Sampling Discretization of Integral Norms

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