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Sampling discretization of the uniform norm and applications

E. D. Kosov, V. N. Temlyakov

TL;DR

This work addresses discretization of the uniform norm for finite-dimensional subspaces, seeking polynomial-in-$N$ sampling instead of the classical exponential sampling required by general bounds. It develops two complementary strategies: weakening the discretization inequality via $N$-dependent factors and imposing structural conditions such as Nikol\'skii inequalities or entropy bounds, with both routes leading to polynomially many sampling points $m$. The paper establishes connections between discretization and bilinear approximations of the Dirichlet kernel, weighted discretization via the Christoffel function, and large-$p$ norm discretization to derive Remez-type inequalities for multivariate trigonometric polynomials. These results yield new Bernstein-type discretization theorems, unconditional discretization bounds, and practical implications for extremal problems in harmonic analysis, including Remez-type inequalities and sampling on frequency-limited subspaces. The work synthesizes classic and modern tools (Dirichlet kernels, de la Vallée Poussin kernels, entropy numbers, and KW-type results) to advance quantitative discretization theory with broad applicability in analysis and approximation.

Abstract

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. Previous known results show that for any $N$-dimensional subspace of the space of continuous functions it is sufficient to use $e^{CN}$ sample points for an accurate upper bound for the uniform norm by the discrete norm and that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. In this paper we focus on two types of results, which allow us to obtain good discretization of the uniform norm with polynomial in $N$ number of points. In the first way we weaken the discretization inequality by allowing a bound of the uniform norm by the discrete norm multiplied by an extra factor, which may depend on $N$. In the second way we impose restrictions on the finite dimensional subspace under consideration. In particular, we prove a general result, which connects the upper bound on the number of sampling points in the discretization theorem for the uniform norm with the best $m$-term bilinear approximation of the Dirichlet kernel associated with the given subspace.

Sampling discretization of the uniform norm and applications

TL;DR

This work addresses discretization of the uniform norm for finite-dimensional subspaces, seeking polynomial-in- sampling instead of the classical exponential sampling required by general bounds. It develops two complementary strategies: weakening the discretization inequality via -dependent factors and imposing structural conditions such as Nikol\'skii inequalities or entropy bounds, with both routes leading to polynomially many sampling points . The paper establishes connections between discretization and bilinear approximations of the Dirichlet kernel, weighted discretization via the Christoffel function, and large- norm discretization to derive Remez-type inequalities for multivariate trigonometric polynomials. These results yield new Bernstein-type discretization theorems, unconditional discretization bounds, and practical implications for extremal problems in harmonic analysis, including Remez-type inequalities and sampling on frequency-limited subspaces. The work synthesizes classic and modern tools (Dirichlet kernels, de la Vallée Poussin kernels, entropy numbers, and KW-type results) to advance quantitative discretization theory with broad applicability in analysis and approximation.

Abstract

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. Previous known results show that for any -dimensional subspace of the space of continuous functions it is sufficient to use sample points for an accurate upper bound for the uniform norm by the discrete norm and that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. In this paper we focus on two types of results, which allow us to obtain good discretization of the uniform norm with polynomial in number of points. In the first way we weaken the discretization inequality by allowing a bound of the uniform norm by the discrete norm multiplied by an extra factor, which may depend on . In the second way we impose restrictions on the finite dimensional subspace under consideration. In particular, we prove a general result, which connects the upper bound on the number of sampling points in the discretization theorem for the uniform norm with the best -term bilinear approximation of the Dirichlet kernel associated with the given subspace.
Paper Structure (6 sections, 28 theorems, 154 equations)

This paper contains 6 sections, 28 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Discretization of the uniform norm under the Nikol'skii inequality assumption
  3. Discretization and bilinear approximations
  4. Some results on the Bernstein discretization problem
  5. Remez-type inequalities for the multivariate trigonometric polynomials
  6. Some comments

Key Result

Theorem 1.1

Let $X_N$ be an $N$-dimensional subspace of ${\mathcal{C}}(\Omega)$. There exists a set $\xi=\{\xi^\nu\}_{\nu=1}^m$ of $m\le 9^N$ points such that for any $f\in X_N$ we have

Theorems & Definitions (46)

  • Theorem 1.1: KKT
  • Theorem 1.2: VT168,DPTT
  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.3
  • proof
  • Corollary 2.1
  • ...and 36 more