Doubling the rate -- improved error bounds for orthogonal projection with application to interpolation

TL;DR

This paper proves that under certain conditions, the convergence rate of orthogonal projections in $L_2$ can be doubled for smoother functions, improving error bounds in interpolation methods.

Contribution

It generalizes and extends the doubling of convergence rates to a broad class of Hilbert space projections and interpolation techniques.

Findings

Doubling of $L_2$ convergence rates for smoother functions.

Improved error bounds in the $H$-norm for orthogonal projections.

Application to kernel and radial basis function interpolation methods.

Abstract

Convergence rates for $L_2$ approximation in a Hilbert space $H$ are a central theme in numerical analysis. The present work is inspired by Schaback (Math. Comp., 1999), who showed, in the context of best pointwise approximation for radial basis function interpolation, that the convergence rate for sufficiently smooth functions can be doubled, compared to the best rate for functions in the "native space" $H$. Motivated by this, we obtain a general result for $H$-orthogonal projection onto a finite dimensional subspace of $H$: namely, that any known $L_2$ convergence rate for all functions in $H$ translates into a doubled $L_2$ convergence rate for functions in a smoother normed space $B$, along with a similarly improved error bound in the $H$-norm, provided that $L_2$, $H$ and $B$ are suitably related. As a special case we improve the known $L_2$ and $H$-norm convergence rates for kernel interpolation in reproducing kernel Hilbert spaces, with particular attention to a recent study (Kaarnioja, Kazashi, Kuo, Nobile, Sloan, Numer. Math., 2022) of periodic kernel-based interpolation at lattice points applied to parametric partial differential equations. A second application is to radial basis function interpolation for general conditionally positive definite basis functions, where again the $L_2$ convergence rate is doubled, and the convergence rate in the native space norm is similarly improved, for all functions in a smoother normed space $B$.