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RELATIVE WIDTH OF SMOOTH CLASSES OF MULTIVARIATE PERIODIC FUNCTIONS WITH RESTRICTIONS ON ITERATED LAPLACE DERIVATIVES IN THE L2-METRIC

刘永平; 杨连红   

  1. 北京师范大学数学学院, 北京 100875
  • 收稿日期:2004-11-30 修回日期:1900-01-01 出版日期:2006-10-20 发布日期:2006-10-20
  • 通讯作者: 刘永平
  • 基金资助:

    Supported partly by National Natural Science Foundation of China (10471010) and partly by the project ``Representation Theory and Related Topics"
    of the ``985 Program" of Beijing Normal University

RELATIVE WIDTH OF SMOOTH CLASSES OF MULTIVARIATE PERIODIC FUNCTIONS WITH RESTRICTIONS ON ITERATED LAPLACE DERIVATIVES IN THE L2-METRIC

Liu Yongping; Yang Lianhong   

  1. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
  • Received:2004-11-30 Revised:1900-01-01 Online:2006-10-20 Published:2006-10-20
  • Contact: Liu Yongping

摘要:

For two subsets W and V of a Banach space X, let Kn(W,V,X) denote the relative Kolmogorov n-width of W relative to V defined by

Kn(W,V,X):= infLn supf∈W inf g∈ V∩ Ln || f-g||X,
where the infimum is taken over all n-dimensional linear subspaces Ln of X. Let W2(△r) denote the class of 2π-periodic functions f with d-variables satisfying
[-π,π]d |△r f(x)|2,dx≤ 1,
while △r is the r-iterate of Laplace operator △. This article discusses the relative Kolmogorov n-width of W2(△r)$ relative to W2(△r) in Lq([-π,π]d),(1≤ q≤∞), and obtain its weak asymptotic result.

关键词: Multivariate function classes, width, relative width

Abstract:

For two subsets W and V of a Banach space X, let Kn(W,V,X) denote the relative Kolmogorov n-width of W relative to V defined by

Kn(W,V,X):= infLn supf∈W inf g∈ V∩ Ln || f-g||X,
where the infimum is taken over all n-dimensional linear subspaces Ln of X. Let W2(△r) denote the class of 2π-periodic functions f with d-variables satisfying
[-π,π]d |△r f(x)|2,dx≤ 1,
while △r is the r-iterate of Laplace operator △. This article discusses the relative Kolmogorov n-width of W2(△r)$ relative to W2(△r) in Lq([-π,π]d),(1≤ q≤∞), and obtain its weak asymptotic result.

Key words: Multivariate function classes, width, relative width

中图分类号: 

  • 41A46