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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2019, Vol. 36 ›› Issue (5): 551-556.doi: 10.3969/j.issn.1005-3085.2019.05.006

• • 上一篇    下一篇

$|x|^{\alpha} (1\leq \alpha <2)$ 在调整的正切节点组的有理逼近

程一元1,  张永全2,  查星星1   

  1. 1- 巢湖学院数学与统计学院，合肥  238000
    2- 浙江财经大学数据科学学院，杭州 310018
  • 收稿日期:2017-06-13 接受日期:2018-05-08 出版日期:2019-10-15 发布日期:2019-12-15
  • 通讯作者: 张永全 E-mail: zyqmath@163.com
  • 基金资助:
    国家自然科学基金(61573326)；安徽省高校优秀青年人才支持计划(gxyq2019082)；浙江省一流学科A类(浙江财经大学统计学)资助.

The Rational Approximation to $|x|^{\alpha} (1\leq \alpha<2)$ at the Adjusted Tangent Nodes

CHENG Yi-yuan1,  ZHANG Yong-quan2,  ZHA Xing-xing1   

  1. 1- School of Mathematics and Statistics, Chaohu College, Hefei 238000
    2- School of Data Sciences, Zhejiang University of Finance & Economics, Hangzhou 310018
  • Received:2017-06-13 Accepted:2018-05-08 Online:2019-10-15 Published:2019-12-15
  • Contact: Y. Zhang. E-mail address: zyqmath@163.com
  • Supported by:
    The National Natural Science Foundation of China (61573326); the Program in the Youth Elite Support Plan in Universities of Anhui Province (gxyq2019082); the First Class Discipline of Zhejiang-A (Zhejiang University of Finance and Economics-Statistics).

PDF (PC)

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摘要： 由于 Newman 有理算子对 $|x|$ 逼近效果较好，我们考虑 Newman-$\alpha$ 型有理算子对 $|x|^{\alpha}$ 的逼近．本文主要在结点组 $X=\{\tan^{2}\frac{k\pi}{4n}\}_{k=1}^{n}$ 情形下，讨论了 Newman-$\alpha$ 算子逼近 $|x|^{\alpha}$ 的收敛速度，最后得到确切的逼近阶为 $O(\frac{1}{n^{2\alpha}})$．该结果不仅包含了 $\alpha=1$ 时的逼近结果，而且优于结点组取作第一、二类 Chebyshev 结点组、等距结点组等情形时的结论．

关键词: 有理逼近, Newman-$\alpha$ 型有理算子, 逼近阶

Abstract: Since Newman's rational operator has a good approximation for $|x|$, we consider the approximation of $|x|^{\alpha}$ by a Newman-$\alpha$ rational operator. In this paper, we discuss the convergence rate of the operator Newman-$\alpha$ at the adjusted tangent nodes $X=\{\tan^{2}\frac{k\pi}{4n}\}_{k=1}^{n}$, and finally obtain the exact approximation order $O(\frac{1}{n^{2\alpha}})$. The result not only contains the approximation result in the case of $\alpha=1$, but it is better than the conclusion when the node group is selected for the first and the second type of Chebyshev nodes, equidistant nodes etc.

Key words: rational approximation, Newman-$\alpha$ type rational operator, approximation order

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