Some approximation results by Bernstein-Kantorovich operators based on (p,q)-integers

TL;DR

This paper introduces (p,q)-Bernstein-Kantorovich operators, a new generalization of classical operators, and studies their approximation properties, convergence rates, and local approximation behavior, providing more flexible tools for function approximation.

Contribution

The paper develops a novel (p,q)-analogue of Bernstein-Kantorovich operators and analyzes their approximation properties, extending previous q-analogues with greater flexibility.

Findings

Operators converge to functions as shown by Korovkin's theorem

Convergence rate is quantified using modulus of continuity

Operators exhibit favorable local approximation properties

Abstract

In this paper, First we have given the modified form of (p,q)-analogues of Bernstein and Bernstein operators [21-23] and then we introduce a new analogue of Bernstein-Kantorovich operators which we call as (p,q)-Bernstein-Kantorovich operators. We discuss approximation properties for these operators based on Korovkin's type approximation theorem and we compute the order of convergence using usual modulus of continuity and also the rate of convergence when f is a Lipschitz function. Moreover, we also study the local approximation property of the (p,q)-Kantorovich operators . We show comparisons and some illustrative graphics for the convergence of operators to a function. In comparison to q-analogoue of Bernstein-Kantorovich operators, our generalization gives more flexibility for the convergence of operators to a function.