Characterizing discrete time function spaces [Elektronische Ressource] / Marie Wild

Description

Technische Universit¨at Munc¨ henZentrum MathematikCharacterizing Discrete-Time FunctionSpacesMarie WildVollst¨andiger Abdruck der von der Fakult¨at fur¨ Mathematik der TechnischenUniversit¨at Mun¨ chen zur Erlangung des akademischen Grades einesDoktors der Naturwissenschaften (Dr.rer.nat.)genehmigten Dissertation.Vorsitzender: Univ.-Prof. Dr. Michael UlbrichPrufer¨ der Dissertation: 1. Priv.-Doz. Dr. Hartmut Fuhr¨2. Univ.-Prof. Dr. Rupert Lasser3. Prof. Dr. Rodolfo H. Torres,University of Kansas / USA(schriftliche Beurteilung)Die Dissertation wurde am 30. Ma¨rz 2006 bei der Technischen Universit¨ateingereicht und durch die Fakult¨at fur¨ Mathematik am 17. Juli 2006 angenommen.2PrefaceThis thesis was carried out at the GSF Institute of Biomathematics and Biometry(IBB), Neuherberg, under the supervision of PD Dr. Hartmut Fuhr,¨ whom I wantto thank first and foremost. Without his help and encouragement, this work wouldnot have been possible.IwanttothankallthecolleaguesattheIBB,especiallytheheadofthegroup‘Math-ematical Modelling in Ecology and the Biosciences’, Prof. Dr. Gerhard Winkler,and the head of the institute, Prof. Dr. Rupert Lasser, for their support.Last but not least many thanks to my parents and all of my friends –Dominik, min¨a rakastan sinua.Munich, March 2006 Marie WildFrom May 2002 to July 2005, the thesis was funded by the GSF and from August2005 to March 2006 by the Munich University of Technology.

Sujets

• Mathematics

Informations

• Publié par: technische_universitat_munchen
• Publié le: 01 janvier 2006
• Nombre de lectures: 44
• Langue: English

Extrait

TechnischeUniversit¨atM¨unchen
ZentrumMathematik

CharacterizingDiscrete-TimeFunction
Spaces

WildMarie

Vollst¨andigerAbdruckdervonderFakult¨atf¨urMathematikderTechnischen
Universit¨atM¨unchenzurErlangungdesakademischenGradeseines
DoktorsderNaturwissenschaften(Dr.rer.nat.)
tion.Dissertatengenehmig

Vorsitzender:Univ.-Prof.Dr.MichaelUlbrich
Pr¨uferderDissertation:1.Priv.-Doz.Dr.HartmutF¨uhr
2.Univ.-Prof.Dr.RupertLasser
3.Prof.Dr.RodolfoH.Torres,
UniversityofKansas/USA
(schriftlicheBeurteilung)

DieeingereicDissherttatioundndurcwurdehdieamFa30.kult¨Mat¨afrz¨ur2M00a6beithematikderTaecmhnisc17.heJulin2Univ006ersit¨angenoatmmen.

2

Preface

ThisthesiswascarriedoutattheGSFInstituteofBiomathematicsandBiometry
(IBB),Neuherberg,underthesupervisionofPDDr.HartmutF¨uhr,whomIwant
tothankfirstandforemost.Withouthishelpandencouragement,thisworkwould
nothavebeenpossible.
IwanttothankallthecolleaguesattheIBB,especiallytheheadofthegroup‘Math-
ematicalModellinginEcologyandtheBiosciences’,Prof.Dr.GerhardWinkler,
andtheheadoftheinstitute,Prof.Dr.RupertLasser,fortheirsupport.
Lastbutnotleastmanythankstomyparentsandallofmyfriends–
Dominik,min¨arakastansinua.

Munich,March2006

WildMarie

FromMay2002toJuly2005,thethesiswasfundedbytheGSFandfromAugust
2005toMarch2006bytheMunichUniversityofTechnology.Additionallyfunding
camethroughtheEuropeanResearchTrainingNetworkHASSIP.

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4

Contents

efacerP

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Introduction7
Preliminaries..................................12

1BesovSpacesonR15
1.1BesovSpacesonRandtheirCharacterizations.............16
1.1.1ModuliofSmoothness......................19
1.1.2Littlewood-PaleyTypeCharacterization............21
1.1.3ϕ-transformCharacterization..................23
1.1.4WaveletCharacterization.....................26
1.2NonlinearWaveletApproximationandBesovSpaces.........28
1.2.1NonlinearApproximationofDiscrete-TimeSignals......29
1.3AimsofThisThesis............................30

2WaveletAnalysisofDiscrete-DomainSignals31
22.1WaveletBasesforL(R).........................32
2.1.1WaveletsandFilters.......................36
2.1.2PropertiesofWaveletBases...................40
2.1.3BiorthogonalBases........................42
22.2Discrete-TimeWaveletBasesfor(Z).................44
2.2.1TheDiscrete-TimeWaveletTransform.............45
2.2.2RegularityofDiscrete-TimeWavelets..............52
22.2.3ConnectiontoWaveletBasesforL(R).............53

