Approximation by Szász-Jakimovski-Leviatan-Type Operators via Aid of Appell Polynomials

Md Nasiruzzaman,A F Aljohani

doi:10.1155/2020/9657489

Abstract

The main purpose of the present article is to construct a newly Szász-Jakimovski-Leviatan-type positive linear operators in the Dunkl analogue by the aid of Appell polynomials. In order to investigate the approximation properties of these operators, first we estimate the moments and obtain the basic results. Further, we study the approximation by the use of modulus of continuity in the spaces of the Lipschitz functions, Peetres K-functional, and weighted modulus of continuity. Moreover, we study A-statistical convergence of operators and approximation properties of the bivariate case.

Highlights

In 1969, Jakimovski and Leviatan introduced a sequence of positive linear operators fLn gn≥1 [1], by using Appell polynos mials
If we take h ∈ E1⁄20,∞Þ, an analogue of Szász operators was proved by Jakimovski and Leviatan, where E1⁄20, ∞Þ
We prove this theorem by applying Korovkin’s theorem so it is sufficient to show that lim J ∗n,λ φ j ; y − y j n→∞

Summary

Introduction

In 1969, Jakimovski and Leviatan introduced a sequence of positive linear operators fLn gn≥1 [1], by using Appell polynos mials [2] FðvÞevy = ∑∞. S=0 Ps ðyÞv and defined as Ln ðh ; yÞ = s e−ny ∞ 〠 Ps ðnyÞh n F ð1Þ s=0 ð1Þ ∞. S s s−i where Fð1Þ ≠ 0, FðvÞ = ∑∞. S=0 cs v , and Ps ðyÞ = ∑i=0 ci ðy /ðs −. For all n ∈ N and ci /Fð1Þ ≥ 0, the positive linear operators Ln are defined on 1⁄20, 1Þ given by Wood in [3]. If we take h ∈ E1⁄20,∞Þ, an analogue of Szász operators was proved by Jakimovski and Leviatan, where E1⁄20, ∞Þ denotes the set of functions on 1⁄20, ∞Þ} such that ∣hðyÞ ∣ ≤

Objectives

Results

Conclusion