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

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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.

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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.

3

4

Contents

efacerP

3

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

5

6

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

....

..

62777785

898991

97

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

8

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

9

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

10

thisway,weobtainthatthediscrete-timewaveletsassociatedtoamultiresolution
analysisareunconditionalbasesforawholefamilyofdiscrete-timesignalspaces.
ItmaynotbetoosurprαisingthattheresultingspacesareagainBesovspaces,the
discreteBesovspacesBp,q(Z)introducedbyTorres[31].
Whilethisthesisstrictlyavoidsanyappealtothecontinuoustimetheory,wewill
adopttheproofstrategyandtechniquesfromthecontinuoussetting,aspresented
inthebookofFrazier,JawerthandWeiss[17]or,inaslightlymoregeneralcontext,
byKyriazis[14].
Chapter1providesashortintroductiontoBesovspacesonR,andtheirvarious
characterizations,usingmoduliofsmoothness(Section1.1.1),Littlewood-Paleythe-
ory(Section1.1.2),ϕ-transforms(Section1.1.3)orwavelets(Section1.1.4).Thisis
ontheonehandintendedasanintroduction,providingafirstglimpseontechniques
thatareusedandadaptedlateron,andshowingtheversatilityofBesovspaces.But
itisalsoaprecursorofthediscrete-timeresultsweestablishlateron:Analogstoall
ofthesecharacterizationswillbeestablishedinChapter3fordiscrete-timesignals.
InSection1.2,weelaborateonourpreviouslymadestatementthatBesovspacethe-
oryandwaveletapproximationtheorycanbeconsideredasidentical,bydiscussing
therelationshipbetweenBesovspacesandnonlinearwaveletapproximation.With
mostofthecentralnotionsanddefinitionsestablished,wethendescribetheaimsof
thisthesisinsomewhatgreaterdetail(Section1.3).
Chapter2isdevotedtoanintroductiontowavelets.Eventhoughthefocusof
thisworkisondiscrete-timewavelets,westartoutwithashortrundownofthe
basicfactsconcerningwaveletorthonormalbasesandmultiresolutionanalysisin
continuoustime.Inparticular,weexplaintheoriginofthefastwavelettransform
fromacontinuoustimemultiresolutionanalysis(Section2.1.1),andshortlydiscuss
additionaldesirablepropertiesofwaveletsinL2(R)suchasvanishingmoments,
smoothnessandcompactsupport(Section2.1.2).Weshortlycommentonbiorthog-
onalwavelets,asallofourresultslateroncanbeestablishedwithoutanyadditional
effortforthebiorthogonalsetting(Section2.1.3).
Wethenturnattentiontodiscrete-timewavelets.Discrete-timewaveletsystems
ariseasabyproductofthefastwavelettransform(Section2.2.1),butcanbeunder-
stoodassystemsofoscillatorybuildingblocksindexedbytranslationparameters
and(dyadic)scaleparameters(Theorem2.2.2).OfcrucialimportanceforChapter
3willbeadditionalpropertiesofthewaveletsystem,suchasvanishingmoments,
supportpropertiesand–somewhatunexpectedly–regularity(Section2.2.2).This
latterpropertyhasbeenintroducedbyRioul[26];itconnectsdiscrete-timewavelets
tocontinuous-timewaveletsvialargescalelimits(Section2.2.3).
Chapter3finallydealswiththecentralpurposeofthisthesis,namelythecharac-
terizationofBesovspacesindiscretetime.Forthedefinitionofthesespaces,the
startingpointistheso-calledϕ-transformcharacterizationofBαp,q(Z),establishedby
Torres.Justasinthecontinuouscase,thistransformisinmanywaysquitesimilar

11

toingawavoff-diageletonatralnsfodecarm,yofandcertathisinsimilainfiniterityamatllowsriceas.proThisofofdecatheybmainehavresultiorisbyderivstudy-ed
thefromdisscruiteteable-timeconditiowavensletsregaasprordingvidedsuppbyort,Sectiovanisn2hing.2.2.momentsandsmoothnessof
Chapter1showedthattheBesovspacesonRhavequiteanumberofdifferent,but
equivalentdescriptions.Usingourwaveletcharacterization,wecanprovethatthe
sameisvalidfortheircounterpartsonZ.Weobtainfurtherdescriptionsofthese
spaces,whichcanbeviewedasmore‘intrinsic’incontrasttotheLittlewood-Paley
typedefinitionin[31],suchasintermsofiterateddifferencesandmeanoscillation
properties(Section3.4).
defineMimicfokingr1the<p<definition∞,t∈ofR+mo,tduliheorf-thsmooroderthnessmofordulusconoftinsmoothnuous-timeessofffunctioinnslp,(wZe)
ybωpr(f,t)=m∈Zs,u|mp|<tΔrmf(∙)p,

whereΔrmf(n)isthedifferenceoperatorΔmf(n)=f(n+m)−f(n)iteratedrtimes.
Forα>0,1<p,q<∞,r=α+1,wedefinethespaceBqα(lp(Z))asthespaceof
sequences(f(n)),suchthat

fBα(lp(Z)):=((2−jαωpr(f,2j))q)1/q<∞
q1≥j

Weshowthatforα>0,1<p,q<∞thespacesBqα(lp(Z))coincidewiththeBαp,q(Z)
withequivalentnorms(Theorem3.4.6).
warAnotdhemarcnnerhaisracterizaationdescriptioforntheindisctermsretofe-timemeanBesovoscillatiospacnesovweerginettervinaalsstra(Theoightforemr-
.9).3.4Fandorcinon[8].tinuo[30]us-cotimnetainsBesoavspaces,descriptionthisoftypeodiscrete-timefcharinacterizattermsionofwameansgivoescniinllatio[9n]
proThepreretiessultsofofseSequectionnces3.4forsoexhibitmesptheecialusecasesfulnessoftofheourparwavameterseletcα,harp,q.acterizationin
anaTheoremlogsof3.3.8their,andcontintheyempuous-timehasizecountheterparviewts.ofdiscrete-timeBesovspacesasworthy
Wepresenatlsoatreatdiscrete-timeanotherwsacavleeletofchafunctioracterizanspationcesforinthethisdiscrete-thesis:timeinTChariebel-pterL4,izorkinwe
ααFbyp,qQ(.Z)Sunspainces[29]anainlogtermstoourofsmoresultothfoarBtomicp,q(Zdec).ompTheseositiospacesns.wereTheoremalready4.2.6prodiscusvidessed
anewcharacterizationofthesespaces,notcontainedinSun’sresults.

12

Preliminaries
Thissectionismainlythoughttofixnotation.Italsoprovidesashortreviewof
basesforHilbertandBanachspaces,butthisiskeptshortandisjustmeanttoto
providethelanguageweuselateroninthisthesis.

For1≤p<∞,thespaceLp(R)isdefinedasthespaceofmeasurablefunctionsf
onR,forwhich1/p
fLp(R)=|f(t)|p<∞.
REquippedwiththeabovenorm,Lp(R)isaBanachspace,providedthatalmost
everywhereagreeingfunctionsareidentified.Forp=∞,theintegralaboveis
replacedbyesssupintheusualway.
p(Z)denotesthecorrespondingp-summablesequencespace.

TheFouriertransformonL1(R)isdefinedby
Rf(ω)=f(t)e−iωtdt,
andisextendedtothePlanchereltransformonL2(R).Withthechosennormaliza-
tion,onehas1
fL2(R)=2πfL2(R).
S(R)denotestheSchwartzspaceofrapidlydecreasingfunctions,andS(R)
itsdual,thespaceoftempereddistributions.
Byf,g,wedenotethestandardinnerproductinaHilbertspace,aswellasthe
pairingofatempereddistributionwithaSchwartzfunctiong.

Theinvolutionforasequencereadsasg∗(n)=g(−n),wheregstandsforcomplex
n.atioconjug

Theconvolutionproductoff∈S(R)withaSchwartzfunctionh∈S(R)is
ybdefinedf∗h,g=f,h∗∗g,
whereforh,g∈S(R),
h∗g(∙)=h(t)g(∙−t)dt.
Theconvolutionproductoftwosequencesg,hisdefinedby
g∗h(n)=g(k)h(n−k).
k

13

ForafunctionfdefinedonR,thesupportoffisdefinedassuppf={t:f(t)=0}.
Forasequenceg=(g(n))n∈Z,suppgwillbethesmallestintervalcontainingthose
nforwhichh(n)=0.

ForN∈N,theupsamplingoperatoractingonasequencegisdefinedby
↑Ng(n)=g(N−1n),ifN−1n∈Zand0otherwise;
thedownsamplingoperatorisdefinedby↓Ng(n)=g(Nn).
Toavoidclutterednotation,Cdenotesaconstantwhichisallowedtochangewithin
anargument.ThenotionABmeansthatthereexistconstantsC1,C2>0,such
thatC1A≤B≤C2B.
BasesforHilbertandBanachspaces

Inthisthesis,wefrequentlyusedifferentconceptsofbasesinHilbertandBanach
spaces.Wegivesomeofthemostelementarydefinitionsandresults,takenfrom[4].
LetinthefollowingHbeaseparableHilbertspaceandIacountableindexset.We
assumeItobesuitablynumbered,thusintroducingasummationorderonI.
Definition0.1.Afamilyofvectors(en)n∈I⊆Hisanorthonormalsystem
if,(ONS)en,em=δn,m,foralln,m∈I,
whereδn,m=1forn=mand0otherwise.

Definition0.2.AnONS(en)n∈I⊆Hisanorthonormalbasis(ONB)ifitis
:inompletecHH=span{en}.(0.0.1)
Equation(0.0.1)isequivalentto
f2H=|f,en|2forallf∈H.
I∈n

Definition0.3.Afamilyofvectors(fn)n∈I⊆HisaRieszBasisforH,ifthere
existsanONB(en)n∈IforHandaboundedinvertiblemappingT:H→H,such
thatTen=fnforalln∈I.
AssociatedtoanyRieszbasisisadualfamily,whichisalsoaRieszBasis.

14

˜Theorem0.4.If(fn)n∈IisaRieszBasisforH,thereexistsauniquefamily
(fn)n∈I⊆H,suchthatforeveryf∈H
f=n∈If,f˜nfn.
(f˜n)isalsoaRieszbasisand(fn),(f˜n)arebiorthogonal,i.e.
fn,f˜m=δn,m,foralln,m∈I.

Inthissense,wecallthefamilies(fn),(f˜n)biorthogonalbases.
ForaRieszBasis(fn),thereexistconstantsA,B>0,suchthatforeveryf∈H,
Af2H≤|f,fn|2≤Bf2H.
nnnTheconstantsthereinarecalledRieszbounds.Ofcourse,anyONBisaRiesz
basiswithRieszboundsA=B=1.
LetusnowconsiderthemoregeneralsituationinaBanachspaceB,wherethe
conceptoforthogonalityisnotapplicableanymore.
Definition0.5.Afamilyofvectors(en)n∈I⊆BisaSchauderbasisifforeach
f∈B,thereexistuniquecoefficients(cn)n∈I,suchthat
f=n∈Icnen(0.0.2)

withconvergenceinB.
Theabovedefinitiondependsontheorderofsummation:itcanhappenthatthe
sumisdivergentforacertainpermutationofsummands.Wethusintroducethe
notionofunconditionalconvergence.
Definition0.6.ASchauderBasis(en)n∈I⊆Bisanunconditionalbasis,if
theseries(0.0.2)convergesunconditionally,i.e.n∈Icσ(n)eσ(n)convergesforall
permutationsσofI.Inthiscase,thelimitisthesame,regardlessoftheorderof
summation.ThebasisinconceptaHilbofertanspaceunconditH.Inionalfact,basisanyinRieszaBabasisnachforspaHceisanextendsunconditiothatofnalaRieszbasis
.forH

1hapterC

noSpacesvBeso

RAsfunctionstatedinspacestheinwithtroafoduction,cusotnhisdiscrete-thesistdeaimelsBewithsovcsharpaces.acterizationsofdiscrete-time
Inthischapter,wegiveanoverviewoftheircontinuous-timecounterparts,theBesov
spacesonR.Roughlyspoken,theBesovclasscanbeviewedasageneralizationof
classicalsmoothnessspaces,suchasH¨olderorSobolevspaces,tospacesoffunctionsp
anddistributionspossessingsmoothnessoforderα∈R,measuredindifferentL
spaces.Athirdparameterqallowsfinerdistinctions.
bIneliteratfound,ure,theiramdescrultitudeiptionofdepdifferenendingtbuotnthequivealendevicestdefinitiowhichnsaorefBusedesovtsopamecesacasuren
smoothness.Inthefirstsection,werecallsomeofthesecharacterizations.
Furthermore,Besovspacesarerelatedtononlinearapproximationbehaviorofwavelet
expansions.Simplistically,approximatingafunctionwhichiscontainedinaBesov
inspaceO(Nof−αor).derThisαbisysue,Ntermswhichofinafawcatveletcouldbseries,eregtheardedapproasanotximatherionchaerrorracterizadecreasestion
forcertainBesovspaces,isdescribedmorepreciselyinSection1.2ofthischapter.
Inthenextsection,weswitchfromcontinuoustodiscretetime:Inwaveletappli-
cationssuchassignalorimageprocessing,thedataunderconsiderationaregiven
discretely.Passingsuchadiscrete-timefunctionthroughawaveletfilterbank,decay
propertiesofthearisingcoefficientsstillreflectnonlinearapproximationproperties
ofthefunction,whereasthecontinuous-timetheory,thoughservingasabasisof
heuristics,doesnotprovideasatisfactorycharacterizationofthesesignalsviafunc-
spaces.ntioSo,ontheonehand,thischapterservestomotivatethestudyofdiscrete-timeBesov
spacesandtospecifytheaimsofthisthesisinSection1.3.Ontheotherhand,itis
alsomeantasashortoverviewoncontinuous-timeBesovspacesandaquickguided
tourthroughthemultitudeoftheirdifferentdescriptions,whicharespreadamong
ture.litera

