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ValerySerov

Fourier Series,

Fourier Transform and

their Applications to

Mathematical Physics

ValerySerov

Department ofMathematical Sciences

UniversityofOulu

Oulu

Finland

ISSN 0066-5452ISSN 2196-968X(electronic)

Applied MathematicalSciences

ISBN 978-3-319-65261-0I SBN978-3-319-65262-7(eBook)

DOI: 10.1007/978-3-319-65262-7

Library ofCongress ControlNumber: 2017950249

cSpringer InternationalPublishing AG2017

Preface

The moderntheory ofanalysisand differentialequations ingeneral certainlyin- cludes theF ouriertransform,F ourierseries,inte graloperators,spectralt heoryof differentialoperators,harmonic analysisand muchmore. This bookcombines all these subjectsbasedon aunied approachthat usesmodern vie wo na llthese themes. Thebookconsists offour parts:F ouriers eriesand thed iscreteFourier transform, Fouriertransform anddistributions, Operator theoryandintegralequa- tions andIntroduction to partialdifferential equationsand itoutgrewfromthehalf- semester courseso fthesame namegiv enby theauthor atUniversityofOulu,Fin- land during2005-2015. Each partf ormsaself-contained text (althoughtheyare linkedb ya common approach) andcan beread independently. The bookisdesignedto beamodern introduction toqualitativ emethodsusedinharmonicanalysis andpartial dif ferential equations (PDEs).I tcanbenoted thata surv ey ofthe state oftheartforallparts of this bookcan befound ina very recent andfundamental workof B.Simon[35]. This bookcontains about250 ex ercises thatareanintegral parto ft hetext.E ach part containsits ow ncollectionofex ercises withown numeration.Theyarenotonly an integralparto ft hebook,bu talsoi ndispensablef ortheunderstandingofallparts whose collectioni sthecontent ofthisbook.It canbe expected thata carefulreader will completea lltheseex ercises. This bookis intendedfor graduate le velstudentsm ajoringinpureandapplied mathematics butev enanadvancedresearchercan ndhere veryuseful information which previouslycouldonlyb edetected inscientic articlesor monographs. Each parto fthebook beginswithits ow nintroductionw hichcontains thef acts (mostly) fromf unctionalanalysisusedthereinafter .Some of themarep rov edwhile the othersare not. The rstpart,F ouriers eriesandthed iscreteFouriert ransform,i sd evotedto the classicalone-dimensionaltrigonometric Fourier series withsomeapplications to PDEsand signalprocessing. Thispart pro videsa self-contained treatmentofall well knownresults(b utnot only)attheb eg inninggraduate le vel. Comparedwith some knowntexts (see[12, 18,29,35,38, 44,45])this partu sesm any function spaces suchas Sobolev ,Besov,N ikol"skiiandH¨older spaces.A llthesespaces are introduced byspecial mannervia the Fourier coefcientsandthe yareused inthe proofs ofmain results.Samedenition ofSobolev spacescan befound in[35].The advantageof suchapproach isthat weare ablet op rov equite easilythe preciseem- beddings fort hesespacesthat arethesame asin classicalfunction theory(see[ 1,3,

26, 42]).In theframe ofthis parts omev erydelicate properties ofthe trigonometric

