Orthogonal polynomials of several variables / Charles F. Dunkl, University of Virginia, Yuan Xu, University of Oregon.

Format:

Book

Author/Creator:

Dunkl, Charles F., 1941- author.

Xu, Yuan, 1957- author.

Series:

Encyclopedia of mathematics and its applications ; v. 155.

Encyclopedia of mathematics and its applications ; volume 155

Language:

English

Subjects (All):

Orthogonal polynomials.

Functions of several real variables.

Physical Description:

1 online resource (xvii, 420 pages) : digital, PDF file(s).

Edition:

Second edition.

Place of Publication:

Cambridge : Cambridge University Press, 2014.

Language Note:

English

Summary:

Serving both as an introduction to the subject and as a reference, this book presents the theory in elegant form and with modern concepts and notation. It covers the general theory and emphasizes the classical types of orthogonal polynomials whose weight functions are supported on standard domains. The approach is a blend of classical analysis and symmetry group theoretic methods. Finite reflection groups are used to motivate and classify symmetries of weight functions and the associated polynomials. This revised edition has been updated throughout to reflect recent developments in the field. It contains 25% new material, including two brand new chapters on orthogonal polynomials in two variables, which will be especially useful for applications, and orthogonal polynomials on the unit sphere. The most modern and complete treatment of the subject available, it will be useful to a wide audience of mathematicians and applied scientists, including physicists, chemists and engineers.

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

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.

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and indexes.

ISBN:

1-316-08317-9

1-316-05717-8

1-316-05480-2

1-316-08081-1

1-316-07135-9

1-316-07608-3

1-316-07371-8

1-316-07845-0

1-107-78613-4