15

16

1.1BesovSpacesonRandtheirCharacterizations

1.1BesovSpacesonRandtheirCharacteriza-
onsti

ThereisnouniquewaytodefineBesovspaces.Thereisalargevarietyofdescrip-
tions,whichareessentiallyequivalenttoeachother.ThispropertyofBesovspaces
canbethesourceofconfusion,inparticularasnotationsandnormalizationstendto
varybetweendifferentsources,butitalsoreflectstheirstatusasanimportantclass
offunctionspaces,locatedattheintersectionofvariousmathematicalsubdisciplines.
Thecharacterizationsgivenherecanbegroupedroughlyintotwocategories.The
firstclassofapproachescharacterizesfunctionsanddistributionsbytheirsmooth-
nessintermsofderivativesordifferences,whereastheotherwaytomeasuresmooth-
nesspresentedhereusesFourieranalyticaldevices.
Toclarifythesenotions,westartbyrecallingaclassofwell-knownsmoothness
spaces,theclassofSobolevspacesinL2(R).Wewillintroducetwotypesofspaces:
thehomogeneousandtheinhomogeneousspaces.
Definition1.1.1.Letk∈N.ThehomogeneousSobolevspaceW˙2k(R)is
definedasthespaceofktempered2distributionsfonR,forwhichthe(distributional)
derivativeoforderk,Df,isinL(R).
Forf∈S(R),|f|W˙2k(R):=DkfL2(R)definesasemi-normonW˙2k(R).
Notethatthesesemi-normsarenotnormsingeneral:|f|W˙k2(R)=0forfapolyno-
mialoforderlessthank.Theybecomenormsfortempereddistributionsmodulo
polynomials.WewillalsoencounterthissituationintheBesovcaselateron,sothis
remarkshouldjustserveasafirstwarningtothereader.

Definition1.1.2.Letk∈N.TheinhomogeneousSobolevspaceW2k(R)is
definedasthespaceoftempereddistributionsonRhavingalltheir(weak)derivatives
uptoorderkinL2(R).
EquippedwiththenormfW2k(R):=i≤kDifL2(R),W2k(R)isaHilbertspace.
Intheintroductiontothischapter,weclaimedthatonecouldregardBesovspaces
ascertaingeneralizationsofclassicalfunctionspacessuchastheSobolevspaces.
AfirstattemptinthisdirectioncouldbetoaskforspacesW2α(R),whereαis
nonintegral.H.Triebel[33]describesthisas‘fillingthegaps’betweeenthespaces
L2(R)=:W20(R),W21(R),W22(R),....
Afirst‘filling’canbeobtainedviadifferences,inspiredbythedefinitionofH¨older
spaces:

1.1BesovSpacesonRandtheirCharacterizations

17

LetB2α(R),0<α<1,bethecollectionoff∈S(R),suchthat
|f(x)−f(y)|2dxdy1/2
fB2α(R):=fL2(R)+R×R|x−y|α|x−y|<∞.
Clearly,thiscanbeeasilyextendedtolargervaluesofα,usingdifferencesonthe
derivativesorworkingwithhigherorderdifferences.
OnecouldalsoconsidertoreplacetheL2(R)-normsbyLp(R)-normswherep=2.
ThiswillleadtothedefinitionofBesovspacesviaiterateddifferencesinLp(R),
decayinginLq(R)inacertainwaytowardssmalldifferences.Thiswillbethetopic
ofSubsection1.1.1.TheSlobodeckijspacesB2α(R)abovewillcoincidewiththe
spacesB2α(L2(R))intheseterms.
Forasecondwaytofillthegaps,letusgobacktotheSobolevspacesW2k(R),k∈N.
Usingthefactthatdifferentiationcorrespondstopointwisemultiplicationwiththe
argumentontheFourierside,itiseasytoseethattheyadmitacharacterizationin
Fourieranalyticalterms.Theirnormisequivalentto
fW2k(R)(1+|ω|2)k/2f(ω)L2(R).
Replacingtheintegerexponentbyamoregeneralα∈R+leadsto
fW2α(R):=(1+|ω|2)α/2f(ω)L2(R),
andinfact,wehaveW2α(R)=B2α(R):thetwofillingproceduresleadtothesame
spaces.AlsothehomogeneousSobolevspacesemi-normscanbedescribedbythe
decayoftheFouriertransform:
|f|W˙2k(R)|ω|kfˆ(ω)L2(R).
Again,inbothcases,onecouldthinkofgeneralizingthesespacestosomeLp(R),
yieldingthesocalledLiouvilleorBesselpotentialspaces,see[33].
AmoregeneralapproachusesLittlewood-Paleytheory.LittlewoodandPaleyob-
tainedacharacterizationofLp(T),1<p<∞intermsoftrigonometricseries.
Furthermore,theyshowedthatthep-normisequivpalenttoacertainnormonthe
Fouriercoefficients.The2π-periodicfunctionsinLaretherebyfullycharacterized
bythebehavioroftheirFourierexpansions.
So,thebasicideaistocharacterizefunctionspacesbycertainsetsoffunctionswhich
‘span’thesespacesandmoreover,giverisetoanequivalentdescriptionintermsof
theassociated‘expansioncoefficients’.
Thepreciseformulationrequiresanumberoftechnicalassumptions:

18

1.1BesovSpacesonRandtheirCharacterizations

Letϕ∈S(R),satisfying
suppϕ⊆{ω:π/4<|ω|<π},(1.1.1)
forsomeC,ε>0,|ϕ(ω)|>Con{ω:π/4+ε<|ω|<π−ε}.(1.1.2)
Further,letϕbesuchthatfor(ϕν)ν∈Z=(2−νϕ(2−ν∙))ν∈Z,wehave
|ϕν(ω)|2=1forω∈R\{0}.(1.1.3)
Z∈νAsthe(ϕˆν)formapartitionofunity(1.1.3),andas|ω|2k2−2νkonsuppϕν,this
yieldsfortheSobolevsemi-norm
R|f|2W˙2k(R)=DkfL22(R)|ω|2k|f(ω)|2dω
R=|ω|2k|f(ω)|2|ϕν(ω)|2dω
Z∈ν2−2νk|f(ω)|2|ϕν(ω)|2dω
RZ∈ν=(2−νkf∗ϕνL2(R))2.(1.1.4)
Z∈νTheunderlyingideaisthatanyL2(R)-functioncanbedecomposedas
f=f∗ϕν∗ϕν∗,(1.1.5)
Z∈νwhereϕν∗(x)=ϕν(−x).(1.1.5)iscalledtheCalder´onreproducingformula.
Themapf→(f∗ϕν)ν∈Zcanbeunderstoodasadecompositionoffintosignal
componentsf∗ϕνwhoseFouriertransformsarelocalizedindyadicallyspaced‘fre-
quencybands’.ThesmoothnessoffisthenrelatedtothedecayoftheL2-normsof
thedifferentcomponents.Ontheonehand,thisdefinitionreflectsthewell-known
characterizationofglobalsmoothnessviatheFouriertransform,buttheuseofa
smoothpartitionofunityinFourierdomain,resultinginarapidlydecayingwindow
ϕ,allowstomeasuresmoothnessofafunctionlocally.
OnecannowconsiderthisdecayinothernormsthanL2,whichwillleadtoanother
equivalentdefinitionofBesovandrelatedspaces.Wedescribethismoredetailed
inSubsection1.1.2.Ofcourse,themeaningof(1.1.5)hasthentobehandledwith
greatcare.Forgeneralf∈S(R),theexpansionwillconverge(intheweak-∗-sense)
only‘modulopolynomials’.ThisissueisalsoatopicofSubsection1.1.2.

.1.4)(1

1.1BesovSpacesonRandtheirCharacterizations

91

TheSpaces.nextBydisSubsectcretizingion,1.1.3,Calder´odealsn’swithformulathe,Fϕrazier-transforandmJacwharaerth[1cterizatio5],[16]nofobtaBesoinedv
thatforanyfinaBesovspace,
f=ν,k∈Zf,ϕν,kϕν,k,(1.1.6)
wherefork∈Z,ϕν,k(x)=2−ν/2ϕ(2−νx−k).Moreover,themembershipof
finaBesovspaceisfullycharacterizedbythemembershipofthecoefficients
(f,ϕν,k)ν,k∈Zinasequencespace.
Thisexpansiodecompnintoositioannorforthonormaldistributiowavnseletintbahesis,disctrehoughte-timenon-Boesorthovgospacenalisandwsimilaithrto‘basisan
elements’thatarecompactlysupportedinFourierdomain.InSubsection1.1.4,we
shortcertalyinaddexplainitionalhowcondittoionsderiv-earethatuncLo2(Rnditiona)-ortlhonobasermalsforwatheveletBesobavsesspaces-assatisfyingwell.
traAlthonsfoughrmcwhaefocusracterizaonttionshederivLittlewedoofromd-Pita,lewyedesstartcriptwithiontheandBesothevϕ-spaceandwdefinitioaveletn
basedoniterateddifferences,whichisfrequentlyusedinliterature.

1.1.1ModuliofSmoothness
Inthissubsection,wedefineBesovspacesasspacesoffunctionswithacommon
orderofsmoothness,whichismeasuredviaiterateddifferences.Thesespaceswill
bedenotedbyBqα(Lp(R))andB˙qα(Lp(R)),respectively.Asthissubsectionismainly
introductory,werestrictourdiscussiontospaceswhereα>0,1<p,q<∞in
ordertoavoidspecialcasetreatment.Foramoredetaileddiscussion,werefere.g.
to[25]or[34].
Leth∈R.ForafunctionfdefinedonR,the(forward)differenceoperatorofstep
hisgivenby
Δhf(x)=f(x+h)−f(x),
andforr∈N+,definethedifferenceoperatoroforderr,steph,inductivelyby
Δhrf(x)=Δh(Δhr−1f(x)).
Notethatther-thdifferenceoperatorinexplicitformisgivenby
rΔrhf(x)=kr(−1)r−kf(x+kh).
=0kDefinition1.1p.3.For1<p<∞,t∈R+,ther-thordermodulusofsmooth-
nessoffinL(R)isdefinedby
rrωp(f,t)=h∈Rs,|uph|<tΔhf(∙)p.

20

1.1BesovSpacesonRandtheirCharacterizations

.1.7)(1

Letf,gbedefinedonR.Then,foreacht∈R+,
ωrp(f+g,t)≤ωpr(f,t)+ωpr(g,t),(1.1.7)
andforfmultipliedbyascalarα,
ωpr(αf,t)≤|α|ωpr(f,t).(1.1.8)
Asωpr(f,t)vanishesforpolynormialsofdegree≤r−1,ωpr(∙,t)isasemi-normonthe
setoffunctionsforwhichωp(f,t)<∞.
ωpr(f,t)isincreasingforeachpandr,furthermore,forM∈N,
ωpr(f,M∙t)≤Mrωpr(f,t).(1.1.9)
Soifωpr(f,t)<∞forsomet>0,itisfiniteforallt∈R+.
Forf∈Lp(R),wehave
ωpr(f,t)≤2rfp.(1.1.10)
pvTherefersely,ore,functiofunctionsnswithinLfinite(R)mohavdulieoffinitesmomoodulithnessofarsemonotothness,necessabutrilynoteinLptha(Rt).con-
Forf∈Lp(R),wehaveωpr(f,t)→0monotonicallyast→0.Generallyspeaking,
thefasterthisconvergence,thesmootherf.
NotethattheLp-norminthedefinitionofωpr(f,t)allowsaratherwildbehaviorof
fto,leasrancelongoafsBestheovspacesexceptionaB˙lqα(setLp(hasR))wsmalledefinemeasure.below,Thiswithproprespertecyttoimpliesjumps.acertain
Definition1.1.4.Forα>0,1<p,q<∞,r=α˙α+1p,afunctionfdefinedonR
issaidtobeinthehomogeneousBesovspaceBq(L(R)),if
∞dt|f|B˙qα(lp(Z)):=(0(t−αωpr(f,t))qt)1/q<∞.(1.1.11)

proThep|er∙|tiesBqα(ofLp(Rthe))armoedulisemi-ofnosmormsointhnessgene;raltheybbecausecomeeofnothermspmoolynodulomialpolynocancellatiomialsonf
degree≤r−1.Furthermore,theB˙qα(Lp(R))-seminormsareallequivalentmodulo
polynomialsusingdifferentmoduliofsmoothnessr>αinthedefinition.
WedefinethecorrespondinginhomogeneousBesovspacesBqα(Lp(R))bythenorm
fBqα(Lp(R)):=fLp(R)+|f|B˙qα(Lp(R)).

