  We study the recovery of multivariate functions from reproducing kernel
Hilbert spaces in the uniform norm. Our main interest is to obtain
preasymptotic estimates for the corresponding sampling numbers. We obtain
results in terms of the decay of related singular numbers of the compact
embedding into $L_2(D,\varrho_D)$ multiplied with the supremum of the
Christoffel function of the subspace spanned by the first $m$ singular
functions. Here the measure $\varrho_D$ is at our disposal. As an application
we obtain near optimal upper bounds for the sampling numbers for periodic
Sobolev type spaces with general smoothness weight. Those can be bounded in
terms of the corresponding benchmark approximation number in the uniform norm,
which allows for preasymptotic bounds. By applying a recently introduced
sub-sampling technique related to Weaver's conjecture we mostly lose a
$\sqrt{\log n}$ and sometimes even less. Finally we point out a relation to the
corresponding Kolmogorov numbers.
