Smallest and largest generalized eigenvalues of large moment matrices and some applications

TL;DR

This paper introduces a novel approach to compare measures via their moment matrices using generalized eigenvalues, providing insights into measure support localization and applications in polynomial approximation.

Contribution

It develops a new method based on generalized eigenvalues of moment matrices to analyze measure support and applies it to polynomial approximation problems.

Findings

Support localization related to Jordan curves

Convex envelope description via Rayleigh quotients

Applications to polynomial approximation in mean square

Abstract

The main aim of this work is to compare two Borel measures thorough their moment matrices using a new notion of smallest and largest generalized eigenvalues. With this approach we provide information in problems as the localization of the support of a measure. In particular, we prove that if a measure is comparable in an algebraic way with a measure in a Jordan curve then the curve is contained in its support. We obtain a description of the convex envelope of the support of a measure via certain Rayleigh quotients of certain infinite matrices. Finally some applications concerning polynomial approximation in mean square are given, generalizing the results in [9].