1.1BesovSpacesonRandtheirCharacterizations

12

Usingthemonotonicityofωpr(f,∙)and(1.1.9),thesemi-normcanbediscretized,
yieldinganequivalentsemi-norm
|f|B˙qα(Lp(R))((2−jαωpr(f,2j))q)1/q.(1.1.12)
Z∈jTheabovespacesB˙qα(Lp(R))andBqα(Lp(R)),respectively,generalizetheconsider-
ationsfromtheintroductiontothissection:choosingp=q=2andα=k∈N,
thesespacescoincidewiththehomogeneousresp.inhomogeneousSobolevspacesof
orderk.Fornonintegralα,wehaveB2α(L2(R))=B2α(R),theSlobodeckijspaces
].[33

1.1.2Littlewood-PaleyTypeCharacterization
Ininhomthissogeneousubsection,smowegothnessiveaspaLitcestlewB˙oαp,qo(d-PR)aleyandtBypαp,qe(R),definitioα∈nRofandhomo0<p,geneousq≤a∞nd.
pαThesneouseaspacesnalogs,willrespcoectivincideely,withwhicthhewBeesovdefinespadcesinB˙thqe(Llast(Rs))andubsectiontheirviainhoiteramoge-ted
s.differenceTheprinciplebehindthistypeofcharacterizationofBesovspacesistodecompose
distributionsintoseriesofsmoothcomponents:
Let(ϕν)ν∈Zbeafamilyofrapidlydecreasingfunctions,satisfying(1.1.1)-(1.1.3).
Recallfromabovethatforf∈L2(R)(see(1.1.5)):
f=f∗ϕν∗ϕν∗,
Z∈νwithconvergenceinL2(R)(Calder´onformula)and
fL22(R)=f∗ϕνL22(R).(1.1.13)
Z∈νThereby,L2(R)-functionsarecharacterizedbythesizeofthe‘smoothparts’f∗ϕν.
inWethewanbttoeginninggeneraoflizethisthiscthaypptereofconresultcerningtootheSobroslepavces(spacesa)first.Asexaamplefirstwasstepgivenin
thisdirection,wewillinvestigatetheconvergenceofCalder´on’sformulainspaces
differentfromL2(R).
Forf∈S(R),Calder´on’sformuladoesingeneralnotconvergeinthedistributional
sense.Thesumoverν≥1maydiverge,seetheexamplesin[23]and[14].
However,itcanbeshownthattheseries
Di(f∗ϕν∗ϕν∗)(1.1.14)
Z∈ν

22

1.1BesovSpacesonRandtheirCharacterizations

convergesforsomei∈N.
Thisisinfactequivalenttotheexistenceofasequenceofpolynomials{Pk}k≥1of
degreelessthani,suchthat
kg:=k→lim∞(f∗ϕν∗ϕν∗+Pk),
−∞=νinS(R).
Furthermore,thelimitabovediffersfromfbyapolynomial,assupp(f−g)=0.
Altogether,wehaveforf∈S(R)
f=f∗ϕν∗ϕν∗,(1.1.15)
Z∈νwithconvergenceinS/P(R),theequivalenceclassoftempereddistributionsmodulo
mials.olynopWenowdefinesmoothnessspacesbygeneralizing(1.1.13).
Definition1.1.5.Forα∈R,0<p,q≤∞,thespaceB˙αp,q(R)isthecollectionof
allf∈S/P(R),suchthat
fB˙αp,q(R):=((2−ναf∗ϕνLp(R))q)1/q<∞.
Z∈νThisdefinitionisindependentofthechoiceofthefamily(ϕν)ν∈Z.
TheB˙αp,q(R)spacesareBanachspacesfor1≤p,q<∞andquasi-Banachspaces
e.wisrheot

(1The.1.12),underlyingbutinfact,conceptB˙αp,qof(R)ameasuringndBqα(Lsmop(otR))hneagrsseaes(inmo(1.1.5dulo)pisoquitlynomiaediffels)renattfroleastm
forEspαecially>0,w1e<sapw,qin<(∞1.1.4,)thetharatngeB˙2k,2of(R)para=meW˙2tke(rRs)forforkwhic∈hN.weOndefineaccoundthetoflatthis,ter.
wewillalsocalltheB˙αp,q(R)spaceshomogeneousBesovspaces,withanowextended
rangeofparameters.
OnealsocandefineinhomogeneousspacesBαp,q(R)byreplacingthelowfrequency
partsintheCalder´onformulabyasinglefunction:
LetΦ∈S(R),wheresuppΦ⊆]−π,π[andforsomeC,ε>0,|Φ(ω)|>Con
{ω:π+ε<|ω|<π+ε}.
Let(ϕν)ν∈Zagainsatisfy(1.1.1)and(1.1.2),suchthat
|Φ(ω)|2+|ϕν(ω)|2=1.
1≤−ν

1.1BesovSpacesonRandtheirCharacterizations

32

Wehavenowforf∈S(R)
f=f∗Φ∗Φ∗+f∗ϕν∗ϕν∗.(1.1.16)
1≤−νDefinition1.1.6.Forα∈R,0<p,q≤∞,thespaceBαp,q(R)isthecollectionof
allf∈S(R),suchthat
fBαp,q(R):=f∗ΦLp(R)+((2−ναf∗ϕνLp(R))q)1/q<∞.
1≤−ν

Again,thesespaceagreewiththeBesovspacesfromSubsection1.1.1forthepa-
rametersα,p,q,forwhichthelatterisdefined.Thereby,wewillcalltheBαp,q(R)
spacesinhomogeneousBesovspacesaswell.
Thetermshomogeneousandinhomogeneousoriginatefromthebehaviorofbothof
thespacesconcerningdilation,asforthehomogeneousBesovspacesB˙αp,q(R),l∈Z,
f(2l∙)B˙αp,q(R)=C2l(α−1/p)fB˙αp,q(R),
whereasfortheinhomogeneousspaces,thisequalityisgenerallynottrue[32].

1.1.3ϕ-transformCharacterization
In[15],FrazierandJawerthintroducedtheso-calledϕ-transform,whichcanbe
viewedasaacriticallysampledversionof(1.1.5).Thefunctionspaceswhich
aredescribedbyLittlewood-Paleyexpressionscanalsobecharacterizedbythe
ϕ-transform.Moreprecisely,theϕ-transformcoefficientscarryallthenecessary
informationtoconcludethemembershipofadistributioninaBesovspace.More-
over,theconditiononthecoefficientsisjustasizecondition,whichmaysimplify
applicationssuchasthestudyoflinearoperatorsontheB˙αp,q(R)(andtheBαp,q(R))
andrelatedspaces.Amoredetaileddescripitionoftheseresultscanbefoundin
[15],[16]or[17].
Consideragainafunctionϕ∈S(R),satisfying(1.1.1),(1.1.2)and(1.1.3).
Forν,k∈Zlet
ϕν,k(x)=2−ν/2ϕ(2−νx−k).
Startingfromtheformula(1.1.5)andusingtechniquessimilartoShannonsampling,
FrazierandJawerthderivedthatforanyf∈S(R)
f=f,ϕν,kϕν,k,(1.1.17)
ν∈Zk∈Z

24

1.1BesovSpacesonRandtheirCharacterizations

withconvergenceinS/P(R),whichisadiscretizationof1.1.15.
Furthermore,theBesovspacenormscanbeequivalentlyexpressedintermsofthe
ts.efficiencoDefinition1.1.7.Forα∈R,0<p,q≤∞,letthecoefficientspaces˙bαp,q(R)bethe
collectionofallcomplex-valuedsequencess=(sν,k)ν,k∈Z,satisfying
sb˙α(R):=(((2−ν(α+1/2−1/p)|sν,k|)p)q/p)1/q<∞.(1.1.18)
p,qν∈Zk∈Z

sνLet,k=thef,ϕϕν,k-transfor,andmfoSrϕaforfcomplex-v∈S(R)aluedbedefinesequencedbty=S(ϕtfν,k=)ν,ks∈Z=(sdefineν,k)ν,kthe∈Z,invwhereerse
ϕ-transformbyTϕbyTϕt=ν,ktν,kϕν,k.
In[15],FrazierandJawerthprovethefollowingresult:
Theorem1.1.8.Letα∈R,0<p,q≤∞.
BothoftheoperatorsSϕ:B˙αp,q(R)→b˙αp,q(R)andTϕ:b˙αp,q(R)→B˙αp,q(R)arebounded
withfB˙αp,q(R)Sϕfb˙pα,q(R)andTϕ◦Sϕ=idB˙αp,q(R).
Inotherwords,underthesemaps,B˙αp,q(R)isaretractofb˙αp,q(R),andB˙αp,q(R)can
Observethatthewell-definednessandunconditionalconvergenceofTϕt=ν,ktν,kϕν,k,
beidentifiedwiththeclosedsubspaceSϕ(B˙αp,q(R))ofb˙αp,q(R).
whichwasnotclearinitially,followsfromthetheorem.
Thereisananalogousresultconcerningtheinhomogeneousspaces,startingfrom
theidentity(1.1.16).
LetagainΦ∈S(R),withsuppΦ⊆]−π,π[andforsomeC,ε>0,|Φ(ω)|>Con
{ω:π+ε<|ω|<π+ε}andlet(ϕν)ν∈Zsatisfy(1.1.1)and(1.1.2),suchthat
|Φ(ω)|2+|ϕν(ω)|2=1.
1≤−νWriteagainϕν,k(x)=2−ν/2ϕ(2−νx−k)forν,k∈ZandΦk(x)=Φ(x−k).
Theϕ-transformidentityforf∈S(R)nowreadsas
f=f,ΦkΦk+f,ϕν,kϕν,k,(1.1.19)
k∈Zν≤−1k∈Z
convergingintheweak-∗sense.
theThehosizeofmogeneousthecocasefficiee.nAntsareflecpprotspriatethesfomormulatothnessionofoftthehisdistrsizeibutionconditiofnisjustgivlikeneinin
thenextdefinition.

1.1BesovSpacesonRandtheirCharacterizations

52

Definition1.1.9.Forα∈R,0<p,q≤∞,letthespacesbαp,q(R)bethecollection
ofallcomplex-valuedsequencess=(sν,k)ν,k∈Z,satisfying
sbαp,q(R):=(((2−ν(α+1/2−1/p)|sν,k|)p)q/p)1/q<∞.(1.1.20)
ν≤0k∈Z
Letnowtheϕ-transformforf∈S(R)bedefinedbythemappingonthecoefficients
(sν,k)ν≤0,k∈Z,wheresν,k=f,ϕν,kifν≤−1ands0,k=f,Φk.Definingtheinverse
ϕ-transformintheobviousway,wehave,analogouslytothehomogeneousspaces,
thattheBαp,q(R)spacesareretractsofbαp,q(R)andthecoefficientssatisfy
fBαp,q(R)(sν,k)bαp,q(R).(1.1.21)
TheseresultscanbeusedfortheanalysisoflinearoperatorsactingontheBesov
spaces.Letf∈B˙αp,q(R).ByTheorem1.1.8,f=ν≥1,k∈Zsν,kϕν,k,where(sν,k)=(f,ϕν,k).
ApplyingalinearoperatorTleadsto
Tf=sν,kTϕν,k
,kν=sν,k(Tϕν,k,ϕµ,lϕµ,l)
µ,l,kν=(Tϕν,k,ϕµ,lsν,k)ϕµ,l.(1.1.22)
,kνµ,lDefinethematrixAbyA:=(aµ,l,ν,k),whereaµ,l,ν,k=Tϕν,k,ϕµ,l.
Thus,using(1.1.22)
Tf=(As)µ,lϕµ,l,
µ,lwhere(As)µ,l=ν,kaµ,l,ν,ksν,k.
ByTheorem1.1.8,ifAisboundedonthesequencespace,
TfB˙αp,q(R)≤CAsb˙αp,q(R)≤Csb˙αp,q(R)≤CfB˙αp,q(R),(1.1.23)
andthereby,thisimpliesboundednessofT.Thus,thestudyofboundedlinear
operatorsontheBesovspacereducestostudyingmatricesontheb˙αp,q(R)spaces.
Moreformally,foralinearoperatorTonB˙αp,q(R),define
Sϕ∗(T)=Sϕ◦T◦Tϕ

).1.22(1

26

1.1BesovSpacesonRandtheirCharacterizations

andsimilarly,forAaboundedoperatoronb˙αp,q(R)
Tϕ∗(A)=Tϕ◦A◦Sϕ.
Asseenin(1.1.23),T∗isbounded.TheboundednessofS∗andthefactthat
Tϕ∗◦Sϕ∗(T)=Tϕ◦Sϕ◦ϕT◦Tϕ◦Sϕ=Timplythatthespaceϕofboundedlinear
ααoptheeraretratorsctonpropB˙p,q(ertRy)isisalifteretrdtactoofthetheoperaspacetoroflevboel.undeThisdallolinearwsoptoeratorstudysonopb˙erp,qato(Rrs):
ontheBesovspacesbystudyingpropertiesofcertainmatrices[17].
One‘almostconditiodiagnonalitforymaproptriceserty’.tobeRoboughlyundedspooknen,thethiscoconefficienditiontspacdesceisribethesshoow-cafalledst
thesizeoftheentriesaµ,l,ν,kdecaysawayfromthediagonal,whereµ=ν,l=k.An
exactdefinitioncanbefoundagaine.g.in[17].Wewilldealwithmatricesofthis
ttheypeinconditmoionsreodentailthesewhenquencdeaeslingthereinwithcathenbeviewdiscrete-edtiasmealmospaces,stdiagseeonalLemmaconditio3.3.7ns.:
Anwillabethepplicatioissuenofofthestheeuprescoultsmingisthesubsewavctioeletn.characterizationofBesovspaces,which

