New Definitions about A I -Statistical Convergence with Respect to a Sequence of Modulus Functions and Lacunary Sequences

Ömer Kişi,Ekrem Savas,Hafize Gümüş

doi:10.3390/axioms7020024

Ömer Kişi, Ekrem Savas + Show 1 more

Open Access

https://doi.org/10.3390/axioms7020024

Copy DOI

Journal: Axioms	Publication Date: Apr 13, 2018
Citations: 3	License type: CC BY 4.0

Affiliation: Üsküdar University, Necmettin Erbakan University

Abstract

In this paper, using an infinite matrix of complex numbers, a modulus function and a lacunary sequence, we generalize the concept of I -statistical convergence, which is a recently introduced summability method. The names of our new methods are A I -lacunary statistical convergence and strongly A I -lacunary convergence with respect to a sequence of modulus functions. These spaces are denoted by S θ A I , F and N θ A I , F , respectively. We give some inclusion relations between S A I , F , S θ A I , F and N θ A I , F . We also investigate Cesáro summability for A I and we obtain some basic results between A I -Cesáro summability, strongly A I -Cesáro summability and the spaces mentioned above.

Highlights

As is known, convergence is one of the most important notions in mathematics
After giving the definition of statistical convergence, we can show that any convergent sequence is statistically convergent, but not
It has become an active research area in recent years. This concept has applications in different fields of mathematics such as number theory [7], measure theory [8], trigonometric series [1], summability theory [9], etc. Following this very important definition, the concept of lacunary statistical convergence was defined by Fridy and Orhan [10]

Summary

Introduction

Convergence is one of the most important notions in mathematics. Statistical convergence extends the notion. In 1935, statistical convergence was given by Zygmund in the first edition of his monograph [1] It was formally introduced by Fast [2], Fridy [3], Salat [4], Steinhaus [5] and later was reintroduced by Schoenberg [6]. This concept has applications in different fields of mathematics such as number theory [7], measure theory [8], trigonometric series [1], summability theory [9], etc Following this very important definition, the concept of lacunary statistical convergence was defined by Fridy and Orhan [10]. A -lacunary statistically convergence with respect to a sequence of modulus functions

Definitions and Notations

Cesàro Summability for AI

Conclusions