Difference between revisions of "Book:Gabor Szegő/Orthogonal Polynomials/Fourth Edition"

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{{Book|Orthogonal Polynomials|1939|American Mathematical Society|0-8218-1023-5|Gabor Szegő}}
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{{Book|Orthogonal Polynomials (fourth edition)|1975|American Mathematical Society|0-8218-1023-5|Gabor Szegő}}
  
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===Online versions===
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===Online copies===
 
[https://people.math.osu.edu/nevai.1/SZEGO/szego=szego1975=ops=OCR.pdf hosted by The Ohio State University]<br />
 
[https://people.math.osu.edu/nevai.1/SZEGO/szego=szego1975=ops=OCR.pdf hosted by The Ohio State University]<br />
  
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[[Category:Books]]
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===Contents===
 +
:PREFACE
 +
:PREFACE TO THE REVISED EDITION
 +
:PREFACE TO THE THIRD EDITION
 +
:PREFACE TO THE FOURTH EDITION
 +
:CHAPTER I. PRELIMINARIES
 +
::1.1. Notation
 +
::[[Signum|$(1.1.1)$]]
 +
::[[Signum|$(1.1.2)$]]
 +
::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
 +
:::[[Jacobi P|page 58]]
 +
:::Theorem 4.1.
 +
::4.2. Differential equation
 +
::::[[Differential equation for Jacobi P|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
 +
 
 +
[[Category:Book]]

Latest revision as of 02:58, 15 July 2018


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
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