Generalized Statistical Convergence and Convergence Free Spaces

Abstract

Many results about statistical convergence in a general setting can be derived by consideration of the convergence free space of null sequences and its related cofilter and filter. We derive algebraic and order properties, preservation under uniform convergence, Cauchy properties, and properties of cluster points. We also show that for certain types of statistical convergence stronger convergence properties also hold. Finally we discuss the relationship between generalized statistical convergence and subsets of the Stone᎐Cech compactification of the integers.

Ideal Limit Theorems and Their Equivalence in $(\ell)$-Group Setting

“…Ideal convergence was introduced by Kostyrko,Šalát, andWilczyński (2000/2001) and independently by Nuray and Ruckle (2000) with the name of "cofilter convergence", while filter convergence was introduced by Katětov (1968). Ideal convergence includes as a particular case the statistical convergence, introduced by Fast (1951) and Steinhaus (1951) (see also Connor, 1992;Fridy & Miller, 1991;Kolk, 1993;Šalát, 1980).…”

Two valued measure and summability of double sequences

“…Investigations in this line was initiated by Fast [8] and independently by Schoenberg [17] who introduced the idea of statistical convergence. Since then this concept was studied byŠalát [16], Fridy [9], Connor ([2], [3]) and many others (see [5], [6], [10], [12], [13]) where more references can be found about related works). In particular, in [2] and [3] Connor proposed two very interesting extensions of the concept of statistical convergence using a complete {0, 1} valued measure µ defined on an algebra of subsets of N which form the basis of many more recent works ( [4] where more references can be found).…”

On Zweier generalized difference ideal convergent sequences in a locally convex space defined by Musielak-Orlicz function

“…Nowadays many authors to use an equivalent dual notion of the ideal convergence. Kostyrko et al [24] and Nuray and Ruckle [30] independently studied in detalis about the notion of ideal convergence which is based on the structure of the admissible ideal I of subsets of natural numbers N. Later on it was further investigated by many authors, e.g. Tripathy and Hazarika [36,37], Hazarika [12], Hazarika and Savaş [11] and references therein.…”