Mathematics of Approximation

418 Pages·2012·2.953 MB·English

by  Johan de De Villiers

MATHEMATICSTEXTBOOKSFORSCIENCEANDENGINEERING
VOLUME1
SERIESEDITOR: C.K.CHUI
Mathematics Textbooks for Science and
Engineering
SeriesEditor:
C.K.Chui
StanfordUniversity,USA
(ISSN:2213-0853)
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Textbooks in the series ‘Mathematics Textbooks for Science and Engineering’ will be
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AMSTERDAM–PARIS–BEIJING
(cid:2)c ATLANTISPRESS
Mathematics of Approximation
Johan de Villiers
DepartmentofMathematicalSciences
StellenboschUniversity
SouthAfrica
AMSTERDAM–PARIS–BEIJING
AtlantisPress
8,squaredesBouleaux
75019Paris,France
ForinformationonallAtlantisPresspublications,visitourwebsiteat:www.atlantis-press.com
Copyright
Thisbook,oranypartsthereof,maynotbereproducedforcommercialpurposesinanyformorby
anymeans,electronicormechanical,includingphotocopying,recordingoranyinformationstorage
andretrievalsystemknownortobeinvented,withoutpriorpermissionfromthePublisher.
ISBNs
Print: 978-94-91216-49-7
E-Book: 978-94-91216-50-3
ISSN: 2213-0853
(cid:2)c 2012ATLANTISPRESS
To my mother, and
in remembrance of my father
MTSE : Editorial for Volume I
Recent years have witnessed an extraordinarily rapid advance in the direction of infor-
mationtechnologywithinboththescientificandengineeringdisciplines. Inaddition,the
currentprofoundtechnologicaladvancesofdataacquisitiondevicesandtransmissionsys-
temscontributeenormouslytothecontinuingexponentialgrowthofdatainformationthat
requiresmuchbetterdataprocessingtools.Tomeetsuchurgentdemands,innovativemath-
ematicaltheory,methods,andalgorithmsmustbedeveloped,withemphasison suchap-
plicationareasascomplexdataorganization,contaminatednoiseremoval,corrupteddata
repair, lost data recovery, reduction of data volume, data dimensionality reduction, data
compression,dataunderstandingandvisualization,aswellasdatasecurityandencryption.
Therevolutionofthedatainformationexplosionasmentionedabovedemandsearlymath-
ematicaltrainingandemphasisondatamanipulationatthecollegelevelandbeyond.This
newAtlantis/Springerbookseries,“MathematicsTextbooksforScienceandEngineering
(MTSE)”isfoundedtomeettheneedsofsuchmathematicstextbooksthatcanbeusedfor
bothclassroom teachingand self-study. For thebenefitof studentsand readersfrom the
interdisciplinaryareasofmathematics,computerscience,physicalandbiologicalsciences,
andvariousengineeringspecialities,contributingauthorsarerequestedtokeepinmindthat
thewritingsfor the MTSEbookseries shouldbe elementaryand relativelyeasy to read,
withsufficientexamplesandexercises.Wewelcomesubmissionofsuchbookmanuscripts
fromallwhoagreewithusonthispointofview.
Thisfirstvolumeis intendedto beanelementarytextbookfor“Mathematicsof Approx-
imation”, with emphasison constructivemethodsand derivationof errorformulas. This
bookiselementary,self-contained,andfriendlytowardsteacherandreader.Itisasuitable
textbookforteachinginavarietyofcoursesbothattheundergraduateandbeginninggrad-
uatelevels. Theauthor,ProfessorJohandeVilliers,iscongratulatedforcontributingthis
verynicetextbook.
vii
Foreword
Mathematicsofapproximationplaysakeyroleinbridgingthegapbetweenabstractmath-
ematical theory and numerical implementation. With the recent exponentialincrease of
available data that are easily accessible and relevant to our daily lives, understandingof
suchdataoftenrequiressophisticatedmathematicalmanipulationbytakingadvantageof
the continuing rapid technological advancement of computational capability. However,
