Table Of ContentCHEBYSHEV
POLYNOMIALS
J.C. MASON
D.C. HANDSCOMB
CHAPMAN & HALL/CRC
A CRC Press Company
Boca Raton London New York Washington, D.C.
© 2003 by CRC Press LLC
Library of Congress Cataloging-in-Publication Data
Mason, J.C.
Chebyshev polynomials / John C. Mason, David Handscomb.
p. cm.
Includes bibliographical references and index.
ISBN 0-8493-0355-9 (alk. paper)
1. Chebyshev polynomials. I. Handscomb, D. C. (David Christopher) II.
Title..
QA404.5 .M37 2002
515′.55—dc21 2002073578
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© 2003 by John C. Mason and David Handscomb
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© 2003 by CRC Press LLC
In memory of recently departed friends
Geoff Hayes, Lev Brutman
© 2003 by CRC Press LLC
Preface
Over thirty years have elapsed since the publication of Fox & Parker’s 1968
text Chebyshev Polynomials in Numerical Analysis. This was preceded by
Snyder’s briefbut interesting1966textChebyshev Methods in Numerical Ap-
proximation. The only significant later publication on the subject is that by
Rivlin (1974, revised and republished in 1990) — a fine exposition of the
theoretical aspects of Chebyshev polynomials but mostly confined to these
aspects. An up-to-date but broader treatment of Chebyshev polynomials is
consequently long overdue, which we now aim to provide.
The idea that there are really four kinds of Chebyshev polynomials, not
justtwo,hasstronglyaffectedthecontentofthisvolume.Indeed,theprop-
ertiesofthefourkindsofpolynomialsleadtoanextendedrangeofresultsin
many areas such as approximation, series expansions, interpolation, quadra-
ture andintegralequations,providingaspurtodevelopingnew methods. We
do not claim the third- and fourth-kind polynomials as our own discovery,
butwedoclaimtohavefollowedcloseontheheelsofWalterGautschiinfirst
adopting this nomenclature.
Ordinaryandpartialdifferentialequationsarenowmajorfieldsofapplica-
tionforChebyshevpolynomialsand,indeed,therearenowfarmorebookson
‘spectral methods’ — at least ten major works to our knowledge — than on
Chebyshev polynomials per se. This makes it more difficult but less essential
todiscussthefullrangeofpossibleapplicationsinthisarea,andherewehave
concentrated on some of the fundamental ideas.
We are pleased with the range of topics that we have managed to in-
clude. However, partly because each chapter concentrates on one subject
area,we have inevitably left a greatdeal out — for instance: the updating of
theChebyshev–Pad´etableandChebyshevrationalapproximation,Chebyshev
approximationonsmallintervals,Faberpolynomialsoncomplexcontoursand
Chebyshev (L∞) polynomials on complex domains.
For the sake of those meeting this subject for the first time, we have
included a number of problems at the end of each chapter. Some of these, in
the earlier chapters in particular, are quite elementary; others are invitations
to fill in the details of working that we have omitted simply for the sake of
brevity; yet others are more advanced problems calling for substantial time
and effort.
We have dedicated this book to the memory of two recently deceased col-
leaguesand friends,who haveinfluenced us inthe writing ofthis book. Geoff
Hayes wrote (with Charles Clenshaw) the major paper on fitting bivariate
polynomialstodatalyingonafamilyofparallellines. Theiralgorithmretains
its place in numerical libraries some thirty-seven years later; it exploits the
idea that Chebyshev polynomials form a well-conditioned basis independent
© 2003 by CRC Press LLC
ofthe spacing ofdata. Lev Brutman specialisedinnear-minimax approxima-
tions and related topics and played a significant role in the development of
this field.
In conclusion, there are many to whom we owe thanks, of whom we can
mention only a few. Among colleagues who helped us in various ways in the
writingofthisbook(butshouldnotbeheldresponsibleforit),wemustname
Graham Elliott, Ezio Venturino, William Smith, David Elliott, Tim Phillips
andNickTrefethen;forgettingthebookstartedandkeepingitoncourse,Bill
MortonandElizabethJohnstoninEngland,BobStern,JamieSigalandothers
at CRC Press in the United States; for help with preparing the manuscript,
Pam Moore and Andrew Crampton. We must finally thank our wives, Moya
and Elizabeth, for the blind faith in which they have encouraged us to bring
this workto completion,without evidencethat itwasevergoingto getthere.
