On the Approximation Properties of  q − Analogue Bivariate  λ -Bernstein Type Operators

Edmond Aliaga,Behar Baxhaku,Gestur Lafsson

doi:10.1155/2020/4589310

On the Approximation Properties of q − Analogue Bivariate λ -Bernstein Type Operators

Edmond Aliaga, Behar Baxhaku + Show 1 more

Open Access

https://doi.org/10.1155/2020/4589310

Copy DOI

Abstract

In this article, we establish an extension of the bivariate generalization of the q -Bernstein type operators involving parameter λ and extension of GBS (Generalized Boolean Sum) operators of bivariate q -Bernstein type. For the first operators, we state the Volkov-type theorem and we obtain a Voronovskaja type and investigate the degree of approximation by means of the Lipschitz type space. For the GBS type operators, we establish their degree of approximation in terms of the mixed modulus of smoothness. The comparison of convergence of the bivariate q -Bernstein type operators based on parameters and its GBS type operators is shown by illustrative graphics using MATLAB software.

Highlights

Let h ∈ CðSÞ with S = 1⁄20, 1,λ ∈ 1⁄2−1, 1, and m ∈ N: In 2018, Chen et al [1] proposed a new generalization of Bernstein operators based on a fixed real parameter λ ∈ 1⁄2−1, 1 as
Kajla and Miclacus [16] studied the rate of approximation of Bögel continuous and Bögel differentiable functions by the Generalized Boolean Sum (GBS) operators of Bernstein-Durrmeyer type operators
Denote E ∗λ m1,m2 = ∣GBS m1,m2,qm1 ðhðs1, s2 Þ ; y1, y2 Þ − hðy1, y2 Þ ∣, the error function of approximation by operators. This example explains the convergence of the operators GB

Summary

Introduction

The purpose of this article is to present an extension of the bivariate λ, q-Bernstein type operators involving parameters and obtain the degree of approximation by means of the Lipschitz type space for two variables. We consider the associated Generalized Boolean Sum (GBS) operators and study their degree of approximation in terms of the mixed modulus of smoothness for bivariate functions. Let C ′ ðS2 Þ denote the space of continuous functions hðy , y2 Þ on S2 whose first-order partial derivatives g′y1 and g′y2 are continuous on S2. Our result yields us the rate of approximation for continuously differentiable functions on S2 by the operators λ 1 ,λ 2 ,q ðμm1 ,λ1 ,2 ðqm , y1 ÞÞ1/2 and δ2 = ðμm2 ,λ2 ,2 ðqm , y2 ÞÞ1/2 , we obtain the desired result.

Construction of GBS Operator of Generalized

Degree of Approximation by GBS m

Numerical Results and Discussions