  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.
