Doubling the rate – improved error bounds for orthogonal projection in Hilbert spaces

Abstract

Convergence rates for L2 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 general 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 L2 convergence rate for all functions in H translates into a doubled L2 convergence rate for functions in a smoother normed space B, along with a similarly improved error bound in the H-norm, provided that L2, H and B are suitably related. As a special case we improve the known L2 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 L2 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.