Fourier series, Fourier transform and their applications to mathematical physics

519 Pages·2017·2.333 MB·English

by  Serov V

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
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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