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arXiv · 2603.02459

A survey of sampling discretization of integral and uniform norms

F. Dai·E. Kosov·V. Temlyakov
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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}.

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F. Dai, E. Kosov, V. Temlyakov. 2026-03-02. A survey of sampling discretization of integral and uniform norms. https://arxiv.org/abs/2603.02459

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