Duration and bandwidth limiting : prolate functions, sampling, and applications PDF
Preview Duration and bandwidth limiting : prolate functions, sampling, and applications
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 Technische Universität Berlin LosAngeles,CA,USA Berlin, Germany DouglasCochran MauroMaggioni ArizonaStateUniversity DukeUniversity Phoenix,AZ,USA Durham,NC,USA HansG.Feichtinger ZuoweiShen UniversityofVienna NationalUniversityofSingapore Vienna,Austria Singapore ChristopherHeil ThomasStrohmer GeorgiaInstituteofTechnology UniversityofCalifornia Atlanta,GA,USA Davis,CA,USA StéphaneJaffard YangWang UniversityofParisXII MichiganStateUniversity Paris,France EastLansing,MI,USA Jeffrey A. Hogan • Joseph D. Lakey Duration and Bandwidth Limiting Prolate Functions, Sampling, and Applications Jeffrey A. Hogan Joseph D. Lakey School of Mathematical Department of Mathematical Sciences and Physical Sciences New Mexico State University University of Newcastle Las Cruces, NM 88003-8001 Callaghan, NSW 2308 USA Australia [email protected] [email protected] ISBN 978-0-8176-8306-1 e-ISBN 978-0-8176-8307-8 DOI 10.1007/978-0-8176-8307-8 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011942155 Mathematics Subject Classification (2010): 42A10, 42C05, 65T50, 94A12, 94A20 © Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhäuser Boston, c/o Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Birkhäuser Boston is part of Springer Science+Business Media (www.birkhauser.com) Inmemory ofourfathers,RonHoganandFrankLakey 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 ofcreativecross-fertilizationwithdiverseareas.Theintricateandfundamentalre- lationship 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–frequencyanaly- sis,andfractalgeometry,aswellasthediversetopicsthatimpingeonthem. Forexample,wavelettheorycanbeconsideredanappropriatetooltodealwith somebasicproblemsindigitalsignalprocessing,speechandimageprocessing,geo- physics, pattern recognition, biomedical engineering, and turbulence. These areas implement the latest technology from 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 ofANHA.Weintendtopublishthescopeandinteractionthatsuchahostofissues demands. Alongwithourcommitmentto publishmathematicallysignificantworksatthe frontiersofharmonicanalysis,wehaveacomparablystrongcommitmenttopublish vii viii ANHASeriesPreface majoradvancesinthefollowingapplicabletopicsinwhichharmonicanalysisplays asubstantialrole: Antennatheory Predictiontheory Biomedicalsignal processing Radarapplications Digitalsignal processing Samplingtheory Fastalgorithms Spectralestimation Gabortheoryandapplications Speechprocessing Imageprocessing Time–frequencyand Numericalpartialdifferentialequations time-scaleanalysis Wavelettheory TheabovepointofviewfortheANHAbookseriesisinspiredbythehistoryof Fourieranalysisitself,whosetentaclesreachintosomanyfields. Inthelasttwo centuries,Fourieranalysishashada majorimpactonthe 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 propertiesof trigonometricseries. 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,e.g.,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 adaptivemodelinginherentin time–frequency-scalemethodssuch as wavelet the- ory. The coherent states of mathematical physics are translated and modulated ANHASeriesPreface ix Fouriertransforms,andtheseareused,inconjunctionwiththeuncertaintyprinciple, fordealingwithsignalreconstructionincommunicationstheory.Wearebacktothe raisond’eˆtreoftheANHAseries! UniversityofMaryland JohnJ.Benedetto CollegePark SeriesEditor
The list of books you might like
