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Sampling projections in the uniform norm

David Krieg, Kateryna Pozharska, Mario Ullrich, Tino Ullrich

TL;DR

The paper proves that every $n$-dimensional subspace of $B(D)$ admits a sampling projection using $m=2n$ samples with operator norm bounded by $\|P\|\le C\sqrt{n}$, providing a constructive Kadets–Snobar-type result for the uniform norm and linking it to optimal recovery in $L_p$. The authors develop a discretization framework based on maximizing Gram determinants (Kiefer–Wolfowitz) and a BSU subsampling step to realize explicit, least-squares sampling projections via an unweighted estimator, with uniform-in-$p$ error bounds. They extend the discretization to $L_p$ norms by interpolation and give sharp, dimension-dependent bounds; these yield nontrivial relations between linear sampling numbers and Kolmogorov/Gelfand widths, showing that slight oversampling yields substantial improvements. The results are shown to be essentially optimal in their polynomial rates, with numerical constants made explicit, and have broad implications for sampling recovery and the comparison of linear versus nonlinear information in functional approximation. The methods blend discrete design theory (D-optimal designs) with constructive discretizations, offering a practical framework for normed-function approximation from finite samples.

Abstract

We show that there are sampling projections on arbitrary $n$-dimensional subspaces of $B(D)$ with at most $2n$ samples and norm of order $\sqrt{n}$, where $B(D)$ is the space of complex-valued bounded functions on a set $D$. This gives a more explicit form of the Kadets-Snobar theorem for the uniform norm and improves upon Auerbach's lemma. We discuss consequences for optimal recovery in $L_p$.

Sampling projections in the uniform norm

TL;DR

The paper proves that every -dimensional subspace of admits a sampling projection using samples with operator norm bounded by , providing a constructive Kadets–Snobar-type result for the uniform norm and linking it to optimal recovery in . The authors develop a discretization framework based on maximizing Gram determinants (Kiefer–Wolfowitz) and a BSU subsampling step to realize explicit, least-squares sampling projections via an unweighted estimator, with uniform-in- error bounds. They extend the discretization to norms by interpolation and give sharp, dimension-dependent bounds; these yield nontrivial relations between linear sampling numbers and Kolmogorov/Gelfand widths, showing that slight oversampling yields substantial improvements. The results are shown to be essentially optimal in their polynomial rates, with numerical constants made explicit, and have broad implications for sampling recovery and the comparison of linear versus nonlinear information in functional approximation. The methods blend discrete design theory (D-optimal designs) with constructive discretizations, offering a practical framework for normed-function approximation from finite samples.

Abstract

We show that there are sampling projections on arbitrary -dimensional subspaces of with at most samples and norm of order , where is the space of complex-valued bounded functions on a set . This gives a more explicit form of the Kadets-Snobar theorem for the uniform norm and improves upon Auerbach's lemma. We discuss consequences for optimal recovery in .
Paper Structure (10 sections, 12 theorems, 67 equations)

This paper contains 10 sections, 12 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. On maximizers of the Gram determinant
  3. Discretization via subsampling
  4. Discretization results for the sup-norm
  5. Discretization results for the $L_p$-norm
  6. $L_p$-errors of least-squares algorithms
  7. Implications for optimal sampling recovery
  8. Sampling vs. Kolmogorov numbers
  9. Sampling vs. Gelfand numbers
  10. Sharpness of our results

Key Result

Theorem 1

There is an absolute constant $C>0$ such that the following holds. Let $D$ be a set and $V_n$ be an $n$-dimensional subspace of $B(D)$. Then there are $2n$ points $x_1,\dots,x_{2n}\in D$ and functions $\varphi_1,\dots,\varphi_{2n} \in V_n$ such that $P\colon B(D)\to V_n$ with $Pf=\sum_{i=1}^{2n} f(x

Theorems & Definitions (33)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 4: Auerbach's lemma and interpolating projections
  • Remark 5: Discretization of the uniform norm
  • Remark 6: Nikol'skii inequalities
  • Remark 7: Constant
  • Remark 8
  • Proposition 9
  • Lemma 10
  • ...and 23 more