Approximation in Hilbert spaces of the Gaussian and other weighted power series kernels

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 decays as $\alpha_n^{{ -1/2}}$ up to an exponential factor. For the Gaussian kernel the dominating factor in the bounds is $(n!)^{-1/2}$.