Fouriers eries(Chapter10) areconsideredusing quiteelementary proofs (seea lso [46]). Theunied approachallo wsus alsotoconsidernaturally thed iscreteF ourier transform andestablish itsdeep connectionswith thecontinuous Fourier transform. As aconsequence wepro ve thefamousWhittaker -Shannon-Boas theoremaboutthe reconstruction ofband-limiteds ignalvia thetrigonometricF ouriers eries(seeChap- heat, waveandLaplaceequationa represented inChapter 14. Itis accompaniedbya largenumberof ve ryuseful exercisesande xampleswith applicationsinPDEs(see also [10,17]). in thisbook andit isconcerned withdistrib ution theory of L.Schwartzandits ap- plications tothe Schr¨odinger andmagnetic Schr¨odinger operators(see Chapter 32). The estimatesforL aplacianand Hamiltonianthatgeneralizew ellkno wnAgmon" s estimates onthe continuousspectrum are presentedin thispart(see Chapter 23). This partcan beconsidered asone ofthe most important becauseo fnumerous ap- plications inthe scattering theoryandinv erse problems.Here wehaveconsidered for thersttime some classical directscatteringproblemsf ortheSchr¨odinger op- erator andfor the magneticSchr¨odinger operatorwith singular(locally unbounded) coefcientsi ncludingthemathematical foundationsof theclassicalapproximation of M.Born. Also, thepropertieso fR iesztransformandRiesz potentials(seeChap- ter21) areinv estigatedverycarefullyinthispart. Beforethismaterialcouldonlyb e found inscientic journals ormonographs butnot intextbooks.T hereisagoodcon- nection ofthis partwithOperator theoryand integralequations.T hemain technique applied herei stheF ouriertransform. The thirdpart,Operator theoryand integral equations,i sd evotedmostly tothe self-adjoint butunboundedoperatorsi nH ilberts pacesandtheirapplications toin- tegralequations insuch spaces.T headv antageof thispartis thatmany important results ofJ. vo nNeumann"stheory ofsymmetricoperatorsa recollectedtogether. J. vonNeumann"s spectraltheoremallows us,fore xample,tointroducethe heat kernelwithoutsolving the heatequation. Moreover ,we show applicationsofthe spectral theoremofJ. vo nN eumann(fortheseoperators)to thespectralt heoryo f elliptic differentialoperators.Inparticular ,t hee xistenceofFriedrichs extensionf or these operatorsw ithdiscretespectrumi sp rovided. Specialattentionis devotedto the Schr¨odinger andthe magnetic Schr¨odinger operators.The famous diamagnetic inequality ispro vedhere.Wefollow in thisconsiderationB .Simon[35] (slightly differentapproachcan befound in[28]). We recommend(in additiont ot hispart) the readergetacquainted withthe books[4, 13,15, 24,41]. As aconsequence of the spectraltheory ofelliptic differentialoperators theintegral equationswithweak singularities areconsideredin quitesimple mannernot onlyi nH ilberts pacesb ut also insome Banachspaces, e.g. in thespaceof continuousfunctionsonclosed manifolds. Thecentral pointof thisconsideration is the Rieszt heoryofcompact (not necessarilyself-adjoint) operatorsi nH ilbertandBanachspaces. Inorder to keepthis parts hort,some proofswillnot begi ven, norwillallt heoremsb ep roved in completegenerality .Forman ydetailsofthese integral equationswe recommend [22]. Weare abletoi nv estigatein quitesimplemannerone-dimensionalVolterrain- tegralequations withweak singularitiesi nL (a,b)and singularinte gralequations in theperiodic H¨older spacesC [Ša,a]. Concerningapproximationm ethodsour considerations uset hegeneraltheoryo fbounded orcompact operatorsinH ilbert spaces andwe follow mostlythem onographofKress[ 22]. The fourthpart,Introduction to partiald ifferentialequations,s erve sasani n- troduction tomodern methodsforc lassicaltheoryof partiald ifferentialequations. Fouriers eriesandFourier transform playcrucialrolehere too.Animportant (and quite independent)s egmentofthispartis the self-containedtheory of quasi-linear partial differentialequationsoforder one.The main attention in this partisd evoted to ellipticboundary value problemsinSobole vand H¨older spaces.I nparticular,t he unique solvabilityofdirect scatteringproblem forH elmholtzequation is pro vided. Wein vestigateverycarefullythemapping anddiscontinuitypropertieso fdouble and singlel ayerpotentialswithcontinuous densities.W ealso refert os imilarprop- erties ofdoubleand singlel ayerpotentials withdensities inSobolev spacesH 1/2 (S) andH

