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）积分范数的采样离散化研究中取得了重大进展。研究表明，采样离散化结果在诸多应用中具有重要价值，尤其在采样恢复问题中展现出实用价值。典型的采样离散化研究成果通常能证明存在用于离散化的优质采样点。本文的主要目的是证明：在通用离散化问题中，定义域上根据给定概率测度独立同分布生成的随机点，能以高概率成为优质采样点。此外，我们还将展示基于通用离散化优质采样点的简单贪婪型算法，能够在平方范数下实现良好的信号恢复效果。