MATHEMATICSTEXTBOOKSFORSCIENCEANDENGINEERING VOLUME1 SERIESEDITOR: C.K.CHUI Mathematics Textbooks for Science and Engineering SeriesEditor: C.K.Chui StanfordUniversity,USA (ISSN:2213-0853) Aimsandscopeoftheseries Textbooks in the series ‘Mathematics Textbooks for Science and Engineering’ will be aimedatthebroadmathematics,scienceandengineeringundergraduateandgraduatelev- els, coveringall areasof appliedandapplicablemathematics, interpretedin thebroadest sense. Formoreinformationonthisseriesandourotherbookseries,pleasevisitourwebsiteat: www.atlantis-press.com/publications/books 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
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