Š1/2

(S), respectively,butwill notproveany ofthese results, referringfortheir proofs tomonographs [22]and[25].H ere( andelse wherei nthebook) Sdenotes the boundary ofabounded domaini nR n and ifthe smoothness ofSis notspecied explicitlythen itis assumedt ob es uchthatSobolev embeddingtheoremholds. Compared withwell known textson partialdifferential equationssomedirectand inversescatteringproblems forHelmholtz,Schr ¨odinger andmagnetic Schr¨odinger operators areconsideredin thispart. Asit wa sm entionedearlier this typeofmater- ial couldnot befound inte xtbooks.T hepresentation inman yplacesof thispart has been stronglyi nuencedbythem onographs[6, 7, 11](seealso [8, 16,24,36,40]). In closingw enotethat thisbook isnotascomprehensi ve as the fundamental workofB. Simon[ 35].B utthe bookcanbeconsidered asagoodintroduction to modern theoryof analysisanddiff erentialequations andmight beusefulnotonly to studentsand PhDstudents but alsoto allresearchersw hoha ve applicationsin mathematical physicsandengineering sciences. This bookcouldnotha veappeared without thes trongparticipation,bothi ncontent andtypesetting, ofmycolleague Adj. Prof.M arkusHarju.Finally, aspecial thankstoprofessorD av idColton from UniversityofDela wa re(USA)whoencouragedthewriting of thisbookand who has supportedt heauthorve rymuch overthe years.

