Orthogonal Polynomials

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