Lectures on Constructive Approximation : Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and the Ball

324 Pages·2013·6.12 MB·English

by  Volker Michel

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