3Discrete-TimeBesovSpacesandtheirCharacterizations57
3.1Littlewood-PaleyTypeDefinitionofBαp,q(Z)..............58
3.2ϕ-transformDecompositionofBαp,q(Z)..................59

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α3.3WaveletCharacterizationofB(Z)..................
p,qα3.4‘Intrinsic’CharacterizationsofB(Z)................
p,q3.4.1Discrete-TimeModuliofSmoothness.............
α3.4.2MeanOscillationCharacterizationofB(Z).........
p,q

4Discrete-TimeTriebel-LizorkinSpaces
4.1Definitionandϕ-transformDecomposition..............
4.2WaveletCharacterizationofDiscrete-TimeTriebel-LizorkinSpaces.

DiscussionAndOutlook

Bibliography

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62777785

898991

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99

Introduction

foThiscusonthesisBesovdealsspaceswithocnharaZandcterizatiotheirnswavofeletspaccesharaofcterizdiscrete-ation.timefunctions,witha
Itisawidelyacceptedfactthatthesuccessofwaveletsinapplicationsisbased
ontheirabilitytoefficientlyrepresent‘realistic’signals.Thisefficiencyistwofold:
Computationalefficiencyisguaranteedbyfastfilterbankalgorithmsassociated
toawaveletbasis,theso-calledfastwavelettransform.
Anequallyimportantpropertyofwaveletsistheirapproximation-theoreticef-
ficiency,thatis,theabilityofwaveletstocapturesalientfeaturesofasignalina
fewlargecoefficients.
Thissignalspr:opGivertenyoafwwaavveleteletswitishbestsufficientlyexemplifiedmanybyvmeansanishingofmopiemencewisets,thepolynonzeronomial
waveletcoefficientswillbelocatedatthejumpsofthesignal.
Acanmobereelaformboratulatede(ainndtermormseopfowBesoerful)vdescspacesr.iptionDesofpitweavtheeletfactapprothatximaBesotionvspacestheory
wspaceerethconceiveoryedandsomewav25eletyearsapproprioximartotionwavtheoryelets,itariseproidentbablyicalf;aseiretoCsahaypterthat1foBresoanv
explanationofthisstatement.
Hence,orthonormalwaveletbasesprovideaclassofsignaltransformsthatareeasily
implemented,withfastalgorithmsandcompletelyunderstoodapproximationthe-
ory,andmuchusehasbeenmadeofthesefeatures,bothfortheoreticalandapplied
purposes[5,7,15,19,24].
Hoertieswevofer,wavdespiteeletstahereofactftenthatusedthecosimultmputaaneouslytional,aondneapproshouldximationoten-thattheoreticthereprisop-a,
somewhatsubtle,gapseparatingthetwo:Strictlyspeaking,thecomputational
wapartveletonlytransfoapplrm,iestowhereasdiscrethetelattteimrearesigonlynalsaandpplicabletheirtodecoconmptinuoositionus-timebythesignafastls.
Thisclosegit.apInhasthisbeentheacsis,knowewledgpresenedteaarlydison,butcrete-timenotvmersiouchnhaofswbaeevneletdoneapprosincexima-to
tioexpnect,theoritisy,agathatinislinkspedtoecificallyascaletunedofBtoesovthespacfastes,wavprevioeletuslydtransforefinedm.byAsToneorresmigh[31].t

