Approximation Properties ofλ-Gamma Operators Based onq-Integers

Wen-Tao Cheng,Xiao-Jun Tang

doi:10.1155/2020/5710510

Wen-Tao Cheng, Xiao-Jun Tang

Open Access

https://doi.org/10.1155/2020/5710510

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Abstract

In the present paper, we will introduceλ-Gamma operators based onq-integers. First, the auxiliary results about the moments are presented, and the central moments of these operators are also estimated. Then, we discuss some local approximation properties of these operators by means of modulus of continuity and PeetreK-functional. And the rate of convergence and weighted approximation for these operators are researched. Furthermore, we investigate the Voronovskaja type theorems including the quantitativeq-Voronovskaja type theorem andq-Grüss-Voronovskaja theorem.

Highlights

Gamma operators are very important positive linear operators and have been widely used in probability theory and computational mathematics
For f ∈ CðR+Þ, n = 1, 2, 3, ⋯ where R+ = ð0,∞Þ and CðR+Þ be the space of all continuous functions f on the interval R+, the Gamma operators were introduced in [1] by
We can learn some properties of Gamma operators and their modified operators in [2,3,4,5,6,7]

Summary

Introduction

Gamma operators are very important positive linear operators and have been widely used in probability theory and computational mathematics. For f ∈ CðR+Þ, n = 1, 2, 3, ⋯ where R+ = ð0,∞Þ and CðR+Þ be the space of all continuous functions f on the interval R+, the Gamma operators were introduced in [1] by. In [8], Qi et al defined new Gamma operators as follows: Gn,λð f. In order to preserve the constant, we defined λ-Gamma operators as follows: Definition 1. For f ∈ CðR+Þ, λ ∈ N, n = λ, λ + 1, ⋯, the λ -Gamma operators are defined by. The paper is organized as follows: In Section 1, we introduce the history of Gamma operators, recall some basic notations about the q-calculus, and construct λ-Gamma operators based on q-integers with q-Gamma function. We firstly prove quantitative q-Voronovskaja type theorems in terms of weighted modulus of continuity, and the q-GrüssVoronovskaja theorem in the quantitative mean is presented (for the quantitative q-Voronovskaja type theorem0 and the q-Grüss-Voronovskaja theorem for the other operators, see [12, 13])

Auxiliary Results

Local Approximation

Rate of Convergence

Weighted Approximation

Voronovskaja Type Theorems

D2qn ðgðxÞÞA ðf