Sampling recovery in $L_2$ and other norms

TL;DR

This paper investigates function recovery in multiple norms using sampling, providing error bounds and demonstrating the near-optimality of linear algorithms in certain spaces, especially for the supremum norm.

Contribution

It introduces new worst-case error bounds for function recovery across various norms and shows linear sampling algorithms are nearly optimal in many RKHS settings.

Findings

Derived error bounds in $L_p$ norms based on $L_2$ approximation and Christoffel functions.

Proved linear sampling algorithms are optimal up to a constant factor for $p=\infty$ in many RKHS.

Extended the understanding of sampling recovery in different function spaces.

Abstract

We study the recovery of functions in various norms, including $L_p$ with $1\le p\le\infty$, based on function evaluations. We obtain worst case error bounds for general classes of functions in terms of the best $L_2$-approximation from a given nested sequence of subspaces and the Christoffel function of these subspaces. In the case $p=\infty$, our results imply that linear sampling algorithms are optimal up to a constant factor for many reproducing kernel Hilbert spaces.