Lectures on Constructive Approximation : Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and the Ball PDF
Preview Lectures on Constructive Approximation : Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and the Ball
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 Volker Michel Lectures on Constructive Approximation Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and the Ball VolkerMichel GeomathematicsGroup DepartmentofMathematics UniversityofSiegen Siegen,Germany ISBN978-0-8176-8402-0 ISBN978-0-8176-8403-7(eBook) DOI10.1007/978-0-8176-8403-7 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2012948340 Mathematics Subject Classification (2010): 33C45, 33C50, 33C55, 33F05, 41-01, 41A05, 41A10, 41A15,41A30,41A50,41A52,41A63,42A10,42A15,42A82,42A85,42C05,42C10,42C40,43A90, 65D05,65D07,65D20,65D30,65D32,65R32,65T40,65T50,65T60,86-08,86A22 ©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. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.birkhauser-science.com) Thisbookisdedicatedto theFatherof Geomathematics, Prof.Dr.WilliFreeden, apassionateteacher,avisionaryscientist, anda wisementor,inacknowledgement of hisinfinitesupportandin thehopeformany furtherjointprojectsto come. 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 algorithmsand computervision methods. The underlyingmathematics of wavelet theorydependsnotonlyonclassicalFourieranalysisbutalsoonideasfromabstract harmonicanalysis,includingvonNeumannalgebrasandtheaffinegroup.Thisleads toastudyoftheHeisenberggroupanditsrelationshiptoGaborsystemsandofthe 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. vii viii 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 Fourier analysis not only characterizes the behavior of the prime numbers, but it alsoprovidesthepropernotionofspectrumforphenomenasuchaswhitelight;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 the adaptivemodelinginherentintime-frequency-scalemethodssuchaswavelettheory. ANHASeriesPreface ix The coherentstates of mathematicalphysicsare translated and modulatedFourier transforms, and these are used, in conjunctionwith the uncertainty principle, for dealing with signal reconstruction in communications theory. We are back to the raisond’eˆtreoftheANHAseries! CollegePark,MD JohnJ.Benedetto SeriesEditor
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