Fourier series, Fourier transform and their applications to mathematical physics PDF
Preview Fourier series, Fourier transform and their applications to mathematical physics
Applied Mathematical Sciences Valery Serov Fourier Series, Fourier Transform and Their Applications to Mathematical Physics Applied Mathematical Sciences Volume 197 Editors S.S. Antman, Institute for Physical Science and Technology, University of Maryland, College Park, MD,USA [email protected] Leslie Greengard, Courant Institute of Mathematical Sciences, New York University, New York, NY,USA [email protected] P.J.Holmes,DepartmentofMechanicalandAerospaceEngineering,PrincetonUniversity,Princeton, NJ,USA [email protected] Advisors J.Bell,LawrenceBerkeleyNationalLab,CenterforComputationalSciencesandEngineering,Berkeley, CA,USA P.Constantin,DepartmentofMathematics,PrincetonUniversity,Princeton,NJ,USA R.Durrett,DepartmentofMathematics,DukeUniversity,Durham,NC,USA R.Kohn,CourantInstituteofMathematicalSciences,NewYorkUniversity,NewYork,NY,USA R.Pego,DepartmentofMathematicalSciences,CarnegieMellonUniversity,Pittsburgh,PA,USA L.Ryzhik,DepartmentofMathematics,StanfordUniversity,Stanford,CA,USA A.Singer,DepartmentofMathematics,PrincetonUniversity,Princeton,NJ,USA A.Stevens,DepartmentofAppliedMathematics,UniversityofMu¨nster,Mu¨nster,Germany A.Stuart,CaliforniaInstituteofTechnology,Pasadena,CA,USA S.Wright,ComputerSciencesDepartment,UniversityofWisconsin,Madison,WI,USA FoundingEditors FritzJohn,JosephP.LaSalle,LawrenceSirovich Moreinformationaboutthisseriesathttp://www.springer.com/series/34 Valery Serov Fourier Series, Fourier Transform and Their Applications to Mathematical Physics ValerySerov DepartmentofMathematicalSciences UniversityofOulu Oulu Finland ISSN0066-5452 ISSN2196-968X (electronic) AppliedMathematicalSciences ISBN978-3-319-65261-0 ISBN978-3-319-65262-7 (eBook) DOI:10.1007/978-3-319-65262-7 LibraryofCongressControlNumber:2017950249 MathematicsSubjectClassification(2010):26A16,26A45,35A08,35F50,35J05,35J08,35J10,35J15, 35K05,35K08,35L05,35P25,35R30 (cid:2)c SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface The modern theory of analysis and differential equations in general certainly in- cludes the Fourier transform, Fourier series, integral operators, spectral theory of differential operators, harmonic analysis and much more. This book combines all these subjects based on a unified approach that uses modern view on all these themes. The book consists of four parts: Fourier series and the discrete Fourier transform, Fourier transform and distributions, Operator theory and integral equa- tionsandIntroductiontopartialdifferentialequationsanditoutgrewfromthehalf- semestercoursesofthesamenamegivenbytheauthoratUniversityofOulu,Fin- landduring2005–2015. Each part forms a self-contained text (although they are linked by a common approach) and can be read independently. The book is designed to be a modern introductiontoqualitativemethodsusedinharmonicanalysisandpartialdifferential equations(PDEs).Itcanbenotedthatasurveyofthestateoftheartforallpartsof thisbookcanbefoundinaveryrecentandfundamentalworkofB.Simon[35]. Thisbookcontainsabout250exercisesthatareanintegralpartofthetext.Each partcontainsitsowncollectionofexerciseswithownnumeration.Theyarenotonly anintegralpartofthebook,butalsoindispensablefortheunderstandingofallparts whosecollectionisthecontentofthisbook.Itcanbeexpectedthatacarefulreader willcompletealltheseexercises. This book is intended for graduate level students majoring in pure and applied mathematicsbutevenanadvancedresearchercanfindhereveryusefulinformation whichpreviouslycouldonlybedetectedinscientificarticlesormonographs. Eachpartofthebookbeginswithitsownintroductionwhichcontainsthefacts (mostly)fromfunctionalanalysisusedthereinafter.Someofthemareprovedwhile theothersarenot. The first part, Fourier series and the discrete Fourier transform, is devoted to the classical one-dimensional trigonometric Fourier series with some applications toPDEsandsignalprocessing.Thispartprovidesaself-containedtreatmentofall well known results (but not only) at the beginning graduate level. Compared with some known texts (see [12, 18, 29, 35, 38, 44, 45]) this part uses many function spaces such as Sobolev, Besov, Nikol’skii and Ho¨lder spaces. All these spaces are v vi Preface introduced by special manner via the Fourier coefficients and they are used in the proofsofmainresults.SamedefinitionofSobolevspacescanbefoundin[35].The advantageofsuchapproachisthatweareabletoprovequiteeasilythepreciseem- beddingsforthesespacesthatarethesameasinclassicalfunctiontheory(see[1,3, 