Orthogonal Polynomials

1st Edition - January 1, 1971
Author: Géza Freud
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 1 - 5 9 4 0 - 9

Orthogonal Polynomials contains an up-to-date survey of the general theory of orthogonal polynomials. It deals with the problem of polynomials and reveals that the sequence of… Read more

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Orthogonal Polynomials contains an up-to-date survey of the general theory of orthogonal polynomials. It deals with the problem of polynomials and reveals that the sequence of these polynomials forms an orthogonal system with respect to a non-negative m-distribution defined on the real numerical axis. Comprised of five chapters, the book begins with the fundamental properties of orthogonal polynomials. After discussing the momentum problem, it then explains the quadrature procedure, the convergence theory, and G. Szegő's theory. This book is useful for those who intend to use it as reference for future studies or as a textbook for lecture purposes

Preface

Notations

Chapter I Fundamental Properties of Orthogonal Polynomial

§ I.1. Definition of Systems of Orthogonal Polynomials

§ I.2. Recursion Formula. Preliminaries Concerning the Position of the Zeros

§ I.3. The Gauss—Jacobi Quadrature Formula

§ I.4. Consequences of the Quadrature Formula

§ I.5. The Markov—Stieltjes Inequality

§ I.6. The Chebyshev and the Legendre Polynomials

§ I.7. Some Elementary Estimations of the Orthogonal Polynomials

§ I.8. The Jacobi Polynomials

Problems and Remarks on Chapter I

Chapter II Elements of the Theory of the Hamburger—Stieltjes Momentum Problem

§ II.1. On the Solvability of the Momentum Problem

§ II.2. Conditions for the Uniqueness of the Solution

§ II.3. Connection Between the Uniqueness of the Solution of the Momentum Problem and the Approximation by Polynomials

§ II.4. The Completeness of the System of the Orthogonal Polynomials in Ldα2

§ II.5. A Uniqueness Criterion of M. Riesz

Problems and Remarks on Chapter II

Chapter III Quadrature Procedure and Interpolation Over the Zeros of the Orthogonal Polynomials

§ III.1. On the Convergence of the Quadrature Procedure

§ III.2. Convergence in the Quadratic Mean of the Interpolation Polynomials

§ III.3. Estimations of the Christoffel Numbers

§ III.4. An Estimation of the Speed of the Convergence of the Quadrature Procedure

§ III.5. Estimation of the Distance Between Two Successive Zeros of ψn(xf ξ)

§ III.6. Point-by-Point and Uniform Convergence of the Interpolation Procedure

§ III.7. Behavior of the Orthogonal Polynomials on the Complex Plane

§ III.8. Interpolation of Analytic Functions

§ III.9. The Distribution Function of the Zeros

Problems And Remarks on Chapter III

Chapter IV Convergence Theory of the Series of Orthogonal Polynomials

§ IV.1. Fundamental Ideas. Absolute Convergence of the Series of Orthogonal Polynomials

§ IV.2. The Lebesgue Points of the Functions Belonging to Ldαp

§ IV.3. Strong (C, L)-Summability of the Orthogonal Polynomials

§ IV.4. Approximation Properties of the Partial Sums

§ IV.5. Convergence Criteria

§ IV.6. Remarks on "Convergence almost Everywhere"

Problems And Remarks On Chapter Iv

Chapter V The Theory of G. Szegȍ

§ V.1. The Orthogonal Polynomials on the Unit Circle

§ V.2. Szegȍ's Extremum Problem

§ V.3. The Szegȍ Function and the Function Class Haμ2

§ V.4. Asymptotic Properties of the Orthogonal Polynomials (First Part)

§ V.5. Asymptotic Properties of the Orthogonal Polynomials (Continued). The Class Lip (1/2, 2). Localization Of The Validity Of The Asymptotic Properties

§ V.6. Asymptotic Formula for the Christoffel Numbers

§ V.7. Supplement to the Convergence Theory of Series of Orthogonal Polynomials

§ V.8. Asymptotic Value of the Distance Between Neighboring Zeros

Problems and Remarks on Chapter V

Some Unsolved Problems

Bibliography

Author Index

Subject Index