Generalized Bernstein-Durrmeyer Operators of Blending Type

TL;DR

This paper introduces a new class of generalized Bernstein-Durrmeyer operators of blending type, analyzing their approximation properties, convergence behavior, and asymptotic formulas, supported by computational illustrations.

Contribution

It develops a Durrmeyer variant of generalized Bernstein operators with non-negative parameter ?, providing new approximation theorems and convergence results.

Findings

Operators preserve constant functions.

Established global and local approximation theorems.

Demonstrated convergence through computational examples.

Abstract

In this article we present the Durrmeyer variant of generalized Bernstein operators that preserve the constant functions involving non-negative parameter ?. We derive the approximation behaviour of these operators including global approximation theorem via Ditzian-Totik modulus of continuity, the order of convergence for the Lipschitz type space. Furthermore, we study a Voronovskaja type asymptotic formula and local approximation theorem by means of second order modulus of smoothness. Furthermore, we obtain the rate of approximation for absolutely continuous functions having a derivative equivalent with a function of bounded variation. In the last section of the article, we illustrate the convergence of these operators for certain functions using Maple software.