Rate of Convergence and Tractability of the Radial Function Approximation Problem

This article studies the problem of approximating functions belonging to a Hilbert space $H_d$ with an isotropic or anisotropic Gaussian reproducing kernel, $$ K_d(bx,bt) = expleft(-sum_{ell=1}^dgamma_ell^2(x_ell-t_ell)^2right) mbox{for all} bx,btinreals^d. $$ The isotropic case corresponds to using the same shape parameters for all coordinates, namely $gamma_ell=gamma>0$ for all $ell$, whereas the anisotropic case corresponds to varying shape parameters $gamma_ell$. We are especially interested in moderate to large $d$.