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 Lp 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 ε Lp, 1 ≤ p ≤∞.
© Society for Industrial and Applied Mathematics
Least squares approximation, Manifold, Positive definite kernels, Sobolev spaces
Hangelbroek, Thomas, Francis J. Narcowich, Xingping Sun, and Joseph D. Ward. "Kernel Approximation on Manifolds II: The L_∞ Norm of the L_2 Projector." SIAM journal on mathematical analysis 43, no. 2 (2011): 662-684.
SIAM Journal on Mathematical Analysis