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A survey of sampling discretization of integral and uniform norms

F. Dai, E. Kosov, V. 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 \cite{DPTT, KKLT, LMT}.

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}.
Paper Structure (16 sections, 32 theorems, 223 equations)

This paper contains 16 sections, 32 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Connections with other areas
  3. Marcinkiewicz discretization of $L_2$ norms
  4. Characterization of (\ref{['4-1:2024']}) via spectral norms
  5. Discretization using random samples
  6. The minimum number of points required for $L_2$-discretization
  7. Marcinkiewicz discretization theorems
  8. A conditional theorem
  9. Preliminary discretizations
  10. Sampling discretization of $L_p$ norms
  11. Sampling discretization with weights
  12. Universal discretization
  13. Some improved bounds in sampling discretization of integral norms
  14. Sampling discretization in the uniform norm
  15. General bounds of $m(X_N;\infty; C)$ with $C$ being independent of $N$
  16. ...and 1 more sections

Key Result

Lemma 3.1

Let $\xi^1, \ldots, \xi^m\in \Omega$, $\lambda_1,\ldots, \lambda_m\in \mathbb{R}$ and $\epsilon\in (0, 1)$. Then $X_N$ admits a Marcinkiewicz-type discretization theorem of the form 4-1:2024 if and only if where $I$ denotes the $N\times N$ identity matrix, and $\|\cdot\|$ denotes the spectral norm.

Theorems & Definitions (54)

  • Definition 2.1: DKT
  • Lemma 3.1
  • Theorem 3.2: Te18
  • Remark 3.3
  • Lemma 3.4: Tr
  • Remark 3.5
  • proof : Proof of Theorem \ref{['Thm-4-1']}
  • Remark 3.6
  • Corollary 3.7
  • Corollary 3.8
  • ...and 44 more