Orthogonal Polynomials of Several Variables / Edition 1

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Overview

"This is the first modern book on orthogonal polynomials of several variables, which are interesting both as objects of study and as tools used in multivariate analysis, including approximations and numerical integration. The book, which is intended both as an introduction to the subject and as a reference, 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, or those of Gaussian type, for which fairly explicit formulae exist. The approach is a blend of classical analysis and symmetry-group-theoretic methods. Reflection groups are used to motivate and classify symmetries of weight functions and the associated polynomials. Many results come from current research literature. The book will be welcomed by research mathematicians and applied scientists, including applied mathematicians, physicists, chemists and engineers."--BOOK JACKET.
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Editorial Reviews

From the Publisher
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 research on generalizations of various classical cases.' Mathematical Reviews

Review of the first edition: 'This book is very impressive and shows the richness of the theory.' Vilmos Totik, Acta Scientiarum Mathematicarum

'This is a valuable book for anyone with an interest in special functions of several variables.' Marcel de Jeu, American Mathematical Society

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Product Details

Meet the Author

Charles F. Dunkl is Professor Emeritus of Mathematics at the University of Virginia. Among his work one finds the seminal papers containing the construction of differential-difference operators associated to finite reflection groups and related integral transforms. Aspects of the theory are now called Dunkl operators, the Dunkl transform, and the Dunkl kernel. Dunkl is a Fellow of the Institute of Physics, and a member of SIAM and of its Activity Group on Orthogonal Polynomials and Special Functions, which he founded in 1990 and then chaired from 1990 to 1998.

Yuan Xu is Professor of Mathematics at the University of Oregon. His work covers topics in approximation theory, harmonic analysis, numerical analysis, orthogonal polynomials and special functions, and he works mostly in problems of several variables. Xu is currently on the editorial board of five international journals and has been a plenary or invited speaker in numerous international conferences. He was awarded a Humboldt research fellowship in 1992–93 and received a Faculty Excellence Award at the University of Oregon in 2009. He is a member of SIAM and of its Activity Group on Orthogonal Polynomials and Special Functions.

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Table of Contents

Preface
1 Background 1
2 Examples of Orthogonal Polynomials in Several Variables 30
3 General Properties of Orthogonal Polynomials in Several Variables 63
4 Root Systems and Coxeter groups 137
5 Spherical Harmonics Associated with Reflection Groups 175
6 Classical and Generalized Classical Orthogonal Polynomials 225
7 Summability of Orthogonal Expansions 255
8 Orthogonal Polynomials Associated with Symmetric Groups 287
9 Orthogonal Polynomials Associated with Octahedral Groups and Applications 337
Bibliography 372
Author index 384
Symbol index 387
Subject index 389
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