Oulu, FinlandVa lerySerov

June 2017

Contents

PartI Fourier SeriesandtheDiscr eteF ourierT ransform

1 Introduction................................................... 3

2 FormulationofF ourierSeries ................................... 11

3 FourierCoefÞcientsand TheirPr operties......................... 17

4 ConvolutionandParse valÕ sEquality.............................23

5Fej"er Meansof Fourier Series.Uniquenessoft heF ourierSeries. .... 27

6 TheRiemannÐLebesgue Lemma................................. 33

7 TheF ourierSeriesofa Square-Integrable Function.The

RieszÐFischer Theorem......................................... 37

8 BesovandH ¬older Spaces........................................ 45

9 Absolutecon vergence.BernsteinandPeetr eTheorems.............. 53

10 DirichletK ernel.PointwiseandUnif ormConver gence.............. 59

11 Formulationofthe Discrete Fourier Transformand ItsProperties.... 77

12 ConnectionBetween theDiscr eteF ourierTransform

and theF ourierTransform. ..................................... 85

13 SomeA pplicationsoftheDiscr eteF ourierT ransform............... 93

14 ApplicationstoSolving SomeModel Equations.................... 99

14.1 TheOne-Dimensional HeatEquation .........................99

14.2 TheOne-Dimensional Wa veEquation........................113

14.3 TheLaplace Equationin aRectangle andin aDisk .............121

PartII Fourier Transformand Distributions

15 Introduction.................................................. 131

16 TheF ourierTransform inSchwartzSpace........................ 133

17 TheF ourierTransform inL

p (R n ),1p2....................... 143

18 TemperedDistributions......................................... 153

19 ConvolutionsinSandS

........................................ 167

20 Sobolevspaces................................................. 175

20.1 Sobolevspaceson boundeddomains .........................188

21 HomogeneousDistrib utions..................................... 193

22 FundamentalSolution of theHelmholtzOperator .................. 207

23 Estimatesf ortheLaplacian andHamiltonian ..................... 217

PartIII OperatorTheory andIntegral Equations

24 Introduction................................................... 247

25 InnerPr oductSpacesandHilbert Spaces......................... 249

26 SymmetricOperators inHilbert Spaces........................... 261

27 Johnvo nNeumannÕsspectral theorem............................ 279

28 Spectraof Self-AdjointOperators ................................ 295

29 QuadraticF orms.FriedrichsExtension........................... 313

30 EllipticD ifferentialOperators................................... 319

31 SpectralFunctions ............................................. 331

32 TheSchr ¬odinger Operator...................................... 335

33 TheM agneticSchr¬odinger Operator............................. 349

34 IntegralOperators withW eakSingularities. IntegralEquations

of theF irstandSecondKinds. ................................... 359

35 VolterraandSingularIntegral Equations......................... 371

36 ApproximateMethods.......................................... 379

PartIV Partial DifferentialE quations

37 Introduction................................................... 393

38 LocalExistence Theory........................................ 405

39 TheL aplaceOperator.......................................... 421

40 TheD irichletandNeumannPr oblems............................ 437

41 LayerPotentials............................................... 451

42 EllipticB oundaryValue Problems............................... 471

43 TheD irectScatteringProblemf or theHelmholtzEquation ......... 485

44 SomeI nverseScatteringProblemsfor theSchr ¬odinger Operator.... 493

45 TheH eatOperator............................................. 507

46 TheW aveOperator............................................ 517

References........................................................ 529 Index............................................................ 531

Part I

Fourier Series and the Discrete Fourier

Transform

Chapter 1

Introduction

Definition 1.1.A functionf(x)of one variablexis said to beperiodicwith period T>0 if the domainD(f)offcontainsx+Twhenever it containsxand if for every x?D(f), one has f(x+T)=f(x).(1.1)

Remark 1.2.If alsox-T?D(f), then

f(x-T)=f(x). It follows that ifTis a period off, thenmTis also a period for every integerm>0. The smallest value ofT>0forwhich(1.1) holds is called thefundamental period off.

For example, the functions

sin m πx

L,cosm

πx L,e i mπx L ,m=1,2,... are periodic with fundamental periodT= 2L m . Note also that they are periodic with common period 2L. If some functionfis defined on the interval[a,a+T], withT>0 andf(a)= f(a+T), thenfcan beextended periodicallywith periodTto the whole line as Therefore, we may assume from now on that every periodic function is defined on the whole line. c?Springer International Publishing AG 2017 V. Serov,Fourier Series, Fourier Transform and Their Applications to Mathematical Physics, Applied Mathematical Sciences 197,

DOI 10.1007/978-3-319-65262-7

13

4 Part I: Fourier Series and the Discrete Fourier Transform

b a |f(x)| p dx<∞.

The set of all such functions is denoted byL

p (a,b). Whenp=1, we say thatfis integrable.

The following "continuity" in the sense ofL

p f?L p (a,b)andε>0, there is a continuous functiongon[a,b]such that b a |f(x)-g(x)| p dx? 1/p (see e.g., Corollary5.3). Iffisp-integrable andgisp -integrable on[a,b], where 1 p+1p =1,1F(x,y)dy? dx= d c b a

F(x,y)dx?

dy= b a d c

F(x,y)dxdy,

whereF(x,y)?L 1 ((a,b)×(c,d)). Iff 1 ,f 2 ,...,f n nj=1 f j , and b a n j=1 f j (x)? p dx? 1/p n j=1 b a |f j (x)| p dx? 1/p .(1.2) This inequality is calledMinkowski"s inequality. As a consequence of H¨older's inequality we obtain thegeneralized Minkowski inequality b a d c

F(x,y)dy????

p dx? 1/p d c b a |F(x,y)| p dx? 1/p dy.(1.3)

1 Introduction5

Hint.Prove first H¨older's inequality for sums, i.e., n j=1 a j b j n j=1 |a j p 1/p n j=1 |b j p 1/p where 1Exercise 1.2.Prove (1.2) and (1.3). Lemma 1.3.If f is periodic with period T>0and if it is integrable on every finite interval, then a+T a f(x)dx= T 0 f(x)dx(1.4) for every a?R.

Proof.Let firsta>0. Then

a+T a f(x)dx= a+T 0 f(x)dx- a 0 f(x)dx Tquotesdbs_dbs8.pdfusesText_14

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