数学物理学报(英文版) ›› 2012, Vol. 32 ›› Issue (2): 471-482.doi: 10.1016/S0252-9602(12)60030-7

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MULTIVARIATE WEIGHTED BERNSTEIN-TYPE INEQUALITY AND ITS APPLICATIONS

曹飞龙|林绍波   

  1. Department of Information and Mathematics Sciences, China Jiliang University, Hangzhou 310018, China; Institute of Information and System Sciences, Xi’an Jiaotong University, Xi’an 710049, China
  • 收稿日期:2008-10-08 修回日期:2011-02-05 出版日期:2012-03-20 发布日期:2012-03-20
  • 基金资助:

    The research was supported by the National Natural Science Foundation of China (90818020, 60873206) and the Foundation of Innovation Team of Science and Technology of Zhejiang Province of China (2009R50024).

MULTIVARIATE WEIGHTED BERNSTEIN-TYPE INEQUALITY AND ITS APPLICATIONS

 CAO Fei-Long, LIN Shao-Bo   

  1. Department of Information and Mathematics Sciences, China Jiliang University, Hangzhou 310018, China; Institute of Information and System Sciences, Xi’an Jiaotong University, Xi’an 710049, China
  • Received:2008-10-08 Revised:2011-02-05 Online:2012-03-20 Published:2012-03-20
  • Supported by:

    The research was supported by the National Natural Science Foundation of China (90818020, 60873206) and the Foundation of Innovation Team of Science and Technology of Zhejiang Province of China (2009R50024).

摘要:

Bernstein inequality played an important role in approximation theory and Fourier analysis. This article first introduces a general system of functions and the so-called multivariate weighted Bernstein, Nikol’skiÏ, and Ul’yanov-type inequalities. Then, the relations among these three inequalities are discussed. Namely, it is proved that a family of functions equipped with Bernstein-type inequality satisfies Nikol’skiÏ-type and Ul’yanov-type inequality. Finally, as applications, some classical inequalities are deduced from the obtained results.

关键词: Bernstein-type inequality, Nikol’skiÏ-type inequality, Ul’yanov-type inequal-ity, approximation

Abstract:

Bernstein inequality played an important role in approximation theory and Fourier analysis. This article first introduces a general system of functions and the so-called multivariate weighted Bernstein, Nikol’skiÏ, and Ul’yanov-type inequalities. Then, the relations among these three inequalities are discussed. Namely, it is proved that a family of functions equipped with Bernstein-type inequality satisfies Nikol’skiÏ-type and Ul’yanov-type inequality. Finally, as applications, some classical inequalities are deduced from the obtained results.

Key words: Bernstein-type inequality, Nikol’skiÏ-type inequality, Ul’yanov-type inequal-ity, approximation

中图分类号: 

  • 41A17