Rate of Convergence and Tractability of the Radial Function Approximation Problem

Details Rate of Convergence and Tractability of the Radial Function Approximation Problem Rate of Convergence and Tractability of the Radial Function Approximation Problem

2010-12-13

arxiv.org/abs/1012.2605v1

Mathematics

Numerical Analysis

Scientific paper

This article studies the problem of approximating functions belonging to a Hilbert space $H_d$ with an isotropic or anisotropic Gaussian reproducing kernel, $$ K_d(\bx,\bt) = \exp\left(-\sum_{\ell=1}^d\gamma_\ell^2(x_\ell-t_\ell)^2\right) \ \ \ \mbox{for all}\ \ \bx,\bt\in\reals^d. $$ 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 $d$.

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