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E-grāmata: Orthogonal Polynomials of Several Variables

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(University of Virginia), (University of Oregon)
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Orthogonal Polynomials of Several VariablesPalielināt
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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.

Serving both as an introduction to the subject and as a reference, this book covers the general theory and emphasizes the classical types of orthogonal polynomials, or those of Gaussian type. Containing 25% brand new material, this revised edition reflects progress made in the field over the past decade.

Recenzijas

Review of the first edition: 'This book is the first modern treatment of orthogonal polynomials of several real variables. It presents not only a general theory, but also detailed results of recent research on generalizations of various classical cases.' Mathematical Reviews Review of the first edition: 'This book is very impressive and shows the richness of the theory.' Vilmos Totik, Acta Scientiarum Mathematicarum 'This is a valuable book for anyone with an interest in special functions of several variables.' Marcel de Jeu, American Mathematical Society

Papildus informācija

Updated throughout, this revised edition contains 25% new material covering progress made in the field over the past decade.
Preface to the Second Edition xiii
Preface to the First Edition xv
1 Background
1(27)
1.1 The Gamma and Beta Functions
1(2)
1.2 Hypergeometric Series
3(3)
1.2.1 Lauricella series
5(1)
1.3 Orthogonal Polynomials of One Variable
6(7)
1.3.1 General properties
6(3)
1.3.2 Three-term recurrence
9(4)
1.4 Classical Orthogonal Polynomials
13(9)
1.4.1 Hermite polynomials
13(1)
1.4.2 Laguerre polynomials
14(2)
1.4.3 Gegenbauer polynomials
16(4)
1.4.4 Jacobi polynomials
20(2)
1.5 Modified Classical Polynomials
22(5)
1.5.1 Generalized Hermite polynomials
24(1)
1.5.2 Generalized Gegenbauer polynomials
25(2)
1.5.3 A limiting relation
27(1)
1.6 Notes
27(1)
2 Orthogonal Polynomials in Two Variables
28(29)
2.1 Introduction
28(1)
2.2 Product Orthogonal Polynomials
29(1)
2.3 Orthogonal Polynomials on the Unit Disk
30(5)
2.4 Orthogonal Polynomials on the Triangle
35(2)
2.5 Orthogonal Polynomials and Differential Equations
37(1)
2.6 Generating Orthogonal Polynomials of Two Variables
38(7)
2.6.1 A method for generating orthogonal polynomials
38(2)
2.6.2 Orthogonal polynomials for a radial weight
40(1)
2.6.3 Orthogonal polynomials in complex variables
41(4)
2.7 First Family of Koornwinder Polynomials
45(3)
2.8 A Related Family of Orthogonal Polynomials
48(2)
2.9 Second Family of Koornwinder Polynomials
50(4)
2.10 Notes
54(3)
3 General Properties of Orthogonal Polynomials in Several Variables
57(57)
3.1 Notation and Preliminaries
58(2)
3.2 Moment Functionals and Orthogonal Polynomials in Several Variables
60(10)
3.2.1 Definition of orthogonal polynomials
60(4)
3.2.2 Orthogonal polynomials and moment matrices
64(3)
3.2.3 The moment problem
67(3)
3.3 The Three-Term Relation
70(12)
3.3.1 Definition and basic properties
70(3)
3.3.2 Favard's theorem
73(3)
3.3.3 Centrally symmetric integrals
76(3)
3.3.4 Examples
79(3)
3.4 Jacobi Matrices and Commuting Operators
82(5)
3.5 Further Properties of the Three-Term Relation
87(9)
3.5.1 Recurrence formula
87(7)
3.5.2 General solutions of the three-term relation
94(2)
3.6 Reproducing Kernels and Fourier Orthogonal Series
96(7)
3.6.1 Reproducing kernels
97(4)
3.6.2 Fourier orthogonal series
101(2)
3.7 Common Zeros of Orthogonal Polynomials in Several Variables
103(4)
3.8 Gaussian Cubature Formulae
107(5)
3.9 Notes
112(2)
4 Orthogonal Polynomials on the Unit Sphere
114(23)
4.1 Spherical Harmonics
114(5)
4.2 Orthogonal Structures on Sd and on Bd
119(6)
4.3 Orthogonal Structures on Bd and on Sd+m-1
125(4)
4.4 Orthogonal Structures on the Simplex
129(4)
4.5 Van der Corput--Schaake Inequality
133(3)
4.6 Notes
136(1)
5 Examples of Orthogonal Polynomials in Several Variables
137(37)
5.1 Orthogonal Polynomials for Simple Weight Functions
137(4)
5.1.1 Product weight functions
138(1)
5.1.2 Rotation-invariant weight functions
138(1)
5.1.3 Multiple Hermite polynomials on Rd
139(2)
5.1.4 Multiple Laguerre polynomials on Rd+
141(1)
5.2 Classical Orthogonal Polynomials on the Unit Ball
141(9)
5.2.1 Orthonormal bases
142(1)
5.2.2 Appell's monic orthogonal and biorthogonal polynomials
