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Orthogonal polynomials of several variables

Author: Charles F Dunkl; Yuan Xu
Publisher: Cambridge : Cambridge University Press, 2014.
Series: Encyclopedia of mathematics and its applications, 155.
Edition/Format:   Print book : English : Second editionView all editions and formats
Summary:

Serving both as an introduction to the subject and as a reference, this book covers the general theory and emphasizes the classical types of orthogonal polynomials, or those of Gaussian type.  Read more...

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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Charles F Dunkl; Yuan Xu
ISBN: 9781107071896 1107071895
OCLC Number: 869066543
Description: xvii, 420 pages : illustrations ; 24 cm.
Contents: 1. Background --
The gamma and beta functions --
Hypergeometric series --
Orthogonal polynomials of one variable --
Classical orthogonal polynomials --
Modified classical polynomials --
Notes --
2. Orthogonal polynomials in two variables --
Introduction --
Product orthogonal polynomials --
Orthogonal polynomials on the unit disk --
Orthogonal polynomials on the triangle --
Orthogonal polynomials and differential equations --
Generating orthogonal polynomials of two variables --
First family of koornwinder polynomials --
A related family of orthogonal polynomials --
Second family of koornwinder polynomials --
3. General properties of orthogonal polynomials in several variables --
Notation and preliminaries --
Moment funtionals and orthogonal polynomials --
The three-term relation --
Jacobi matrices and commuting operators --
Further properties of the three-term relation --
Reproducing kernels and fourier orthogonal series --
Common zeros of orthogonal polynomials in several variables --
Gaussian cubature formulae --
Notes --
4. Orthogonal polynomials on the unit sphere --
Spherical harmonics --
Orthoginal structures on Sd and on Bd --
Orthogonal structures on Bd and on Sd+m-1 --
Orthogonal structure on the simplex --
Van der corput --
Schaake inequality --
5. Examples of orthogonal polynomials in several variables --
Orthogonal polynomials for simple weight functions --
Classical orthogonal polynomials on the unit ball --
Classical orthogonal polynomials on the simplex --
Orthogonal polynomials via symmetric functions --
Chebyshev polynomials to Type Ad --
Sobolev orthogonal polynomials on the unit ball --
6. Root systems and coxeter groups --
Introduction and overview --
Root systems --
Invariant polynomials --
Differential-difference operators --
The intertwining operator --
The K-analogue of the exponential --
Invariant differential operators --
7. Spherical harmonics associated with reflection groups --
h-Harmonic polynomials --
Inner products on polynomials --
Reproducing kernels and the poisson kernel --
Integration of the intertwining operator --
Example: Abelian group Z d/2 --
Example: Dihedral groups --
The dunk1 transform --
8. Generalized classical orthogonal polynomials --
Generalized classical orthogonal polynomials on the ball --
Generalized classical orthogonal polynomials on the simplex --
Generalized hermite polynomials --
Generalized laguerre polynomials --
9. Summability of orthogonal expansions --
General results on orthogonal expansions --
Orthogonal expansion on the sphere --
Orthogonal expansion on the ball --
Orthogonal expansion on the simplex --
Orthogonal expansion of Laguerre and Hermite polynomials --
Multiple Jacobi expansion --
10. Orthogonal polynomials associated with symmetric groups --
Partitions, compositions and orderings --
Commuting self-adjoint operators --
The dual polynomials basis --
Sd-invariant subspaces --
Degree-changing recurrences --
Norm formulae --
Symmetric functions and jack polynomials --
Miscellaneous topics --
11. Orthogonal polynomials associated with octahedral groups, and applications --
Operators of Type B --
Polynomial eigenfunctions of Type B --
Generalized binomial coefficients --
Hermite polynomials of Type B --
Calogero-Sutherland systems.
Series Title: Encyclopedia of mathematics and its applications, 155.
Responsibility: Charles F. Dunkl, University of Virginia, Yuan Xu, University of Oregon.
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Review of the first edition: 'This book is the first modern treatment of orthogonal polynomials of several real variables. It presents not only a general theory, but also detailed results of recent Read more...

 
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