Subspaces and Orthogonal Decompositions Generated by Bounded Orthogonal Systems

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• 作者：Olivier Guédon ; Shahar Mendelson ; Alain Pajor and Nicole Tomczak-Jaegermann
• 关键词：Empirical processes ; generic chaining ; bounded orthogonal systems ; orthogonal decompositions
• 刊名：Positivity
• 出版年：2007
• 出版时间：May, 2007
• 年：2007
• 卷：11
• 期：2
• 页码：269-283
• 全文大小：237 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Fourier Analysis
  Operator Theory
  Potential Theory
  Calculus of Variations and Optimal Control
  Econometrics
  
• 出版者：Birkh盲user Basel
• ISSN：1572-9281

文摘

We investigate properties of subspaces of L₂ spanned by subsets of a finite orthonormal system bounded in the L_∞ norm. We first prove that there exists an arbitrarily large subset of this orthonormal system on which the L₁ and the L₂ norms are close, up to a logarithmic factor. Considering for example the Walsh system, we deduce the existence of two orthogonal subspaces of L₂n, complementary to each other and each of dimension roughly n/2, spanned by ± 1 vectors (i.e. Kashin’s splitting) and in logarithmic distance to the Euclidean space. The same method applies for p > 2, and, in connection with the Λ_p problem (solved by Bourgain), we study large subsets of this orthonormal system on which the L₂ and the L_p norms are close (again, up to a logarithmic factor).