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.

本文致力于从给定的有限维子空间离散化函数的积分范数。尽管这个问题在应用中极为重要，但它的系统研究才刚刚开始。在本文中，我们获得了所有积分范数\（L_q \），\（1 \ le q <\ infty \）的条件定理，这是\（q = 1 \）的已知结果的扩展。为了成功地使积分范数离散化，我们引入了一种新技术，该技术是将概率技术与统一范数中的熵数相结合的结果。作为一般条件定理的应用，我们针对双曲交叉频率的多元三角多项式推导了新的Marcinkiewicz型离散化。