Approximation in Hilbert spaces of the Gaussian and related analytic kernels

Abstract

We consider linear approximation based on function evaluations in reproducing kernel Hilbert spaces of certain analytic weighted power series kernels and stationary kernels on the interval $[-1,1]$. Both classes contain the popular Gaussian kernel $K(x, y) = \exp (-\tfrac{1}{2}\varepsilon ^{2}(x-y)^{2})$. For weighted power series kernels we derive almost matching upper and lower bounds on the worst-case error. When applied to the Gaussian kernel our results state that, up to a sub-exponential factor, the $n$th minimal error decays as $(\varepsilon /2)^{n} (n!)^{-1/2}$. The proofs are based on weighted polynomial interpolation and classical polynomial coefficient estimates that we use to bound the Hilbert space norm of a weighted polynomial fooling function.