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Motivated by asymptotic questions related to the spectral theory of complex random matrices, this work focuses on the asymptotic analysis of orthogonal polynomials with respect to quasi-harmonic potentials in the complex plane. The ultimate goal is to develop new techniques to obtain strong asymptotics (asymptotic expansions valid uniformly on compact subsets) for planar orthogonal polynomials and use these results to understand the limiting behavior of spectral statistics of matrix models as their size goes to infinity. For orthogonal polynomials on the real line the powerful Riemann-Hilbert approach is the main analytic tool to derive asymptotics for the eigenvalue correlations in Hermitian matrix models. As yet, no such method is available to obtain asymptotic information about planar orthogonal polynomials, but some steps in this direction have been taken. The results of this thesis concern the connection between the asymptotic behavior of orthogonal polynomials and the corresponding equilibrium measure. It is conjectured that this connection is established via a quadrature identity: under certain conditions the weak-star limit of the normalized zero counting measure of the orthogonal polynomials is a quadrature measure for the support of the equilibrium measure of the corresponding two-dimensional electrostatic variational problem of the underlying potential. Several results are presented on equilibrium measures, quadrature domains, orthogonal polynomials and their relation to matrix models. In particular, complete strong asymptotics are obtained for the simplest nontrivial quasi-harmonic potential by a contour integral reduction method and the Riemann-Hilbert approach, which confirms the above conjecture for this special case.
This book establishes bounds and asymptotics under almost minimal conditions on the varying weights, and applies them to universality limits and entropy integrals. Orthogonal polynomials associated with varying weights play a key role in analyzing random matrices and other topics. This book will be of use to a wide community of mathematicians, physicists, and statisticians dealing with techniques of potential theory, orthogonal polynomials, approximation theory, as well as random matrices.
"Orthogonal polynomials satisfy a three-term recursion relation irrespective of the weight function with respect to which they are defined. This gives a simple formula for the kernel function, known in the literature as the Christoffel-Darboux sum. The availability of asymptotic results of orthogonal polynomials and the simple structure of the Christoffel-Darboux sum make the study of unitary ensembles of random matrices relatively straightforward. In this book, the author develops the theory of skew-orthogonal polynomials and obtains recursion relations which, unlike orthogonal polynomials, depend on weight functions. After deriving reduced expressions, called the generalized Christoffel-Darboux formulas (GCD), he obtains universal correlation functions and non-universal level densities for a wide class of random matrix ensembles using the GCD. The author also shows that once questions about higher order effects are considered (questions that are relevant in different branches of physics and mathematics) the use of the GCD promises to be efficient. Titles in this series are co-published with the Centre de Recherches Mathématiques."--Publisher's website.
Publisher description
This two-part volume gives a comprehensive overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. A major theme involves the connections between the Verblunsky coefficients (the coefficients of the recurrence equation for the orthogonal polynomials) and the measures, an analog of the spectral theory of one-dimensional Schrödinger operators. Among the topics discussed along the way are the asymptotics of Toeplitz determinants (Szegő's theorems), limit theorems for the density of the zeros of orthogonal polynomials, matrix representations for multiplication by (CMV matrices), periodic Verblunsky coefficients from the point of view of meromorphic functions on hyperelliptic surfaces, and connections between the theories of orthogonal polynomials on the unit circle and on the real line. The book is suitable for graduate students and researchers interested in analysis.
The present volume contains the Proceedings of the Seventh Iberoamerican Workshop in Orthogonal Polynomials and Applications (EIBPOA, which stands for Encuentros Iberoamericanos de Polinomios Ortogonales y Aplicaciones, in Spanish), held at the Universidad Carlos III de Madrid, Leganés, Spain, from July 3 to July 6, 2018. These meetings were mainly focused to encourage research in the fields of approximation theory, special functions, orthogonal polynomials and their applications among graduate students as well as young researchers from Latin America, Spain and Portugal. The presentation of the state of the art as well as some recent trends constitute the aim of the lectures delivered in the EIBPOA by worldwide recognized researchers in the above fields. In this volume, several topics on the theory of polynomials orthogonal with respect to different inner products are analyzed, both from an introductory point of view for a wide spectrum of readers without an expertise in the area, as well as the emphasis on their applications in topics as integrable systems, random matrices, numerical methods in differential and partial differential equations, coding theory, and signal theory, among others.
This volume contains invited lectures and selected contributions from the International Workshop on Orthogonal Polynomials and Approximation Theory, held at Universidad Carlos III de Madrid on September 8-12, 2008, and which honored Guillermo Lopez Lagomasino on his 60th birthday. This book presents the state of the art in the theory of Orthogonal Polynomials and Rational Approximation with a special emphasis on their applications in random matrices, integrable systems, and numerical quadrature. New results and methods are presented in the papers as well as a careful choice of open problems, which can foster interest in research in these mathematical areas. This volume also includes a brief account of the scientific contributions by Guillermo Lopez Lagomasino.
In this present paper, the authors consider the normal matrix model with cubic plus linear potential.
The first chapter lists the basic results of orthogonal polynomials, Jacobi, Laguerre, and Hermite polynomials, and collects some frequently used theorems and formulas. As a base and useful tool, the representation and quantitative theory of Hermite interpolation is the subject of Chapter 2. The theory of power orthogonal polynomials begins in Chapter 3: existence, uniqueness, Characterisations, properties of zeros, and continuity with respect to the measure and the indices are all considered. Chapter 4 deals with Gaussian quadrature formulas and their convergence. Chapter 5 is devoted to the theory of Christo®el type functions, which are related to Gaussian quadrature formulas and is one of the important contents of power orthogonal polynomials. The explicit representation of power orthogonal polynomials is an interesting problem and is discussed in Chapter 6. Chapter 7 is a detailed treatment of zeros in power orthogonal polynomials. Chapter 8 is devoted to bounds and inequalities of power orthogonal polynomials. In Chapters 9 and 10 we study asymptotics of general polynomials and power orthogonal polynomials, respectively. In Chapter 11 we discuss convergence of power orthogonal series, Lagrange and Hermite interpolation, and two positive operators constructed by power orthogonal polynomials. In Chapter 12 we investigate Gaussian quadrature formulas for extended Chebyshev spaces. In Chapter 13 we give construction methods for power orthogonal polynomials and Gaussian quadrature formulas; we also provide numerical results and numerical tables.