Majorizing measures and proportional subsets of bounded orthonormal systems

We study some natural empirical processes determined by uniformly convex Banach spaces with modulus of convexity of power type 2. Results of this kind were extensively studied in a Hilbertian setting, and became an important tool for investigations, for example, of the behaviour of various random sets of vectors (as in [14, 15, 7, 16, 2, 6]). We then apply these results to address a problem of selecting a subset of a bounded orthonormal system (for example, a set of characters), in the spirit of a result of Bourgain (see [20]) and of Talagrand [20], that also has applications to a sparse reconstruction ([2, 16, 6]). A particular case of this result, formulated in Theorem 3, is a so called Kashin’s splitting of a set of 2k orthonormal vectors in L2 that is bounded in L∞.