Subspaces and orthogonal decompositions generated by bounded orthogonal systems

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 L2 , 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). 0 Introduction In this note we consider a space L2 of functions on a probability space and we investigate properties of its subspaces spanned by a finite subset of an orthonormal system which consists of functions bounded in L∞. Typical examples of such systems are the trigonometric and the Walsh systems. The question we study is whether there is a subspace spanned by a large subset of the orthonormal system on which the L2 and Lp norms are close. The two cases we focus on are when p > 2 and p = 1. Formally we address 2000 MSC-classification: 46B07, 41A45, 94B75, 52B05, 62G99 Partially supported by an Australian Research Council Discovery grant. This author holds the Canada Research Chair in Geometric Analysis.