Approximation by Sz醩z Type Operators Involving Apostol-Genocchi Polynomials

Mine Menekşe Yılmaz

doi:10.32604/cmes.2022.017385

Abstract

The goal of this paper is to give a form of the operator involving the generating function of Apostol-Genocchi polynomials of order α. Applying the Korovkin theorem, we arrive at the convergence of the operator with the aid of moments and central moments. We determine the rate of convergence of the operator using several tools such as -functional, modulus of continuity, second modulus of continuity. We also give a type of Voronovskaya theorem for estimating error. Moreover, we investigate some results about convergence properties of the operator in a weighted space. Finally, we give numerical examples to support our theorems by using the Maple.

Highlights

The Weierstrass approximation theorem shows that the polynomials are uniformly dense in the space of continuous functions on a compact interval equipped with supremum norm [1]
We give some examples to obtain an upper bound for the error f (x) − An(α,β,m) (f ; x) in the terms of the modulus of continuity
We have introduced a form of the operator using the generating function of Apostol-Genocchi polynomials of order α and obtained the approximation properties and rate of convergence of this operator

Summary

Introduction

The Weierstrass approximation theorem shows that the polynomials are uniformly dense in the space of continuous functions on a compact interval equipped with supremum norm [1]. Since Bernstein [2] proved the Weierstrass theorem using a polynomial class in 1911, some authors [3,4,5,6], defined linear positive operators for the same purpose. One of these operators is Szász operators that generalization of Bernstein polynomials to infinite interval [7]: Sn (f ; x) = e−nx ∞ (nx)k f k! In [8], assuming that g (z) is analytic

Objectives

Conclusion