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

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.