set of statistically convergent sequences does not generate a locally convex $FK-$space. Recently, generalizations of statistical convergence have started to arise in many articles by several authors. Mursaleen [32] gave the concept of $\lambda -$statistical convergence in 2000, while Savaş [36] examined the relationship between strong almost convergence and almost $\lambda -$statistical convergence in the same year. Afterward, in 2010, Colak [5] made a new approach to the concept by studying the notion of statistical convergence of order $\alpha$ where $\alpha \in (0,1]$. Nowadays statistical convergence has been studied by many mathematicians such as Bhardwaj et al. [3, 4], Colak and Bektas [7] , Connor [8, 9], Et et al. [10,11,12,13,14, 37], Fridy and Orhan [18,19,20], Gadjiev and Orhan [21], Hazarika et al. [22,23,24], Isik et al. [25,26,27], Kolk [28], Mohiuddine et al. [30, 31], Rath and Tripathy [34], Sengul et al. [39,40,41] and many others. In 1932 Agnew [1] presented deferred Cesàro mean by modifying Cesàro mean to obtain more useful methods including stronger features which do not belong to nearly all methods. Kucukaslan and Yılmaztürk [29] came up with the idea of combining the deferred Cesàro mean and the concept of statistical convergence. This gave them the opportunity to generalize both strong $s-$Cesàro summability and statistical convergence with the sense of deferred Cesàro mean. In this study, we introduce the concepts of deferred statistical convergence of order $\alpha \beta$ and strongly $s-$deferred Cesàro summability of order $\alpha \beta$ of complex (or real) sequences.