Subspaces and Orthogonal Decompositions Generated by Bounded Orthogonal Systems

Subspaces and Orthogonal Decompositions Generated by Bounded Orthogonal Systems We investigate properties of subspaces of L 2 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 L 1 and the L 2 norms are close, up to a logarithmic factor. Considering for example the Walsh system, we deduce the existence of two orthogonal subspaces of L 2 n , 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 L 2 and the L p norms are close (again, up to a logarithmic factor). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Subspaces and Orthogonal Decompositions Generated by Bounded Orthogonal Systems

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Publisher
Springer Journals
Copyright
Copyright © 2007 by Birkhäuser Verlag, Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-006-2059-1
Publisher site
See Article on Publisher Site

Abstract

We investigate properties of subspaces of L 2 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 L 1 and the L 2 norms are close, up to a logarithmic factor. Considering for example the Walsh system, we deduce the existence of two orthogonal subspaces of L 2 n , 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 L 2 and the L p norms are close (again, up to a logarithmic factor).

Journal

PositivitySpringer Journals

Published: Apr 6, 2007

References

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