ThisbookwastypesetatOxfordUniversityComputingLaboratory,using
Lamport’s LATEX2ε package.
John Mason
David Handscomb
April 2002
© 2003 by CRC Press LLC
Contents
1Definitions
1.1Preliminaryremarks
1.2Trigonometricdefinitionsandrecurrences
1.2.1Thefirst-kindpolynomial Tn
1.2.2Thesecond-kindpolynomial Un
1.2.3Thethird-an dfou rth-kindpolynomials Vn and Wn
(theairfoilpolynomials)
1.2.4Connectionsbetweenthefourkindsofpolynomial
1.3ShiftedChebyshevpolynomials
1.3.1Theshiftedpolynomials T∗, U∗, V∗, W∗
n n n n
1.3.2Chebyshevpolynomialsforthegeneralrange[ a,b]
1.4Chebyshevpolynomialsofacomplexvariable
1.4.1Conform almappingofacircletoandfromanellipse
1.4.2Chebyshevpolynomialsin z
1.4.3Shabatpolynomials
1.5ProblemsforChapter1
2BasicPropertiesandFormulae
2.1Introduction
2.2Chebyshevpolynomialzerosandextrema
2.3RelationsbetweenChebyshevpolyno mialsa ndpowersof x
2.3.1Powersof x intermsof {Tn(x)}
2.3.2 Tn(x)intermsofpowersof x
2.3.3Ratiosofcoefficientsin Tn(x)
2.4EvaluationofChebyshevsums,products,integralsandderiva-
tives
2.4.1EvaluationofaChebyshevsum
2.4.2StabilityoftheevaluationofaChebyshevsum
2.4.3Evaluationofaproduct
2.4.4Evaluationofanintegral
2.4.5Evaluationofaderivative
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2.5ProblemsforChapter2
3TheMinimaxPropertyandItsApplications
3.1Approximation—theoryandstructure
3.1.1Theappr oximationproblem
3.2Bestandminimaxapproximation
3.3Theminimaxp ropertyoftheChebyshevpolynomials
3.3.1WeightedChebyshevpolynomialsofsecond,thirdand
fourthkinds
3.4TheChebyshevsemi-iterativemethodforlinearequations
3.5Telescopingproceduresforpowerseries
3.5.1ShiftedChebyshevpolynomialson[0 ,1]
3.5.2Implementationofefficientalgorithms
3.6Thetaumethodforseriesandrationalfunctions
3.6.1Theextendedtaumethod
3.7ProblemsforChapter3
4OrthogonalityandLeast-SquaresApproximation
4.1Introduction—fromminimaxtoleastsquares
4.2Orthogonalityo fChebyshevpo lynomials
4.2.1Orthogonalpolynomialsandweightfunctions
4.2.2Chebyshevpo lynomialsa sorthogonalpo lynomials
4.3Orthogonalpolynomialsandbest L2 approximations
4.3.1Orthogonalpolyno mialexpansions
4.3.2Convergencein L2 oforthogonalexpansions
4.4Recurrencerelations
4.5Rodrigues’formulaeanddifferentialequations
4.6DiscreteorthogonalityofChebyshevpolynomials
4.6.1First-kindpolynomials
4.6.2Second-kindpolynomials
4.6.3Third-andfourth-kindpolynomials
4.7DiscreteChebyshevtransformsandthefastFouriertransform
4.7.1ThefastFouriertransform
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4.8Discretedatafittingbyorthogonalpolynomials:theForsythe–
Clenshawmethod
4.8.1Bivariatediscretedatafittingonornearafamilyof
linesorcurves
4.9Orthogonalityinthecomplexplane
4.10ProblemsforChapter4
5ChebyshevSeries
5.1Introduction—Chebyshevseriesandotherexpansions
5.2SomeexplicitChebyshevseriesexpansions
5.2.1Generatingfunctions
5.2.2Approximateseriesexpa nsions
5.3Fourier–ChebyshevseriesandFouriertheory
5.3.1 L2-convergence
5.3.2Pointwiseanduniformconvergence
