Application of f-lacunary statistical convergence to approximation theorems

Vinod K Bhardwaj,Shweta Dhawan

doi:10.1186/s13660-018-1871-z

Vinod K Bhardwaj, Shweta Dhawan

Open Access

https://doi.org/10.1186/s13660-018-1871-z

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Abstract

The concept of f-lacunary statistical convergence which is, in fact, a generalization of lacunary statistical convergence, has been introduced recently by Bhardwaj and Dhawan (Abstr. Appl. Anal. 2016:9365037, 2016). The main object of this paper is to prove Korovkin type approximation theorems using the notion of f-lacunary statistical convergence. A relationship between the newly established Korovkin type approximation theorems via f-lacunary statistical convergence, the classical Korovkin theorems and their lacunary statistical analogs has been studied. A new concept of f-lacunary statistical convergence of degree β (0 < beta< 1) has also been introduced, and as an application a corresponding Korovkin type theorem is established.

Highlights

1.1 Density by moduli and statistical convergence The idea of statistical convergence, which is, a generalization of the usual notion of convergence, was first introduced by Fast [14] and Steinhaus [37] independently in 1951 and since several generalizations and applications of this concept have been investigated by various authors, namely Salát [35], Fridy [16], Aizpuru et al [1], Aktuğlu [2], Gadjiev and Orhan [18], Mursaleen and Alotaibi [28], and many others.Statistical convergence depends on the natural density of subsets of the set N = {1, 2, 3, . . .}
We study a relationship between the f -lacunary statistical analog and the f statistical analog of the Korovkin first theorem
We prove an f -lacunary statistical analog of the Korovkin second theorem, from which the lacunary statistical analog is obtained as a particular case

Summary

Introduction

1.1 Density by moduli and statistical convergence The idea of statistical convergence, which is, a generalization of the usual notion of convergence, was first introduced by Fast [14] and Steinhaus [37] independently in 1951 and since several generalizations and applications of this concept have been investigated by various authors, namely Salát [35], Fridy [16], Aizpuru et al [1], Aktuğlu [2], Gadjiev and Orhan [18], Mursaleen and Alotaibi [28], and many others. Statistical convergence depends on the natural density of subsets of the set N = {1, 2, 3, . The natural density d(K) of a set K ⊆ N A sequence (xk) in X is said to be statistically convergent to some x ∈ X, if for each > 0 the set {k ∈ N : xk – x ≥ } has natural

Discussion

Conclusion