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Orthogonal Polynomials of Several Variables, Second Edition

Book Description

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

  1. Cover
  2. Half-title page
  3. Series page
  4. Title page
  5. Copyright page
  6. Dedication
  7. Contents
  8. Preface to the Second Edition
  9. Preface to the First Edition
  10. 1. Background
    1. 1.1 The Gamma and Beta Functions
    2. 1.2 Hypergeometric Series
      1. 1.2.1 Lauricella series
    3. 1.3 Orthogonal Polynomials of One Variable
      1. 1.3.1 General properties
      2. 1.3.2 Three-term recurrence
    4. 1.4 Classical Orthogonal Polynomials
      1. 1.4.1 Hermite polynomials
      2. 1.4.2 Laguerre polynomials
      3. 1.4.3 Gegenbauer polynomials
      4. 1.4.4 Jacobi polynomials
    5. 1.5 Modified Classical Polynomials
      1. 1.5.1 Generalized Hermite polynomials
      2. 1.5.2 Generalized Gegenbauer polynomials
      3. 1.5.3 A limiting relation
    6. 1.6 Notes
  11. 2. Orthogonal Polynomials in Two Variables
    1. 2.1 Introduction
    2. 2.2 Product Orthogonal Polynomials
    3. 2.3 Orthogonal Polynomials on the Unit Disk
    4. 2.4 Orthogonal Polynomials on the Triangle
    5. 2.5 Orthogonal Polynomials and Differential Equations
    6. 2.6 Generating Orthogonal Polynomials of Two Variables
      1. 2.6.1 A method for generating orthogonal polynomials
      2. 2.6.2 Orthogonal polynomials for a radial weight
      3. 2.6.3 Orthogonal polynomials in complex variables
    7. 2.7 First Family of Koornwinder Polynomials
    8. 2.8 A Related Family of Orthogonal Polynomials
    9. 2.9 Second Family of Koornwinder Polynomials
    10. 2.10 Notes
  12. 3. General Properties of Orthogonal Polynomials in Several Variables
    1. 3.1 Notation and Preliminaries
    2. 3.2 Moment Functionals and Orthogonal Polynomials in Several Variables
      1. 3.2.1 Definition of orthogonal polynomials
      2. 3.2.2 Orthogonal polynomials and moment matrices
      3. 3.2.3 The moment problem
    3. 3.3 The Three-Term Relation
      1. 3.3.1 Definition and basic properties
      2. 3.3.2 Favard’s theorem
      3. 3.3.3 Centrally symmetric integrals
      4. 3.3.4 Examples
    4. 3.4 Jacobi Matrices and Commuting Operators
    5. 3.5 Further Properties of the Three-Term Relation
      1. 3.5.1 Recurrence formula
      2. 3.5.2 General solutions of the three-term relation
    6. 3.6 Reproducing Kernels and Fourier Orthogonal Series
      1. 3.6.1 Reproducing kernels
      2. 3.6.2 Fourier orthogonal series
    7. 3.7 Common Zeros of Orthogonal Polynomials in Several Variables
    8. 3.8 Gaussian Cubature Formulae
    9. 3.9 Notes
  13. 4. Orthogonal Polynomials on the Unit Sphere
    1. 4.1 Spherical Harmonics
    2. 4.2 Orthogonal Structures on S[sup(d)] and on B[sup(d)]
    3. 4.3 Orthogonal Structures on B[sup(d)] and on S[sup(d+m−1)]
    4. 4.4 Orthogonal Structures on the Simplex
    5. 4.5 Van der Corput–Schaake Inequality
    6. 4.6 Notes
  14. 5. Examples of Orthogonal Polynomials in Several Variables
    1. 5.1 Orthogonal Polynomials for Simple Weight Functions
      1. 5.1.1 Product weight functions
      2. 5.1.2 Rotation-invariant weight functions
      3. 5.1.3 Multiple Hermite polynomials on R[sup(d)]
      4. 5.1.4 Multiple Laguerre polynomials on R[sup(d)][sub(+)]
    2. 5.2 Classical Orthogonal Polynomials on the Unit Ball
      1. 5.2.1 Orthonormal bases
      2. 5.2.2 Appell’s monic orthogonal and biorthogonal polynomials
      3. 5.2.3 Reproducing kernel with respect to W[sup(B)][sub(μ)] on B[sup(d)]
    3. 5.3 Classical Orthogonal Polynomials on the Simplex
    4. 5.4 Orthogonal Polynomials via Symmetric Functions
      1. 5.4.1 Two general families of orthogonal polynomials
      2. 5.4.2 Common zeros and Gaussian cubature formulae
    5. 5.5 Chebyshev Polynomials of Type A[sub(d)]
    6. 5.6 Sobolev Orthogonal Polynomials on the Unit Ball
      1. 5.6.1 Sobolev orthogonal polynomials defined via the gradient operator
