Approximation properties of weighted Kantorovich type operators in compact disks

Nazim I Mahmudov,Mustafa Kara

doi:10.1186/s13660-015-0571-1

Nazim I Mahmudov, Mustafa Kara

Open Access

https://doi.org/10.1186/s13660-015-0571-1

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Abstract

In this paper, we discuss the approximation properties of the complex weighted Kantorovich type operators. Quantitative estimates of the convergence, the Voronovskaja type theorem, and saturation of convergence for complex weighted Kantorovich polynomials attached to analytic functions in compact disks will be given. In particular, we show that for functions analytic in $\{ z\in C:\vert z\vert < R \} $ , the rate of approximation by the weighted complex Kantorovich type operators is $1/n$ .

Highlights

1 Introduction The first constructive proof of Weierstrass approximation theorem was given by Bernstein [ ]
He gave an alternative proof to the Weierstrass approximation theorem and introduced the following polynomial: n
The theorem gives a Voronovskaja type result in compact disks, for complex weighted Kantorovich type operators attached to an analytic function in DR, where R >, and with center

Summary

Introduction

The first constructive (and simple) proof of Weierstrass approximation theorem was given by Bernstein [ ]. He gave an alternative proof to the Weierstrass approximation theorem and introduced the following polynomial: n. For any f ∈ Lp([ , ]), ≤ p ≤ ∞, Ditzian and Totik (see [ ]) introduced the Kantorovich-. Let w(x) = xα( – x)β , α, β > – , ≤ x ≤ , be the classical Jacobi weights. In [ ], Ditzian and Totik studied the case of weighted approximation properties of Kn(f ; x) in

Lpw under the restrictions

Secondly using the known inequality m

We apply