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The present paper deals with the Durrmeyer construction of operators based on a class of orthogonal polynomials called Apostol-Genocchi polynomials. For the proposed operators, we first establish a global approximation result followed by its convergence estimate in terms of usual, r-th and weighted modulus of continuity. We further study the asymptotic type results such as the Voronovskaya theorem and quantitative Voronovskaya theorem. Moreover, we estimate the rate of pointwise convergence of the proposed operators for functions of bounded variation defined on the interval (0, ∞). Finally, the results are validated through graphical representations and an absolute error table. [ABSTRACT FROM AUTHOR]

This paper proposes the extended form of the α$$ \alpha $$‐Bernstein operators with the Pólya–Eggenberger distribution, also known as contagion distribution, using the integral operators for m≥2$$ m\ge 2 $$. We will refer to these sequence of operators as the “Parametric‐Bernstein operators based on contagion distribution” or simply the modified form of α$$ \alpha $$‐Bernstein operators. These operators are defined with the help of two parameters, namely, α$$ \alpha $$ and γ$$ \gamma $$. 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 h$$ h $$. We give some basic properties, study the Voronovskaya type asymptotic result and discuss the A$$ A $$‐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 e0$$ {e}_0 $$ and e2$$ {e}_2 $$. Further, using modulus of continuity, we compare the properties and behavior of both the operators. [ABSTRACT FROM AUTHOR]

We propose certain Durrmeyer-type operators for Apostol-Genocchi polynomials in this research. We explore these operators' approximation attributes and measure the rate of convergence. In addition, we present a direct approximation theorem based on first and second-order modulus of continuity, local approximation findings for Lipschitz class functions and a direct theorem based on the typical modulus of continuity. Finally, we showed a graph illustrating the convergence of the suggested operators and an error table. [ABSTRACT FROM AUTHOR]

In this paper, we are dealing with Cheney-Sharma Chlodovsky Durrmeyer operators and studying their approximation properties. The Bohman-Korovkin theorem is verified and estimated the convergence properties using of modulus of continuity, Lipschitz- type space, and Ditzian-Totik modulus of continuity. After that, the weighted approximation result is also given. Finally, some results related to the A-statistical convergence of the operators are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider Szász-Durrmeyer type operators based on Charlier polynomials associated with Srivastava-Gupta operators [17]. For the considered operators, we discuss error of estimation by using first and second order modulus of continuity, Lipchtiz-type space, Ditzian-Totik modulus of smoothness, Voronovskaya type asymptotic formula and weighted modulus of continuity. [ABSTRACT FROM AUTHOR]

Gupta operators are a modified form of Srivastava-Gupta operators and we are concerned about investigating the difference of operators and we estimate the difference of Mastroianni operators with Gupta operators in terms of modulus of continuity of first order. We also study the weighted approximation of functions and obtain the rate of convergence with the help of the moduli of continuity as well as Peetre's K-functional of Gupta operators. [ABSTRACT FROM AUTHOR]