The B\xe9zier variant of Kantorovich type \u03bb-Bernstein operators

Qing-Bo Cai

doi:10.1186/s13660-018-1688-9

Abstract

In this paper, we introduce the Bézier variant of Kantorovich type λ-Bernstein operators with parameter lambdain[-1,1]. We establish a global approximation theorem in terms of second order modulus of continuity and a direct approximation theorem by means of the Ditzian–Totik modulus of smoothness. Finally, we combine the Bojanic–Cheng decomposition method with some analysis techniques to derive an asymptotic estimate on the rate of convergence for some absolutely continuous functions.

Highlights

We propose the Kantorovich type λ-Bernstein operators n k+1 n+1
We have presented a Bézier variant of Kantorovich type λ-Bernstein operators Ln,λ,α(f ; x), and established approximation theorems by using the usual second order modulus of smoothness and the Ditzian–Totik modulus of smoothness
From Theorem 3.3 of Sect. 3, we know that the rate of convergence of operators Ln,λ,α(f ; x) for f ∈ DB

Summary

Introduction

In 1912, Bernstein [1] proposed the famous polynomials, nowadays called Bernstein polynomials, to prove the Weierstrass approximation theorem as follows: nk. And Bernstein basis functions bn,k(x) are defined as follows: bn,k(x) =. In 2010, Ye et al [14] defined the following new Bernstein bases with shape parameter λ:. We propose the Kantorovich type λ-Bernstein operators n k+1 n+1. K k=0 n+1 and the Bézier variant of Kantorovich type λ-Bernstein operators n k+1. Lemma 2.2 Let ei = ti, i = 0, 1, 2, and n > 1, for the Kantorovich type λ-Bernstein operators Kn,λ(f ; x), we have the following equalities: Kn,λ(e0; x) = 1;. Lemma 2.4 For the Bézier variant of Kantorovich type λ-Bernstein operators Ln,λ,α(f ; x) and f ∈ C[0,1] with the sup-norm f := supx∈[0,1] |f (x)|, we have.

Main results

Conclusion