Subspaces and Orthogonal Decompositions Generated by Bounded Orthogonal Systems |
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| Authors: | Olivier Guédon Shahar Mendelson Alain Pajor Nicole Tomczak-Jaegermann |
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| Institution: | (1) Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, Paris 6, 4 place Jussieu, 75005 Paris, France;(2) Centre for Mathematics and its Applications, The Australian National University, Canberra, ACT, 0200, Australia;(3) Department of Mathematics, Technion I.I.T, Haifa, 3200, Israel;(4) équipe d'Analyse et Mathématiques Appliquées, Université de Marne-la-Vallée, 5 boulevard Descartes, Champs sur Marne, 77454, Marne-la-Vallée Cedex 2, France;(5) Dept. of Math. and Stat. Sciences, University of Alberta, Edmonton, T6G 2G1, Alberta, Canada |
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| Abstract: | 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 Λp problem (solved by Bourgain), we study large subsets of this orthonormal system on which the L2 and the Lp norms are close (again, up to a logarithmic factor).
Partially supported by an Australian Research Council Discovery grant.
This author holds the Canada Research Chair in Geometric Analysis. |
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| Keywords: | 46B07 46B09 42A05 42A61 94B75 62G99 |
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