Orthogonal Polynomials of Several Variables (Encyclopedia of Mathematics and its Applications, Series Number 81)

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. 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.--jacket. Examples Of Orthogonal Polynomials In Several Bariables -- General Properties Of Orthogonal Polynomialsin Several Variables -- Root Systems And Coxeter Groups -- Sperical Harmonics Associated With Reflection Groups -- Classical And Generalized Classical Orthogonal Polynomials -- Summability Of Orthogonal Expansions -- Orthogonal Polynomials Associated With Symmetric Groups -- Orthogonal Polynomials Associated With Octahedral Groups And Applications. Charles F. Dunkl And Yuan Xu. Includes Bibliographical References (p. 372-383) And Indexes. This is the first modern book on orthogonal polynomials of several variables, blending classical analysis and symmetry-group-theoretic methods. It is intended both as an introduction to the subject and as a reference, covering the general theory and emphasizing the classical types of orthogonal polynomials, or those of Gaussian type. The theory of orthogonal polynomials of several variables, especially those of classical type, uses a significant amount of analysis in one variable.