A survey of sampling discretization of integral and uniform norms

TL;DR

This survey reviews recent advances in sampling discretization of integral and uniform norms for finite-dimensional function spaces, emphasizing techniques and results related to classical inequalities and their generalizations.

Contribution

It consolidates recent developments in sampling discretization, highlighting new techniques and generalizations beyond classical inequalities for finite-dimensional spaces.

Findings

Generalizes Marcinkiewicz-Zygmund inequalities for broader function spaces

Highlights key techniques used in recent proofs and results

Provides a comprehensive overview of current research trends

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}.