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, $$K_d(\text{bx},\text{bt})=\exp (-\sum _{\mathrm{\ell }=1}^d{\gamma }_{\mathrm{\ell }}^2(x_{\mathrm{\ell }}-t_{\mathrm{\ell }}{)}^2)forall\text{bx},\text{bt}\in {\mathbb{R}}^d\mathrm{.}$$ The isotropic case corresponds to using the same shape parameters for all coordinates, namely ${\gamma }_{\mathrm{\ell }}=\gamma >0$ for all $\mathrm{\ell }$, whereas the anisotropic case corresponds to varying shape parameters ${\gamma }_{\mathrm{\ell }}$. We are especially interested in moderate to large $d$.