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This article considers linear approximation based on function evaluations inreproducing kernel Hilbert spaces of the Gaussian kernel and a more generalclass of weighted power series kernels on the interval $[-1, 1]$. We derivealmost matching upper and lower bounds on the worst-case error, measured bothin the uniform and $L^2([-1,1])$-norm, in these spaces. The results show thatif the power series kernel expansion coefficients $\alpha_n^{-1}$ decay atleast 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 toa 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 anexponential factor. For the Gaussian kernel the dominating factor in the boundsis $(n!)^{-1/2}$.

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