Lower bounds for integration and recovery in L2

Function values are, in some sense, almost as good as general linear information for L2-approximation (optimal recovery, data assimilation) of functions from a reproducing kernel Hilbert space. This was recently proved by new upper bounds on the sampling numbers under the assumption that the singular values of the embedding of this Hilbert space into L2 are square-summable. Here we mainly prove new lower bounds. In particular we prove that the sampling numbers behave worse than the approximation numbers for Sobolev spaces with small smoothness. Hence there can be a logarithmic gap also in the case where the singular numbers of the embedding are square-summable. We first prove new lower bounds for the integration problem, again for rather classical Sobolev spaces of periodic univariate functions.(c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Automated summary

Function values are almost as informative as general linear information for L2-approximation of functions, and this paper mainly focuses on proving new lower bounds for this behavior. It is shown that sampling numbers can behave worse than approximation numbers, especially for Sobolev spaces with low smoothness. Additionally, new lower bounds for the integration problem are proven.