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Sampling discretization in Orlicz spaces

Egor Kosov, Sergey Tikhonov

TL;DR

This work develops a comprehensive framework for discretizing integral Orlicz norms on finite-dimensional subspaces of continuous functions. By combining Nikolskii-type inequalities, entropy numbers, and Talagrand-type chaining (via GMPT07), the authors prove that, for a broad class of ${\bf \Phi}$-functions $\Phi$, a number of sampling points $m$ of order $\Phi((BN)^{1/\min\{p,2\}})(\log BN)^3$ suffices to preserve the $\|\cdot\|_{\Phi}$-norm on an $N$-dimensional subspace up to a relative error $\varepsilon$. They extend these results to one-sided discretization for arbitrary subspaces, develop weighted discretization schemes, and apply the findings to sampling recovery, deriving upper bounds on sampling numbers in Orlicz norms in terms of Kolmogorov widths, with explicit forms for model Phi-functions such as ${\Phi_{p,\alpha,\beta}}$ and ${\Phi_q}$. The results provide a bridge between continuous Orlicz norms and discrete sampling schemes, enabling effective reconstruction and error control in norms near $L^2$ and beyond. Overall, the paper advances the theory of discretization in Orlicz spaces and offers practical implications for sampling and recovery in high-dimensional approximation.

Abstract

We obtain new sampling discretization results in Orlicz norms on finite dimensional spaces. As applications, we study sampling recovery problems, where the error of the recovery process is calculated with respect to different Orlicz norms. In particular, we are interested in the recovery by linear methods in the norms close to $L^2$.

Sampling discretization in Orlicz spaces

TL;DR

This work develops a comprehensive framework for discretizing integral Orlicz norms on finite-dimensional subspaces of continuous functions. By combining Nikolskii-type inequalities, entropy numbers, and Talagrand-type chaining (via GMPT07), the authors prove that, for a broad class of -functions , a number of sampling points of order suffices to preserve the -norm on an -dimensional subspace up to a relative error . They extend these results to one-sided discretization for arbitrary subspaces, develop weighted discretization schemes, and apply the findings to sampling recovery, deriving upper bounds on sampling numbers in Orlicz norms in terms of Kolmogorov widths, with explicit forms for model Phi-functions such as and . The results provide a bridge between continuous Orlicz norms and discrete sampling schemes, enabling effective reconstruction and error control in norms near and beyond. Overall, the paper advances the theory of discretization in Orlicz spaces and offers practical implications for sampling and recovery in high-dimensional approximation.

Abstract

We obtain new sampling discretization results in Orlicz norms on finite dimensional spaces. As applications, we study sampling recovery problems, where the error of the recovery process is calculated with respect to different Orlicz norms. In particular, we are interested in the recovery by linear methods in the norms close to .
Paper Structure (17 sections, 26 theorems, 241 equations)

This paper contains 17 sections, 26 theorems, 241 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: Orlicz space and chaining bounds
  3. $\Phi$-functions
  4. Orlicz space
  5. Nikolskii's inequality
  6. Entropy numbers and chaining bound
  7. Discretization with equal weights for $\Phi\in {\rm (aInc)}_p\cap{\rm (Dec)}$
  8. Main theorems and discussion
  9. Proof of Theorems \ref{['cor4.1']} and \ref{['t3.2']}: the key step
  10. Proof of Theorem \ref{['cor4.1']}: the case $p\geqslant 2$
  11. Proof of Theorem \ref{['cor4.1']}: the case $p\in (1, 2)$
  12. Proof of Theorem \ref{['t3.2']}
  13. Descritization under relaxed assumptions on ${\bf \Phi}$-functions
  14. One-sided weighted discretization for an arbitrary subspace
  15. Bounds of integral Orlicz norms by discrete Orlicz norms
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $p\in [1, \infty)$, $\varepsilon\in(0, 1)$. Let $\Phi$ be a ${\bf \Phi}$-function such that the function $t\mapsto \Phi(t)t^{-p}$ is increasing and assume that there is $q\in[p, \infty)$ such that the function $t\mapsto \Phi(t)t^{-q}$ is decreasing. There is a constant $C_{\Phi, \varepsilon}>0$ and a subset ${\bf x}:=\{x_1, \ldots, x_m\}\subset \Omega$ of cardinality $m$ such that Here $\|\c

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5: see HH19
  • Lemma 2.6
  • Lemma 2.7
  • ...and 50 more