F_{(lambda,mu)}-statistical convergence of order \u03b1\u0303 for double sequences

Mahmut Işik,Yavuz Altin

doi:10.1186/s13660-017-1512-y

Abstract

New concepts of f_{lambda,mu }-statistical convergence for double sequences of order α̃ and strong f_{lambda,mu }-Cesàro summability for double sequences of order α̃ are introduced for sequences of (complex or real) numbers. Furthermore, we give the relationship between the spaces w_{tilde{alpha },0}^{2} ( f,lambda,mu ), w_{tilde{alpha }}^{2} ( f,lambda,mu ) and w_{tilde{alpha},infty }^{2} ( f,lambda,mu ). Then we express the properties of strong f_{lambda,mu }-Cesàro summability of order β̃ which is related to strong f_{lambda,mu }-Cesàro summability of order α̃. Also, some relations between f_{lambda,mu }-statistical convergence of order α̃ and strong f_{lambda,mu }-Cesàro summability of order α̃ are given.

Highlights

The first idea of statistical convergence goes back to the first edition of the famous Zygmund’s monograph [ ]
F f (j, k) ∈ In × Im : |xjk – | ≥ ε. In this case we write Sα (f, λ, μ)- limj,k xjk =, and we denote the set of all fλ,μ-statistically convergent double sequences of order α by Sα (f, λ, μ), where f is an unbounded modulus function
5 Conclusions In this study, we define fλ,μ-statistical convergence for double sequences of order α, where f is an unbounded modulus function. Besides this we study strong fλ,μ-Cesàro summability for double sequences of order α and give inclusion relations

Summary

Introduction

The first idea of statistical convergence goes back to the first edition of the famous Zygmund’s monograph [ ]. A double sequence x = (xjk)j,k∈N is said to be statistically convergent to if for every ε > the set {(j, k) : j ≤ m, k ≤ n : |xjk – | ≥ ε} has double natural density zero [ ] In this case, one can write st - lim x = , and we denote the collection of all statistically convergent double sequences by st. (xjk) is said to be fλ,μ-statistically convergent of order α if there is a complex number such that, for every ε > , In this case we write Sα ̃ (f , λ, μ)- limj,k xjk = , and we denote the set of all fλ,μ-statistically convergent double sequences of order α by Sα ̃ (f , λ, μ), where f is an unbounded modulus function.

Following the proof of Proposition of

Conclusions