Recent applications of harmonic analysis to function spaces, differential equations, and data science. V.2 PDF
Preview Recent applications of harmonic analysis to function spaces, differential equations, and data science. V.2
Applied and Numerical Harmonic Analysis Isaac Pesenson Quoc Thong Le Gia Azita Mayeli Hrushikesh Mhaskar Ding-Xuan Zhou Editors Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science Novel Methods in Harmonic Analysis, Volume 2 Applied and Numerical Harmonic Analysis SeriesEditor JohnJ.Benedetto UniversityofMaryland CollegePark,MD,USA EditorialAdvisoryBoard AkramAldroubi GittaKutyniok VanderbiltUniversity TechnischeUniversitätBerlin Nashville,TN,USA Berlin,Germany DouglasCochran MauroMaggioni ArizonaStateUniversity DukeUniversity Phoenix,AZ,USA Durham,NC,USA HansG.Feichtinger ZuoweiShen UniversityofVienna NationalUniversityofSingapore Vienna,Austria Singapore,Singapore ChristopherHeil ThomasStrohmer GeorgiaInstituteofTechnology UniversityofCalifornia Atlanta,GA,USA Davis,CA,USA StéphaneJaffard YangWang UniversityofParisXII MichiganStateUniversity Paris,France EastLansing,MI,USA JelenaKovacˇevic´ CarnegieMellonUniversity Pittsburgh,PA,USA Moreinformationaboutthisseriesathttp://www.springer.com/series/4968 Isaac Pesenson (cid:129) Quoc Thong Le Gia Azita Mayeli (cid:129) Hrushikesh Mhaskar Ding-Xuan Zhou Editors Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science Novel Methods in Harmonic Analysis, Volume 2 Editors IsaacPesenson QuocThongLeGia DepartmentofMathematics SchoolofMathematicsandStatistics TempleUniversity UniversityofNewSouthWales Philadelphia,PA,USA Sydney,NSW,Australia AzitaMayeli HrushikeshMhaskar DepartmentofMathematics InstituteofMathematicalSciences TheGraduateCenter,CUNY ClaremontGraduateUniversity NewYork,NY,USA Claremont,CA,USA Ding-XuanZhou DepartmentofMathematics CityUniversityofHongKong KowloonTong,HongKong ISSN2296-5009 ISSN2296-5017 (electronic) AppliedandNumericalHarmonicAnalysis ISBN978-3-319-55555-3 ISBN978-3-319-55556-0 (eBook) DOI10.1007/978-3-319-55556-0 LibraryofCongressControlNumber:2017939351 Mathematics Subject Classification (2010): 05E15, 11G15, 11R04, 11R09, 11R47, 11R56, 14F05, 33C10,33C55,33E12,35A25,35J25,41A15,42A16,42A38,42B10,42B35,42B37,42A99,42C99, 46L10,46L40,46L53,46L54,47B34,60G15,60G60,62D99,62-07,68P99,86A20,86A99,94A12, 94A20 ©SpringerInternationalPublishingAG2017 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,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisbookispublishedunderthetradenameBirkhäuser,www.birkhauser-science.com TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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 rela- tionship between harmonic analysis and fields such as signal processing, 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 theorydependsnotonlyonclassicalFourieranalysisbutalsoonideasfromabstract 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. v vi ANHASeriesPreface Alongwithourcommitmenttopublishmathematicallysignificantworksatthe frontiersofharmonicanalysis,wehaveacomparablystrongcommitmenttopublish majoradvancesinthefollowingapplicabletopicsinwhichharmonicanalysisplays asubstantialrole: Antennatheory Predictiontheory Biomedicalsignalprocessing Radarapplications Digitalsignalprocessing Samplingtheory Fastalgorithms Spectralestimation Gabortheoryandapplications Speechprocessing Imageprocessing Time-frequencyand Numericalpartialdifferentialequations time-scaleanalysis Wavelettheory TheabovepointofviewfortheANHAbookseriesisinspiredbythehistoryof 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 Fourieranalysisnotonlycharacterizesthebehavioroftheprimenumbersbutalso provides the proper notion of spectrum for phenomena such as white light; this latterprocess leads totheFourier analysis associated withcorrelationfunctions 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 adaptivemodelinginherentintime-frequency-scalemethodssuchaswavelettheory. ANHASeriesPreface vii 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! CollegePark,MD,USA JohnJ.Benedetto Preface Wepresentthesecondoftwovolumes,whicharecomposedofmorethan30articles related to harmonic analysis. Harmonic analysis is a very old topic, which still continuestodrawtheinterestofmanymathematicians.Modernresearchinthisarea ismotivated both bydeeper and new theoretical questions and numerous practical applications. These volumes aim to provide a sample of some of these directions. All the authors were selectively invited and comprise both senior and junior mathematicians. We are pleased to have received an unexpectedly enthusiastic responsetoourinvitations. In response to the number of papers we received, it was suggested by Birkhäuser/Springer to split our book into two volumes. Chapters in each volume are organized into parts according to their topics, and the order of chapters in each part is alphabetical. This first volume, entitled “Frames and Other Bases in Abstract and Function Spaces” consists of 16 chapters. It is quite homogeneous mathematicallysinceeverychapterrelatestothenotionofframesorbasesofother types.Theintroductiontothisvolumecontainssomebasicnotionsofthetheoryof frames and underlines the way the chapters fit into the general theme. The second volume, which is called “Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science,” consists of 15 chapters and is verydiverse.Itsintroductionisjustacollectionofextendedabstracts. Wewereluckytoreceiveexcellentcontributionsbytheauthors,andweenjoyed working with them. We deeply appreciate the generous help of many of our colleagues who were willing to write very professional and honest reviews on submissions to our volumes. We are very thankful to John Benedetto, who is the series editor of the Birkhäuser Applied and Numerical Harmonic Analysis Series, for his constant and friendly support. We appreciate the constant assistance of Birkhäuser/SpringereditorsDanielleWalkerandBenjaminLevitt.Wearethankful toMeyerPesensonandAlexanderPowellfortheirconstructivecommentsregarding introductions. We acknowledge our young colleague Hussein Awala for his help withorganizingfilesandtemplates. ix x Preface We hope these volumes will be useful for people working in different fields of harmonicanalysis. Philadelphia,PA,USA IsaacPesenson Sydney,NSW,Australia QuocThongLeGia NewYork,NY,USA AzitaMayeli Claremont,CA,USA HrushikeshMhaskar KowloonTong,HongKong Ding-XuanZhou
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