On Durrmeyer Typeλ-Bernstein Operators via (p,q)-Calculus

Qing-Bo Cai,Guorong Zhou,Mohammad Mursaleen

doi:10.1155/2020/8832627

Abstract

In the present paper, Durrmeyer typeλ-Bernstein operators via (p,q)-calculus are constructed, the first and second moments and central moments of these operators are estimated, a Korovkin type approximation theorem is established, and the estimates on the rate of convergence by using the modulus of continuity of second order and Steklov mean are studied, a convergence theorem for the Lipschitz continuous functions is also obtained. Finally, some numerical examples are given to show that these operators we defined converge faster in someλcases than Durrmeyer type (p,q)-Bernstein operators.

Highlights

In 2016, Mursaleen et al [1] proposed the following (p, q)-analogue of Bernstein operators: nBn,p,q ð f ; xÞ = 〠 bn,k ðx ; p, qÞ f k=01⁄2kp,q pk−n 1⁄2np,q !, x ∈ 1⁄20, 1, ð1Þ where bn,k ðx ; p, qÞðk = 0, 1, ⋯, nÞ are (p, q)-Bernstein basis functions and defined as bn,k ðx ; p, qÞ =Bλn,p,q ð f ; xÞ = 〠 bλn,k ðx ; p, qÞ f " # n pnðn−1Þ/2 kThey introduced and studied some important approximation properties of the Stancu type of operators (1) in [2]
This paper is mainly organized as follows: in Section 2, we estimate some moments and central moments of Dλn,p,q ð f ; xÞ in order to obtain our main results; in Section 3, we study a Korovkin type approximation theorem and estimate the rate of convergence of Dλn,p,q ð f Þ to f by using the second order modulus of smoothness, Peetre’s K-functional, Steklov mean function, and Lipschitz class function; in Section 4, we give some numerical experiments to verify our theoretical results; in the final section, a conclusion is given
The fact that bn,k ðxÞ = pnðn−1Þ/2 bn,k ðx ; p, qÞ, we get the proof of Lemma 1

Summary

Introduction

X ∈ 1⁄20, 1, ð1Þ where bn,k ðx ; p, qÞðk = 0, 1, ⋯, nÞ are (p, q)-Bernstein basis functions and defined as bn,k ðx ; p, qÞ =. They introduced and studied some important approximation properties of the Stancu type of operators (1) in [2]. This paper is mainly organized as follows: in Section 2, we estimate some moments and central moments of Dλn,p,q ð f ; xÞ in order to obtain our main results; in Section 3, we study a Korovkin type approximation theorem and estimate the rate of convergence of Dλn,p,q ð f Þ to f by using the second order modulus of smoothness, Peetre’s K-functional, Steklov mean function, and Lipschitz class function; in Section 4, we give some numerical experiments to verify our theoretical results; in the final section, a conclusion is given

Some Lemmas

Data Availability n

Numerical Examples

Conclusion