Book contents
- Frontmatter
- Contents
- Contributors
- Preface
- 1 Preliminaries
- 2 General Orthogonal Polynomials
- 3 Jacobi and Related Polynomials
- 4 Recursively Defined Polynomials
- 5 Wilson and Related Polynomials
- 6 Discrete Orthogonal Polynomials
- 7 Some q-Orthogonal Polynomials
- 8 The Askey–Wilson Family of Polynomials
- 9 Orthogonal Polynomials on the Unit Circle
- 10 Zeros of Orthogonal Polynomials
- 11 The Moment Problem
- 12 Matrix-Valued Orthogonal Polynomials and Differential Equations
- 13 Some Families of Matrix-Valued Jacobi Orthogonal Polynomials
- References
- Index
9 - Orthogonal Polynomials on the Unit Circle
Published online by Cambridge University Press: 14 September 2020
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Book contents
- Frontmatter
- Contents
- Contributors
- Preface
- 1 Preliminaries
- 2 General Orthogonal Polynomials
- 3 Jacobi and Related Polynomials
- 4 Recursively Defined Polynomials
- 5 Wilson and Related Polynomials
- 6 Discrete Orthogonal Polynomials
- 7 Some q-Orthogonal Polynomials
- 8 The Askey–Wilson Family of Polynomials
- 9 Orthogonal Polynomials on the Unit Circle
- 10 Zeros of Orthogonal Polynomials
- 11 The Moment Problem
- 12 Matrix-Valued Orthogonal Polynomials and Differential Equations
- 13 Some Families of Matrix-Valued Jacobi Orthogonal Polynomials
- References
- Index
Summary
One way to generalize orthogonal polynomials on subsets of ℝ is to consider orthogonality on curves in the complex plane. Among these generalizations, the most developed theory is the general theory of orthogonal polynomial on the unit circle T. The basic sources for this chapter are Grenander and Szegő (1958), Szegő ([1939] 1975), Geronimus (1961, 1962), Simon (2004a,b), Ismail (2005b, Chapters 8 and 17), and recent papers which will be cited in the appropriate places.
In what follows we shall use Simon’s abbreviation OPUC for orthogonal polynomials on the unit circle.
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- Encyclopedia of Special Functions: The Askey-Bateman Project , pp. 199 - 241Publisher: Cambridge University PressPrint publication year: 2020
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