Orthogonal Polynomials in Two Variables

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ISBN: 9789056991678 | 9056991671
Cover: Hardcover
Copyright: 8/19/1999

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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.

Preface

Preface to the English Edition

Notation

xix

General properties of polynomials orthogonal over a domain

(36)

Polynomials in two variables orthogonal over a domain

(5)

The existence theorem and criteria of orthogonality

(4)

Algebraic properties

(8)

Monic orthogonal polynomials

(6)

Normal biorthogonal systems

(4)

Fourier series of orthogonal polynomials in two variables

(3)

Fourier series for differentiable functions

(6)

Some typical examples and special cases of orthogonality over a domain

(26)

Different products of classical orthogonal polynomials

(5)

Various cases of connection between orthogonality over a domain and orthogonality on an interval

(6)

Some theorems in the case of a weight function with separating variables

(4)

Conditions of interconnection between the weight function and the domain of orthogonality

(5)

Examples of computations of weight function moments

(6)

Classical Appell's orthogonal polynomials

(24)

Rodrigues formula for Appell's polynomials

(6)

Representation of the Appell polynomials via the hypergeometric function of two variables

(3)

Differential equation for the Appell polynomials

(3)

Orthogonality of eigenfunctions of the Appell equation

(4)

Normal biorthogonal Appell system

(4)

Series in the Appell polynomials

(4)

Admissible differential equation for polynomials orthogonal over a domain

(44)

The main differential operator and a theorem on orthogonality

(5)

Admissibility conditions for the main differential equation

(5)

Some examples and properties of admissible differential equations

(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

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Orthogonal Polynomials in Two Variables

Summary

Table of Contents

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Orthogonal Polynomials in Two Variables

Summary

Table of Contents

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