Statistical weighted (Nλ,p,q)(Eλ,1)A-summability with application to Korovkin's type approximation theorem

doi:10.1016/j.bulsci.2022.103146

Bulletin des Sciences Mathématiques

Volume 178, September 2022, 103146

https://doi.org/10.1016/j.bulsci.2022.103146 Get rights and content

Abstract

In this paper, we first introduce the notion of statistical weighted $(N_{λ}, p, q) (E_{λ}, 1)$ A-summability and then investigate its relation with the weighted $(N_{λ}, p, q) (E_{λ}, 1) A$ -statistical convergence. We also study the weighted $(N_{λ}, p, q) (E_{λ}, 1)$ -regular matrix and find the conditions for the matrix A to be weighted $(N_{λ}, p, q) (E_{λ}, 1)$ -regular. Furthermore, as an application of the results presented in this paper, we prove a Korovkin type approximation theorem involving the statistical weighted $(N_{λ}, p, q) (E_{λ}, 1)$ A-summability, which we have introduced in this paper. Relevant connections of the proposed investigation with other recent developments on the subject are also indicated.

Section snippets

Introduction and motivation

A new type of convergence, which is known as statistical convergence, was first introduced by Fast [14] and Zygmund [41] (see also [42]). It was later studied by Schoenberg [36], Šalát [35], Fridy [15] and Connor [7]. In the year 2013, Belen and Mohiuddine [2] presented a generalization of this notion, which was subsequently modified by Ghosal [16]. Various applications and other related developments based upon the notion of statistical convergence can be found in the works by (for example)

Statistical weighted $(N_{λ}, p, q) (E_{λ}, 1)$ A-summability

In this section, we use the notion of weighted Nörlund-Euler λ-statistical convergence which was given by Loku and Aljimi [23] as follows.

Let $λ = (λ_{n})$ be a non-decreasing sequence of positive numbers tending to ∞ such that $λ_{n + 1} ≦ λ_{n} + 1 (n \in N; λ_{1} = 1) .$ The collection of all such sequences will be denoted here by Δ.

We denote by $I_{n}$ the closed interval given by $I_{n} = [n - λ_{n} + 1, n] .$ We also define the intervals $I_{k, 1}^{\infty}$ and $I_{k, m - 1}^{\infty}$ by $I_{k, 1}^{\infty} = [k - λ_{k} + 1, \infty]$ and $I_{k, m - 1}^{\infty} = [k - λ_{k} + m - 1, \infty],$ respectively.

Let $(p_{n})$ and $(q_{n})$ be the two

Application to approximation theorems

In this section, we apply the $A_{S}^{(N_{λ}, E_{λ})}$ -summability in order to establish a Korovkin type approximation theorem. The study of this class of widely- and extensively-investigated approximation theorems was initiated by Korovkin [22], so they are called Korovkin's type approximation theorems. In recent years, many researchers investigated and extended this kind of approximation theorems by using various test functions in several different fields of mathematics. Statistical version of such

Conclusion

In our present investigation, we have first introduced a (presumably new) notion of statistical weighted $(N_{λ}, p, q) (E_{λ}, 1)$ A-summability. Then, by using this notion, we have investigated its relation with the weighted $(N_{λ}, p, q)$ $(E_{λ}, 1) A$ -statistical convergence. We have also studied the weighted $(N_{λ}, p, q)$ $(E_{λ}, 1)$ -regular matrix and derived the conditions for the matrix A to be weighted $(N_{λ}, p, q) (E_{λ}, 1)$ -regular. As an application of our findings, we have established a Korovkin type approximation theorem

Declaration of Competing Interest

The authors declare that they have no conflicts of interest.

