论文标题
对整体规范进行离散化
Sampling discretization of integral norms
论文作者
论文摘要
该论文致力于从给定有限维尺寸的子空间中离散功能的积分规范。即使该问题在应用中非常重要，但其系统的研究最近已经开始。在本文中，我们获得了所有积分规范$ l_q $，$ 1 \ le q <\ infty $的有条件定理，这是已知结果的扩展，以$ q = 1 $。为了成功地离散整体规范，我们引入了一种新技术，这是概率技术与统一规范中熵数的结果的组合。作为一般条件定理的应用，我们为多元三角多项式多项式提供了新的Marcinkiewicz类型离散化，并带有来自双曲线交叉的频率。
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 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 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.
