Approximation properties of certain operator-induced norms on Hilbert spaces

Abstract

We consider a class of operator-induced norms, acting as finite-dimensional surrogates to the L 2 norm, and study their approximation properties over Hilbert subspaces of L 2 . The class includes, as a special case, the usual empirical norm encountered, for example, in the context of nonparametric regression in a reproducing kernel Hilbert space (RKHS). Our results have implications to the analysis of M -estimators in models based on finite-dimensional linear approximation of functions, and also to some related packing problems.