Excursions in Harmonic Analysis, Volume 4: The February Fourier Talks at the Norbert Wiener Center PDF
Preview Excursions in Harmonic Analysis, Volume 4: The February Fourier Talks at the Norbert Wiener Center
Applied and Numerical Harmonic Analysis Radu Balan, Matthew Begué John J. Benedetto, Wojciech Czaja Kasso A. Okoudjou, Editors Excursions in Harmonic Analysis, Volume 4 The February Fourier Talks at the Norbert Wiener Center Applied and Numerical Harmonic Analysis SeriesEditor JohnJ.Benedetto UniversityofMaryland CollegePark,MD,USA EditorialAdvisoryBoard AkramAldroubi GittaKutyniok VanderbiltUniversity TechnischeUniversita¨tBerlin 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 Ste´phaneJaffard YangWang UniversityofParisXII MichiganStateUniversity Paris,France EastLansing,MI,USA JelenaKovacˇevic´ CarnegieMellonUniversity Pittsburgh,PA,USA Moreinformationaboutthisseriesathttp://www.springer.com/series/4968 Radu Balan •Matthew Begue´ • John J. Benedetto Wojciech Czaja • Kasso A. Okoudjou Editors Excursions in Harmonic Analysis, Volume 4 The February Fourier Talks at the Norbert Wiener Center Editors RaduBalan MatthewBegue´ DepartmentofMathematics DepartmentofMathematics NorbertWienerCenter NorbertWienerCenter UniversityofMaryland UniversityofMaryland CollegePark,MD,USA CollegePark,MD,USA JohnJ.Benedetto WojciechCzaja DepartmentofMathematics DepartmentofMathematics NorbertWienerCenter NorbertWienerCenter UniversityofMaryland UniversityofMaryland CollegePark,MD,USA CollegePark,MD,USA KassoA.Okoudjou DepartmentofMathematics NorbertWienerCenter UniversityofMaryland CollegePark,MD,USA ISSN2296-5009 ISSN2296-5017 (electronic) AppliedandNumericalHarmonicAnalysis ISBN978-3-319-20187-0 ISBN978-3-319-20188-7 (eBook) DOI10.1007/978-3-319-20188-7 LibraryofCongressControlNumber:2012951313 Mathematics Subject Classification (2010): 26-XX, 35-XX, 40-XX, 41-XX, 42-XX, 43-XX, 44-XX, 46-25XX,47-XX,58-XX,60-XX,62-XX,65-XX,68-XX,78-XX,92-XX,93-XX,94-XX SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2015 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. Printedonacid-freepaper SpringerInternational PublishingAGSwitzerlandispartofSpringerScience+Business Media(www. springer.com) Dedicated to ArneBeurling suigenerisamongharmonicanalysts ANHA Series Preface TheAppliedandNumericalHarmonicAnalysis(ANHA)bookseriesaimstoprovide theengineering,mathematical,andscientificcommunitieswithsignificantdevelop- mentsinharmonicanalysis,rangingfromabstractharmonicanalysistobasicappli- cations.Thetitleoftheseriesreflectstheimportanceofapplicationsandnumerical implementation,butrichnessandrelevanceofapplicationsandimplementationde- pendfundamentallyonthestructureanddepthoftheoreticalunderpinnings.Thus, fromourpointofview,theinterleavingoftheoryandapplicationsandtheircreative symbioticevolutionisaxiomatic. 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 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 somebasicproblemsindigitalsignalprocessing,speechandimageprocessing,geo- physics, pattern recognition, biomedical engineering, and turbulence. These areas implement the latest technologyfrom sampling methods on surfaces to fast algo- rithmsandcomputervisionmethods.Theunderlyingmathematicsofwavelettheory dependsnotonlyonclassicalFourieranalysis,butalsoonideasfromabstracthar- monic analysis, includingvon Neumannalgebrasand the affine group.This leads 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 of ANHA. We intend to publish with the scope and interactionthat such a hostof issuesdemands. 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 centuriesFourier analysis has had a major impacton the devel- opment of mathematics, on the understanding of many engineering and scientific phenomena,andonthesolutionofsomeofthemostimportantproblemsinmathe- maticsandthesciences.Historically,Fourierseriesweredevelopedintheanalysis of some of the classical PDEs of mathematical physics;these series were used to solve such equations. In order to understandFourier series and the kinds of solu- tionstheycouldrepresent,someofthemostbasicnotionsofanalysisweredefined, e.g.,theconceptof“function”.SincethecoefficientsofFourierseriesareintegrals, it is no surprise that Riemann integrals were conceived to deal with uniqueness propertiesoftrigonometricseries. Cantor’sset theorywas also developedbecause ofsuchuniquenessquestions. 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 providesthepropernotionofspectrumforphenomenasuchaswhitelight;thislatter processleadstotheFourieranalysisassociatedwithcorrelationfunctionsinfilter- ingandpredictionproblems,andtheseproblems,inturn,dealnaturallywithHardy spacesinthetheoryofcomplexvariables. 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 unimodu- lar 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 conjunction with the uncertainty principle, for dealing with signal reconstruction in communications theory. We are back to the raisond’eˆtreoftheANHAseries! UniversityofMaryland JohnJ.Benedetto CollegePark SeriesEditor
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