A Characterization of Polynomial Density on Curves via Matrix Algebra

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.