  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.