26,42]).Intheframeofthispartsomeverydelicatepropertiesofthetrigonometric Fourier series (Chapter 10) are considered using quite elementary proofs (see also [46]).TheunifiedapproachallowsusalsotoconsidernaturallythediscreteFourier transformandestablishitsdeepconnectionswiththecontinuousFouriertransform. AsaconsequenceweprovethefamousWhittaker-Shannon-Boastheoremaboutthe reconstructionofband-limitedsignalviathetrigonometricFourierseries(seeChap- ter13).ManyapplicationsofthetrigonometricFourierseriestotheone-dimensional heat,waveandLaplaceequationarepresentedinChapter14.Itisaccompaniedbya largenumberofveryusefulexercisesandexampleswithapplicationsinPDEs(see also[10,17]). Thesecondpart,Fouriertransformanddistributions,probablytakesacentralrole inthisbookanditisconcernedwithdistributiontheoryofL.Schwartzanditsap- plicationstotheSchro¨dingerandmagneticSchro¨dingeroperators(seeChapter32). TheestimatesforLaplacianandHamiltonianthatgeneralizewellknownAgmon’s estimates on the continuous spectrum are presented in this part (see Chapter 23). Thispartcanbeconsideredasoneofthemostimportantbecauseofnumerousap- plications in the scattering theory and inverse problems. Here we have considered forthefirsttimesomeclassicaldirectscatteringproblemsfortheSchro¨dinger op- eratorandforthemagneticSchro¨dingeroperatorwithsingular(locallyunbounded) coefficientsincludingthemathematicalfoundationsoftheclassicalapproximation ofM.Born.Also,thepropertiesofRiesztransformandRieszpotentials(seeChap- ter21)areinvestigatedverycarefullyinthispart.Beforethismaterialcouldonlybe foundinscientificjournalsormonographsbutnotintextbooks.Thereisagoodcon- nectionofthispartwithOperatortheoryandintegralequations.Themaintechnique appliedhereistheFouriertransform. The third part, Operator theory and integral equations, is devoted mostly to the self-adjointbutunboundedoperatorsinHilbertspacesandtheirapplicationstoin- tegral equations in such spaces. The advantage of this part is that many important results of J. von Neumann’s theory of symmetric operators are collected together. J. von Neumann’s spectral theorem allows us, for example, to introduce the heat kernel without solving the heat equation. Moreover, we show applications of the spectral theorem of J. von Neumann (for these operators) to the spectral theory of ellipticdifferentialoperators.Inparticular,theexistenceofFriedrichsextensionfor these operators with discrete spectrum is provided. Special attention is devoted to theSchro¨dingerandthemagneticSchro¨dingeroperators.Thefamousdiamagnetic inequality is proved here. We follow in this consideration B. Simon [35] (slightly different approach can be found in [28]). We recommend (in addition to this part) the reader get acquainted with the books [4, 13, 15, 24, 41]. As a consequence of thespectraltheoryofellipticdifferentialoperatorstheintegralequationswithweak singularities are considered in quite simple manner not only in Hilbert spaces but also in some Banach spaces, e.g. in the space of continuous functions on closed Preface vii manifolds. The central point of this consideration is the Riesz theory of compact (not necessarily self-adjoint) operators in Hilbert and Banach spaces. In order to keepthispartshort,someproofswillnotbegiven,norwillalltheoremsbeproved incompletegenerality.Formanydetailsoftheseintegralequationswerecommend [22].Weareabletoinvestigateinquitesimplemannerone-dimensionalVolterrain- tegralequationswithweaksingularitiesinL∞(a,b)andsingularintegralequations in the periodic Ho¨lder spaces Cα[−a,a]. Concerning approximation methods our considerations use the general theory of bounded or compact operators in Hilbert spacesandwefollowmostlythemonographofKress[22]. The fourth part, Introduction to partial differential equations, serves as an in- troductiontomodernmethodsforclassicaltheoryofpartialdifferentialequations. Fourier series and Fourier transform play crucial role here too. An important (and quiteindependent)segmentofthispartistheself-containedtheoryofquasi-linear partialdifferentialequationsoforderone.Themainattentioninthispartisdevoted toellipticboundaryvalueproblemsinSobolevandHo¨lderspaces.Inparticular,the uniquesolvabilityofdirectscatteringproblemforHelmholtzequationisprovided. We investigate very carefully the mapping and discontinuity properties of double andsinglelayerpotentialswithcontinuousdensities.Wealsorefertosimilarprop- ertiesofdoubleandsinglelayerpotentialswithdensitiesinSobolevspacesH1/2(S) andH−1/2(S),respectively,butwillnotproveanyoftheseresults,referringfortheir