Sampling recovery in $L_2$ and other norms

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.