References (42)

C. Belen et al.
Generalized weighted statistical convergence and application
Appl. Math. Comput.
(2013)
N.L. Braha et al.
$Λ^{2}$ -Weighted statistical convergence and Korovkin and Voronovskaya type theorems
Appl. Math. Comput.
(2015)
N.L. Braha et al.
A Korovkin's type approximation theorem for periodic functions via the statistical summability of the generalized de la Vallèe Poussin mean
Appl. Math. Comput.
(2014)
O.H.H. Edely et al.
On statistical A-summability
Math. Comput. Model.
(2009)
O.H.H. Edely et al.
Korovkin type approximation theorems obtained through generalized statistical convergence
Appl. Math. Lett.
(2010)
O.H.H. Edely et al.
Approximation for periodic functions via weighted statistical convergence
Appl. Math. Comput.
(2013)
S. Ghosal
Generalized weighted random convergence in probability
Appl. Math. Comput.
(2014)
U. Kadak et al.
Statistical weighted $B$ -summability and its applications to approximation theorems
Appl. Math. Comput.
(2017)
S.A. Mohiuddine
An application of almost convergence in approximation theorems
Appl. Math. Lett.
(2011)
M. Mursaleen et al.
Weighted statistical convergence and its application to Korovkin type approximation theorem
Appl. Math. Comput.
(2012)

H.M. Srivastava et al.

Generalized equi-statistical convergence of positive linear operators and associated approximation theorems

Math. Comput. Model.

(2012)

T. Acar et al.

Korovkin-type theorems in weighted $L^{p}$ -spaces via summation process

Sci. World J.

(2013)

N.L. Braha et al.

A parametric generalization of the Baskakov-Schurer-Szász-Stancu approximation operators

Symmetry

(2021)

N.L. Braha et al.

Some weighted statistical convergence and associated Korovkin and Voronovskaya type theorems

J. Appl. Math. Comput.

(2021)

J.S. Connor

The statistical and strong p-Cesàro convergence of sequences

Analysis

(1988)

R.G. Cooke

Infinite Matrices and Sequence Spaces

(1950)

Ö. Çakar et al.

On uniform approximation by Bleimann, Butzer and Hahn on all positive semiaxis

Trans. Acad. Sci. Azerbaijan Ser. Phys. Tech. Math. Sci.

(1999)

E. Erkuş et al.

A-Statistical extension of the Korovkin type approximation theorem

Proc. Indian Acad. Sci. Math. Sci.

(2003)

H. Fast

Sur la convergence statistique

Colloq. Math.

(1951)

J.A. Fridy

On statistical convergence

Analysis

(1985)

A.D. Gadjiev et al.

Some approximation theorems via statistical convergence

Rocky Mt. J. Math.

(2002)

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Statistical weighted (Nλ,p,q)(Eλ,1)A-summability with application to Korovkin's type approximation theorem

Abstract

Section snippets

Introduction and motivation

Statistical weighted (Nλ,p,q)(Eλ,1)A-summability

Application to approximation theorems

Conclusion

Declaration of Competing Interest

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Math. Comput.

Math. Comput. Model.

Appl. Math. Lett.

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Math. Lett.

Appl. Math. Comput.

Math. Comput. Model.

Korovkin-type theorems in weighted Lp-spaces via summation process

Sci. World J.

A parametric generalization of the Baskakov-Schurer-Szász-Stancu approximation operators

Symmetry

Some weighted statistical convergence and associated Korovkin and Voronovskaya type theorems

J. Appl. Math. Comput.

The statistical and strong p-Cesàro convergence of sequences

Analysis

Infinite Matrices and Sequence Spaces

On uniform approximation by Bleimann, Butzer and Hahn on all positive semiaxis

Trans. Acad. Sci. Azerbaijan Ser. Phys. Tech. Math. Sci.

A-Statistical extension of the Korovkin type approximation theorem

Proc. Indian Acad. Sci. Math. Sci.

Sur la convergence statistique

Colloq. Math.

On statistical convergence

Analysis

Some approximation theorems via statistical convergence

Rocky Mt. J. Math.

Statistical weighted $(N_{λ}, p, q) (E_{λ}, 1) A$ -summability with application to Korovkin's type approximation theorem

Statistical weighted $(N_{λ}, p, q) (E_{λ}, 1)$ A-summability

Korovkin-type theorems in weighted $L^{p}$ -spaces via summation process