Sampling Discretization of Integral Norms

The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. Even though this problem is extremely important in applications, its systematic study has begun only recently. In this paper we obtain a conditional theorem for all integral norms $$L_q$$ , $$1\le q<\infty $$ , which is an extension of known results for $$q=1$$ . To discretize the integral norms successfully, we introduce a new technique, which is a combination of a probabilistic technique with results on the entropy numbers in the uniform norm. As an application of the general conditional theorem, we derive a new Marcinkiewicz-type discretization for the multivariate trigonometric polynomials with frequencies from the hyperbolic crosses.