      2. 5.6.2 Sobolev orthogonal polynomials defined via the Laplacian operator
    7. 5.7 Notes
  15. 6. Root Systems and Coxeter Groups
    1. 6.1 Introduction and Overview
    2. 6.2 Root Systems
      1. 6.2.1 Type A[sub(d−1)]
      2. 6.2.2 Type B[sub(d)]
      3. 6.2.3 Type I[sub(2)](m)
      4. 6.2.4 Type D[sub(d)]
      5. 6.2.5 Type H[sub(3)]
      6. 6.2.6 Type F[sub(4)]
      7. 6.2.7 Other types
      8. 6.2.8 Miscellaneous results
    3. 6.3 Invariant Polynomials
      1. 6.3.1 Type A[sub(d−1)] invariants
      2. 6.3.2 Type B[sub(d)] invariants
      3. 6.3.3 Type D[sub(d)] invariants
      4. 6.3.4 Type I[sub(2)](m) invariants
      5. 6.3.5 Type H[sub(3)] invariants
      6. 6.3.6 Type F[sub(4)] invariants
    4. 6.4 Differential–Difference Operators
    5. 6.5 The Intertwining Operator
    6. 6.6 The k-Analogue of the Exponential
    7. 6.7 Invariant Differential Operators
    8. 6.8 Notes
  16. 7. Spherical Harmonics Associated with Reflection Groups
    1. 7.1 h-Harmonic Polynomials
    2. 7.2 Inner Products on Polynomials
    3. 7.3 Reproducing Kernels and the Poisson Kernel
    4. 7.4 Integration of the Intertwining Operator
    5. 7.5 Example: Abelian Group Z[sup(d)][sub(2)]
      1. 7.5.1 Orthogonal basis for h-harmonics
      2. 7.5.2 Intertwining and projection operators
      3. 7.5.3 Monic orthogonal basis
    6. 7.6 Example: Dihedral Groups
      1. 7.6.1 An orthonormal basis of H[sub(n)](h[sup(2)][sub(α,β)])
      2. 7.6.2 Cauchy and Poisson kernels
    7. 7.7 The Dunkl Transform
    8. 7.8 Notes
  17. 8. Generalized Classical Orthogonal Polynomials
    1. 8.1 Generalized Classical Orthogonal Polynomials on the Ball
      1. 8.1.1 Definition and differential–difference equations
      2. 8.1.2 Orthogonal basis and reproducing kernel
      3. 8.1.3 Orthogonal polynomials for Z[sup(d)][sub(2)]-invariant weight functions
      4. 8.1.4 Reproducing kernel for Z[sup(d)][sub(2)]-invariant weight functions
    2. 8.2 Generalized Classical Orthogonal Polynomials on the Simplex
      1. 8.2.1 Weight function and differential–difference equation
      2. 8.2.2 Orthogonal basis and reproducing kernel
      3. 8.2.3 Monic orthogonal polynomials
    3. 8.3 Generalized Hermite Polynomials
    4. 8.4 Generalized Laguerre Polynomials
    5. 8.5 Notes
  18. 9. Summability of Orthogonal Expansions
    1. 9.1 General Results on Orthogonal Expansions
      1. 9.1.1 Uniform convergence of partial sums
      2. 9.1.2 Cesàro means of the orthogonal expansion
    2. 9.2 Orthogonal Expansion on the Sphere
    3. 9.3 Orthogonal Expansion on the Ball
    4. 9.4 Orthogonal Expansion on the Simplex
    5. 9.5 Orthogonal Expansion of Laguerre and Hermite Polynomials
    6. 9.6 Multiple Jacobi Expansion
    7. 9.7 Notes
  19. 10. Orthogonal Polynomials Associated with Symmetric Groups
    1. 10.1 Partitions, Compositions and Orderings
    2. 10.2 Commuting Self-Adjoint Operators
    3. 10.3 The Dual Polynomial Basis
    4. 10.4 S[sub(d)]-Invariant Subspaces
    5. 10.5 Degree-Changing Recurrences
    6. 10.6 Norm Formulae
      1. 10.6.1 Hook-length products and the pairing norm
      2. 10.6.2 The biorthogonal-type norm
      3. 10.6.3 The torus inner product
      4. 10.6.4 Monic polynomials
      5. 10.6.5 Normalizing constants
    7. 10.7 Symmetric Functions and Jack Polynomials
    8. 10.8 Miscellaneous Topics
    9. 10.9 Notes
  20. 11. Orthogonal Polynomials Associated with Octahedral Groups, and Applications
    1. 11.1 Introduction
    2. 11.2 Operators of Type B
    3. 11.3 Polynomial Eigenfunctions of Type B
    4. 11.4 Generalized Binomial Coefficients
    5. 11.5 Hermite Polynomials of Type B
    6. 11.6 Calogero–Sutherland Systems
      1. 11.6.1 The simple harmonic oscillator
      2. 11.6.2 Root systems and the Laplacian
      3. 11.6.3 Type A models on the line
      4. 11.6.4 Type A models on the circle
      5. 11.6.5 Type B models on the line
    7. 11.7 Notes
  21. References
  22. Author Index
  23. Symbol Index
  24. Subject Index