Generalized Kantorovich-Szász type operations involving Charlier polynomials

Abstract

The purpose of this paper is to introduce a new kind of Kantorovich-Szász type operators based on Charlier polynomials and study its various approximation properties. We establish some local direct theorems, e.g., Voronovskaja type asymptotic theorem and an estimate of error by means of the Lipschitz type maximal function and the Peetre’s K-functional. We also discuss the weighted approximation properties. Next, we construct a bivariate case of the above operators and study the degree of approximation with the aid of the complete and partial moduli of continuity. A Voronovskaja type asymptotic theorem and the order of convergence by considering the second order modulus of continuity are also proved. We define the associated Generalized Boolean Sum (GBS) operators and discuss the degree of approximation by using mixed modulus of smoothness for Bögel continuous and Bögel differentiable functions. Furthermore, by means of a numerical example it is shown that the proposed operators provide us a better approximation than the operators corresponding to the particular case ℘ = 1. We also illustrate the convergence of the bivariate operators and the associated GBS operators to a certain function and show that the GBS operators enable us a better error estimation than the bivariate operators using Matlab algorithm.