7

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Tobemoreprecise,letusdefineforα∈R,0<p,q<∞,thecoefficientspaces
b˙αp,q(R)asthecollectionofallcomplex-valuedsequencest=(tj,l)j,l∈Z,satisfying
tb˙αp,q(R):=(((2−j(α+1/2−1/p)|tj,l|)p)q/p)1/q<∞.
j∈Zl∈Z
Thesenormsareusedtomeasurethedecayoftheexpansioncoefficientsofasignal
finawaveletorthonormalbasis.Suchabasisisasystem(ψj,l)j,l∈Zoffunctions
arisingfromasuitable‘motherwavelet’ψ∈L2(R)bytranslationanddilation,
ψj,l(x)=2−j/2ψ(2−jx−l).
AshortsurveyofwaveletbasesinL2(R)andtheirconstructioncanbefoundin
.2.1SectionTypically,waveletsfulfilladditionaldesirableproperties,besidesgeneratinganor-
thonormalbasis,suchas
•smoothness,i.e.ψ∈CM,forM∈N
•vanishingmoments:Rψ(x)xidx=0,fori=0,...,K−1.
•compactsupport.
Itisknownthatifthewaveletfamily(ψj,l)j,l∈ZhastheabovepropertieswithM
andKlargeenough,thenafunctionfisinaBesovspaceB˙αp,q(R),ifandonlyif
(f,ψj,l)isinthecorrespondingcoefficientspaceb˙αp,q(R);seee.g.[17].Inaddition,
wehavethenormequivalence
fB˙αp,q(R)(f,ψj,l)b˙αp,q(R).
Earlyon,thesenormequivalenceshavebeenrelatedtothenonlinearapproximation
behaviorofwaveletexpansionsandtowaveletapplications:adecayofcoefficients
likeinb˙αp,q(R)islinkedtothedecayoftheapproximationerrorofwaveletexpansions
byN>0terms(see[7]orSection1.2.1below).
Theseresultsarewidelyusedinsignalandimageprocessing:Asmalllistofref-
erencesthatusetherelationshipbetweenwaveletsandBesovspacetoderivealgo-
rithmsfordiverseproblemssuchasdenoising,compression,deconvolutionorRadon
inversion,is[5,10,1,2,24,13,21].
Inmostapplicationshowever,thedataunderconsiderationaregivendiscretely,and
areprocessedbythefastwavelettransform.Thisalgorithmarisesnaturallyfroma
multiresolutionanalysis,whichcanbeassociatedtomostorthonormalwavelet
bases(inparticulartoallsmoothwaveletswithcompactsupport,see[22]).
Thus,adiscreteseries(f(n))n∈Zismappedtothefamilyof(dj,l)j≥1,l∈Zofdiscrete
waveletcoefficients.ObservethatbytheFischer-Riesztheorem,eachcoefficientdj,l

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isgivenbythescalarproductoffwithasuitablediscrete-timewavelethj,l.

ααThis∈R,sugg0<estps,qto<∞,considerasthethecollectionspaceofof(trcomplex-vuncated)coaluedefficsientequencesfamilies,s=(bsjp,q,l)(jZ≥),1,l∈Zfor,
hwhicforsbαp,q(Z):=(((2−j(α+1/2−1/p)|sj,l|)p)q/p)1/q<∞.
j≥1l∈Z
Thiscoefficientdecaystillreflectsthenon-linearapproximationbehaviorof(f(n)).
Itisthereforenaturaltoaskwhethertheproperty(dj,l)j≥1,l∈Z∈bαp,q(Z)canbechar-
ofacterithezedfilter-inabankosimilarr,equivsatisfactoalently,rywtheayasindiscrete-timetheconwtinaveuoletuscasefamily-(frohm)propertiesand
ofthesequencef.j,lj≥1,l∈Z
Somewhatsurprisingly,literaturesofardoesnotseemtoprovideasimpleanswerto
thisbasisqofuestion.heuristicsNonetwhichehless,aretheappliednormtoeqtheuivadisclencreeteinthesetting,continwhereuousonlytime-thecasetruncaistedthe
coefficientseriesareavailable.
assoThecciatonedtintouousthemtheoryultiresoluthastheionfolloanalysiswing,toandoffer:defineLettheφconbetintheuous-scatimelingfunctiofunctionn
Fcien=tsofn∈FZf(n)coincideτnφ,withwhere(dτjn,l)φj≥1,l∈denotesZforjthe≥tr1,anslateandvaofnisφ.hforThenscalesthejwa≤v0.eletHeconce,effi-
assumingsufficientvanishingmoments,smoothnessanddecayoftheassociatedcon-
tinuoustimewavelets,
(dj,l)j≥1,l∈Z∈bαp,q(Z)⇔F∈B˙αp,q(R).
However,Fisnoteasilyaccessible.Theproblemispresentedbythescalingfunction
φthe:Fsorcalingmanyfuncwatvioneletisbasesonly,kandnowninpaimplicrticularitly,aforsthethecoresultmpactlyofasuplimitpoprortedcess.wavelets,
Hence,membershipofFinaBesovspaceisnoteasilychecked,andtheequivalence
less.uselmostaisWehavenotbeenabletolocateanyresultinliteraturedealingwiththisproblem,
theHenceconwtinearruous-ivetatimethesettingfollodowingesnot(somewhatgiveaovconclusiverstated)eanswerconclusion:toourAnquestioalgon.rithm
usispacengtcheharactcascadeeristalicsgoinricothmnti,nbutuousderivtime,edifsronomttheheuristoretiicscallyusingjtustihefieBd.esov

yarSummαarThegumenmaintspurthaptosedoofnotthisusethesisanyeismbtogiveddingeincriteriatotheforcon(djtin,l)juo≥1us,l∈Z-time∈bsep,qt(Zting.),withIn

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thisway,weobtainthatthediscrete-timewaveletsassociatedtoamultiresolution
analysisareunconditionalbasesforawholefamilyofdiscrete-timesignalspaces.
ItmaynotbetoosurprαisingthattheresultingspacesareagainBesovspaces,the
di