Orthogonal Polynomials of Several Variables: Charles F. Dunkl, Yuan Xu: 9780521800433: Book: Computers & Technology: eCampus.com

This is the first modern book on orthogonal polynomials of several variables, which are valuable tools used in multivariate analysis, including approximations and numerical integration. The book presents the theory in elegant form and with modern concepts and notation. It introduces the general theory and emphasizes the classical types of orthogonal polynomials whose weight functions are supported on standard domains such as the cube, the simplex, the sphere and the ball. It also focuses on those of Gaussian type, for which fairly explicit formulae exist. The authors' approach blends classical analysis and symmetry-group-theoretic methods. The book will be welcomed by research mathematicians and applied scientists, including applied mathematicians, physicists, chemists and engineers.

Treatise presenting an overview of special functions, specifically orthogonal polynomials of several variables. Provides organizing ideas, motivation, and historical background for the study of these functions, for students in mathematical physics, special functions, and related areas.

Orthogonal polynomials of several variables, approximation theory, symmetry-group methods.

Preface

xiii

Background

(29)

The Gamma and Beta Functions

(2)

Hypergeometric Series

(4)

Orthogonal Polynomials of One Variable

(6)

General properties

(2)

Three term recurrence

(4)

Classical Orthogonal Polynomials

(10)

Hermite polynomials

(1)

Laguerre polynomials

(2)

Gegenbauer polynomials

(4)

Jacobi polynomials

(2)

Modified Classical Polynomials

(5)

Generalized Hermite polynomials

(1)

Generalized Gegenbauer polynomials

(2)

A limiting relation

(1)

Notes

(2)

Examples of Orthogonal Polynomials in Several Variables

(33)

Notation and Preliminary

(3)

Spherical Harmonics

(4)

Classical Orthogonal Polynomials

(14)

Multiple Jacobi polynomials on the cube

(1)

Classical orthogonal polynomials on the unit ball

(8)

Classical orthogonal polynomials on the simplex

(3)

Multiple Hermite polynomials on Rd

(2)

Multiple Laguerre polynomials on Rd+

(1)

Other Examples of Orthogonal Polynomials

(7)

Two general families of orthogonal polynomials

(4)

A method for generating orthogonal polynomials of two variables

(2)

Disc polynomials

(1)

Van der Corput-Schaake Inequality

(2)

Notes

(3)

General Properties of Orthogonal Polynomials in Several Variables

(74)

Moment Functionals and Orthogonal Polynomials in Several Variables

(11)

Definition of orthogonal polynomials

(5)

Orthogonal polynomials and moment matrices

(3)

The moment problem

(3)

The Three Term Relation

(13)

Definition and basic properties

(4)

Favard's theorem

(3)

Centrally symmetric integrals

(3)

Examples

(3)

Jacobi Matrices and Commuting Operators

(7)

Further Properties of the Three Term Relation

(10)

Recurrence formula

(7)

General solutions of the three-term relation

102

(3)

Reproducing Kernels and Fourier Orthogonal Series

105

(8)

Reproducing kernels

106

(4)

Fourier orthogonal series

110

(3)

Common Zeros of Orthogonal Polynomials in Several Variables

113

(4)

Gaussian Cubature Formulae

117

(9)

Characterization of Gaussian cubature formulae

118

(5)

Examples of Gaussian cubature formulae

123

(3)

Orthogonal Polynomials on the Unit Sphere

126

(8)

Orthogonal structures on Sd and on Bd

126

(6)

Orthogonal structure on Bd and on Sd+m

132

(2)

Notes

134

(3)

Root Systems and Coxeter groups

137

(38)

Introduction and Overview

137

(2)

Root Systems

139

(8)

Type Ad-1

142

(1)

Type Bd

143

(1)

Type I2(m)

144

(1)

Type Dd

144

(1)

Type H3

145

(1)

Type F4

146

(1)

Other types

146

(1)

Miscellaneous results

146

(1)

Invariant Polynomials

147

(4)

Type Ad-1 invariants

149

(1)

Type Bd invariants

150

(1)

Type Dd invariants

150

(1)

Type I2 (m) invariants

151

(1)

Type H3 invariants

151

(1)

Type F4 invariants

151

(1)

Differential-Difference Operators

151

(6)

The Intertwining Operator

157

(10)

The k-Analog of the Exponential

167

(2)

Invariant Differential Operators

169

(4)

Notes

173

(2)

Spherical Harmonics Associated with Reflection Groups

175

(50)

h-Harmonic Polynomials

175

(10)

Inner Products on Polynomials

185

(4)

Reproducing Kernels and the Poisson Kernel

189

(3)

Integration of the Intertwining Operator

192

(5)

Example: Abelian Group Zd2

197

(8)

Example: Dihedral Groups

205

(11)

An orthonormal basis of Hn (h2α,β)

206

(8)

Cauchy and Poisson kernels

214

(2)

The Fourier Transform

216

(8)

Notes

224

(1)

Classical and Generalized Classical Orthogonal Polynomials

225

(30)

Generalized Classical Orthogonal Polynomials on the Ball

225

(11)

Definition and differential-difference equations

225

(6)

Bases, reproducing kernels, and the Funk-Hecke formula

231

(5)

Orthogonal Polynomials on the Simplex

236

(7)

General weight functions on Td

236

(3)

Generalized classical orthogonal polynomials

239

(4)

Generalized Hermite Polynomials

243

(6)

Generalized Laguerre Polynomials

249

(4)

Notes

253

(2)

Summability of Orthogonal Expansions

255

(32)

General Results on Orthogonal Expansions

255

(7)

Uniform convergence of partial sums

255

(4)

Cesaro means of the orthogonal expansion

259

(3)

Orthogonal Expansion on the Sphere

262

(4)

Orthogonal Expansion on the Ball

266

(5)

Orthogonal Expansions on the Simplex

271

(3)

Orthogonal Expansion of Laguerre and Hermite Polynomials

274

(5)

Multiple Jacobi Expansion

279

(5)

Notes

284

(3)

Orthogonal Polynomials Associated with Symmetric Groups

287

(50)

Introduction

287

(1)

Partitions, Compositions and Orderings

288

(1)

Commuting Self-Adjoint Operators

289

(3)

The Dual Polynomial Basis

292

(7)

Sd Invariant Subspaces

299

(5)

Degree Changing Recurrences

304

(4)

Norm Formulae

308

(13)

Hook length products and the pairing norm

308

(3)

The biorthogonal type norm

311

(2)

The torus inner product

313

(3)

Normalizing constants

316

(5)

Symmetric Functions and Jack Polynomials

321

(9)

Miscellaneous Topics

330

(5)

Notes

335

(2)

Orthogonal Polynomials Associated with Octahedral Groups and Applications

337

(35)

Introduction

337

(1)

Operators of Type B

338

(3)

Polynomial Eigenfunctions of Type B

341

(9)

Generalized Binomial Coefficients

350

(9)

Hermite Polynomials of Type B

359

(1)

Calogero-Sutherland Systems

360

(10)

The simple harmonic oscillator

361

(1)

Root systems and the Laplacian

362

(1)

Type A models on the line

363

(2)

Type A models on the circle

365

(3)

Type B models on the line

368

(2)

Notes

370

(2)

Bibliography

372

(12)

Author index

384

(3)

Symbol index

387

(2)

Subject index

389




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