proofstomonographs[22]and[25].Here(andelsewhereinthebook)Sdenotesthe boundary of a bounded domain in Rn and if the smoothness of S is not specified explicitly then it is assumed to be such that Sobolev embedding theorem holds. Compared with well known texts on partial differential equations some direct and inversescatteringproblemsforHelmholtz,Schro¨dingerandmagneticSchro¨dinger operatorsareconsideredinthispart.Asitwasmentionedearlierthistypeofmater- ialcouldnotbefoundintextbooks.Thepresentationinmanyplacesofthisparthas beenstronglyinfluencedbythemonographs[6,7,11](seealso[8,16,24,36,40]). In closing we note that this book is not as comprehensive as the fundamental work of B. Simon [35]. But the book can be considered as a good introduction to modern theory of analysis and differential equations and might be useful not only to students and PhD students but also to all researchers who have applications in mathematicalphysicsandengineeringsciences.Thisbookcouldnothaveappeared without the strong participation, both in content and typesetting, of my colleague Adj.Prof.MarkusHarju.Finally,aspecialthankstoprofessorDavidColtonfrom University of Delaware (USA) who encouraged the writing of this book and who hassupportedtheauthorverymuchovertheyears. Oulu,Finland ValerySerov June2017 Contents PartI FourierSeriesandtheDiscreteFourierTransform 1 Introduction................................................... 3 2 FormulationofFourierSeries ................................... 11 3 FourierCoefficientsandTheirProperties......................... 17 4 ConvolutionandParseval’sEquality ............................. 23 5 Feje´rMeansofFourierSeries.UniquenessoftheFourierSeries. .... 27 6 TheRiemann–LebesgueLemma ................................. 33 7 The Fourier Series of a Square-Integrable Function. The Riesz–FischerTheorem. ........................................ 37 8 BesovandHo¨lderSpaces........................................ 45 9 Absoluteconvergence.BernsteinandPeetreTheorems.............. 53 10 DirichletKernel.PointwiseandUniformConvergence. ............. 59 11 FormulationoftheDiscreteFourierTransformandItsProperties.... 77 12 ConnectionBetweentheDiscreteFourierTransform andtheFourierTransform. ..................................... 85 13 SomeApplicationsoftheDiscreteFourierTransform............... 93 14 ApplicationstoSolvingSomeModelEquations .................... 99 14.1 TheOne-DimensionalHeatEquation ......................... 99 14.2 TheOne-DimensionalWaveEquation ........................ 113 14.3 TheLaplaceEquationinaRectangleandinaDisk ............. 121 ix x Contents PartII FourierTransformandDistributions 15 Introduction .................................................. 131 16 TheFourierTransforminSchwartzSpace ........................ 133 17 TheFourierTransforminLp(Rn),1≤p≤2....................... 143 18 TemperedDistributions......................................... 153 (cid:3) 19 ConvolutionsinSandS ........................................ 167 20 Sobolevspaces ................................................. 175 20.1 Sobolevspacesonboundeddomains ......................... 188 21 HomogeneousDistributions ..................................... 193 22 FundamentalSolutionoftheHelmholtzOperator.................. 207 23 EstimatesfortheLaplacianandHamiltonian ..................... 217 PartIII OperatorTheoryandIntegralEquations 24 Introduction................................................... 247 25 InnerProductSpacesandHilbertSpaces ......................... 249 26 SymmetricOperatorsinHilbertSpaces........................... 261 27 JohnvonNeumann’sspectraltheorem............................ 279 28 SpectraofSelf-AdjointOperators................................ 295 29 QuadraticForms.FriedrichsExtension. .......................... 313 30 EllipticDifferentialOperators ................................... 319 31 SpectralFunctions ............................................. 331 32 TheSchro¨dingerOperator ...................................... 335 33 TheMagneticSchro¨dingerOperator ............................. 349 34 IntegralOperatorswithWeakSingularities.IntegralEquations oftheFirstandSecondKinds.................................... 359 35 VolterraandSingularIntegralEquations ......................... 371 36 ApproximateMethods .......................................... 379 PartIVPartialDifferentialEquations 37 Introduction................................................... 393 38 LocalExistenceTheory ........................................ 405
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