Modulation Spaces: With Applications to Pseudodifferential Operators and Nonlinear Schrödinger Equations PDF
Preview Modulation Spaces: With Applications to Pseudodifferential Operators and Nonlinear Schrödinger Equations
Applied and Numerical Harmonic Analysis Árpád Bényi Kasso A. Okoudjou Modulation Spaces With Applications to Pseudodifferential Operators and Nonlinear Schrödinger Equations Applied and Numerical Harmonic Analysis SeriesEditor JohnJ.Benedetto UniversityofMaryland CollegePark,MD,USA AdvisoryEditors AkramAldroubi GittaKutyniok VanderbiltUniversity TechnicalUniversityofBerlin Nashville,TN,USA Berlin,Germany DouglasCochran MauroMaggioni ArizonaStateUniversity JohnsHopkinsUniversity Phoenix,AZ,USA Baltimore,MD,USA HansG.Feichtinger ZuoweiShen UniversityofVienna NationalUniversityofSingapore Vienna,Austria Singapore,Singapore ChristopherHeil ThomasStrohmer GeorgiaInstituteofTechnology UniversityofCalifornia Atlanta,GA,USA Davis,CA,USA StéphaneJaffard YangWang UniversityofParisXII HongKongUniversityof Paris,France Science&Technology Kowloon,HongKong JelenaKovacˇevic´ CarnegieMellonUniversity Pittsburgh,PA,USA Moreinformationaboutthisseriesathttp://www.springer.com/series/4968 Árpád Bényi (cid:129) Kasso A. Okoudjou Modulation Spaces With Applications to Pseudodifferential Operators and Nonlinear Schrödinger Equations ÁrpádBényi KassoA.Okoudjou DepartmentofMathematics DepartmentofMathematics WesternWashingtonUniversity UniversityofMaryland Bellingham,WA,USA CollegePark,MD,USA ISSN2296-5009 ISSN2296-5017 (electronic) AppliedandNumericalHarmonicAnalysis ISBN978-1-0716-0330-7 ISBN978-1-0716-0332-1 (eBook) https://doi.org/10.1007/978-1-0716-0332-1 Mathematics Subject Classification: 42-02, 42B15, 42B35, 42B37, 47G30, 35Q55, 46E30, 46E35, 35-XX,46-XX ©SpringerScience+BusinessMedia,LLC,partofSpringerNature2020 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered companySpringerScience+BusinessMedia,LLC Theregisteredcompanyaddressis:1NewYorkPlaza,NewYork,NY10004,U.S.A. PentruMona,Alexs¸iBasti,cumulta˘ dragoste-Á.B. ÁRouky,Shadeh,Shola,etFemi,avecamour etaffection-K.A.O. ANHA Series Preface The Applied and Numerical Harmonic Analysis (ANHA) book series aims to providetheengineering,mathematical,andscientificcommunitieswithsignificant developments in harmonic analysis, ranging from abstract harmonic analysis to basic applications. The title of the series reflects the importance of applications and numerical implementation, but richness and relevance of applications and implementation depend fundamentally on the structure and depth of theoretical underpinnings. Thus, from our point of view, the interleaving of theory and applicationsandtheircreativesymbioticevolutionisaxiomatic. Harmonicanalysisisawellspringofideasandapplicabilitythathasflourished, developed, and deepened over time within many disciplines and by means of creative cross-fertilization with diverse areas. The intricate and fundamental relationshipbetweenharmonicanalysisandfieldssuchassignalprocessing,partial differentialequations(PDEs),andimageprocessingisreflectedinourstate-of-the- artANHAseries. Our vision of modern harmonic analysis includes mathematical areas such as wavelettheory,Banachalgebras,classicalFourieranalysis,time-frequencyanalysis, andfractalgeometry,aswellasthediversetopicsthatimpingeonthem. Forexample,wavelettheorycanbeconsideredanappropriatetooltodealwith some basic problems in digital signal processing, speech and image processing, geophysics, pattern recognition, biomedical engineering, and turbulence. These areas implement the latest technology from sampling methods on surfaces to fast algorithms and computer vision methods. The underlying mathematics of wavelet theorydependsnotonlyonclassicalFourieranalysis,butalsoonideasfromabstract harmonicanalysis,includingvonNeumannalgebrasandtheaffinegroup.Thisleads toastudyoftheHeisenberggroupanditsrelationshiptoGaborsystems,andofthe metaplectic group for a meaningful interaction of signal decomposition methods. Theunifyinginfluenceofwavelettheoryintheaforementionedtopicsillustratesthe justification for providing a means for centralizing and disseminating information fromthebroader,butstillfocused,areaofharmonicanalysis.Thiswillbeakeyrole of ANHA. We intend to publish with the scope and interaction that such a host of issuesdemands. vii viii ANHASeriesPreface Alongwithourcommitmenttopublishmathematicallysignificantworksatthe frontiersofharmonicanalysis,wehaveacomparablystrongcommitmenttopublish majoradvancesinthefollowingapplicabletopicsinwhichharmonicanalysisplays asubstantialrole: Antennatheory Predictiontheory Biomedicalsignalprocessing Radarapplications Digitalsignalprocessing Samplingtheory Fastalgorithms Spectralestimation Gabortheoryandapplications Speechprocessing Imageprocessing Time-frequencyand Numericalpartialdifferentialequations Time-scaleanalysis Wavelettheory The above point of view for the ANHA book series is inspired by the