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Applied and Numerical Harmonic Analysis Simon Foucart Holger Rauhut A Mathematical Introduction to Compressive Sensing Applied and Numerical Harmonic Analysis SeriesEditor JohnJ.Benedetto UniversityofMaryland CollegePark,MD,USA EditorialAdvisoryBoard AkramAldroubi JelenaKovacˇevic´ VanderbiltUniversity CarnegieMellonUniversity Nashville,TN,USA Pittsburgh,PA,USA AndreaBertozzi GittaKutyniok UniversityofCalifornia TechnischeUniversita¨tBerlin LosAngeles,CA,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 Ste´phaneJaffard YangWang UniversityofParisXII MichiganStateUniversity Paris,France EastLansing,MI,USA Forfurthervolumes: http://www.springer.com/series/4968 Simon Foucart (cid:129) Holger Rauhut A Mathematical Introduction to Compressive Sensing SimonFoucart HolgerRauhut DepartmentofMathematics LehrstuhlCfu¨rMathematik(Analysis) DrexelUniversity RWTHAachenUniversity Philadelphia,PA,USA Aachen,Germany ISBN978-0-8176-4947-0 ISBN978-0-8176-4948-7(eBook) DOI10.1007/978-0-8176-4948-7 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2013939591 MathematicsSubjectClassification(2010):46B09,68P30,90C90,94A08,94A12,94A20 ©SpringerScience+BusinessMediaNewYork2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.birkhauser-science.com) 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. Harmonic analysis is a wellspring of ideas and applicability that has flour- ished, developed, and deepened over time within many disciplines and by means of creative cross-fertilization with diverse areas. The intricate and fundamental relationship between harmonic analysis and fields such as signal processing, partial differential equations (PDEs), and image processing is reflected in our state-of-the-artANHAseries. Our vision of modern harmonic analysis includes mathematical areas such as wavelet theory, Banach algebras, classical Fourier analysis, time–frequency analysis, andfractalgeometry,aswell asthe diversetopicsthatimpingeon them. For example, wavelet theory can be considered an appropriate tool to deal with 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 algorithmsand computervision methods. The underlyingmathematics of wavelet theorydependsnotonlyonclassicalFourieranalysis,butalsoonideasfromabstract harmonicanalysis,includingvonNeumannalgebrasandtheaffinegroup.Thisleads toastudyoftheHeisenberggroupanditsrelationshiptoGaborsystems,andofthe metaplectic group for a meaningful interaction of signal decomposition methods. Theunifyinginfluenceofwavelettheoryintheaforementionedtopicsillustratesthe justification for providinga means for centralizing and disseminating information fromthebroader,butstillfocused,areaofharmonicanalysis.Thiswillbeakeyrole ofANHA.Weintendtopublishthescopeandinteractionthatsuchahostofissues demands. v vi ANHASeriesPreface Alongwithourcommitmentto publishmathematicallysignificantworksatthe 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,e.g.,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 latterprocessleadsto theFourieranalysisassociatedwithcorrelationfunctionsin 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 integraloperators.Problemsin antenna theoryare 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 ANHASeriesPreface vii the adaptive modeling inherent in time–frequency-scale methods such as wavelet theory. The coherentstates of mathematical physics are translated and modulated Fouriertransforms,andtheseareused,inconjunctionwiththeuncertaintyprinciple, fordealingwithsignalreconstructionincommunicationstheory.Wearebacktothe raisond’eˆtreoftheANHAseries! UniversityofMaryland JohnJ.Benedetto CollegePark SeriesEditor Dedicatedtoour families Pour Jeanne Fu¨r Daniela,Niels, PaulinaundAntonella