Approximation in Hilbert spaces of the Gaussian and related analytic kernels

TL;DR

This paper analyzes the approximation capabilities of certain analytic kernels, including the Gaussian, in Hilbert spaces, providing bounds on the minimal error decay rate for function approximation.

Contribution

It derives nearly tight bounds on approximation errors for Gaussian and related kernels, using polynomial interpolation and coefficient estimates.

Findings

Error decays as $(rac{ ext{epsilon}}{2})^n (n!)^{-1/2}$ for Gaussian kernel

Provides upper and lower bounds for weighted power series kernels

Uses polynomial interpolation techniques for error analysis

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.