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 L n2 , 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).
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Partially supported by an Australian Research Council Discovery grant.
This author holds the Canada Research Chair in Geometric Analysis.
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Guédon, O., Mendelson, S., Pajor, A. et al. Subspaces and Orthogonal Decompositions Generated by Bounded Orthogonal Systems. Positivity 11, 269–283 (2007). https://doi.org/10.1007/s11117-006-2059-1
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DOI: https://doi.org/10.1007/s11117-006-2059-1