Lower bounds for integration and recovery in $L_2$

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.