A Note on New Construction of Meyer-König and Zeller Operators and Its Approximation Properties

Qing-Bo Cai,Fuat Usta,Khursheed J Ansari

doi:10.3390/math9243275

Qing-Bo Cai, Fuat Usta + Show 1 more

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https://doi.org/10.3390/math9243275

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Abstract

The topic of approximation with positive linear operators in contemporary functional analysis and theory of functions has emerged in the last century. One of these operators is Meyer–König and Zeller operators and in this study a generalization of Meyer–König and Zeller type operators based on a function τ by using two sequences of functions will be presented. The most significant point is that the newly introduced operator preserves {1,τ,τ2} instead of classical Korovkin test functions. Then asymptotic type formula, quantitative results, and local approximation properties of the introduced operators are given. Finally a numerical example performed by MATLAB is given to visualize the provided theoretical results.

Highlights

Introduction and PreliminariesApproximation theory is a significant tool especially for the solution of problems put forward in functional analysis theory
Afterwards, new linear positive operators were defined by many researchers and their approximation properties were examined with the help of the Korovkin type theorem
We introduced a generalization of Meyer–König and Zeller operators which depends on a function τ ( x ) by using two sequences of functions, rm ( x ) and sm ( x )

Summary

Introduction

Introduction and PreliminariesApproximation theory is a significant tool especially for the solution of problems put forward in functional analysis theory. Weierstrass showed the existence of polynomials that converge properly to functions that are continuous in a closed interval [ a, b]. This theorem was later proved by Bernstein in the interval [0, 1] with the help of polynomials named after him [2]. Bohman in 1952 and Korovkin in 1953, based on this theorem, proved an outstanding theorem regarding the uniform convergence of linear positive operators to continuous functions. It has been proven by this theorem that only three conditions should be investigated to achieve uniform convergence in the finite interval.

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