Approximation of functions of two variables by certain linear positive operators

TL;DR

This paper introduces new linear positive operators for approximating functions of two variables, analyzes their approximation properties, and extends these operators to higher orders, demonstrating convergence and rates of approximation.

Contribution

The paper presents novel linear positive operators for bivariate functions, explores their approximation behavior, and generalizes them to higher orders with convergence analysis.

Findings

Operators effectively approximate continuous functions on compact sets.

Higher-order operators improve approximation accuracy.

Convergence rates are established using modulus of continuity.

Abstract

We introduce certain linear positive operators and study some approximation properties of these operators in the space of functions, continuous on a compact set, of two variables. We also find the order of this approximation by using modulus of continuity. Moreover we define an $r$th order generalization of these operators and observe its approximation properties. Furthermore, we study the convergence of the linear positive operators in a weighted space of functions of two variables and find the rate of this convergence using weighted modulus of continuity.