Orthogonal Polynomials of Several Variables

by Charles F. Dunkl , Yuan Xu

• ISBN13: 9780521800433
• ISBN10: 0521800439
• Format: Hardcover
• Copyright: 2001-03-19
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

Preface	xiii
Background	1	(29)
The Gamma and Beta Functions	1	(2)
Hypergeometric Series	3	(4)
Orthogonal Polynomials of One Variable	7	(6)
General properties	7	(2)
Three term recurrence	9	(4)
Classical Orthogonal Polynomials	13	(10)
Hermite polynomials	14	(1)
Laguerre polynomials	15	(2)
Gegenbauer polynomials	17	(4)
Jacobi polynomials	21	(2)
Modified Classical Polynomials	23	(5)
Generalized Hermite polynomials	25	(1)
Generalized Gegenbauer polynomials	26	(2)
A limiting relation	28	(1)
Notes	28	(2)
Examples of Orthogonal Polynomials in Several Variables	30	(33)
Notation and Preliminary	30	(3)
Spherical Harmonics	33	(4)
Classical Orthogonal Polynomials	37	(14)
Multiple Jacobi polynomials on the cube	37	(1)
Classical orthogonal polynomials on the unit ball	38	(8)
Classical orthogonal polynomials on the simplex	46	(3)
Multiple Hermite polynomials on Rd	49	(2)
Multiple Laguerre polynomials on Rd+	51	(1)
Other Examples of Orthogonal Polynomials	51	(7)
Two general families of orthogonal polynomials	51	(4)
A method for generating orthogonal polynomials of two variables	55	(2)
Disc polynomials	57	(1)
Van der Corput-Schaake Inequality	58	(2)
Notes	60	(3)
General Properties of Orthogonal Polynomials in Several Variables	63	(74)
Moment Functionals and Orthogonal Polynomials in Several Variables	64	(11)
Definition of orthogonal polynomials	64	(5)
Orthogonal polynomials and moment matrices	69	(3)
The moment problem	72	(3)
The Three Term Relation	75	(13)
Definition and basic properties	75	(4)
Favard's theorem	79	(3)
Centrally symmetric integrals	82	(3)
Examples	85	(3)
Jacobi Matrices and Commuting Operators	88	(7)
Further Properties of the Three Term Relation	95	(10)
Recurrence formula	95	(7)
General solutions of the three-term relation	102	(3)
Reproducing Kernels and Fourier Orthogonal Series	105	(8)
Reproducing kernels	106	(4)
Fourier orthogonal series	110	(3)
Common Zeros of Orthogonal Polynomials in Several Variables	113	(4)
Gaussian Cubature Formulae	117	(9)
Characterization of Gaussian cubature formulae	118	(5)
Examples of Gaussian cubature formulae	123	(3)
Orthogonal Polynomials on the Unit Sphere	126	(8)
Orthogonal structures on Sd and on Bd	126	(6)
Orthogonal structure on Bd and on Sd+m	132	(2)
Notes	134	(3)
Root Systems and Coxeter groups	137	(38)
Introduction and Overview	137	(2)
Root Systems	139	(8)
Type Ad-1	142	(1)
Type Bd	143	(1)
Type I2(m)	144	(1)
Type Dd	144	(1)
Type H3	145	(1)
Type F4	146	(1)
Other types	146	(1)
Miscellaneous results	146	(1)
Invariant Polynomials	147	(4)
Type Ad-1 invariants	149	(1)
Type Bd invariants	150	(1)
Type Dd invariants	150	(1)
Type I2 (m) invariants	151	(1)
Type H3 invariants	151	(1)
Type F4 invariants	151	(1)
Differential-Difference Operators	151	(6)
The Intertwining Operator	157	(10)
The k-Analog of the Exponential	167	(2)
Invariant Differential Operators	169	(4)
Notes	173	(2)
Spherical Harmonics Associated with Reflection Groups	175	(50)
h-Harmonic Polynomials	175	(10)
Inner Products on Polynomials	185	(4)
Reproducing Kernels and the Poisson Kernel	189	(3)
Integration of the Intertwining Operator	192	(5)
Example: Abelian Group Zd2	197	(8)
Example: Dihedral Groups	205	(11)
An orthonormal basis of Hn (h2α,β)	206	(8)
Cauchy and Poisson kernels	214	(2)
The Fourier Transform	216	(8)
Notes	224	(1)
Classical and Generalized Classical Orthogonal Polynomials	225	(30)
Generalized Classical Orthogonal Polynomials on the Ball	225	(11)
Definition and differential-difference equations	225	(6)
Bases, reproducing kernels, and the Funk-Hecke formula	231	(5)
Orthogonal Polynomials on the Simplex	236	(7)
General weight functions on Td	236	(3)
Generalized classical orthogonal polynomials	239	(4)
Generalized Hermite Polynomials	243	(6)
Generalized Laguerre Polynomials	249	(4)
Notes	253	(2)
Summability of Orthogonal Expansions	255	(32)
General Results on Orthogonal Expansions	255	(7)
Uniform convergence of partial sums	255	(4)
Cesaro means of the orthogonal expansion	259	(3)
Orthogonal Expansion on the Sphere	262	(4)
Orthogonal Expansion on the Ball	266	(5)
Orthogonal Expansions on the Simplex	271	(3)
Orthogonal Expansion of Laguerre and Hermite Polynomials	274	(5)
Multiple Jacobi Expansion	279	(5)
Notes	284	(3)
Orthogonal Polynomials Associated with Symmetric Groups	287	(50)
Introduction	287	(1)
Partitions, Compositions and Orderings	288	(1)
Commuting Self-Adjoint Operators	289	(3)
The Dual Polynomial Basis	292	(7)
Sd Invariant Subspaces	299	(5)
Degree Changing Recurrences	304	(4)
Norm Formulae	308	(13)
Hook length products and the pairing norm	308	(3)
The biorthogonal type norm	311	(2)
The torus inner product	313	(3)
Normalizing constants	316	(5)
Symmetric Functions and Jack Polynomials	321	(9)
Miscellaneous Topics	330	(5)
Notes	335	(2)
Orthogonal Polynomials Associated with Octahedral Groups and Applications	337	(35)
Introduction	337	(1)
Operators of Type B	338	(3)
Polynomial Eigenfunctions of Type B	341	(9)
Generalized Binomial Coefficients	350	(9)
Hermite Polynomials of Type B	359	(1)
Calogero-Sutherland Systems	360	(10)
The simple harmonic oscillator	361	(1)
Root systems and the Laplacian	362	(1)
Type A models on the line	363	(2)
Type A models on the circle	365	(3)
Type B models on the line	368	(2)
Notes	370	(2)
Bibliography	372	(12)
Author index	384	(3)
Symbol index	387	(2)
Subject index	389