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A note on sampling recovery of multivariate functions in the uniform norm

Kateryna Pozharska, Tino Ullrich

TL;DR

The recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm is studied to obtain preasymptotic estimates for the corresponding sampling numbers and a relation to the corresponding Kolmogorov numbers is pointed out.

Abstract

We study the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. Our main interest is to obtain preasymptotic estimates for the corresponding sampling numbers. We obtain results in terms of the decay of related singular numbers of the compact embedding into $L_2(D,\varrho_D)$ multiplied with the supremum of the Christoffel function of the subspace spanned by the first $m$ singular functions. Here the measure $\varrho_D$ is at our disposal. As an application we obtain near optimal upper bounds for the sampling numbers for periodic Sobolev type spaces with general smoothness weight. Those can be bounded in terms of the corresponding benchmark approximation number in the uniform norm, which allows for preasymptotic bounds. By applying a recently introduced sub-sampling technique related to Weaver's conjecture we mostly lose a $\sqrt{\log n}$ and sometimes even less. Finally we point out a relation to the corresponding Kolmogorov numbers.

A note on sampling recovery of multivariate functions in the uniform norm

TL;DR

The recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm is studied to obtain preasymptotic estimates for the corresponding sampling numbers and a relation to the corresponding Kolmogorov numbers is pointed out.

Abstract

We study the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. Our main interest is to obtain preasymptotic estimates for the corresponding sampling numbers. We obtain results in terms of the decay of related singular numbers of the compact embedding into multiplied with the supremum of the Christoffel function of the subspace spanned by the first singular functions. Here the measure is at our disposal. As an application we obtain near optimal upper bounds for the sampling numbers for periodic Sobolev type spaces with general smoothness weight. Those can be bounded in terms of the corresponding benchmark approximation number in the uniform norm, which allows for preasymptotic bounds. By applying a recently introduced sub-sampling technique related to Weaver's conjecture we mostly lose a and sometimes even less. Finally we point out a relation to the corresponding Kolmogorov numbers.

Paper Structure

This paper contains 11 sections, 13 theorems, 133 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Reproducing kernel Hilbert spaces
  3. Least squares approximation in the uniform norm
  4. Improved bounds for sampling numbers
  5. Sampling and Kolmogorov numbers
  6. The power of standard information in the uniform norm
  7. Examples
  8. Sobolev type spaces
  9. Sobolev Hilbert spaces with mixed smoothness
  10. A univariate example
  11. Acknowledgment.

Key Result

Theorem 3.1

Let $H(K)$ be a reproducing kernel Hilbert space of complex-valued functions defined on a compact domain $D \subset \mathds{R}^d$ with a continuous and bounded kernel $K(\mathbf{x}, \mathbf{y})$, $K\colon D\times D \rightarrow \mathds{C}$, i.e., $\|K\|_{\infty} := \sup_{\mathbf{x} \in D} \sqrt{K(\ma with $r>1$, the reconstruction operator $\widetilde{S}_{\mathbf{X}}^m$ (see Algorithm algo1:reweigh

Theorems & Definitions (38)

  • Theorem 3.1
  • proof
  • Remark 3.2
  • Corollary 3.3
  • proof
  • Remark 3.4: Other density
  • Remark 3.5: Non-weighted version
  • Theorem 4.1: Ni_Sc_Ut20NiOlUl16LimTe2020
  • Theorem 4.2
  • proof
  • ...and 28 more