Orthogonal Polynomials of Several Variables.
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| Author / Creator: | Dunkl, Charles F. |
|---|---|
| Edition: | 2nd ed. |
| Imprint: | Cambridge : Cambridge University Press, 2014. |
| Description: | 1 online resource (440 pages) |
| Language: | English |
| Series: | Encyclopedia of Mathematics and its Applications ; v. 155 Encyclopedia of mathematics and its applications. |
| Subject: | |
| Format: | E-Resource Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/12015594 |
Table of Contents:
- Cover; Half-Title page; Series page; Title page; Copyright page; Dedication page; Contents; Preface to the Second Edition; Preface to the First Edition; 1 Background; 1.1 The Gamma and Beta Functions; 1.2 Hypergeometric Series; 1.2.1 Lauricella series; 1.3 Orthogonal Polynomials of One Variable; 1.3.1 General properties; 1.3.2 Three-term recurrence; 1.4 Classical Orthogonal Polynomials; 1.4.1 Hermite polynomials; 1.4.2 Laguerre polynomials; 1.4.3 Gegenbauer polynomials; 1.4.4 Jacobi polynomials; 1.5 Modified Classical Polynomials; 1.5.1 Generalized Hermite polynomials.
- 1.5.2 Generalized Gegenbauer polynomials1.5.3 A limiting relation; 1.6 Notes; 2 Orthogonal Polynomials in Two Variables; 2.1 Introduction; 2.2 Product Orthogonal Polynomials; 2.3 Orthogonal Polynomials on the Unit Disk; 2.4 Orthogonal Polynomials on the Triangle; 2.5 Orthogonal Polynomials and Differential Equations; 2.6 Generating Orthogonal Polynomials of Two Variables; 2.6.1 A method for generating orthogonal polynomials; 2.6.2 Orthogonal polynomials for a radial weight; 2.6.3 Orthogonal polynomials in complex variables; 2.7 First Family of Koornwinder Polynomials.
- 2.8 A Related Family of Orthogonal Polynomials2.9 Second Family of Koornwinder Polynomials; 2.10 Notes; 3 General Properties of Orthogonal Polynomials in Several Variables; 3.1 Notation and Preliminaries; 3.2 Moment Functionals and Orthogonal Polynomials in Several Variables; 3.2.1 Definition of orthogonal polynomials; 3.2.2 Orthogonal polynomials and moment matrices; 3.2.3 The moment problem; 3.3 The Three-Term Relation; 3.3.1 Definition and basic properties; 3.3.2 Favard''s theorem; 3.3.3 Centrally symmetric integrals; 3.3.4 Examples; 3.4 Jacobi Matrices and Commuting Operators.
- 3.5 Further Properties of the Three-Term Relation3.5.1 Recurrence formula; 3.5.2 General solutions of the three-term relation; 3.6 Reproducing Kernels and Fourier Orthogonal Series; 3.6.1 Reproducing kernels; 3.6.2 Fourier orthogonal series; 3.7 Common Zeros of Orthogonal Polynomials in Several Variables; 3.8 Gaussian Cubature Formulae; 3.9 Notes; 4 Orthogonal Polynomials on the Unit Sphere; 4.1 Spherical Harmonics; 4.2 Orthogonal Structures on S[sup(d)] and on B[sup(d)]; 4.3 Orthogonal Structures on B[sup(d)] and on S[sup(d+m-1)]; 4.4 Orthogonal Structures on the Simplex.
- 4.5 Van der Corput
- Schaake Inequality4.6 Notes; 5 Examples of Orthogonal Polynomials in Several Variables; 5.1 Orthogonal Polynomials for Simple Weight Functions; 5.1.1 Product weight functions; 5.1.2 Rotation-invariant weight functions; 5.1.3 Multiple Hermite polynomials on R[sup(d)]; 5.1.4 Multiple Laguerre polynomials on R[sub(+)sup(d)]; 5.2 Classical Orthogonal Polynomials on the Unit Ball; 5.2.1 Orthonormal bases; 5.2.2 Appell''s monic orthogonal and biorthogonalpolynomials; 5.2.3 Reproducing kernel with respect to W[sup(B) sub[Mu] on B[sup(d)].