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One-sided discretization inequalities and sampling recovery

Irina Limonova, Yuri Malykhin, Vladimir Temlyakov

TL;DR

The paper surveys one-sided discretization inequalities—Left Discretization Inequality (LDI) and Right Discretization Inequality (RDI)—and their use in sampling recovery. It develops a matrix-analytic framework linking discretization properties to submatrix norms, establishes lower bounds on the sample size $m$, and discusses when one-sided conditions suffice for recovery guarantees. It also covers universal discretization and nonlinear recovery via $v$-term approximations, providing both conditional and unconditional results under Nikol’skii-type inequalities and related assumptions. The work highlights the interplay between discretization, approximation theory, and recovery performance, with emphasis on how one-sided bounds can drive effective sampling strategies in finite-dimensional function spaces. Overall, it consolidates key methods and results, identifies gaps and open problems, and connects classical discretization theory with contemporary sampling-recovery approaches.

Abstract

Recently, in a number of papers it was understood that results on sampling discretization and on the universal sampling discretization can be successfully used in the problem of sampling recovery. Moreover, it turns out that it is sufficient to only have a one-sided discretization inequality for some of those applications. This motivates us to write the present paper as a survey/research paper with the focus on the one-sided discretization inequalities and their applications in the sampling recovery. In this sense the paper complements the two existing survey papers on sampling discretization.

One-sided discretization inequalities and sampling recovery

TL;DR

The paper surveys one-sided discretization inequalities—Left Discretization Inequality (LDI) and Right Discretization Inequality (RDI)—and their use in sampling recovery. It develops a matrix-analytic framework linking discretization properties to submatrix norms, establishes lower bounds on the sample size , and discusses when one-sided conditions suffice for recovery guarantees. It also covers universal discretization and nonlinear recovery via -term approximations, providing both conditional and unconditional results under Nikol’skii-type inequalities and related assumptions. The work highlights the interplay between discretization, approximation theory, and recovery performance, with emphasis on how one-sided bounds can drive effective sampling strategies in finite-dimensional function spaces. Overall, it consolidates key methods and results, identifies gaps and open problems, and connects classical discretization theory with contemporary sampling-recovery approaches.

Abstract

Recently, in a number of papers it was understood that results on sampling discretization and on the universal sampling discretization can be successfully used in the problem of sampling recovery. Moreover, it turns out that it is sufficient to only have a one-sided discretization inequality for some of those applications. This motivates us to write the present paper as a survey/research paper with the focus on the one-sided discretization inequalities and their applications in the sampling recovery. In this sense the paper complements the two existing survey papers on sampling discretization.
Paper Structure (13 sections, 32 theorems, 158 equations)

This paper contains 13 sections, 32 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. The Marcinkiewicz-type discretization problem.
  3. The Bernstein-type discretization problem.
  4. One-sided discretization problems.
  5. The structure of the paper.
  6. Right Discretization Inequalities (RDI)
  7. RDI with injective sampling operator.
  8. Left Discretization Inequalities (LDI)
  9. Matrix setting
  10. An application to sampling recovery
  11. A comment on the sampling recovery in the uniform norm.
  12. Universal discretization and nonlinear recovery
  13. Some remarks on the discretization of the uniform norm

Key Result

Lemma 2.1

Let $2<p<\infty$, $1\leq q<\infty$ and let $X_N\in\mathrm{NI}(2,p,M)$ with some constant $M$. Assume that $\{\xi^j\}_{j=1}^m\subset \Omega$ and nonnegative weights $\{\lambda_j\}_{j=1}^m$ are such that for any $f\in X_N$ we have Then for any orthonormal basis $\{u_i\}_{i=1}^N$ of $X_N$ and any $j\in \{1,\ldots,m\}$ we have

Theorems & Definitions (56)

  • Lemma 2.1
  • proof
  • Remark 2.1
  • Proposition 2.1
  • Corollary 2.1
  • Remark 2.2
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.3
  • ...and 46 more