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SAT 9 (2021)

Surveys in Approximation Theory, 10 (2026), 1-42.

A survey of sampling discretization of integral and uniform norms

Feng Dai, Egor D. Kosov, and Vladimir N. Temlyakov

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 [18, 32, 45].

E-print: arXiv:2603.02459

Published: 22 October 2026.

Feng Dai
Department of Mathematical and Statistical Sciences
University of Alberta
Edmonton, Canada
E-mail: fdai@ualberta.ca

Egor D. Kosov
Centre de Recerca Matemàtica
Campus de Bellaterra
Barcelona, Spain
E-mail: kosoved09@gmail.com

Vladimir N. Temlyakov
Steklov Mathematical Institute
Russian Academy of Sciences
Moscow, Russia
and
Department of Mechanics and Mathematics
Lomonosov Moscow State University
Moscow, Russia
and
Department of Mathematics
University of South Carolina
Columbia, USA
E-mail: temlyakovv@gmail.com