Relative entropy and the multi-variable multi-dimensional moment problem

Entropy-like functionals on operator algebras have been studied since the pioneering work of von Neumann, Umegaki, Lindblad, and Lieb. The most well-known are the von Neumann entropy $trace (\rho\log \rho)$ and a generalization of the Kullback-Leibler distance $trace (\rho \log \rho - \rho \log \sigma)$, refered to as quantum relative entropy and used to quantify distance between states of a quantum system. The purpose of this paper is to explore these as regularizing functionals in seeking solutions to multi-variable and multi-dimensional moment problems. It will be shown that extrema can be effectively constructed via a suitable homotopy. The homotopy approach leads naturally to a further generalization and a description of all the solutions to such moment problems. This is accomplished by a renormalization of a Riemannian metric induced by entropy functionals. As an application we discuss the inverse problem of describing power spectra which are consistent with second-order statistics, which has been the main motivation behind the present work.Comment: 24 pages, 3 figure

Fej\'er-Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle

We give a complete characterization of the positive trigonometric polynomials Q(\theta,\phi) on the bi-circle, which can be factored as Q(\theta,\phi)=|p(e^{i\theta},e^{i\phi})|^2 where p(z,w) is a polynomial nonzero for |z|=1 and |w|\leq 1. The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight 1/(4\pi^2Q(\theta,\phi)) on the bi-circle. We use this result to describe how specific factorizations of weights on the bi-circle can be translated into identities relating the recurrence coefficients for the corresponding polynomials and vice versa. In particular, we characterize the Borel measures on the bi-circle for which the coefficients multiplying the reverse polynomials associated with the two operators: multiplication by z in lexicographical ordering and multiplication by w in reverse lexicographical ordering vanish after a particular point. This can be considered as a spectral type result analogous to the characterization of the Bernstein-Szeg\H{o} measures on the unit circle

Wide sense stationary processes on Z2: structural properties and parametric models

In this paper, we give a survey of results on the structure of 2-D wide-sense stationary processes with special emphasis on finiteorder 'models. We give firstly the extension of the linear prediction theory to the 2-D case . We study then the following three models: ARMA, wide-sense Markovian fields and state-space representation .Dans cet article, nous synthétisons divers travaux concernant la structure des processus stationnaires au sens large à deux indices discrets en insistant tout particulièrement sur les modèles d'ordre fini . Après avoir présenté les principaux résultats de la généralisation au cas 2-D de la théorie de la prédiction linéaire, nous étudions les modèles ARMA, les champs markoviens stationnaires au sens large et les modèles à représentation d'état qui sont les trois types de modèles paramétriques linéaires et stationnaires les plus utilisés

Spectral Density Functions and Their Applications

The Bernstein-Szegő measure moment problem asks when a given finite list of complex numbers form the Fourier coefficients of the spectral density function of a stable polynomial in the one-variable case. Szegő proved that it is possible if and only if the Toeplitz matrix form by these numbers is positive definite. Bernstein later proved a real line analog of the problem. The question remained open in two variables until Geronimo and Woerdeman stated and proved the necessary and sufficient conditions. Unlike the solution in one variable, it does not suffice to write down a single matrix and check whether it is positive definite. A positive definite completion condition is also required. In this thesis, we further pursue the moment problem in two variables and beyond. We first enhance the two-variable results by identifying the eigenstructure of matrices that arise from the theory. We then create a method that allows us to compute the Fourier coefficients in a given infinite region by using a finite portion of the coefficients. Use is made of determinantal representations of stable polynomials. In addition, we compute the asymptotics for the Fourier coefficients and later generalize the result to higher dimensions. In the final chapter, we draw a connection between offset words and a particular type of spectral density functions and compute the asymptotics of the number of offset words as different parameter changes.Ph.D., Mathematics -- Drexel University, 201