143(5)
5.2.3 Reproducing kernel with respect to WBμ on Bd
148(2)
5.3 Classical Orthogonal Polynomials on the Simplex
150(4)
5.4 Orthogonal Polynomials via Symmetric Functions
154(5)
5.4.1 Two general families of orthogonal polynomials
154(2)
5.4.2 Common zeros and Gaussian cubature formulae
156(3)
5.5 Chebyshev Polynomials of Type Ad
159(6)
5.6 Sobolev Orthogonal Polynomials on the Unit Ball
165(6)
5.6.1 Sobolev orthogonal polynomials defined via the gradient operator
165(3)
5.6.2 Sobolev orthogonal polynomials defined via the Laplacian operator
168(3)
5.7 Notes
171(3)
6 Root Systems and Coxeter Groups
174(34)
6.1 Introduction and Overview
174(2)
6.2 Root Systems
176(7)
6.2.1 Type Ad-1
179(1)
6.2.2 Type Bd
179(1)
6.2.3 Type I2(m)
180(1)
6.2.4 Type Dd
181(1)
6.2.5 Type H3
181(1)
6.2.6 Type F4
182(1)
6.2.7 Other types
182(1)
6.2.8 Miscellaneous results
182(1)
6.3 Invariant Polynomials
183(4)
6.3.1 Type Ad-1 invariants
185(1)
6.3.2 Type Bd invariants
186(1)
6.3.3 Type Dd invariants
186(1)
6.3.4 Type I2(m) invariants
186(1)
6.3.5 Type H3 invariants
186(1)
6.3.6 Type F4 invariants
187(1)
6.4 Differential--Difference Operators
187(5)
6.5 The Intertwining Operator
192(8)
6.6 The κ-Analogue of the Exponential
200(2)
6.7 Invariant Differential Operators
202(5)
6.8 Notes
207(1)
7 Spherical Harmonies Associated with Reflection Groups
208(50)
7.1 h-Harmonic Polynomials
208(9)
7.2 Inner Products on Polynomials
217(4)
7.3 Reproducing Kernels and the Poisson Kernel
221(3)
7.4 Integration of the Intertwining Operator
224(4)
7.5 Example: Abelian Group Zd2
228(12)
7.5.1 Orthogonal basis for h-harmonics
228(4)
7.5.2 Intertwining and projection operators
232(3)
7.5.3 Monic orthogonal basis
235(5)
7.6 Example: Dihedral Groups
240(10)
7.6.1 An orthonormal basis of Hn(h2αβ)
241(7)
7.6.2 Cauchy and Poisson kernels
248(2)
7.7 The Dunkl Transform
250(6)
7.8 Notes
256(2)
8 Generalized Classical Orthogonal Polynomials
258(31)
8.1 Generalized Classical Orthogonal Polynomials on the Ball
258(13)
8.1.1 Definition and differential--difference equations
258(5)
8.1.2 Orthogonal basis and reproducing kernel
263(3)
8.1.3 Orthogonal polynomials for Zd2-invariant weight functions
266(2)
8.1.4 Reproducing kernel for Zd2-invariant weight functions
268(3)
8.2 Generalized Classical Orthogonal Polynomials on the Simplex
271(7)
8.2.1 Weight function and differential--difference equation
271(2)
8.2.2 Orthogonal basis and reproducing kernel
273(3)
8.2.3 Monic orthogonal polynomials
276(2)
8.3 Generalized Hermite Polynomials
278(5)
8.4 Generalized Laguerre Polynomials
283(4)
8.5 Notes
287(2)
9 Summability of Orthogonal Expansions
289(29)
9.1 General Results on Orthogonal Expansions
289(7)
9.1.1 Uniform convergence of partial sums
289(4)
9.1.2 Cesaro means of the orthogonal expansion
293(3)
9.2 Orthogonal Expansion on the Sphere
296(3)
9.3 Orthogonal Expansion on the Ball
299(5)
9.4 Orthogonal Expansion on the Simplex
304(2)
9.5 Orthogonal Expansion of Laguerre and Hermite Polynomials
306(5)
9.6 Multiple Jacobi Expansion
311(4)
9.7 Notes
315(3)
10 Orthogonal Polynomials Associated with Symmetric Groups
318(46)
10.1 Partitions, Compositions and Orderings
318(2)
10.2 Commuting Self-Adjoint Operators
320(2)
10.3 The Dual Polynomial Basis
322(7)
10.4 Sd-Invariant Subspaces
329(5)
10.5 Degree-Changing Recurrences
334(3)
10.6 Norm Formulae
337(13)
10.6.1 Hook-length products and the pairing norm
337(4)
10.6.2 The biorthogonal-type norm
341(2)
10.6.3 The torus inner product
343(3)
10.6.4 Monic polynomials
346(1)
10.6.5 Normalizing constants
346(4)
10.7 Symmetric Functions and Jack Polynomials
350(7)
10.8 Miscellaneous Topics
357(5)
10.9 Notes
362(2)
11 Orthogonal Polynomials Associated with Octahedral Groups, and Applications
364(32)
11.1 Introduction
364(1)
11.2 Operators of Type B
365(3)
11.3 Polynomial Eigenfunctions of Type B
368(8)
11.4 Generalized Binomial Coefficients
376(7)
11.5 Hermite Polynomials of Type B
383(2)
11.6 Calogero--Sutherland Systems
385(9)
11.6.1 The simple harmonic oscillator
386(1)
11.6.2 Root systems and the Laplacian
387(1)
11.6.3 Type A models on the line
387(2)
11.6.4 Type A models on the circle
389(3)
11.6.5 Type B models on the line
392(2)
11.7 Notes
394(2)
References 396(17)
Author Index 413(3)
Symbol Index 416(2)
Subject Index 418
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 199293 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.
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