## Approximation Properties of a New Bernstein-Bézier Operators with Parameters

Qi Qiulan,, Guo Dandan

 基金资助: 国家自然科学基金.  11871191河北省教育厅基金.  ZD2019053河北师范大学基金.  L2020Z03

 Fund supported: the NSFC.  11871191the Scientific Research Fund of Hebei Provincial Education Department.  ZD2019053the NSF of Hebei Normal University.  L2020Z03

Abstract

In this paper, a new generalized Bernstein-Bézier type operators is constructed. The estimates of the moments of these operators are investigated. The rate of convergence in terms of modulus of continuity is given. Then, the equivalent theorem of these operators is studied.

Keywords： Bernstein-Bézier type operators ; Convergence theorem ; Modulus of continuity ; Cauchy-Schwarz inequality

Qi Qiulan, Guo Dandan. Approximation Properties of a New Bernstein-Bézier Operators with Parameters. Acta Mathematica Scientia[J], 2021, 41(3): 583-594 doi:

## 1 引言

$B(\cdot , \cdot)$为Beta函数.

2017年, Ren[10]又引入参数$\alpha$, 改进了上述算子为$E_{n, \beta}^{\alpha}(f;x)$, 并研究了这些算子逼近的Jackson型定理. 为了得到该类算子逼近的Bernstein型定理, 我们对这类算子再次进行了修正, 本文所研究的新算子定义如下: 设$\beta\in[0, 1], \alpha\geq1, n\geq 2,$

## 3 定理的证明

$x\in [0, \frac{1}{n})$时, 有

$$$\varphi^{\lambda}(x)\cdot |I_{1}|\leq \alpha\varphi^{\lambda-1}\sqrt{n}\|f\|\cdot\sqrt{n}\cdot\sqrt{x(1-x)}\leq\alpha\varphi^{\lambda-1}(x)\sqrt{n}\|f\|.$$$

$x\in (1-\frac{1}{n}, 1]$时, 有

$$$\varphi^{\lambda}(x)\cdot |I_{1}|\leq \alpha\varphi^{\lambda-1}\sqrt{n}\|f\|\cdot\sqrt{n}\cdot\sqrt{x(1-x)}\leq\alpha\varphi^{\lambda-1}(x)\sqrt{n}\|f\|.$$$

$x\in E_{n} = [\frac{1}{n}, 1-\frac{1}{n}]$时, 有

$$$|I_{1}| = |-\alpha f(0)J_{n, 1}^{\alpha-1}(x)J_{n, 1}'(x)|\leq|-\alpha f(0)\cdot[1-p_{n, 0}(x)]'| = |\alpha f(0)\cdot p_{n, 0}'(x)|,$$$

$\begin{eqnarray} \varphi^{\lambda}(x)\cdot |I_{1}| \leq\alpha\|f\|\cdot\sum\limits^{n}_{k = 0}\left|p_{n, k}'(x)\right|\cdot\varphi^{\lambda-1}(x) \leq\alpha\varphi^{\lambda-1}(x)\sqrt{n}\|f\|. \end{eqnarray}$

$x\in [0, \frac{1}{n})\bigcup(1-\frac{1}{n}, 1]$时, 用估计(4.2)式和(4.3)式的方法可得

$$$\varphi^{\lambda}(x)\cdot |I_{3}|\leq\alpha\varphi^{\lambda-1}(x)\sqrt{n}\|f\|.$$$

$x\in E_{n} = [\frac{1}{n}, 1-\frac{1}{n}]$时, 结合$|p_{n, n}'(x)|\leq \sum\limits^{n}_{k = 0}|p_{n, k}'(x)|$, 用估计(4.5)式的方法也可得相应于(4.6)式的结果.

$$$L_{1}\leq L_{3}+5\leq 8,$$$

$$$|\varphi^{\lambda}(x)H_{2}|\leq 96\alpha\|\varphi^{\lambda} f'\|.$$$

(Ⅱ) 当$x\in E_{n} = [\frac{1}{n}, 1-\frac{1}{n}]$时, 注意到$p_{n, k}'(x) = \frac{n}{\varphi^{2}(x)}(\frac{k}{n}-x)\cdot p_{n, k}(x)$, 应用Cauchy-Schwarz, 不等式可得

$x\in E_{n}^{C}$时, 用相同的方法可得$A\leq 4+B\leq 9,$那么

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