Density by moduli and statistical convergence

Abstract

By using modulus functions we introduce a new concept of density for sets of natural numbers. Consequently, we obtain a generalization of the notion of statistical convergence which is studied and characterized. As an application, we prove that the ordinary convergence is equivalent to the module statistical convergence for every unbounded modulus function. (2010): Primary 40A35, Secondary 46A45. Mathematics Subject Classification

Orlicz–Pettis Theorem through Summability Methods

“…For instance, it applies for the Erdös-Ulam convergence, w p -Cesàro convergence, f -statistical convergence, etc. (see [15,23]).…”

On new f-statistical convergence in probabilistic normed spaces

“…In 2014, A. Aizpuru et al [27] introduced a new concept of density for sets of natural numbers with respect to the modulus function. They studied and characterized the generalization of this notion of f -density with statistical convergence and proved that ordinary convergence is equivalent to the module statistical convergence for every unbounded modulus function.…”

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

“…The work related to the sequence spaces defined by a modulus may be found in, e.g. , [24, 31, 33–37]. …”

“…Aizpuru et al [33] have recently introduced a new concept of density by moduli and consequently obtained a new concept of non-matrix convergence, namely, f -statistical convergence which is, in fact, a generalization of the concept of statistical convergence and intermediate between the ordinary convergence and the statistical convergence. 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].…”