Orthogonal Polynomials of Several Variables

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.

Preface to the second edition; Preface to the first edition; 1. Background; 2. Orthogonal polynomials in two variables; 3. General properties of orthogonal polynomials in several variables; 4. Orthogonal polynomials on the unit sphere; 5. Examples of orthogonal polynomials in several variables; 6. Root systems and Coxeter groups; 7. Spherical harmonics associated with reflection groups; 8. Generalized classical orthogonal polynomials; 9. Summability of orthogonal expansions; 10. Orthogonal polynomials associated with symmetric groups; 11. Orthogonal polynomials associated with octahedral groups and applications; References; Author index; Symbol index; Subject index.

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.