1.1.4WaveletCharacterization
Thespacesϕby-transforthemdecawyeofdisctheussedassoinciatedthecolastcefficienhapterts.ItalloiswsalsoacphosaracsibletertoizadestioncribofeBestheseov
spacesbyothertransforms,especiallybyorthonormalwavelettransforms.
Inmanyrespects,awaveletorthonormalbasisisquitesimilartothesystemof
functionsfunction,wassohichciatedisdtoilatedthebyϕpow-transforersofm.twBoothandtrasystemsnslatedarisebybyinpictegerkingmaultiplessuitableof
thedilationvariable.ArbitraryfunctionsinL2canbeexpandedinbothsystems,
andexpansioadditnsionalremainpropvertalidiesforofottheherϕ-ffunctionunction(spaces.orthewavelet)guaranteethatthese
Amoredetailedexpositionoftheresultscitedbelowcanbefoundin[17,14].
Thediscrete-tectimehniqueswavusedeletstoinproveChapterthese3.resultsprovideablueprintforourtreatmentof
LetconstitutψbeesaanwavortelethonorfunctionmalbasissuchforthatL2(tRhe)fa(seemily(Chapterψj,l)j,l2∈fZor=aq(2−uicj/k2ψin(2tro−jxductio−ln))
tothisissue).Furthermore,letthiswaveletsystemfulfilltheadditionalconditions

i)ψpossesseszeromomentsofacertainorderN>0,
ii)ψisregularoforderN,e.g.ψisNtimescontinuouslydifferentiable,
iii)ψislocalizedintime,e.g.ψhascompactsupport.

1.1BesovSpacesonRandtheirCharacterizations

72

Wediscussthesepropertiesinmoredetailin2.1.2.
Incontrasttotheϕ-functions,whichareband-limitedSchwartz-functionswithall
oftheirmomentsvanishing,theψ-functionsarenowlocalizedintimeratherthan
frequencyandusuallypossessonlyvanishingmomentsandregularityofacertain
degree.finiteNevertheless,startingfromTheorem1.1.8,onecanprovethattheBesovspacescan
bedescribedalsobythewavelettransform,wherethenumberNabovehastobe
chosenaccordingly,dependingontheorderofsmoothnessαofthespace.Notethat
Ndescribesthefrequencylocalizationofψ(seeRemark2.1.9below),whichmakes
theconditionsonψinTheorem1.1.10intuitivelyplausible.
whererj,l=f,ψj,landtheinversetransformTψaccordinglybyTψv=j,lvj,lψj,l
LetthewavelettransformSψforf∈S/P(R)bedefinedbySψf=r=(rj,l)j,l∈Z,
foracomplex-valuedsequencev=(vj,l)j,l∈Z.
TheproofofanananalogofTheorem1.1.8isbasedonestimatesonhowthe
propertiesofψ,quantifiedbythenumberN,influencetheoff-diagonaldecayof
thematrixA=(ψj,l,ϕν,k)andtoconcludeboundednessofthismatrixonthe
coeffientspace.Usingtheretractpropertywementionedinthelastsection,the
followingtheoremcanbeestablished[17].

Theorem1.1.10.Letα∈R,0<p,q<∞.
ProvidedthatN>max{α,1/min{1,p}−1−α},bothoftheoperatorsSψ:B˙αp,q(R)→
b˙αp,q(R)andTψ:b˙αp,q(R)→B˙αp,q(R)areboundedwithfB˙pα,q(R)Sψfb˙αp,q(R).
Furthermore,Tψ◦Sψ=idB˙αp,q(R)aswellasSψ◦Tψ=idb˙αp,q(R).
Inretrathectoprrthoopertynormaoflwtheavϕelet-transforcase,m.thewaUndervelettheψsystem-transforinfactm,B˙αinherits(R)ismoreisomothanrphicthe
p,qtothewholespaceb˙αp,q(R).Inparticular,observethat
f=j,l∈Zf,ψj,lψj,l(1.1.24)
conconditiovergesnsintunconditiohetheonallyreminareB˙αp,q(thereforeR).Waveuncondilettisyonastemlsbasessatisfyingforawtheholseufficienscalet
ofBesovspacessimultaneously.
Ofresultscourse,canbetherefoundisain[17similar],[14]resultor,inforathslighetlyinhomodifferengeneoustcontBexesot,vin[2spaces.0].These

28

1.2NonlinearWaveletApproximationandBesovSpaces

1.2NonlinearWaveletApproximationandBesov
SpacesTheresultswepresentedinSection1.1.4arerelatedtothenonlinearapproxima-
tionbehaviorofwaveletexpansionsandtowaveletapplicationsinsignalandimage
processing,suchasdenoisingandcompression;letusmentionDonohoandJohn-
stone’sWaveletShrinkage(seee.g.[5])or[24]forcompression.
Asamotivatingexample,weconsidertheproblemofapproximatingafunction
f∈L2(R)byN>0termsofawaveletseries.Theresultswegiveinthissection
areborrowedfrom[7].
Letagain(ψj,l)j,l∈ZbeanorthonormalwaveletbasisforL2(R).Thetaskisto
approximatef∈L2(R)byNwaveletcoefficients,i.e.topickanapproximantfrom
ΣN={g∈L2(R):g=cj,lψj,l,Λ⊂N,card(Λ)≤N}.
(j,l)∈Λ
ThisisannonlinearproblemasthespacesΣNarenonlinear:ΣN=ΣN+ΣN⊂Σ2N.
Definetheapproximationerrorby
σN(f)=g∈infΣNf−gL2(R).
Heuristically,onecanconceivethatthe‘smoother’f,thefasterthiserrordecreases.
Toquantifythis,weask:forwhichf∈L2(R)wehaveforagivenα>0
σN(f)≤MN−α,
forsomeM>0,or,slightlystronger,f∈Apα(L2(R),ΣN),where
/p1∞Apα(L2(R),ΣN):={f∈L2(R),(NασN(f))p1<∞}.
N=1NHerethejustificationfordescribingthecondition
∞(NασN(f))p1<∞
N=1nas‘σN(f)decaysasN−α’isprovidedbytheobservationthatNασN(f)hasto
convergetozeroinordertoguaranteefinitenessofthesum.Infact,σNhasto
decayslightlyfaster.Thepparameterplaystheroleofafine-tuningparameter,
similartotheroleofqinthedefinitionofBesovspaces.
Therecipetominimizef−gL2(R)subjecttog∈ΣNistobuildthewaveletseries
upbypickingtheNcoefficientsofthelargestabsolutevalues.Solet(cn)bethe
permutationofthesequence(f,ψj,l)j,l∈Z,suchthat|c1|≥|c2|≥|c3|≥...≥0.

1.2NonlinearWaveletApproximationandBesovSpaces

92

Then,wehaveσN(f)=(j∞=N+1cj2)1/2.Estimatingthesecoefficientsandusingthat
pembedscontinuouslyinto2for0<p<2,onecanshowthatf∈Apα(L2(R),ΣN)
ifandonlyif(f,ψj,l)j,l∈Z∈p(Z×Z),where0<p<2,α=1/p−1/2.
Asp(Z×Z)=b˙αp,p(Z)inthiscase,wecanconcludethat,assumingcertainsupport,
zeromomentandsmoothnesspropertiesofthewaveletsystem,
f∈Aαp(L2(R),ΣN)⇐⇒f∈B˙αp,p(R).
Summingup,wehavethatforf∈L2(R),adecayoftheapproximationerrorlike
O(N−α)correspondstothemembershipoffinaBesovspace.

1.2.1NonlinearApproximationofDiscrete-TimeSignals
Inapplications,dataareusuallygivendiscretely.Usually,thesedataarefedstraightly
intoMallat’scascadealgorithm(seeChapter2),usingthediscretefiltersarisingfrom
acontinuous-timemultiresolutionanalysis:
coThisefficienway,ts,thewhichdiscretecanseberieinsisterpretedmappedastoexapansionfamily(codj,l)jefficien≥1,l∈tsZooffthediscretesignalwavwiteleht
respecttoadiscrete-timewaveletbasis.
Consideringthe2exampleofN-termapproximationinthisdiscretecase,say,of
fdiscus=(fsio(nn))ab∈ove(isZ),validonefocanradneyriveHilbaertssimilaparceresultandnoastoinnlytheforconthetinL2uous(R)case:casewthee
considered.So,roughlysaid,wehavethatthe2-approximationerrordecreaseslikeO(N−α)for
(dj,l)∈p(N×Z),0<p<2,α=1/p−1/2.
Thissuggestsusingthespaceoftruncatedcoefficientfamiliesbα(Z),α∈R,0<
p,q<∞,asthecollectionofcomplex-valuedsequencess=(sj,l)p,qj≥1,l∈Z,forwhich
sbαp,q(Z):=(((2−j(α+1/2−1/p)|sj,l|)p)q/p)1/q<∞.(1.2.1)
j≥1l∈Z
Thenthetaskarisestocharacterize(dj,l)j≥1,l∈Z∈b˙αp,q(Z)inasimilarwayasforthe
thecontinfilteruousbankcaseratfrohemrthpropanonertiestheoftheunderlysingequenceconftinalouone,us-tandimetowavstudyelets.conditionson
Aswealreadydiscussedintheintroduction,fromthecontinuous-timepointofview,
thedecayof(dj,l)asin(1.2.1)dependsonthemembershipofF=n∈Zf(n)τnφin
aBesovspaceBαp,q(R),whichinvolvesthescalingfunctionφassociatedtothefilter
.bankAtthispoint,thereseemstobeagapinthetheoryexistingsofar:exceptfor
proheuristicsvidingabaseddirectondesthecriptcontioninforuous-timediscreteanadatalogue,withwecoecouldfficiennottsinfindbαan(Zy)inliterattermsure
p,qofmembershipinaspaceofsequences.

30

1.3AimsofThisThesis

Thisobservationisrelevantbecauseanyargumentcitingresultsforthecontinuous
timesetting,butapplyingthemtothecoefficientcomputedindiscretetime,implic-
itlyusesthecoefficientnorm(1.2.1)onthewaveletcoefficients.Existingtheorydoes
notevenprovidecriteriaforconsistencyofthesenormswithrespecttochangeof
basis.eletvaw

1.3AimsofThisThesis

Wearenowreadytoformulatethemainobjectivesofthisthesis:

1.tWeimewansettttoingcinawharacterizeaythatwisavaseletcocompleteefficiandentdesatisfactocayinryaaspurtheelcyhardiscreacteriza-te-
tionforcontinuous-timesignals.ThisleadsustothestudyofBesovspaces
ontheintegersBαp,q(Z),whosedefinitionwasgivenbyTorresin[31].
2Especially,westudynecessaryandsufficientconditionsonwaveletbasesfor
l(Z)toconstituteunconditionalbasesforthesespaces.
2.ThespacesBαp,q(Z)aredefinedviaLittlewood-Paley-theory.Anotheraimfor
usistogivefurthercharacterizationsofthesespacessuchasintermsof
iterateddifferencesandmoduliofsmoothness,analogtothecontinuous-time
settingasinSubsection1.1.1.Here,thewaveletcharacterizationwillproveto
beparticularlyuseful.
3.Moreover,wewanttoextendourresultstootαherscalesofdiscrete-timefunc-
tionspaces,suchasTriebel-LizorkinspacesFp,q(Z).

Befo2rewestartwiththisprogram,theupcomingchapterdealswithwaveletanalysis
Z).(for

2hapterC

WaveletAnalysisof
sgnalSiainDiscrete-Dom

Thischapterisconcernedwithwaveletbasesindiscretetime.Theresultsbeloware
consideredtobeknown,thoughthediscrete-timepointofviewistreatedlessoften
thanthecontinuous-timecase.
Anyway,beforewearereadytotreatthediscrete-timetheory,wewilldealwiththe
continuous-timecaseinthefirstsectionofthischapter.Wegiveashortcompendium
ofwaveletbasesforL2(R)associatedtomultiresolutionanalysis,restrictingourselves
tosomeofthemainnotionsandresults.
Therearetworeasonsfordoingthis:Thefirstisthatcontinuous-timewavelets
(atleastthosethatarisefromamultiresolutionanalysis)giverisetodiscrete-time
filterbanksandassociatedwaveletsystemsin2(Z).Moreover,desirableproperties
bofestdiscbryete-cotimemparisowanveletotst,hevconanishingtinuoumosmentimetssaettnding,smoowherthnessetheycanarbeenaturalunderstoaondd
well-understood.
WediscusstherelationsbetweenwaveletfunctionsandfilterbanksinSubsection
2.1.1andshortlydescribecertainusefulpropertiesofwaveletsystemssuchasvan-
ishingmomentsandregularity(22.1.2).Subsection2.1.3givesashortglimpseon
biorthogonalwaveletbasesforL(R).
Thisfirstsectionwillbekeptshortasitmainlyservestofixnotationforthecorre-
spondingdiscrete-timenotionsandresultscomingupinthefollowingsection.Thus,
forproofsandfurtherresults,wedefertotheusualliteratureonwavelettheorysuch
asthebooksbyDaubechies[12],Meyer[20],Wojtaszczyk[34]orMallat[19].
Inthenextsection,2.2,weswitchfromcontinuoustodiscretetime:wegivean
overviewofwaveletsystemsin2(Z)andtheirproperties.Themainsourcesforthis
arethebookbyCohen[3]andthearticlesbyRioul[27,26,28].
Ofthevariousdesirablepropertiesofwavelets,thenotionofregularityisprobably

31

32

2.1WaveletBasesforL2(R)

theleastintuitiveindiscretetime:smoothnessconditionsrelyingonsmall-scale
limitscannotbeimposedonsequences.ThenotionofregularityasgivenbyRioul
inthetwolatterpaperswementionedaboveratherdealswithlargescalelimits.As
thisregularityconditionwillbecrucialfortheproofsinsection3.3,wediscussthis
propertyinanextrasubsection(2.2.2).
Itisalsotheregularitypropertywhichconnectsthediscretetothecontinuousthe-
ory:discretetimewaveletfamiliespossessingsomeorderofregularityconvergein
acertainsensetocontinuous-timewaveletsofthesameregularity.Thiswillbethe
issueofSubsection2.2.3.