history of Fourieranalysisitself,whosetentaclesreachintosomanyfields. In the last two centuries Fourier analysis has had a major impact on the development of mathematics, on the understanding of many engineering and scientificphenomena,andonthesolutionofsomeofthemostimportantproblems in mathematics and the sciences. Historically, Fourier series were developed in the analysis of some of the classical PDEs of mathematical physics; these series were used to solve such equations. In order to understand Fourier series and the kindsofsolutionstheycouldrepresent,someofthemostbasicnotionsofanalysis were defined, e.g., the concept of “function.” Since the coefficients of Fourier seriesareintegrals,itisnosurprisethatRiemannintegralswereconceivedtodeal with uniqueness properties of trigonometric series. Cantor’s set theory was also developedbecauseofsuchuniquenessquestions. A basic problem in Fourier analysis is to show how complicated phenomena, suchassoundwaves,canbedescribedintermsofelementaryharmonics.Thereare twoaspectsofthisproblem:first,tofind,orevendefineproperly,theharmonicsor spectrumofagivenphenomenon,e.g.,thespectroscopyprobleminoptics;second, todeterminewhichphenomenacanbeconstructedfromgivenclassesofharmonics, asdone,forexample,bythemechanicalsynthesizersintidalanalysis. Fourieranalysisisalsothenaturalsettingformanyotherproblemsinengineer- ing, mathematics, and the sciences. For example, Wiener’s Tauberian theorem in Fourieranalysisnotonlycharacterizesthebehavioroftheprimenumbers,butalso provides the proper notion of spectrum for phenomena such as white light; this latter process leads tothe Fourier analysis associated withcorrelation functions in filtering and prediction problems, and these problems, in turn, deal naturally with Hardyspacesinthetheoryofcomplexvariables. Nowadays, some of the theory of PDEs has given way to the study of Fourier integral operators. Problems in antenna theory are studied in terms of unimodular trigonometric polynomials. Applications of Fourier analysis abound in signal processing, whether with the fast Fourier transform (FFT), or filter design, or the ANHASeriesPreface ix adaptivemodelinginherentintime-frequency-scalemethodssuchaswavelettheory. The coherent states of mathematical physics are translated and modulated Fourier transforms, and these are used, in conjunction with the uncertainty principle, for dealing with signal reconstruction in communications theory. We are back to the raisond’êtreoftheANHAseries! UniversityofMaryland JohnJ.Benedetto CollegePark,MD,USA SeriesEditor Preface There are almost two decades since the publication of the by now classical text of Gröchenig [125] on the fundamentals of time-frequency analysis. During this period, a considerable amount of research has been dedicated to the study of “good” function spaces that treat time and frequency simultaneously as well as thoseoperatorsactingcontinuouslyonthem.Insomesense,thisbookbeginswhere [125] ends. Its main goal is to give a flavor of some of the latest developments around the topic of modulation spaces defined on the d-dimensional Euclidean space,byestablishingtheirbasicproperties,equivalenceofvariousdefinitionsthat haveappearedintheliterature,andsomeoftheirapplicationstopseudodifferential operatorsandpartialdifferentialequations.Infact,theseapplicationsaretheraison d’être of this monograph. Indeed, our journey to the subject started from the investigation of the boundedness properties on the modulation spaces of certain multilinear pseudodifferential operators that are known to be unbounded on most classical function spaces. In the process and through the years, we came across a variety of applications of modulation spaces to not only pseudodifferential operators, but also in nonlinear partial differential equations. Therefore, many of theresultsdiscussedinthistextrelateinoneformoranothertosomeofthework done by the authors on the topics presented justifying further that its scope is not tobecomprehensive,asindeeditcannotbe.Thenotesattheendofmostchapters contain connections with other interesting related topics to the ones touched upon withinthetextandsomereferencesusefulinexploringsuchvenues. Becausethismonographisshapedbytheaforementionedapplications,weexpect that it will be useful for researchers in both time-frequency analysis and partial differential equations. Furthermore, our hope is that this book will serve as a standardreferencetograduatestudentsthatareinterestedinthestudyofmodulation spaces,butseasonedresearchersshouldalsofindrelevanttheaccountofsomeofthe maindevelopmentsinthisareaoftime-frequencyanalysisandtheoverarchingsense ofunitybetweenseveralofthesetopics.Thetitlesofeachchapterandthesections withinshouldbeself-explanatorytotheirpurpose. Parts of this book have been used in the first author’s teaching of one quarter graduate(secondyearmasters)leveltopicsinanalysiscourses,buttheintentionof xi
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