Recently, there was a big progress in studying sampling discretization of integral norms of finite dimensional subspaces and collections of such subspaces (universal discretization). It was established that sampling discretization results are useful in a number of applications. In particular, they turn out to be useful in sampling recovery. Typically, recent sampling discretization results provide existence of good points for discretization. The main goal of this paper is to show that in the problem of universal discretization the independent random points on a given domain that are identically distributed according to the given probabilistic measure provide good points with high probability. Also, we demonstrate that a simple greedy type algorithm based on good points for universal discretization provide good recovery in the square norm.

翻译：近期，在有限维子空间及其集合（通用离散化）的积分范数采样离散化研究中取得了重大进展。已有研究表明，采样离散化结果在诸多应用中具有重要价值，特别是在采样恢复中展现出实用性。通常，最新的采样离散化结果证明了离散化优质点的存在性。本文的主要目标是证明：在通用离散化问题中，给定域上依据给定概率测度独立同分布的随机点，能以高概率提供优质离散化点。此外，我们还证明了一种基于通用离散化优质点的简单贪心类型算法，能在平方范数下实现良好的恢复效果。