5.3.3BivariateandmultivariateChebyshevseriesexpansions
5.4Projectionsandnear-bestapproximations
5.5Near-minimaxapproximationbyaChebyshevseries
5.5.1Equalityo fthenormto λn
5.6ComparisonofChebyshevandotherorthogonalpolynomial
expansions
5.7TheerrorofatruncatedChebyshevexpa nsion
5.8Seriesofsecond-,third-an df ourth-kindpolynomial s
5.8.1Seriesofsecond-kindpolynomials
5.8.2Seriesofthird-kindpolynomials
5.8.3MultivariateChebyshevseries
5.9LacunaryChebyshevseries
5.10Chebyshevseriesinthecomplexdomain
5.10.1Chebyshev–Pad´ eapproximations
5.11ProblemsforChapter5
6ChebyshevInterpolation
6.1Polynomia linterpolation
6.2Orthogonalinterpolation
6.3Chebyshevinterpolatio nformulae
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6.3.1Aliasing
6.3.2Second-kindinterpolatio n
6.3.3Third-a ndfo urth-kindinterpolation
6.3.4Conditioning
6.4Best L1 approximationbyChebyshevinterpolatio n
6.5Near-minimaxappr oximationbyChebyshevinterpolatio n
6.6ProblemsforChapter6
7Near-Bets L∞, L1 and Lp Approximations
7.1Near-best L∞ (near-minimax)approximations
7.1.1Second-kindexpansionsin L∞
7.1.2Third -kindexpansionsin L∞
7.2Near-best L1 approximations
7.3Bestandnear-best Lp approximations
7.3.1C omplexvariableresultsforelliptic-typeregions
7.4ProblemsforChapter7
8IntegrationUsingChebyshevPolynomials
8.1IndefiniteintegrationwithChebyshevseries
8.2Gauss–Chebyshevquadrature
8.3QuadraturemethodsofClenshaw–Curtistype
8.3.1Introduction
8.3.2First-kindformulae
8.3.3Second-kindformulae
8.3.4Third-kindformulae
8.3.5Generalremarko nmethodsofClenshaw–Curtistype
8.4ErrorestimationforClenshaw–Curtismethods
8.4.1First-kindpolynomials
8.4.2Fittinganexponentialcurve
8.4.3Otherabscissaeandpolynomials
8.5SomeotherworkonClenshaw–Curtismethods
8.6ProblemsforChapter8
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9SolutionofIntegralEquations
9.1Introduction
9.2Fredholmequationsofthesecondkind
9.3Fredholmequationsofthethirdkind
9.4Fredholmequationsofthefirstkind
9.5Singularkernels
9.5.1Hilbert-typekernelsandrelatedkernels
9.5.2Symm’sintegralequation
9.6Regularisationofintegralequations
9.6.1Discretedatawithsecondderivativeregularisation
9.6.2Detailsofasmoothingalgorithm(secondderivative
regularisation)
9.6.3Asmoothingalgorithmwithweightedfunctionregu-
larisation
9.6.4Evaluationof V(λ)
9.6.5Otherbasisfunctions
9.7Partialdifferentialequationsandboundaryintegralequation
methods
9.7.1Ahypersing ularintegralequa tionderivedfromamixed
boundaryvalueproblemforLaplace’sequation
9.8ProblemsforChapter9
10SolutionofOrdinaryDifferentialEquations
10.1Introduction
10.2Asimpleexample
10.2.1Collocationmethods
10.2.2Errorofthecollocationmethod
10.2.3Projection(tau)methods
10.2.4Erroroftheprecedingprojectionmethod
10.3TheoriginalLanczostau( τ)method
10.4Amoregenerallinearequation
10.4.1Collocationmethod
10.4.2Projectionmethod
10.5Pseudospectralmethods—anotherformofcollocation
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