Book:Gabor Szegő/Orthogonal Polynomials/Fourth Edition

Gabor Szegő: Orthogonal Polynomials (fourth edition)

Published $1975$, American Mathematical Society

ISBN 0-8218-1023-5.

Online copies

hosted by The Ohio State University

Contents

PREFACE

PREFACE TO THE REVISED EDITION

PREFACE TO THE THIRD EDITION

PREFACE TO THE FOURTH EDITION

CHAPTER I. PRELIMINARIES

1.1. Notation

$(1.1.1)$

$(1.1.2)$

1.11. Inequalities

1.12. Polynomials and trigonometric polynomials

1.2. Representation of non-negative trigonometric polynomials

1.2.1. Theorem of Lukacs concerning non-negative polynomials

1.2.2. Theorems of S. Bernstein

1.3. Approximation by Polynomials

1.4. Orthogonality; weight function; vectors in function spaces

1.5. Closure; integral approximations

1.6. Linear functional operations

1.7. The Gamma function

1.7.1. Bessel functions

1.8. Differential equations

1.8.1. Airy's function

1.8.2. Theorems of Sturm's type

1.9. An elementary conformal mapping

1.9.1. The principle of argument; Rouche's theorem; sequences of analytic functions

CHAPTER II. DEFINITION OF ORTHOGONAL POLYNOMIALS; PRINCIPAL EXAMPLES

2.1. Orthogonality

2.2. Orthogonal Polynomials

2.3. Further Examples

2.4. The Classical Orthogonal Polynomials

2.5. A formula of Christoffel

2.6. A class of polynomials considered by S. Bernstein and G. Szegő

2.7. Stieltjes-Wigert polynomials

2.8. Distributions of Stieltjes type; an analogue of Legendre polynomials

2.8.1. Poisson-Charlier polynomials

2.8.2. Krawtchouk's polynomials

2.9. Further special cases

CHAPTER III. GENERAL PROPERTIES OF ORTHOGONAL POLYNOMIALS

CHAPTER IV. JACOBI POLYNOMIALS

4.1. Definition; notation; special cases

page 58

Theorem 4.1.

4.2. Differential equation

Theorem 4.2.1

Theorem 4.2.2.

4.21. Hypergeometric functions

4.22. Generalization

4.23. Second solution

4.24. Transformation of the differential equation

4.3. Rodrigues' formula; the orthonormal set

4.4. Generating function

4.5. Recurrence formula

4.6. Integral representations in general

4.61. Application; functions of the second kind

4.62. Further properties of the functions of the second kind

4.7. Ultraspherical polynomials

4.8. Integral representations for Legendre polynomials

4.81. Legendre functions of the second kind

4.82. Generalizations

4.9. Trigonometric representations

4.10. Further properties of Jacobi polynomials

CHAPTER V. LAGUERRE AND HERMITE POLYNOMIALS

CHAPTER VI. ZEROS OF ORTHOGONAL POLYNOMIALS

CHAPTER VII. INEQUALITIES

CHAPTER VIII. ASYMPTOTIC PROPERTIES OF THE CLASSICAL POLYNOMIALS

CHAPTER IX. EXPANSION PROBLEMS ASSOCIATED WITH THE CLASSICAL POLYNOMIALS

CHAPTER X. REPRESENTATION OF POSITIVE FUNCTIONS

CHAPTER XI. POLYNOMIALS ORTHOGONAL ON THE UNIT CIRCLE

CHAPTER XII. ASYMPTOTIC PROPERTIES OF GENERAL ORTHOGONAL POLYNOMIALS

CHAPTER XIII. EXPANSION PROBLEMS ASSOCIATED WITH GENERAL ORTHOGONAL POLYNOMIALS

CHAPTER XIV. INTERPOLATION

CHAPTER XV. MECHANICAL QUADRATURE

CHAPTER XVI. POLYNOMIALS ORTHOGONAL ON AN ARBITRARY CURVE

PROBLEMS AND EXERCISES

FURTHER PROBLEMS AND EXERCISES

APPENDIX

LIST OF REFERENCES

FURTHER REFERENCES

INDEX