  This article considers linear approximation based on function evaluations in
reproducing kernel Hilbert spaces of the Gaussian kernel and a more general
class of weighted power series kernels on the interval $[-1, 1]$. We derive
almost matching upper and lower bounds on the worst-case error, measured both
in the uniform and $L^2([-1,1])$-norm, in these spaces. The results show that
if the power series kernel expansion coefficients $\alpha_n^{-1}$ decay at
least factorially, their rate of decay controls that of the worst-case error.
Specifically, (i) the $n$th minimal error decays as $\alpha_n^{{ -1/2}}$ up to
a sub-exponential factor and (ii) for any $n$ sampling points in $[-1, 1]$
there exists a linear algorithm whose error is $\alpha_n^{{ -1/2}}$ up to an
exponential factor. For the Gaussian kernel the dominating factor in the bounds
is $(n!)^{-1/2}$.
