Estimates for the Differences of Certain Positive Linear Operators

Ana Maria Acu,Sever Hodiş,Ioan Rașa

doi:10.3390/math8050798

Ana Maria Acu, Sever Hodiş + Show 1 more

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https://doi.org/10.3390/math8050798

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Abstract

The present paper deals with estimates for differences of certain positive linear operators defined on bounded or unbounded intervals. Our approach involves Baskakov type operators, the kth order Kantorovich modification of the Baskakov operators, the discrete operators associated with Baskakov operators, Meyer–König and Zeller operators and Bleimann–Butzer–Hahn operators. Furthermore, the estimates in quantitative form of the differences of Baskakov operators and their derivatives in terms of first modulus of continuity are obtained.

Highlights

The studies of the differences of positive linear operators has as starting point the Lupaş problem proposed in [1] and became an interesting area of research in Approximation Theory
Aral et al [6] obtained some estimates of the differences of positive linear operators defined on unbounded intervals in terms of weighted modulus of continuity
Our study concerns the Baskakov type operators, the kth order Kantorovich modification of the Baskakov operators and the discrete operators associated with Baskakov operators

Summary

Introduction

The studies of the differences of positive linear operators has as starting point the Lupaş problem proposed in [1] and became an interesting area of research in Approximation Theory. Gave a solution to Lupaş’ problem for a more general case in terms of moduli of continuity New results on this topic were given by Gonska et al ([3,4]). Aral et al [6] obtained some estimates of the differences of positive linear operators defined on unbounded intervals in terms of weighted modulus of continuity. The present paper deals with the estimates of the differences of certain positive linear operators (defined on bounded or unbounded intervals) and their derivatives, in terms of the modulus of continuity. Using as measuring tool a K-functional an estimate of the difference between the kth order Kantorovich modification of the Baskakov operators and their associated discrete operators will be established. In what follows k · k will stand for the supremum norm

Baskakov Type Operators

The kth Order Kantorovich Modification of the Baskakov Operators

The Meyer–König and Zeller Operators