Orthogonal Polynomials in Two Variables

by Suetin; P. K.

• ISBN13: 9789056991678
• ISBN10: 9056991671
• Format: Hardcover
• Copyright: 1999-08-19
• Publisher: CRC Press

Summary

Presenting a comprehensive theory of orthogonal polynomials in two real variables and properties of Fourier series in these polynomials, this volume also gives cases of orthogonality over a region and on a contour. The text includes the classification of differential equations which admits orthogonal polynomials as eigenfunctions and several two-dimensional analogies of classical orthogonal polynomials.

Table of Contents

Preface	ix
Preface to the English Edition	xv
Notation	xix
General properties of polynomials orthogonal over a domain	1	(36)
Polynomials in two variables orthogonal over a domain	1	(5)
The existence theorem and criteria of orthogonality	6	(4)
Algebraic properties	10	(8)
Monic orthogonal polynomials	18	(6)
Normal biorthogonal systems	24	(4)
Fourier series of orthogonal polynomials in two variables	28	(3)
Fourier series for differentiable functions	31	(6)
Some typical examples and special cases of orthogonality over a domain	37	(26)
Different products of classical orthogonal polynomials	37	(5)
Various cases of connection between orthogonality over a domain and orthogonality on an interval	42	(6)
Some theorems in the case of a weight function with separating variables	48	(4)
Conditions of interconnection between the weight function and the domain of orthogonality	52	(5)
Examples of computations of weight function moments	57	(6)
Classical Appell's orthogonal polynomials	63	(24)
Rodrigues formula for Appell's polynomials	63	(6)
Representation of the Appell polynomials via the hypergeometric function of two variables	69	(3)
Differential equation for the Appell polynomials	72	(3)
Orthogonality of eigenfunctions of the Appell equation	75	(4)
Normal biorthogonal Appell system	79	(4)
Series in the Appell polynomials	83	(4)
Admissible differential equation for polynomials orthogonal over a domain	87	(44)
The main differential operator and a theorem on orthogonality	87	(5)
Admissibility conditions for the main differential equation	92	(5)
Some examples and properties of admissible differential equations	97	(4)
Affine transformations of the arguments of the main differential equation	101	(4)
Transformation of the coefficients of the characteristic polynomial	105	(10)
Normal forms of the admissible differential equation	115	(8)
Normal forms when reducing the degree of the characteristic polynomial	123	(8)
Potentially self-adjoint equation and Rodrigues formula	131	(32)
Potentially self-adjoint operators	131	(4)
Admissible and potentially self-adjoint equations	135	(11)
Rodrigues formula for polynomials orthogonal over a domain	146	(7)
Weight functions and the Rodrigues formula in the most typical cases	153	(10)
Harmonic polynomials orthogonal over a domain	163	(24)
Homogeneous harmonic polynomials	163	(6)
An analogue of the Christoffel-Darboux formula	169	(4)
Harmonic polynomials orthogonal in the unit disk	173	(3)
Harmonic polynomials orthogonal over a domain in the general case	176	(3)
Harmonic polynomials superorthogonal over a domain	179	(8)
Polynomials in two variables orthogonal on a contour	187	(36)
Main definitions and the simplest properties	187	(4)
Polynomials in two variables orthogonal on an algebraic curve	191	(5)
Harmonic polynomials orthogonal on a contour	196	(4)
Fourier series in harmonic polynomials orthogonal on a contour	200	(6)
Harmonic polynomials superorthogonal on a contour	206	(7)
Examples of superorthogonal systems of harmonic polynomials	213	(10)
Generalized orthogonal polynomials in two variables	223	(30)
Main definitions and the simplest properties	223	(5)
The existence theorem in the most general form	228	(5)
Fourier series in generalized orthogonal polynomials in two variables	233	(8)
Monic orthogonal polynomials under minimal conditions	241	(6)
Generalized generating functions for monic orthogonal polynomials	247	(6)
Other results concerning the connection between orthogonal polynomials and differential equations	253	(32)
Canonical admissible differential equation and monic orthogonal polynomials	253	(5)
Necessary consistency conditions of the canonical operator and the functional	258	(4)
Sufficient conditions of consistency of the canonical operator and the functional	262	(6)
The deduction of the differential equation from the Pearson equation system	268	(8)
An admissible partial differential equation of an arbitrary order	276	(9)
Original results of T. Koornwinder	285	(28)
Examples of the representation of polynomials orthogonal over a domain via the Jacobi polynomials	285	(6)
Orthogonal polynomials in tow conjugate complex variables	291	(5)
The Chebyshev polynomials in two conjugate complex variables for the Steiner domain	296	(12)
Another generalization of the Jacobi polynomials onto the case of two variables	308	(5)
Some recent results	313	(10)
A new generalization of the Appell polynomials	313	(5)
Some properties of the Koornwinder-Steiner polynomials	318	(1)
A two-dimensional analogue of the Chebyshev-Laguerre polynomials	319	(4)
Comments and Supplements	323	(6)
References	329	(14)
Author Index	343	(2)
Subject Index	345