2.1WaveletBasesforL2(R)
2mIntultirhisesosectiolutionn,wanaelysiswill.reWcaellwaantfewtopfactsointoonutwaavgaeinletthabasestwefowrillL(Rrestrict)assoourciateddiscus-to
siontoelementaryresultsandrefertothesourceswementionedintheintroduction
tothischapter.
So,whatisawavelet?
Definition2.1.1.Awaveletsystem2forL2(R)isafamilyoffunctions(ψj,l)j,l∈Z
obtainedfromasinglefunctionψ∈L(R)by
ψj,l(x)=2−j/2ψ(2−jx−l).(2.1.1)
AwaveletbasisisawaveletsystemthatisanorthonormalbasisforL2(R).
theDowamostveletelemenbasestaryexist?wavTheeletsysanswteerm,istheyes-Haaasrawavmotiveletatingsystem.example,wewilltreat
TheHaarwaveletisthefunction
1if0≤x<1/2;
.therwiseo0H(x)=−1if1/2≤x<1;(2.1.2)
Weillustratetheconceptofmultiresolutionanalysis,whichispresentedlateronin
thissection,byshowingthatthefamily(Hj,l)j,l∈Zofdilatedand2translatedversions
(see(2.1.1))oftheHaarwaveletisanorthonormalbasisforL(R).
ThefunctionsHj,laresupportedonthedyadicintervalsDj,l=[l2j,(l+1)2j[.With
thisobservation,itisobviousthattheHaarsystemisorthonormal:
Either,Considerwtewohaovfejthe=fjunctandionslHj=,l,lH.j,lIn.thiscase,thesupportsofthefunctions

2.1WaveletBasesforL2(R)

33

aredisjointandtherebyHj,l,Hj,l=0.Or,wehave,assumingwithoutlossof
generalityj>j,thatthesupportofHj,lisatleasttwiceaslongasthesupportof
Hj,l.Ifthesupportsarenotdisjoint,thesupportofHj,liscontainedinaconstant
intervalofHj,l.Again,Hj,l,Hj,l=0.
Wewanttoshowthat(Hj,l)isanorthonormalbasis,sowhatislefttoproveis
totality,i.e.foranyf∈L2(R),f=j∈Zl∈Zf,Hj,lHj,linthesenseofL2(R):
WeintroducethespacesVjasthespacesoffunctionsinL2(R)whichareconstant
onthedyadicintervalsDj,l,l∈Z.Duetothedyadicstructure,thespacesare
nested,i.e.Vj+1⊂Vj,andwehavef(x)∈Vjifandonlyiff(2jx)∈V0.
Thereby,asthetranslatesofthefunctionφ=χ[0,1[,(φl)l∈Z2=(φ(∙−l))l∈Zobviously
formanorthonormalbasisforthespaceV0,consistingofL(R)-functionsconstant
on[l,l+1[,thefamily(φj,l)l∈Z=(2−j/2φ(2−j∙−l))l∈Zisanorthonormalbasisfor
Vj.Thewholefamily(φj,l)j,l∈ZhoweverisnotanorthonormalbasisforL2(R).
However,wecanderiveanorthonormalbasis,whichwillbetheHaarbasis,from
tions.iderascontheseForf∈L2(R),considertheorthogonalprojectionoffonVj,
Pjf=f,φj,lφj,l=fDj,lχDj,l,(2.1.3)
l∈Zl∈Z
wherefDj,l=2−jx∈Dj,lf(x)dx.Asthisprojectionisthebestapproximationof
ffromVj,andasanyfunctioninL2(R)canbeapproximatedarbitrarilywellby
functionswhicharepiecewiseconstantondyadicintervals,wehave
limj→−∞f−PjfL2(R)=0aswellaslimj→−∞Vj=L2(R).
Lookingatthelimitintheoppositedirection,f∈j∈ZVjimpliesthatfisconstant
onthepositiveaswellasonthenegativerealline,whichforanL2(R)-function
impliesf≡0andalso,limj→∞PjfL2(R)=0.
Usingtheaboveobservations,onecanwriteanyL2(R)asf=j∈ZPjf−Pj+1fand
wedenotePjf−Pj+1fbyQjf.AsPjandPj+1areorthogonalprojectionsonthe
spacesVjandVj+1,respectively,QjistheorthogonalprojectiononWj:=VjVj+1.
ThespacesWj,j∈Z,aremutuallyorthogonalandwecanwrite
L2(R)=Wj.(2.1.4)
Z∈jItmayappearthatwehavelosttrackoftheinitialproblem,buttheHaarwavelets
arecomingintoplayagainrightnow.Itiseasytoseethatthefamily(Hl)l∈Z=
(H(∙−l))l∈ZisanorthonormalbasisforW0andbydilation,(Hj,l)l∈Zisanortho-
normalbasisforWj.
Therefore,wecanwritetheprojectionQjontheWjspacesas
Qjf=f,Hj,lHj,l,(2.1.5)
Z∈l

342.1WaveletBasesforL2(R)
andtogetherwith(2.1.4),thisyields
f=f,Hj,lHj,l,(2.1.6)
Z∈,ljsotheHaarsystemisanorthonormalbasisforL2(R).
Thespaces(Vj)j∈Zweconsideredinthisdiscussionareafirstexampleofamultires-
olutionanalysis,whichwewillnowformallyintroduce.
Definition2.1.2.A2multiresolutionanalysis(MRA)isasequenceofclosed
subspaces(Vj)j∈ZofL(R),suchthat
Vj+1⊂Vj,(2.1.7)
∞jlim→∞Vj=Vj={0},(2.1.8)
−∞=j∞jlim→−∞Vj=Vj=L2(R),(2.1.9)
−∞=jf(x)∈Vjifandonlyiff(2jx)∈V0,(2.1.10)
thereisφ(x)∈V0,suchthat(φ(x−l))l∈ZisanorthonormalbasisforV0.(2.1.11)
Thefunctionφ(x)iscalledthescalingfunctionfor(Vj).
By(2.1.10)and(2.1.11),thesystem
(φj,l)l∈Z=(2−j/2φ(2−j∙−l)))l∈Z
isanorthonormalbasisforVj.
DefinethespacesWjby
Vj−1=Vj⊕Wj.
AnL2functionψiscalledwaveletfunctionassociatedto(Vj),if(ψ(∙−l))l∈Zis
anorthonormalbasisforW0.
case,histIn(ψj,l)l∈Z=(2−j/2ψ(2−j∙−l)))l∈Z
isInaanodditionrtho,wnormaehalvebasisforWj.
L2(R)=Wj
Z∈j

2.1WaveletBasesforL2(R)35
andthesystem(ψj,l)j,l∈ZconstitutesanorthonormalbasisforL2(R).
EveryMRAhasanassociatedwavelet,seethenextsubsection.
ThereareatleasttwowaystogetanMRA,yieldingwaveletbasesasaconsequence.
Asafirstway,onecouldstartbydefiningthespaces(Vj)andtrytofindascaling
functionφ,suchthatthetranslatesformanorthonormalbasisforV0.Herethe
standardexampleisprovidedbythesplinespaces(e.g.[19],Ex.7.3),
Instead,onemayalsonotthat(2.1.10)and(2.1.11)implythateveryMRAis
uniquelydeterminedbyitsscalingfunctionφ,sinceV0istheclosedspanoftranslates
ofφ,andVjisobtainedfromV0bydilation.
MostoftheMRAsusednowadaysareconstructedthisway,i.e.,bypickingasuitable
φ,ataskwhichishoweverhighlynontrivial.
Forf∈L2(R),letPjf,QjfbetheorthogonalprojectionsonVjandWj,respec-
tively.Inparticular,wecanwrite,analogouslyto(2.1.3)and(2.1.5),
Pjf=f,φj,lφj,l,(2.1.12)
Z∈lQjf=f,ψj,lψj,l.(2.1.13)
Z∈lUsingtheaboveprojections,wecanrewritecondition(2.1.8)aslimj→∞PjfL2(R)=
0.Condition(2.1.9)canbereplacedbylimj→−∞f−PjfL2(R)=0.
TheprojectionsonthespacesVjcanbeinterpretedasapproximationsoffat
differentresolutions,whereasthepartialwaveletseriesQjfcanbeviewedasthe
differencebetweentwoapproximationlevels.
Notethatournotationisdifferentfromjmostsourcesinthewaveletliterature:The
scalejcorrespondstodetailsofsize2.Thus,thespacesdecreaseasjincreases.
Weexpansmadeionsthislaterchaon:ngetforoavdiscoidretedeafunclingtwitions,hnegaresoluttiveionindiceisosbviowheusnlyconsideringlimited,soindiscreteour
terms,therewillbenoscalesmallerthan1inthiscase.
Wefinishthissubsectionwithtwoexamples.
Example2.1.3.:HaarMRA
LAs2(Rw)ealrwhiceadyhareprocovensdtinanttheonbtheeginningdyadicofinthistervalsparaDgraj,l,ph,l∈theZ,arespacesamoffunctioultiresolutnsionin
analysiswithscalingfunctionφ=χ[0,1[.Thecorrespondingwaveletbasisisthe
Haarwaveletbasis.
Example2.1.4.:ShannonMRA
Astheaspaceskindooffanofunctionsppositewhicexhtremearetotheband-limited,HaarMRi.eA,withonetheircanFodefineurierantMRAransforams
supportedontheintervals[−2−jπ,2−jπ].Theassociatedscalingfunctionisthesinc
functionφ(x)=πsinπxxandwehaveψ(x)=2sinc(2x)−sinc(x).

36

2.1WaveletBasesforL2(R)

TheMRAsgivenaboveinheritdifferentproperties:
First,theHaarwaveletiscompactlysupportedandtherebywell-localizedintime,
whereastheShannonwavelet,beingcompactlysupportedinfrequency,haspoor
decayintimedomain.
Second,theHaarwaveletisdiscontinuous,whereastheShannonwaveletisC∞.
formaThird,lly,theallHaamormenwatsveofletthehasShannopreciselynwaoveneletvvanisanish.hingmoment,whereas,atleast
TherearecertainlyexamplesofMRAsbetweentheseextremesconcerninglocaliza-
tionintimeandfrequencydomain.TheDaubechiesfamily,startingwiththeHaar
MRA,canbeseenasafamilyofMRAsthatrepresentdifferentcompromisesbe-
tofweenregulatherittywoorderextremes:roughlyForNe/a5,chwithN,thesuppDortaubinecanhiesinterconvsatrlofuctionlengthyields2Na−w1,avaeletnd
Nhavvetoanishingbe‘paymomenedfotr’s.inThusterms,ofdesirablesuppoprortpsize.ertiesThis(smodisothnescussions,visanishingskippedunmomentilwts)e
considermoreindetailusefulpropertiesofwaveletsystemsinSubsection2.1.2.

