Generalized Lupa\c{s}-Durrmeyer type operators involving P\'{o}lya-Eggenberger distribution

Abstract

Approximation by positive linear operators is a mathematical concept that deals with approximating functions using a class of operators that are linear and preserve positivity. These operators are typically defined on function spaces and are commonly used in approximation theory and numerical analysis. Taking this concept further, in this article we introduce a modification to Lupa\c{s} type operators, referred to as Durrmeyer type operators, which are constructed based on the P\'{o}lya-Eggenberger distribution. In the second section, we establish essential auxiliary results pertinent to these newly devised operators. Our subsequent analysis is twofold: firstly, we investigate a Voronovskaja-type asymptotic formula, and secondly, we deduce estimates for the rate of approximation, incorporating both the modulus of smoothness and the Ditzian-Totik modulus of smoothness. Moreover, we determine the rate at which convergence occurs for differential functions characterized by derivatives of bounded variation. Finally, we employ Maple software to visually demonstrate the operators' convergence towards a specific function.