Inverses of moment Hermitian matrices

TL;DR

This paper investigates the conditions for the invertibility of infinite Hermitian moment matrices linked to measures on the complex plane, exploring their eigenvalue behavior and properties of associated Toeplitz matrices.

Contribution

It introduces the concept of weakly asymptotic Toeplitz matrices and relates the inverse of Toeplitz moment matrices to limits of orthogonal polynomial coefficients.

Findings

Inverse of Toeplitz moment matrices can be characterized via orthogonal polynomial coefficients.

The asymptotic behavior of the smallest eigenvalue depends only on the absolutely continuous part of the measure.

Under certain conditions, the inverse of a Toeplitz moment matrix is weakly asymptotic Toeplitz.

Abstract

Motivated by [9] we study the existence of the inverse of infinite Hermitian moment matrices associated with measures with support on the complex plane. We relate this problem to the asymptotic behaviour of the smallest eigenvalues of finite sections and we study it from the point of view of infinite transition matrices associated to the orthogonal polynomials. For Toeplitz matrices we introduce the notion of weakly asymptotic Toeplitz matrix and we show that, under certain assumptions, the inverse of a Toeplitz moment matrix is weakly asymptotic Toeplitz. Such inverses are computed in terms of some limits of the coefficients of the associated orthogonal polynomials. We finally show that the asymptotic behaviour of the smallest eigenvalue of a moment Toeplitz matrix only depends on the absolutely part of the associated measure.