2.1.1WaveletsandFilters
Inthelastsubsectionwedescribedhowtoderivewaveletbasesfrommultiresolution
analyses.WesawthatthescalingfunctionfullydeterminesanMRA.
ThereisyetanotherwaytocharacterizeMRAs,byuseofdiscretetimefilters:the
multiresolutionconditionsyieldadiscretizationwediscussinthissubsection.We
willseethatthisfactalsogivesrisetoafastalgorithmtocomputethewavelet
rm.onsftraThetranslatesofascalingfunctionassociatedtoamultiresolutionanalysis(Vj)
formanorthonormalbasisofV0(2.1.11),andas2−1/2φ(x/2)∈V1⊂V0by(2.1.10),
(2.1.7),weobtainthescalingequation
2−1/2φ(x/2)=g(n)φ(x−n)(2.1.14)
Z∈nwithscalingcoefficientsg(n)=2−1/2φ(x/2),φ(x−n).
Thescalingequationgivesrisetoanumberofinterestingequationsfulfilledbythe
scalingfunctionanditsFouriertransform:
InFourierdomain,theorthonormalityconditionbecomes
|φˆ(ω+2πl)|2=1foralmostallω∈R.(2.1.15)
Z∈lUsingtheFouriertransformof(2.1.14),wehave

.1.15(2)

2.1WaveletBasesforL2(R)

37

2=2|φˆ(2ω+2πl)|2=|φˆ(ω+πl)|2|gˆ(ω+πl)|2
l∈Zl∈Z
=|φˆ(ω+2πl)|2|gˆ(ω)|2+|φˆ(ω+π+2πl)|2|gˆ(ω+π)|2,
l∈Zl∈Z
yieldsllyfinahwhic|gˆ(ω)|2+|gˆ(ω+π)|2=2(a.e.).(2.1.16)
Furthermore,sayforgˆcontinuous,wehave
√gˆ(0)=2andtherebygˆ(π)=0,(2.1.17)
andthosenearω=0arekeptbyapplyingthisfunction.
sowecanregardgasalowpassfilter,asfrequenciesaroundω=πareattenuated
Conditions(2.1.16)and(2.1.17)arenecessaryconditionsonthefilter(g(n)),such
thatthefunctionφin(2.1.14)isascalingfunctionofanMRA.Inasense,they
containarecipeforconstructingφfrom(g(n)):
Iterating(2.1.14)yields

nφˆ(ω)=gˆ(2−kω)φˆ(2−nω).(2.1.18)
=1kHencewemaystartbytakinga2π-periodicgˆ,satisfying(2.1.16)and(2.1.17),and
considering∞φˆ(ω)=gˆ(2−kω).(2.1.19)
=1kThisconstructionhasbeensuccessfullyemployedforconstructingscalingfunctions.
Notehoweverthatwithoutadditionalassumptionsong,theaboveproductwillnot
necessarilyconvergetoascalingfunctionassociatedtoanMRA(see[3]).
Wewillreturntothisdiscussionwhenwedealwithdiscrete-timeMRAsinSection
2.2;therewewillimposeadditionalconditionsonthefilter(referredtoas‘discrete-
timeregularity’)whichwillensureconvergenceofthescheme(2.1.19)toa(then
alsoregular)scalingfunction,seeespeciallySection2.2.3.
Also,thewavelet−1/2functionsarerelatedtodiscretetimefilters.Forawaveletfunction
ψ,necessarily2ψ(x/2)∈W1⊂V0,yieldingthewaveletequation
2−1/2ψ(x/2)=h(n)φ(x−n)(2.1.20)
Z∈nwithcoefficientsh(n)=2−1/2ψ(x/2),φ(x−n).

38

2.1WaveletBasesforL2(R)

Similarcomputationstotheabovegivethatthesecoefficientssatisfy
|hˆ(ω)|2+|hˆ(ω+π)|2=2(a.e.),(2.1.21)
and√hˆ(0)=0andhˆ(π)=2,(2.1.22)
(inthecasewherethispointwisestatementmakessense)suchthathcanbeviewed
asadiscrete-timehighpassfilter.
TheconditionthatthespacesW0andV0areorthogonaltoeachothercanbeex-
pressedintermsofthefiltersby

hˆ(ω)gˆ(ω)+hˆ(ω+π)gˆ(ω+π)=0.(2.1.23)
Itcanbeproven([3,18,20])thatifφisascalingfunctionassociatedtoamultires-
olutionanalysis(Vj)withcorrespondingg,then,thespecialchoice
h(n)=(−1)1−ng(1−n)(2.1.24)
or,equivalently,
hˆ(ω)=gˆ(ω+π)e−iω,(2.1.25)
in(2.1.20)givesawavelet,whosetranslatesareanorthonormalbasisofW0.
Afilterpair(g,h)satisfying(2.1.16),(2.1.21)and(2.1.23)willbecalledperfect
reconstruction(PR)filterpair.Thisnamecomesfromthefactthattherela-
tionbetweenwaveletsandfiltersadmitsafastalgorithmforthewavelettransform.
Beforewediscussthisalgorithminmoredetail,werevisitourexamples:
Example2.1.5.:Haarfilters
Forthemultiresolutionanalysis−1/2withscalingfunctionφ=χ[0,1[,thecorresponding
loByw(pa2.1.2ss4),filterh(0)reads=−gg(n(1))==2−2−1/2fornand=h0(1,1)=andg(00)o=2therwis−1/2.e.
Example2.1.6.:Shannonfilters
ConsideringtheShannonmultiresolutionapproximationwithscalingfunctionφˆ=
χ[−π,π],thefiltersaregivenbygˆ=21/2χ[−π/2,π/2]andhˆ=21/2−gˆ.
Sobases.far,MHereRAsthehasvcaelingservaedndwamainlyveleatsaequattoolionforatppheeaconredvasenienbytproductsconstructiooftnheofincwavlusioelent
propertiesofanMRA.Wehavealreadyseen,however,thatthescalingequation
mayalsoserveasthestartingpointfortheconstructionofanMRA,via(2.1.19).
Similarly,thescalingandthewaveletequationwillalsoserveasthesourceofthe
ctohiefacomputlgoreithmiccoarseconscaletributiowavn,eletthecofaseffictwienatvselofetasigtransfonalrbym.repeThisatedalgoarithmpplicatioallonwosf
discreteconvolutionandsubsamplingsteps.

2.1WaveletBasesforL2(R)

39

Letaj=(aj(l))l∈Z=(f,φj,l)l∈Z.Wewillcallthesequenceajtheapproximation
coefficientsatscalej∈Z.
Letdj=(dj(l))l∈Z=(f,ψj,l)l∈Zbethewaveletcoefficients,whichwillalsobe
calledthedetailcoefficients,atscalej.

Proposition2.1.7.[Mallat]
Thecoefficientscanberecursivelycomputedbythefiltering
aj+1(l)=(aj∗g∗)(2l)=aj(k)g∗(2l−k),(2.1.26)
Z∈kdj+1(l)=(aj∗h∗)(2l)=aj(k)h∗(2l−k).(2.1.27)
Z∈kThereconstructionofafilteringstepisdoneby
aj(l)=aj+1(k)g(l−2k)+dj+1(k)h(l−2k)=(↑2aj+1)∗g+(↑2dj+1)∗h,
k∈Zk∈Z(2.1.28)
thesumofcoefficientsonacoarserscale,upsampledandconvolvedwiththefilters
.h,g

ProofRememberthatφj+1,l∈Vj+1⊂Vj.Expandingφj+1,lintheorthonormal
basis(φj,k)k∈ZofVjyields
φj+1,l=φj+1,l,φj,kφj,k.(2.1.29)
Z∈kComputingtheinnerproductsφj+1,l,φj,kgives
φj+1,l=2−1/2φ(∙/2),φ(∙−k+2l)φj,k
Z∈k=g(k−2l)φj,k.
Z∈kEmployingtheinnerproductonbothsidesof(2.1.29)therebyyields
aj+1,l=aj,kg(k−2l)=(aj∗g∗)(2l),
Z∈kwhichis(2.1.26).
(2.1.27)followsfromanalogouscomputations.

40

2.1WaveletBasesforL2(R)

Forthereconstructionstep,reconsiderthefactthatVj=Vj+1⊕Wj+1.Thereby,
φj,l∈Vjcanbeexpandedintheunionof(φj,k)k∈Zand(ψj,k)k∈Z,whichisan
orthonormalbasisforVj:
φj,l=φj,l,φj+1,kφj+1,k+φj,l,ψj+1,kψj+1,k.(2.1.30)
k∈Zk∈Z
Theaboveconsiderationsgive
φj,l=g(l−2k)φj+1,k+h(l−2k)ψj+1,k.(2.1.31)
k∈Zk∈Z
Takingtheinnerproductonbothsides,
aj,l=aj,kg(l−2k)+dj+1h(l−2k).
k∈Zk∈Z
Iterating(2.1.26),(2.1.27),thefastwavelettransformcomputesthemap(a0,l)l∈Z→
(dj,l)j>0,l∈Zviathecascade
a0→a1→a2→...→aj−1→aj
(2.1.32)
d1d2dj−1dj
whereeachhorizontalarrowrepresentsthesamefilteringandsubsamplingstep
aj+1=↓2(aj∗g∗),andsimilarly,dj+1=↓2(aj∗h∗).
2.1.2PropertiesofWaveletBases
Intheprecedingsubsection,wedealtinageneralwaywithwaveletbasesandtheir
connectiontomultiresolutionanalysisanddiscrete-timefilters.
Theabove-citedresultsconcerningcharacterizationsofBesovspacesviawavelet
coefficients(1.1.4)relyonadditionalproperties,thatistosayvanishingmoments,
smoothnessandcompactsupport,whichwenowattendto.
Thefirstpropertywedealwithisthenotionofvanishingmoments.
Definition2.1.8.ψhasvanishingmomentsoforderN∈Nif
tkψ(t)dt=0for0≤k≤N−1.

aThisfunctiopropnertfywhicensureshiskthat<NψistimesorthogcononatinltououslypolynomiadifferenlstiaofbleoarorderundN−x01..Then,Considerf

2.1WaveletBasesforL2(R)

41

canbeexpandedintoaTaylorpolynomialoveraninterval.Asψcancelsoutthis
polynomial,wehaveforthoseψj,l,whosesupportsarecontainedintheneighborhood
aroundx0,thattheabsolutevaluesofcoefficientsf,ψj,ldecayforsmallscales.
Hence,localsmoothnessleadstoadecayofcoefficients,piecewisesmoothfunctions
canbewellapproximatedbyafewnumberofwaveletcoefficients.
Iffhasasingularityinx1,allthewaveletsψj,lwhichhavex1insidetheirsupport
willfeelthissingularityandmayhavealargecoefficient.Hence,anotherdesirable
propertyistodealwithwaveletswithgooddecayproperties,optimallywithones
ofcompactsupport.
Afurtherdesirablepropertyanalyzingsmoothfunctionsistousewaveletswhich
alsopossesssomeregularityofacertainorder.Wesaythatawaveletisregularof
orderr>0ifitisH¨olderregularoforderr.
9.1.2.Remark1.Theabovepropertiesarenotindependent,forexample,awaveletwithacer-
taindecayandregularitywillhaveacertainorderofzeromoments.
2.Thereisanalternativeinterpretationofsmoothnessandvanishingmoment
propertiesforawaveletψconcerningthelocalizationinFourierdomain:
VanishingmomentsoforderNdescribesthedecayofψ(ω)asω→0:|ψ(ω)|=
O(|ω|N),whereassmoothnessoforderrgives|ψ(ω)|=O(|ω|−r)asω→
∞.TheseobservationsallowtoreadtheconditionsofTheorem1.1.10(and
alsoofTheorem3.3.8)as‘waveletsprovideareasonableapproximationof
ns’.unctio-fϕ3.Notethatthepropertieswediscussedcanusuallybebuiltintothewavelets
bydesigningsuitablefilters.Daubechies[6]constructedwaveletswhichhave
vanishingmomentsofarbitraryorderandareatthesametimeofminimal,
t.orsuppctcompaByconstruction,theDaubechieswaveletswithN∈Nvanishingmoments
haveasupportlengthof2N−1andareforlargeNapproximatelyregularof
order0.2N.Thiswillbeofimportanceintheupcomingsection(2.2),when
wedealwithdiscrete-timewavelets:atleastcompactsupportandvanishing
momentscarryoverimmediatelytothissetting.Theconnectionbetween
filterswithcertainpropertiesleadingtoregularwaveletsgetsmoreclearwhen
wedealwithdiscrete-timemultiresolutionanalysis.

Wefinishthissubsectionbylookingagainatourexamples:
Example2.1.10.:PropertiesofHaarWaveletsTheHaarwavelethascom-
pactsupport,possessesonevanishingmomentandisinfacttheDaubechieswavelet
oforder1.Itisnotcontinuousandtherebynon-regular.

