Lower bounds for integration and recovery in $L_2$

TL;DR

This paper establishes new lower bounds for the performance of integration and recovery methods in Sobolev spaces, highlighting limitations of function value-based information compared to general linear information.

Contribution

It provides the first lower bounds for sampling numbers in Sobolev spaces, showing they can be worse than approximation numbers, especially for small smoothness.

Findings

Sampling numbers behave worse than approximation numbers for small smoothness Sobolev spaces.

Logarithmic gaps exist between sampling and approximation numbers even when singular values are square-summable.

New lower bounds for integration in classical Sobolev spaces of periodic functions.

Abstract

Function values are, in some sense, "almost as good" as general linear information for $L_2$-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 $L_2$ 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.