Kernel Approximation on Manifolds II: The $L_{\infty}$-norm of the $L_2$-projector

TL;DR

This paper investigates kernel approximation on manifolds, focusing on the stability of bases and the boundedness of least squares operators, demonstrating near-best approximation properties for a range of Lp spaces.

Contribution

It establishes the existence of stable local bases and proves Lp-boundedness of the least squares projector, extending classical spline theory results to manifold kernels.

Findings

Existence of local and stable bases for kernel approximation

Lp-boundedness of the least squares operator

Universal near-best approximation for functions in Lp spaces

Abstract

This article addresses two topics of significant mathematical and practical interest in the theory of kernel approximation: the existence of local and stable bases and the L_p--boundedness of the least squares operator. The latter is an analogue of the classical problem in univariate spline theory, known there as the "de Boor conjecture". A corollary of this work is that for appropriate kernels the least squares projector provides universal near-best approximations for functions f\in L_p, 1\le p\le \infty.