42

2.1WaveletBasesforL2(R)

iscoExamplmpactlye2.1.1supp1.or:tedPrinopFerourtieiesrodofmainShannonandthWusavhaelsetaspoTheordecaShannoyinntwimeav.eletIn
contrasttotheHaarwavelet,itisC∞andatleastformally,allofitsmoments
anish.v

2.1.3BiorthogonalBases
ThecharacterizationofBesovspacescanbeextendedtobiorthogonalwaveletsys-
tems,whichiswhyweshortlydiscussthesesystemshere.Forproofsoftheresults
citedbelow,aswellasfurtherdetails,wereferthereaderto[12,19].
AbiorthogonalpairofwaveletbasesisapairofRieszbases(ψj,l)j,land(ψ˜j,l)j,l
arisingintheusualmannerfromfunctionsψ,ψ˜,andfulfillingthebiorthogonality
ontidincoψj,l,ψ˜j,l=δl,lδj,j.
Thisrelation,togetherwiththeRieszbasepropertiesofthesystems,immediately
expansionstheilstaenf=f,ψj,lψ˜j,l=f,ψ˜j,lψj,l.
,lj,ljClearly,thisconceptgeneralizeswaveletorthonormalbases.Itturnsoutthata
convenientmethodfortheconstructionofsuchbasesisprovidedbyintroducing
biorthogonalitytomultiresolutionanalysis:InsteadofasingleMRA,oneconstructs
apair(Vj)j,(V˜j)jofsequencesofspaces,whichhaveallpropertiesofMRAsexcept
for(2.1.11),whichisreplacedbyfunctionsφ,φ˜satisfyingtherequirements
(φ(∙−l))l∈ZisaRieszbasisforV0
(φ˜(∙−l))l∈ZisaRieszbasisforV˜0
φ(∙−l),φ˜(∙−l)=δl,l.
DefiningWjandW˜jasorthogonalcomplements,justasintheorthonormalwavelet
case,onecanprovetheexistenceofwaveletsψ,ψ˜satisfying
(ψ(∙−l))l∈ZisaRieszbasisforW0
(ψ˜(∙−l))l∈ZisaRieszbasisforW˜0
ψ(∙−l),ψ˜(∙−l)=δl,l.
whichentailsthat(ψ˜j,l)j,l,(ψj,l)j,larebiorthogonalwaveletbases.
ThedrawbackofbiorthogonalityisthattheParsevalrelation
f2=|f,ψj,l|2
,lj

2.1WaveletBasesforL2(R)

43

holdingfororthonormalbasesneedstobereplacedbythenormequivalences
f2|f,ψj,l|2|f,ψ˜j,l|2.
,lj,ljFTheorcinshieftance,advaonnetagecanofchobiooserthogonasymmetriclityiswahigveheletrs(whicflexibilithisyinimptohescsiblehoiceinofthewoavrthoelets:g-
onalsetting),oronecandistributedesirablepropertiesbetweenψandψ˜:ψcanbe
chosenwithadesirednumberofvanishingmoments(butlittleregularity),andψ˜
withadesireddegreeofsmoothness.
Thefastwavelettransformeasilyadaptstothebiorthogonalsetting;theonlychange
being˜thatonenowusesonefilterpairg,hforthedecomposition,andadifferent
pairg˜,hforreconstruction.
Coefficientsaj=(aj(l))l∈Z=(f,φ˜j,l)l∈Zonacertainscalej∈Zareusedto
computeapproximationcoefficientsaj+1anddetailcoefficientsdj+1=(dj+1(l))l∈Z=
(f,ψ˜j+1,l˜)l∈Zonacoarser˜˜scalebyconvolutionwithdiscrete-timelowandhighpass
filtersg˜,hassociatedtoφ,ψ,followedbysubsampling:
aj+1(l)=(aj∗g˜)(2l)=aj(k)g˜(2l−k),(2.1.33)
Z∈kdj+1(l)=(aj∗h˜)(2l)=aj(k)h˜(2l−k).(2.1.34)
Z∈kThereconstructionofafilteringstepisdoneby
aj(l)=aj+1(k)g(l−2k)+dj+1(k)h(l−2k),(2.1.35)
k∈Zk∈Z
thesumofcoefficientsonacoarserscale,upsampledandconvolvedwithfiltersg,h
dualtog˜,h˜.

NotethatBesovspaceshaveacharacterizationintermsofbiorthogonalwaveletsas
well:Theorem1.1.10canbeformulatedusingbiorthogonalbases,wherevanishing
momentsandregularityconditionsareimposedseparatelyfortheanalyzingand
synthesizingwavelets,seee.g.[14].
Inthediscrete-timecasewetreatinthisthesis,wewillalsogivethisgeneralized
result,seeTheorem3.3.8,whichobviouslyincludestheorthonormalcase.

44

2.2Discrete-TimeWaveletBasesfor2(Z)

2.2Discrete-TimeWaveletBasesfor2(Z)
Inthissection,wewillfinallydescribewaveletsystemsin2(Z).Thesesystemsare
associatedtothefilterbankalgorithmswhichwedescribedin2.1.1.
Recallthatthefastwavelettransformcomputesthemap(a0,l)l∈Z→(dj,l)j>0,l∈Zvia
thecascade(2.1.32).
Thealgorithmimplementsaunitaryoperator:(a0,l)l∈Zand(dj,l)j>0,l∈Zaretheex-
pansioncoefficientsof
F=a0,lφ(∙−l)∈V0
Z∈lintheONBs(φ0,l)l∈Zand(ψj,l)j>0,l∈ZofV0.
Discretetimewaveletbasesareobtainedbyviewinga0∈2(Z)asinputtoaunitary
transformWd:2(Z)→2(N×Z).Theoutputcanthenbeinterpretedasexpansion
coefficientsofa0inthesystemhj,l=Wd−1(δj,l),thepreimageoftheKroneckerONB
of2(N×Z)underWd.Hence(hj,l)j,lisanONBof2(Z).Weintendtostudybases
ofthiskind,withtheaimofdescribingsignalswithgoodapproximationbehavior.
Thisperspectivemayseemunorthodox,butitisinfactcloselyrelatedtotheway
thatwaveletsareusedonreal-worlddata:Wehaverepeatedlyremarkedthatthese
dataareusuallygivendiscretelyandthatthestandardprocedurefeedsthediscrete
data(f(k))kdirectlyintothewaveletfilterbank.Asaconsequence,thefilterbank
outputconsistsofwaveletcoefficientsdj,l=F,ψj,l,whereF=l∈Zf(l)φ(∙−l).
Theproblemwiththisprocedureisthatitusesthescalingfunctionφ,whichis
notknownexplicitly.Accordingly,thedevelopmentofmodelassumptionsonF,
whichcouldserveasasourceofheuristicsforsignalprocessingalgorithms,becomes
aratherdifficulttasks.
Asamatteroffact,quiteoftenthesesignalmodelsareavailableforfinsteadofF,
say,fisobtainedfromameasuringdevicewithcertainnoisecharacteristics,and
certainexpectedsmoothnessbehaviourinthemeasuredquantity.Havingafully
discretetimetheoryavailableshouldthusallowtodescribeandanalyzewavelet-
basedprocessingalgorithmsinamoretransparentanddirectwaythanviathe
embeddingintoL2(R),whichisobscuredbythescalingfunction.
Inthefollowing,wewillthereforediscardanyreferencetothecontinuous-time
setting,anddescribewaveletsystemsindiscretetime,whicharisefromapairg,hof
perfectreconstructionfilters,andtheassociatedcascade(orfastwavelettransform)
algorithm2(Z)→2(N×Z),asobjectsofindependentinterest.Ourexposition
ofdiscrete-timewaveletsusesideasandresultsfromA.Cohen’sbook[3]andO.
Rioul’spapers[27,28,26].

2.2Discrete-TimeWaveletBasesfor2(Z)

54

2.2.1TheDiscrete-TimeWaveletTransform
Inthissubsection,wewillbeconcernedwiththeconstructionofoperatorsanalogous
22intotthehebaccascadekground.algoritMhmore(prZe)→cisely,(wNe×willZ),cobutnsiderwithoutpairsag,conhtinofuousfilters,timeandMRAthe
followingobjectsconstructedfromg,h.
1.One-stepdecompositionoperatorsa0→(a1,d1)=(↓2(a0∗g∗),↓2(a0∗h∗))
on2(Z).
2.Fulldecompositionoperatorsa0→(dj)j≥1obtainedbycascadingtheone-step
opdecomperatorositiounitan.ry.Ofparticularinterestwillbeconditionsong,hmakingthis
3.Providedthatthefastwavelettransforma0→(dj)j≥1isunitary,itsoutput
canbeundersto2odasexpansioncoefficientsoftheinputsignalwithrespect
tolooankingONBforofexplicit(Z).desThiscriptwillionsbeofthethisbasis.discrete-timewaveletONB,andweare

OneStepDecomposition
Clearly,anecessaryconditionforthefastwavelettransformtobeunitaryisthatthe
one-stepdecompositionisunitary.Thefollowingtheoremgivesaprecisecondition
this.forTheorem2.2.1.Giventwosequencesg,h∈2(Z),considertheoperator
S:2(Z)→2(Z)×2(Z)
f→(↓2(f∗g∗),↓2(f∗h∗))
Thenthefollowingareequivalent:

unitary.isS(i)(ii)Thesystem(g(∙−2l))l∈Z∪(h(∙−2l))l∈ZisanONBof2(Z).
(iii)TheFouriertransformsofgandhfulfilltheperfectreconstruction(PR)con-
ditions(2.1.16),(2.1.21),(2.1.23),i.e.,foralmosteveryω∈R
|gˆ(ω)|2+|gˆ(ω+π)|2=2,(2.2.1)
|hˆ(ω)|2+|hˆ(ω+π)|2=2,(2.2.2)
hˆ(ω)gˆ(ω)+hˆ(ω+π)gˆ(ω+π)=0.(2.2.3)

46

2.2Discrete-TimeWaveletBasesfor2(Z)

Iftheequivalentconditionsarefulfilled,theinverseoperatorisgivenby
S∗:(a,d)→(↑2a)∗g+(↑2d)∗h.

ProofTheequivalenceof(i)and(ii)isobvious.Fortheimplication(ii)⇒(iii)
assumethat(g(∙−2l))l∈Z∪(h(∙−2l))l∈ZisanONB.Theninparticular
π2δ0,l=g(∙−2l),g(∙)
=gˆ(ω)e−iω2lgˆ(ω)dω
0π2=|gˆ(ω)|2e−iω2ldω
0π=e−iω2l|gˆ(ω)|2+|gˆ(ω+π)|2dω.
0Hencetheintegrablefunction
ω→|g(ω)|2+|g(ω+π)|2
on[0,π]hasthesameFouriercoefficientsastheconstantfunctionω→2,andthe
Fourieruniquenesstheoremimplies
|g(ω)|2+|g(ω+π)|2=2(a.e.).
Similarcalculationsprove
|h(ω)|2+|h(ω+π)|2=2(a.e.)
h(ω)g(ω)+h(ω+π)g(ω+π)=0(a.e.),
whicharethe(PR)conditions.
Theconverseisshownsimilarly.
Notehoweverthatfor(ii)⇒(iii)weonlyneededthat(g(∙−2l))l∈Z∪(h(∙−2l))l∈Zis
orthonormal.Somewhatremarkably,orthonormalityofthesystemalreadyimplies
s.mpletenesocitsIn(2.1.1),wealreadyencounteredthe(PR)conditions:intheconstructionof
multiresolutionanalysesonL2(R),theperfectreconstructionpropertyisawell-
knowncondition.Inordertoproperlyappreciateit,recalltheconvolutiontheorem:
(f∗g)∧=f∙g.Lateron,√wewillchoosefiltersh,g,√suchthath(0)=0.Then
exactlyconditions(2.1.22)and(2.1.17).
(PR)entailsthath(π)=2,andconsequentlyg(0)=2andg(π)=0,whichare

2.2Discrete-TimeWaveletBasesfor2(Z)

74

Thus,usingfiltersfroma(continuous-time)MRA,weimmediatelyobtainadiscrete-
ONS.timetioRecallnsforthatthethefilters(PR)tobeconditiorelatensdtoinathenconMRAtinbytheuous-timescalingcaseandwerewavneceletessaryequation.condi-
Wejustprovedthatinthediscrete-timecase,theseconditionsarealsosufficientto
seeyieldthaant(atONB,leastatfolearsfitinnitethefilters)‘one-thisstep’iscaalsoseabtheovce.aseInforthethenext‘fullparadecgraompph,wositioewilln’,
leadingtodiscrete-timeorthonormalwaveletbases.

OrthonormalWaveletBasesfor2(Z)
Thediscretetimewavelettransformisnowobtainedbyiteratingtheonestepde-
ion.itsocompInthefollowingtheorem,thefinitesupportconditioncanbereplacedbytheweaker
conditionthatgisinfinitelydifferentiable[3].Inanycase,theconditionsforthe
existenceofanorthonormalwaveletbasisfor2(Z)a2remuchlessrestrictivethan
fortheexistenceofanassociatedwaveletsysteminL(R);see[3]2forexamplesof
discrete-timewaveletsystemsthatdonotarisefromanMRAinL(R).
Theorem2.2.2.Wavelet-ONBin2(Z)
Letg,h∈2(Z)begivenwith(PR).Assumeinadditionthatgisfinitelysupported.
Givenf∈2(Z),defineinductively
a0=f,aj+1=↓2(aj∗g∗),dj+1=↓2(aj∗h∗).
(a)Thediscretewavelettransform
Wd:f→(dj(l))j≥1,l∈Z
isaunitaryoperator2(Z)→2(N×Z).
(b)Thesequencesdjandajjconsistofexpansioncoefficien2ts:dj(l)=f,hj(∙−2jl)
andaj(l)=f,gj(∙−2l),withsuitablehj,gj∈(Z)(forj≥1).
(c)hjandgjcanbecomputedrecursivelyvia
δ=g00gj+1=gj∗(↑2jg)
hj+1=gj∗(↑2jh)
Byconstruction,theoperatorWdcomputesthecoefficientsoffwithrespecttothe
discrete-timewaveletbasis(hj,l)=(hj(∙−2jl))j∈N,l∈Z.ThissystemisanONB
of2(Z).

