Approximation by bivariate Chlodowsky type Sz\xe1sz\u2013Durrmeyer operators and associated GBS operators on weighted spaces

Reşat Aslan,M Mursaleen

doi:10.1186/s13660-022-02763-7

Abstract

In this article, we consider a bivariate Chlodowsky type Szász–Durrmeyer operators on weighted spaces. We obtain the rate of approximation in connection with the partial and complete modulus of continuity and also for the elements of the Lipschitz type class. Moreover, we examine the degree of convergence with regard to the weighted modulus of continuity and Peetre’s K-functional. Further, we construct the associated GBS type of these operators and estimate the degree of approximation using the mixed modulus of continuity and a class of the Lipschitz of Bögel type continuous functions. Finally, with the help of Maple software, we present the comparisons of the convergence of the bivariate Chlodowsky type Szász–Durrmeyer operators and associated GBS type operators to certain functions with some graphs and error estimation tables.

Highlights

The approximation of the continuous functions via the sequences of linear positive operators, which have many applications in disciplines such as engineering and physics, besides mathematics, has been an important research topic since the last century
A generalization of Bernstein operators on an unbounded set was introduced by Chlodowsky [2]
In 1930, an integral modification of the classical Bernstein operators was presented by Kantorovich [3]

Summary

Introduction

The approximation of the continuous functions via the sequences of linear positive operators, which have many applications in disciplines such as engineering and physics, besides mathematics, has been an important research topic since the last century. We introduce the uniform convergence of these operators and estimate the order of approximation in terms of the partial and complete modulus of continuity for the elements of the Lipschitz type class, weighted modulus of continuity, and Peetre’s K -functional, respectively. 3, we discuss the associated GBS type of these operators and investigate the order of convergence by the mixed modulus of smoothness and the Lipschitz class of the Bögel continuous functions. We present some graphs and error estimation tables to compare the convergence of bivariate and associated GBS type operators to certain functions.

Results

Conclusion