Some improved bounds in sampling discretization of integral norms

Abstract

The paper addresses a problem of sampling discretization of integral norms of elements of finite-dimensional subspaces satisfying some conditions. We prove sampling discretization results under a standard assumption formulated in terms of the Nikol'skii-type inequality. In particular, we obtain some upper bounds on the number of sample points sufficient for good discretization of the integral Lp norms, 1≤p<2, of functions from finite-dimensional subspaces of continuous functions. Our new results improve upon the known results in this direction. We use a new technique based on deep results of Talagrand from functional analysis.