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Orthogonal polynomials of several variables

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Orthogonal polynomials of several variables/ Charles F. Dunkl, Yuan Xu.
Author:	Dunkl, Charles F.,
other author:	Xu, Yuan,
Published:	Cambridge :Cambridge University Press, : 2014.,
Description:	xvii, 420 p. :ill., digital ; : 24 cm.;
Subject:	Orthogonal polynomials. -
Online resource:	https://doi.org/10.1017/CBO9781107786134
ISBN:	9781107786134

Orthogonal polynomials of several variables
Dunkl, Charles F.,1941-

Orthogonal polynomials of several variables[electronic resource] /Charles F. Dunkl, Yuan Xu. - 2nd ed. - Cambridge :Cambridge University Press,2014. - xvii, 420 p. :ill., digital ;24 cm. - Encyclopedia of mathematics and its applications ;155. - Encyclopedia of mathematics and its applications ;v. 82..

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.

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.

ISBN: 9781107786134Subjects--Topical Terms:

527749
Orthogonal polynomials.

LC Class. No.: QA404.5 / .D86 2014

Dewey Class. No.: 515.55

Orthogonal polynomials of several variables
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100 1 $a Dunkl, Charles F., $d 1941- $e author. $3 1239733
245 1 0 $a Orthogonal polynomials of several variables $h [electronic resource] / $c Charles F. Dunkl, Yuan Xu.
250 $a 2nd ed.
260 $a Cambridge : $b Cambridge University Press, $c 2014.
300 $a xvii, 420 p. : $b ill., digital ; $c 24 cm.
490 1 $a Encyclopedia of mathematics and its applications ; $v 155
505 0 $a 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.
520 $a 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.
650 0 $a Orthogonal polynomials. $3 527749
650 0 $a Functions of several real variables. $3 971063
700 1 $a Xu, Yuan, $d 1957- $e author. $3 1239734
830 0 $a Encyclopedia of mathematics and its applications ; $v v. 82. $3 833009
856 4 0 $u https://doi.org/10.1017/CBO9781107786134