Approximation properties of certain operator-induced norms on Hilbert spaces

TL;DR

This paper investigates the approximation capabilities of operator-induced norms on Hilbert spaces, highlighting their relevance to empirical norms in nonparametric regression and implications for M-estimators and packing problems.

Contribution

It introduces a class of operator-induced norms as finite-dimensional surrogates for the L2 norm and analyzes their approximation properties within Hilbert subspaces.

Findings

Operator-induced norms can effectively approximate the L2 norm in Hilbert spaces.

Results inform the analysis of M-estimators in finite-dimensional function models.

Implications extend to packing problems in functional analysis.

Abstract

We consider a class of operator-induced norms, acting as finite-dimensional surrogates to the L2 norm, and study their approximation properties over Hilbert subspaces of L2 . The class includes, as a special case, the usual empirical norm encountered, for example, in the context of nonparametric regression in reproducing kernel Hilbert spaces (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.