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This volume contains the Proceedings of the NATO Advanced Study Institute on "Orthogonal Polynomials and Their Applications" held at The Ohio State University in Columbus, Ohio, U.S.A. between May 22,1989 and June 3,1989. The Advanced Study Institute primarily concentrated on those aspects of the theory and practice of orthogonal polynomials which surfaced in the past decade when the theory of orthogonal polynomials started to experience an unparalleled growth. This progress started with Richard Askey's Regional Confer ence Lectures on "Orthogonal Polynomials and Special Functions" in 1975, and subsequent discoveries led to a substantial revaluation of one's perceptions as to the nature of orthogonal polynomials and their applicability. The recent popularity of orthogonal polynomials is only partially due to Louis de Branges's solution of the Bieberbach conjecture which uses an inequality of Askey and Gasper on Jacobi polynomials. The main reason lies in their wide applicability in areas such as Pade approximations, continued fractions, Tauberian theorems, numerical analysis, probability theory, mathematical statistics, scattering theory, nuclear physics, solid state physics, digital signal processing, electrical engineering, theoretical chemistry and so forth. This was emphasized and convincingly demonstrated during the presentations by both the principal speakers and the invited special lecturers. The main subjects of our Advanced Study Institute included complex orthogonal polynomials, signal processing, the recursion method, combinatorial interpretations of orthogonal polynomials, computational problems, potential theory, Pade approximations, Julia sets, special functions, quantum groups, weighted approximations, orthogonal polynomials associated with rootsystems, matrix orthogonal polynomials, operator theory and group representations.
List of contents
Characterization Theorems for Orthogonal Polynomials.- Orthogonal Polynomials in Coding Theory and Algebraic Combinatorics.- Orthogonal Polynomials, Padé Approximations and Julia Sets.- The Three Term Recurrence Relation and Spectral Properties of Orthogonal Polynomials.- On the Role of Orthogonal Polynomials on the Unit Circle in Digital Signal.- A Survey on the Theory of Orthogonal Systems and Some Open Problems.- Orthogonal Polynomials and Functional Analysis.- Using Symbolic Computer Algebraic Systems to Derive Formulas Involving Orthogonal Polynomials and Other Special Functions.- Computational Aspects of Orthogonal Polynomials.- The Recursion Method and the Schroedinger Equation.- Birth and Death Processes and Orthogonal Polynomials.- Orthogonal Polynomials in Connection with Quantum Groups.- The Approximate Approach to Orthogonal Polynomials for Weights on (-?,?).- Orthogonal Polynomials Associated with Root Systems.- Some Extensions of the Beta Integral and the Hypergeometric Function.- Orthogonal Matrix Polynomials.- Orthogonal Polynomial from a Complex Perspective.- Nth Root Asymptotic Behavior of Orthonormal Polynomials.- An Introduction to Group Representations and Orthogonal Polynomials.- Asymptotics for Orthogonal Polynomials and Three - Term Recurrences.- List of of Participants.- Scientific Program.
