A survey of sampling discretization of integral and uniform norms
Abstract
This paper surveys recent developments in the sampling discretization of integral and uniform norms for functions in general finite-dimensional spaces. These results generalize the classical Marcinkiewicz-Zygmund inequalities for trigonometric and algebraic polynomials, which play a crucial role in Fourier analysis, interpolation, and approximation theory. We focus on the problem in the broad context of finite-dimensional subspaces, where norms defined by general probability measures are approximated by their discrete counterparts. The primary emphasis is on results closely related to the authors' recent research. A key objective is to highlight the main ideas and techniques that form the foundation of the proofs in this area. This survey serves as a complement to three recently published survey papers on sampling discretization \cite{DPTT, KKLT, LMT}.
Keywords
Cite
@article{arxiv.2603.02459,
title = {A survey of sampling discretization of integral and uniform norms},
author = {F. Dai and E. Kosov and V. Temlyakov},
journal= {arXiv preprint arXiv:2603.02459},
year = {2026}
}