Parametric Bernstein operators based on contagion distribution

Abstract

This paper proposes the extended form of the ‐Bernstein operators with the Pólya–Eggenberger distribution, also known as contagion distribution, using the integral operators for . We will refer to these sequence of operators as the “Parametric‐Bernstein operators based on contagion distribution” or simply the modified form of ‐Bernstein operators. These operators are defined with the help of two parameters, namely, and . We observe the significance of both these parameters. Subsequently, we calculate their moments and on the basis of Korovkin's approximation theorem we assert that our defined operators converge for a real valued continuous function . We give some basic properties, study the Voronovskaya type asymptotic result and discuss the ‐statistical approximation of our operators. We also use the modulus of continuity to study the rate of convergence. Alongside our primary work, we have also defined the King‐type modification that preserves the operators at and . Further, using modulus of continuity, we compare the properties and behavior of both the operators.