Orthogonal Polynomials (eBook)

Geza Freud (Autor)

eBook Download: PDF

2014 | 1. Auflage
294 Seiten
Elsevier Science (Verlag)
978-1-4831-5940-9 (ISBN)

Infos

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. Szego'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

Inhalt

Front Cover 1
Orthogonal Polynomials 4
Copyright Page 5
Table of Contents 8
Preface 6
Notations 10
Chapter 1. Fundamental Properties of Orthogonal Polynomials 12
§I.1. Definition of systems of orthogonal polynomials 12
§I.2. Recursion formula. Preliminaries concerning the position of the zeros 16
§I.3. The Gauss—Jacobi quadrature formula 20
§I.4. Consequences of the quadrature formula 24
§I.5. The Markov—Stieltjes inequality 27
§I.6. The Chebyshev and the Legend re polynomials 34
§I.7. Some elementary estimations of the orthogonal polynomials 41
§I.8. The Jacobi polynomials 44
Problems and remarks on Chapter I 48
Chapter 2. Elements of the Theory of the Hamburger—Stieltjes Momentum Problem 55
§II.1. On the solvability of the momentum problem 55
§II.2. Conditions for the uniqueness of the solution 63
§II.3. Connection between the uniqueness of the solution of the momentum problem and the approximation by polynomials 70
§II.4. The completeness of the system of the orthogonal polynomials in L2da 75
§II.5. A uniqueness criterion of M. Riesz 79
Problems and remarks on Chapter II 82
Chapter 3. Quadrature Procedure and Interpolation over the Zeros of the Orthogonal Polynomials 89
§III.1. On the convergence of the quadrature procedure 89
§III.2. Convergence in the quadratic mean of the interpolation polynomials 96
§III.3. Estimations of the Christoffel numbers 101
§III.4. An estimation of the speed of the convergence of the quadrature procedure 106
§III.5. Estimation of the distance between two successive zeros of .n(x.) 112
§III.6. Point-by-point and uniform convergence of the interpolation procedure 113
§III.7. Behaviour of the orthogonal polynomials on the complex plane 117
§III.8. Interpolation of analytic functions 125
§III.9. The distribution function of the zeros 129
Problems and remarks on Chapter III 131
Chapter 4. Convergence Theory of the Series of Orthogonal Polynomials 139
§IV.1. Fundamental ideas. Absolute convergence of the series of orthogonal polynomials 139
§IV.2. The Lebesgue points of the functions belonging t o Lpda 143
§IV.3. Strong (C,1)-summability of the orthogonal polynomials 147
§IV.4. Approximation properties of the partial sums 157
§IV.5. Convergence criteria 164
§IV.6. Remarks on "convergence almost everywhere" 172
Problems and remarks on Chapter IV 182
Chapter 5. The Theory of G. Szego 188
§V.1. The orthogonal polynomials on the unit circle 188
§V.2. Szego's extremum problem 199
§V.3. The Szego function and the function class H2dµ 209
§V.4. Asymptotic properties of the orthogonal polynomials (First part) 220
§V.5. Asymptotic properties of the orthogonal polynomials (continued) The class Lip (1/2, 2). Localization of the validity of the asymptotic properties 230
§V.6. Asymptotic formula for the Christoffel numbers 245
§V.7. Supplement to the convergence theory of series of orthogonal polynomials 258
§V.8. Asymptotic value of the distance between neighbouring zeros 266
Problems and remarks on Chapter V 268
Some unsolved problems 274
Bibliography 280
Author index 292
Subject index 294

Produkt-Details

• Erscheint lt. Verlag: 17.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• ISBN-10: 1-4831-5940-X / 148315940X
• ISBN-13: 978-1-4831-5940-9 / 9781483159409