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).
Positivity – Springer Journals
Published: Apr 6, 2007
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