(Texts in Applied Mathematics 39) Weimin Han, Kendall E. Atkinson (auth.) - Theoretical numerical analysis_ A functional analysis framework-Springer-Verlag New York (2009).pdf
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(Texts in Applied Mathematics 39) Weimin Han, Kendall E. Atkinson (auth.) - Theoretical numerical analysis_ A functional analysis framework-Springer-Verlag New York (2009).pdf
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39EditorsJ.E.MarsdenL.SirovichS.S.AntmanAdvisorsG.IoossP.HolmesD.BarkleyM.DellnitzP.NewtonTextsinAppliedMathematicsForothervolumespublishedinthisseries,goAtkinsonWeiminHanTheoreticalNumericalAnalysisAFunctionalAnalysisFrameworkThirdEditionABCKendallAtkinsonComputerScienceUniversityofIowaUSASeriesEditorsJ.E.MarsdenControlandDynamicalSystems107-81CaliforniaInstituteofTechnologyPasadena,CA91125USAmarsdencds.caltech.eduS.S.AntmanandInstituteforPhysicalScienceandTechnologyUniversityofMarylandUSAssamath.umd.eduWeiminHanUSAwhanmath.uiowa.eduL.SirovichLaboratoryofAppliedMathematicsDepartmentofBiomathematicsMt.SinaiSchoolofMedicineBox1012NewYork,NY10029-6574USAISSN0939-2475ISBN978-1-4419-0457-7e-ISBN978-1-4419-0458-4DOI10.1007/978-1-4419-0458-4SpringerDordrechtHeidelbergLondonNewYorkMathematicsSubjectClassification(2000):
65-01,65-XXcTheuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyPrintedonacid-freepaperSpringerispartofSpringerScience+BusinessMedia()?
SpringerScience+BusinessMedia,LLC2009IowaCity,IA52242IowaCity,IA52242LibraryofCongressControlNumber:
2009926473lawrence.sirovichmssm.eduAllrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrittenpermissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden.arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjecttoproprietaryrights.DepartmentsofMathematics&DepartmentofMathematicsDepartmentofMathematicskendall-atkinsonuiowa.eduUniversityofIowaCollegePark,MD20742-4015DedicatedtoDaisyandClydeAtkinsonHazelandWrayFlemingandDaqingHan,SuzhenQinHuidiTang,ElizabethandMichaelSeriesPrefaceMathematicsisplayinganevermoreimportantroleinthephysicalandbiologicalsciences,provokingablurringofboundariesbetweenscientificdisciplinesandaresurgenceofinterestinthemodernaswellastheclas-sicaltechniquesofappliedmathematics.Thisrenewalofinterest,bothinresearchandteaching,hasledtotheestablishmentoftheseries:
TextsinAppliedMathematics(TAM).Thedevelopmentofnewcoursesisanaturalconsequenceofahighlevelofexcitementontheresearchfrontierasnewertechniques,suchasnumericalandsymboliccomputersystems,dynamicalsystems,andchaos,mixwithandreinforcethetraditionalmethodsofappliedmathematics.Thus,thepurposeofthistextbookseriesistomeetthecurrentandfutureneedsoftheseadvancesandtoencouragetheteachingofnewcourses.TAMwillpublishtextbookssuitableforuseinadvancedundergraduateandbeginninggraduatecourses,andwillcomplementtheAppliedMath-ematicalSciences(AMS)series,whichwillfocusonadvancedtextbooksandresearch-levelmonographs.Pasadena,CaliforniaJ.E.MarsdenProvidence,RhodeIslandL.SirovichCollegePark,MarylandS.S.AntmanPrefaceThistextbookhasgrownoutofacoursewhichweteachperiodicallyattheUniversityofIowa.Wehavebeginninggraduatestudentsinmathematicswhowishtoworkinnumericalanalysisfromatheoreticalperspective,andtheyneedabackgroundinthose“toolsofthetrade”whichwecoverinthistext.Inthepast,suchstudentswouldordinarilybeginwithaone-yearcourseinrealandcomplexanalysis,followedbyaoneortwosemestercourseinfunctionalanalysisandpossiblyagraduatelevelcourseinordi-narydifferentialequations,partialdifferentialequations,orintegralequa-tions.Westillexpectourstudentstotakemostofthesestandardcourses.Thecoursebasedonthisbookallowsthesestudentstomovemorerapidlyintoaresearchprogram.Thetextbookcoversbasicresultsoffunctionalanalysis,approximationtheory,Fourieranalysisandwavelets,calculusanditerationmethodsfornonlinearequations,finitedifferencemethods,Sobolevspacesandweakformulationsofboundaryvalueproblems,finiteelementmethods,ellipticvariationalinequalitiesandtheirnumericalsolution,numericalmethodsforsolvingintegralequationsofthesecondkind,boundaryintegralequationsforplanarregionswithasmoothboundarycurve,andmultivariablepoly-nomialapproximations.Thepresentationofeachtopicismeanttobeanintroductionwithacertaindegreeofdepth.Comprehensivereferencesonaparticulartopicarelistedattheendofeachchapterforfurtherreadingandstudy.Forthisthirdedition,weaddachapteronmultivariablepolynomialapproximationandwerevisenumeroussectionsfromthesecondeditiontovaryingdegrees.Agoodnumberofnewexercisesareincluded.xPrefaceThematerialinthetextispresentedinamixedmanner.Sometopicsaretreatedwithcompleterigour,whereasothersaresimplypresentedwithoutproofandperhapsillustrated(e.g.theprincipleofuniformboundedness).WehavechosentoavoidintroducingaformalizedframeworkforLebesguemeasureandintegrationandalsofordistributiontheory.Insteadweusestandardresultsonthecompletionofnormedspacesandtheuniqueex-tensionofdenselydefinedboundedlinearoperators.ThispermitsustointroducetheLebesguespacesformallyandwithouttheirconcreterealiza-tionusingmeasuretheory.Wedescribesomeofthestandardmaterialonmeasuretheoryanddistributiontheoryinanintuitivemanner,believingthisissufficientformuchofthesubsequentmathematicaldevelopment.Inaddition,wegiveanumberofdeeperresultswithoutproof,citingtheexistingliterature.Examplesofthisaretheopenmappingtheorem,Hahn-Banachtheorem,theprincipleofuniformboundedness,andanumberoftheresultsonSobolevspaces.Thechoiceoftopicshasbeenshapedbyourresearchprogramandinter-estsattheUniversityofIowa.Thesetopicsareimportantelsewhere,andwebelievethistextwillbeusefultostudentsatotheruniversitiesaswell.Thebookisdividedintochapters,sections,andsubsectionsasappropri-ate.Mathematicalrelations(equalitiesandinequalities)arenumberedbychapter,sectionandtheirorderofoccurrence.Forexample,(1.2.3)isthethirdnumberedmathematicalrelationinSection1.2ofChapter1.Defini-tions,examples,theorems,lemmas,propositions,corollariesandremarksarenumberedconsecutivelywithineachsection,bychapterandsection.Forexample,inSection1.1,Definition1.1.1isfollowedbyanexamplelabeledasExample1.1.2.Wegiveexercisesattheendofmostsections.Theexercisesarenumberedconsecutivelybychapterandsection.Attheendofeachchapter,weprovidesomeshortdiscussionsoftheliterature,includingrecommendationsforadditionalreading.Duringthepreparationofthebook,wereceivedhelpfulsuggestionsfromnumerouscolleaguesandfriends.WeparticularlythankP.G.Ciar-let,WilliamA.Kirk,WenbinLiu,andDavidStewartforthefirstedition,B.Bialecki,R.Glowinski,andA.J.Meirforthesecondedition,andYuanXuforthethirdedition.ItisapleasuretoacknowledgetheskillfuleditorialassistancefromtheSeriesEditor,AchiDosanjh.ContentsSeriesPrefaceviiPrefaceix1LinearSpaces11.1Linearspaces.11.2Normedspaces.71.2.1Convergence.101.2.2Banachspaces.131.2.3Completionofnormedspaces.151.3Innerproductspaces.221.3.1Hilbertspaces.271.3.2Orthogonality.281.4Spacesofcontinuouslydifferentiablefunctions.391.4.1Holderspaces.411.5Lpspaces.441.6Compactsets.492LinearOperatorsonNormedSpaces512.1Operators.522.2Continuouslinearoperators.552.2.1L(V,W)asaBanachspace.592.3Thegeometricseriestheoremanditsvariants.602.3.1Ageneralization.64xiiContents2.3.2Aperturbationresult.662.4Somemoreresultsonlinearoperators.722.4.1Anextensiontheorem.722.4.2Openmappingtheorem.742.4.3Principleofuniformboundedness.752.4.4Convergenceofnumericalquadratures.762.5Linearfunctionals.792.5.1Anextensiontheoremforlinearfunctionals.802.5.2TheRieszrepresentationtheorem.822.6Adjointoperators.852.7Weakconvergenceandweakcompactness.902.8Compactlinearoperators.952.8.1CompactintegraloperatorsonC(D).962.8.2Propertiesofcompactoperators.972.8.3IntegraloperatorsonL2(a,b).992.8.4TheFredholmalternativetheorem.1012.8.5AdditionalresultsonFredholmintegralequations.1052.9Theresolventoperator.1092.9.1R()asaholomorphicfunction.1103ApproximationTheory1153.1Approximationofcontinuousfunctionsbypolynomials.1163.2Interpolationtheory.1183.2.1Lagrangepolynomialinterpolation.1203.2.2Hermitepolynomialinterpolation.1223.2.3Piecewisepolynomialinterpolation.1243.2.4Trigonometricinterpolation.1263.3Bestapproximation.1313.3.1Convexity,lowersemicontinuity.1323.3.2Someabstractexistenceresults.1343.3.3Existenceofbestapproximation.1373.3.4Uniquenessofbestapproximation.1383.4Bestapproximationsininnerproductspaces,projectiononclosedconvexsets.1423.5Orthogonalpolynomials.1493.6Projectionoperators.1543.7Uniformerrorbounds.1573.7.1UniformerrorboundsforL2-approximations.1603.7.2L2-approximationsusingpolynomials.1623.7.3Interpolatoryprojectionsandtheirconvergence.1644FourierAnalysisandWavelets1674.1Fourierseries.1674.2Fouriertransform.1814.3DiscreteFouriertransform.187Contentsxiii4.4Haarwavelets.1914.5Multiresolutionanalysis.1995NonlinearEquationsandTheirSolutionbyIteration2075.1TheBanachfixed-pointtheorem.2085.2Applicationstoiterativemethods.2125.2.1Nonlinearalgebraicequations.2135.2.2Linearalgebraicsystems.2145.2.3Linearandnonlinearintegralequations.2165.2.4OrdinarydifferentialequationsinBanachspaces.2215.3Differentialcalculusfornonlinearoperators.2255.3.1FrechetandGateauxderivatives.2255.3.2Meanvaluetheorems.2295.3.3Partialderivatives.2305.3.4TheGateauxderivativeandconvexmi
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