Orthogonal Polynomials of Several Variables

✍ Scribed by Charles F. Dunkl, Yuan Xu

• Publisher: Cambridge University Press
• Year: 2014
• Tongue: English
• Leaves: 439
• Series: Encyclopedia of mathematics and its applications 155
• Edition: 2ed.
• Category: Library

✦ Synopsis

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

✦ Table of Contents

Content: Cover
Half-Title page
Series page
Title page
Copyright page
Dedication page
Contents
Preface to the Second Edition
Preface to the First Edition
1 Background
1.1 The Gamma and Beta Functions
1.2 Hypergeometric Series
1.2.1 Lauricella series
1.3 Orthogonal Polynomials of One Variable
1.3.1 General properties
1.3.2 Three-term recurrence
1.4 Classical Orthogonal Polynomials
1.4.1 Hermite polynomials
1.4.2 Laguerre polynomials
1.4.3 Gegenbauer polynomials
1.4.4 Jacobi polynomials
1.5 Modified Classical Polynomials
1.5.1 Generalized Hermite polynomials. 1.5.2 Generalized Gegenbauer polynomials1.5.3 A limiting relation
1.6 Notes
2 Orthogonal Polynomials in Two Variables
2.1 Introduction
2.2 Product Orthogonal Polynomials
2.3 Orthogonal Polynomials on the Unit Disk
2.4 Orthogonal Polynomials on the Triangle
2.5 Orthogonal Polynomials and Differential Equations
2.6 Generating Orthogonal Polynomials of Two Variables
2.6.1 A method for generating orthogonal polynomials
2.6.2 Orthogonal polynomials for a radial weight
2.6.3 Orthogonal polynomials in complex variables
2.7 First Family of Koornwinder Polynomials. 2.8 A Related Family of Orthogonal Polynomials2.9 Second Family of Koornwinder Polynomials
2.10 Notes
3 General Properties of Orthogonal Polynomials in Several Variables
3.1 Notation and Preliminaries
3.2 Moment Functionals and Orthogonal Polynomials in Several Variables
3.2.1 Definition of orthogonal polynomials
3.2.2 Orthogonal polynomials and moment matrices
3.2.3 The moment problem
3.3 The Three-Term Relation
3.3.1 Definition and basic properties
3.3.2 Favard''s theorem
3.3.3 Centrally symmetric integrals
3.3.4 Examples
3.4 Jacobi Matrices and Commuting Operators. 3.5 Further Properties of the Three-Term Relation3.5.1 Recurrence formula
3.5.2 General solutions of the three-term relation
3.6 Reproducing Kernels and Fourier Orthogonal Series
3.6.1 Reproducing kernels
3.6.2 Fourier orthogonal series
3.7 Common Zeros of Orthogonal Polynomials in Several Variables
3.8 Gaussian Cubature Formulae
3.9 Notes
4 Orthogonal Polynomials on the Unit Sphere
4.1 Spherical Harmonics
4.2 Orthogonal Structures on S[sup(d)] and on B[sup(d)]
4.3 Orthogonal Structures on B[sup(d)] and on S[sup(d+m-1)]
4.4 Orthogonal Structures on the Simplex. 4.5 Van der Corput --
Schaake Inequality4.6 Notes
5 Examples of Orthogonal Polynomials in Several Variables
5.1 Orthogonal Polynomials for Simple Weight Functions
5.1.1 Product weight functions
5.1.2 Rotation-invariant weight functions
5.1.3 Multiple Hermite polynomials on R[sup(d)]
5.1.4 Multiple Laguerre polynomials on R[sub(+)sup(d)]
5.2 Classical Orthogonal Polynomials on the Unit Ball
5.2.1 Orthonormal bases
5.2.2 Appell''s monic orthogonal and biorthogonalpolynomials
5.2.3 Reproducing kernel with respect to W[sup(B) sub[Mu] on B[sup(d)].

✦ Subjects

Orthogonal polynomials;Functions of several real variables;MATHEMATICS -- Calculus;MATHEMATICS -- Mathematical Analysis;Orthogonale reeksen