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. The book will be welcomed by research mathematicians and applied scientists, including applied mathematicians, physicists, chemists and engineers.
• First modern book on subject, with many results drawn from recent literature • Can be used both as an introduction or as a reference, useful to a wide audience • Incorporates classical and modern approaches
1. Background; 2. Examples of orthogonal polynomials; 3. General properties of orthogonal polynomials; 4. Root systems and Coxeter groups; 5. Spherical harmonics associated with reflection groups; 6. Classical and generalized classical orthogonal polynomials; 7. Summability of orthogonal polynomials; 8. Orthogonal polynomials associated with symmetric groups; 9. Orthogonal polynomials associated with octahedral groups; 10. Bibliography; Indexes.
'This book is very impressive and shows the richness of the theory.' Vilmos Totik, Acta Sci. Math.