withoutassuranceofaccuracyinmathematicalmanipulation,resultsfromsimplynumber
crunching by the powerful computer might be meaningless. In addition, for those data
governedby certain physicalphenomenaor biologicalmodels, approximatesolutions of
theassociatedcomplexsystemsintermsofcommonlyusedbasisfunctionsmustguarantee
accuraterepresentationwithinagiventolerance.
This book by a leading expert in the field is intended to meet the need of such mathe-
maticalcontentsfor classroom teaching,particularlyfor theundergraduatecollegelevel.
Approximation and interpolation by algebraic and trigonometric polynomials for global
representation,and splinefunctionsfor localanalysis, arediscussed from first principles
withexamplesandfullsetsofcarefullypreparedexercisesforeachchapter. Convergence
resultsarederivedtoassuremeaningfulmathematicalmanipulation,anderrorestimatesare
developedfordatarepresentation,with aguaranteeofaccuracywithinagiventolerance.
Ontheotherhand,incontrastwiththevastliteratureonApproximationTheoryandCom-
putationalMathematics,ProfessorDeVilliershastakengreatcaretoavoidusingpowerful
advancedmathematicsfromRealAnalysisandFunctionalAnalysis,inthatonlyelemen-
tarytheoryandmethodsfromLinearAlgebraandbasicAdvancedCalculusareappliedin
thederivationsanddiscussionsthroughouttheentirepresentation. Asaresult,thewriting
iselementaryandself-contained.
Furthermore,byincludingallnecessarycomputationaldetails, thewritingofthebookis
friendlyforclassroomteachingaswellasforself-reading. Foradoptionasatextbook,a
ix
x MathematicsofApproximation
varietyofcoursesarementionedwithteachingguidesoutlinedinthePreface. Theauthor
hastaughtfromthepreliminarynotesthatconstitutethecontentsofthismostwelcomed
textbookfromaspanofsometwenty-fiveyearstoundergraduatestudents.
CharlesK.Chui
MenloPark,California
Preface
Theapproximationoffunctionsbyalgebraicpolynomials,trigonometricpolynomials,and
splines,isnotonlyanimportanttopicofmathematicalstudies,butalsoprovidespowerful
mathematicaltoolstosuchapplicationareasasdatarepresentation,signalprocessing,non-
parametrictime-seriesanalysis,computer-aidedgeometricdesign,numericalanalysis,and
solutionsofdifferentialequations.Thisbookisanintroductiontothemathematicalanaly-
sisofsuchapproximation,withastrongemphasisonexplicitapproximationformulations,
correspondingerrorboundsandconvergenceresults,aswellasapplicationsinquadrature.
A mathematically rigorousapproachis adopted throughout,and, apart from an assumed
prerequisite knowledge of advanced calculus and linear algebra, the presentation of the
materialisself-contained.
Thebookissuitableforuseastextbookforcoursesatbothupperundergraduateandgrad-
uate level. Each of the ten chapters is concluded by a set of exercises, with a total of
altogether 220 exercises, some of which are routine, whereas others are concerned with
furtherdevelopmentofthematerial.
Thebookevolvedfromlecturenotescompiledbytheauthorduringapproximately25years
ofteachingcoursesinComputationalMathematicsandApproximationTheoryatDepart-
mentofMathematicalScience,StellenboschUniversity. Severalstandardtextbookswere
consultedandused,somemoreextensivelythanothers,duringthepreparationofthelecture
notesandtheresultingbook,andincludethefollowing,aslistedalphabeticallyaccording
toauthor:
• N.I.Achieser,TheoryofApproximation(translatedbyC.J.Hyman),FrederickUngar
PublishingCo.,NewYork,1956.
• H.Brass,Quadraturverfahren,Vandenhoek&Rupert,Go¨ttingen,1977.
• E.W.Cheney,IntroductiontoApproximationTheory,McGraw-Hill,NewYork,1966.
xi