A Characterization of Polynomial Density on Curves via Matrix Algebra

TL;DR

This paper characterizes when polynomials are dense in certain function spaces on curves using matrix algebra, introducing new indexes related to moment matrices and providing alternative proofs for classical theorems.

Contribution

It introduces three new indexes for positive semidefinite matrices and characterizes polynomial density on Jordan curves using these indexes, offering a novel matrix algebra approach.

Findings

Characterization of polynomial density via the index γ for measures on Jordan curves.

Introduction of three indexes γ, λ, α for positive semidefinite matrices.

Alternative proof of Thomson's theorem using matrix indexes.

Abstract

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces $L^{2}(\mu)$, with $\mu$ a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix to the measure $\mu$. In order to do it, in the more general context of Hermitian positive semidefinite matrices we introduce three indexes $\gamma(\mathbf{M})$, $\lambda(\mathbf{M})$ and $\alpha(\mathbf{M})$ associated with different optimization problems concerning these matrices. Our main result is a characterization of density of polynomials in the case of measures supported on Jordan curves with non empty interior using the index $\gamma$ and other specific index related to it. Moreover, we provide a new point of view of bounded point evaluations associated to a measure in terms of the index $\gamma$ that will allow us to give an alternative proof of Thomson's theorem in \cite{Brennan} by using these matrix indexes. We point out that our techniques are based in matrix algebra tools in the frame of Hermitian Positive Definite matrices and in the computation of certain indexes related to some optimization problems for infinite matrices.