48

2.2Discrete-TimeWaveletBasesfor2(Z)

ProofWefirstprove(b)and(c)byinduction:
Notingthata0=f∗δ0,wefindintheinductionstep
aj+1(k)=↓2(aj∗g∗)(l)
=aj(n)g∗(2l−n)
Z∈nI=Hf,gj(∙−2jn)g∗(2l−n)
Z∈n=f,g∗(2l−n)gj(∙−2jn).
Z∈nNowwecancompute
g∗(2l−n)gj(m−2jn)=gj(m−2j(n+2l))g(n)
n∈Zn∈Z
=gj+1(m−2j+1l),
wheregj+1isgivenas
gj+1(m)=gj(m−2jn)g(n)
Z∈n=gj(m−n)(↑2jg∗)(n)
Z∈n=(gj∗(↑2jg))(m).
Replacinggbyhinthecalculationsyieldtheformulaforhj,andwehaveshown
(b)and(c).
NowTheorem2.2.1impliesthatthemapping
f→(aj,dj,dj−1,dj−2,...,d1)
isunitary,andhencethefamily
(gj,l)l∈Z∪(hi,l)1≤i≤j,l∈Z
isanONBof2(Z),wheregj,l=gj(∙−2jl).Sincethisholdsforallj≥1,weobtain
inparticularthat(hi,l)1≤i≤j,l∈ZisanONSin2(Z).Hencetheonlymissingproperty
istotality.
Forthispurposedefineanalogouslyto(2.1.12)
Pjf:=f,gj,lgj,l,
Z∈l

2.2Discrete-TimeWaveletBasesfor2(Z)

94

whicharetheprojectionsontotheorthogonalcomplement
((hi,l)1≤i≤j,l∈Z)⊥.
HenceweneedtoprovePjf→0,forallf∈2(Z).Sincethespaceoffinitely
supportedsequencesisdensein2(Z),itisenoughtoprovePjf→0forfinite
sequences.Forthispurposeweneedtwoauxiliarystatements
•gj1→0,asj→∞.(Wereferto[3],pages31-32.)
•|supp(gj)|≤|supp(g)|∙2j.Thisiseasilyprovedinductively,usingthat
|supp(↑2jg)|=2j|supp(g)|−2j.
sult,eraAs

Pj(δk)22=|δk,gj(∙−2jm)|2
Z∈m=|gj(k−2jm)|2
Z∈m≤(|supp(g)|+1)gj∞
≤(|supp(g)|+1)gj1→0,
asj→∞.Thisconcludestheproofoftotality,hence(a)isshown.
Forsequencesimplicit(hj)yj,≥1we⊂2will(Z)soamewtavimeseletomitsystem.theThetranslaassotionciatepadrabasismeteriss,athenndobtacallinedthe
byshiftinghjbyintegermultiplesof2j,justasinthetheorem.
3.2.2.Remark1.pairTheg,hfinitenesswithofproptheertiesfilter(sParRe)onlyyieldsausednOforNStointalit2y(Zof).thesystem.Henceany
Anyway,wewillinthefollowingalwaysassumethefilterstobefinite.
2.NotethatthefilterbankpropertiesoftheDWT,i.e.,gaslow-passandhas
high-passfilter,enternowhereintheproof(infact,wecouldaswellexchange
thetwo).Theseareadditionalpropertieswhichwehavetobuildintothe
filters.23.ofTheclosedfamilys(Pubspaj)jc≥es1ofwhicprohjsharectioensmandefinesypropaertiesdecreasingofansMRAequenceinVLj2(=R).PjI((ndeed,Z))
Vj⊃Vj+1isclearbyconstruction.limj→∞Vj={0}hasbeenobservedinthe
prothereofoisftnohemeaprevioningfulustheodefinitrem.ionWofecannodilatiotnoexpnect2(Zan).anaWelogdoofhow(2ev.1.1er0),havesincean

50

2.2Discrete-TimeWaveletBasesfor2(Z)

analogofascalingfunction,intheformofthefamily(gj)j>0:Theprojection
ontoVjisgivenby
Pjf=f,gj(∙−2jk)gj(∙−2jk).
Z∈k4.Thetheoremimpliesthatthemapf→djfactorsintoaconvolutionwithhj∗,
followedbyasubsamplingof2j.Inparticular,wecaninterpretthemapping
f→(d1,d2,d3,...)
asa(subsampled)filterbank.Bytheconvolutiontheorem,(f∗hj∗)∧=fˆ∙hˆj,
whichshowsthatthecoefficientsdjcapturethepartoffsupportedinthe
frequencieswhere|hˆj|islarge.
ObservethatontheFouriertransformsidetherecursionformulaeread
gj+1(ω)=g(2jω)∙gj(ω),hj+1(ω)=h(2jω)∙gj(ω).(2.2.4)
Wenotethesimilaritytotheformula(2.1.19),whichfurtheremphasizesthe
analogyoftherolesofthescalingfunctionφontheonehand,andofthegj
(j∈N)ontheother.In(2.2.3)wewillseethat,undersuitableregularity
conditions,thisanalogyinfacttakestheformofaconvergencestatement:
2j/2gj(n)−φ(2−jn)→0.
Thisproperty(insomewhatsharperformulation)willbeofcrucialimportance
forthestudyofdecayofdiscrete-timewaveletcoefficients.
5.Similarobservationsapplytothediscretetimewavelets.Observethatthe
elementsofthewaveletbasisareagainindexedbyascaleandatranslation
parameter.Insteadofthedilationoperator,whichdoesnotworkproperly
on2(Z),wenowhaverecursivelydefinedwavelets(hj)jofdifferentscales.
Thewaveletbasisinheritstheasymptoticbehaviordescribedin4.,i.e.,under
suitableconditionsonthefilters,wemaythinkofthediscretetimewavelets
hjasapproximatesamplesofacontinuoustimewaveletψ.
6.Thetheoremholdsforbiorthogonalwaveletbasesaswell:aslongasthefilters
satisfythecorrespondingperfectreconstructionconditionsinthebiorthogonal
case,onewillobtaindiscrete-timebiorthogonalwaveletbasesalongthe
linesof2.2.2.Inthefollowingchapter,ourresultswillbeformulatedusing
tion.generalizathis

2.2Discrete-TimeWaveletBasesfor2(Z)

15

PropertiesofDiscrete-TimeWaveletBases
Aswedidinthecontinuouscase,wewilldescribeusefulpropertiesofdiscrete-time
waveletsystemssuchasfinitesupport,vanishingmomentsandregularityproperties.
Abyproductoftheproofof2.2.2isthat|supp(gj)|≤|supp(g)|∙2j,andlikewisefor
hj,ifthefilterhischosenaccordingto(2.1.24).So,startingwithfiltersoffinite
lengthgivesfinitelysupporteddiscrete-timewavelets.
Incontinuoustimetheory,vanishingmomentsarenecessaryrequirementstoensure
decayofwaveletcoefficientsofregularsignals,essentiallybykillingTaylorpolyno-
mials,see(2.1.2).Inthediscretesetting,wewillencounterasimilareffect.The
definitioncarriesoverinaratherstraightforwardway:

Definition2.2.4.Awaveletsystem(hj)j∈N⊂2(Z)hasNvanishingmoments
if∀j≥1,i=0,...,N−1:hj(n)ni=0,
N∈nwherethesumconvergesabsolutely.

Again,awaveletsystemhavingNvanishingmomentskillspolynomialsoforder
<N:IfPisanysuchpolynomial,j∈Nandk∈Z,then
(hj,l)(m)P(m)=0,∀(j,l)∈N×Z.
Z∈mWeobservethatifthehjarefinitesequences(whichisthestandardassumption),
theirFouriertransformsaretrigonometricpolynomials,andhavingNvanishing
momentsisequivalenttothepropertythattheoriginisazerooforderNofhj.The
followingpropositionshowsthatthispropertycanbeeasilycontrolledbychoosing
therightg,viathefactorization(2.2.4):

Proposition√2.2.5.Letg,hbeaperfectreconstructionpairoffinitesequences,
withg(0)=2,andhchosenaccordingto(2.1.24).Theng(π)=0,therefore
g(ω)=(eiω+1)Nm(ω)(2.2.5)
waveletsystem(hj)j∈NconstructedfromgandhhasNvanishingmoments.
with1≤N≤|supp(g)|,andmisatrigonometricpolynomial.Thisimpliesthatthe
√ProofNotethatg(0)=2and(PR)implythatg(π)=0.Then(2.2.5)isa
standardfactaboutpolynomials.Pluggingthisinto(2.1.25)yieldsthathhasa
zerooforderNat0.By(2.2.4),thiszeroisinheritedbyhj+1.
Thenotionofregularityisnotstraightforwardfordiscrete-timesequences.Fur-
thermore,wewillseethatinasense,thispropertylinksthediscretetothecontinuous-
timebases.Wewillthereforetreatthispropertyinanextrasubsection.

52

2.2Discrete-TimeWaveletBasesfor2(Z)

2.2.2RegularityofDiscrete-TimeWavelets
Acontinuous-timefunctionissaidtoberegularifitisatleastcontinuous,oreven
hasseveralcontinuousderivatives.
Thisnotionofregularitydoesnotseemtomakesenseindiscretetime.Certainly,
smoothnessconditionsthatrelyonsmall-scalelimitscannotbeadaptedtothe
discretetimesetting.Nonetheless,thereisauseful(inviewoflaterresults,even
crucial)notionofregularityofawaveletsystem,thathastodowithlargescale
limits.Thefollowingnotionsarebestunderstoodbythinkingofthesequence(gj)j≥1as
discreteapproximationsofacontinuous-timefunction,witheachgjbeingdefined
onthegrid2−jZ.
Regularityfordiscrete-timewaveletswasstudiedbyRioul[26],mimickingHo¨lder-
typeregularityconditionsfordiscrete-timefunctions.
Recallthatacontinuous-timefunctionϕ(x)isLipschitzregularoforderα(ϕ∈C˙α),
0<α≤1,ifforallx,h∈R
|ϕ(x+h)−ϕ(x)|≤C|h|α.
Afunctionϕ(x)issaidtobeH¨olderregularoforderr=N+α(ϕ∈C˙r),0<α≤1,
N∈N,ifitisNtimescontinuouslydifferentiableandtheN-thderivativeisLip-
schitzoforderα.

Letagaing=(g(n))0≤n≤L,L∈N,bealowpassfilteroffinitelengthandconsider
(gj)j≥1,obtainedbytheschemeinTheorem2.2.2:
gj=g∗∗(↑2g∗)∗(↑4g∗)∗∙∙∙∗(↑2j−1g∗).
Wewilldefineregularityofthesequencesgj=(gj(n))n∈ZmimickingLipschitzregu-
lar−jit/2y.Notethatthefollo2wingdefinitionsaretheonesgivenin[26],theextrafactors
2inherearisefrom-normalizationofgj.
Definition2.2.6.(gj)j≥1willbecalledregularoforderα,0<α≤1,ifit
satisfies|gj(n+1)−gj(n)|≤C2−j/2∙2−jα,
whereCisaconstantindependentofjandn.
Inordertoextendthisdefinitiontoregularityofhigherorders,considerthedifference
operatorDappliedtothesequences(gj(n)),
Dgj(n):=(gj(n)−gj(n−1))/2−j.
ForN∈N,letthesequenceofN-thorderdifferencesDNgjbethesequenceob-
tainedbyapplyingDNtimesto(gj(n)).

2.2Discrete-TimeWaveletBasesfor2(Z)

35

Thedifferenceoperatorcanbeseenasadiscretederivationoperator,witha−nojr-
malizationreflectingtheassumptionthatgjisanapproximationonthegrid2Z.
Definition2.2.7.(gj)j≥1willbecalledregularoforderr=N+α,0<α≤1,
N∈N,ifitsatisfies
|DNgj(n+1)−DNgj(n)|≤C2−j/2∙2−jα,
whereCisaconstantindependentofjandn.
Notethatregularityof(gj)j≥1impliesregularityofthefamilyofwaveletsequences
(hj)j≥1iftheyareassociatedto(gj)by(2.1.24).

2.2.3ConnectiontoWaveletBasesforL2(R)
In[26],definitions2.2.6and2.2.7areconceivedtorelatediscrete-timetocontinuous-
timewavelettransformsandtheirproperties.
Letinthefollowingg,hbeapairofPRfilters,satisfyingtherequirementsfor
.2.2.2TheoremRioul[26]definesconvergenceof(gj)-givenbyTheorem2.2.2-forj→∞toa
continuous-timelimitfunctionφ(x)andthenrelatespropertiesofφtoregularity
propertiesofthediscrete-timefunctionsgj.
Definition2.2.1Thesequences(gj)convergeforj→∞pointwisetoalimit
functionφ(x)if,foranysequenceofintegersnjsatisfying
|nj2−j−x|≤C2−j,(2.2.6)
forCaconstantnotdependingonj,wehave
φ(x)=j→lim∞2j/2gj(nj).
Moreover,theconvergenceisuniform,if
xsup|φ(x)−2j/2gj(nj)|→0,asj→∞.
Herewerequire(2.2.6)withaconstantindependentofx.
Theabovedefinitiongivesflexibilityinthewayinterpolationofthesequences(gj(n))
canbedone.Inparticular,convergenceusingstepwiseinterpolationbynj=2jx,
linearinterpolationoreveninterpolationbysmootherfunctionssuchassplinesare
allimpliedbythisdefinition.

54

2.2Discrete-TimeWaveletBasesfor2(Z)

Filterswith(PR)donotnecessarilygiverisetoascalingfunctionandthereby
awaveletbasisinL2(R),butadditionalrequirementsneedtobemettoensure
convergenceintheconstructionofacontinuous-timescalingfunctionφ.Moreover,
inthestudyofsmoothnesspropertiesofthisscalingfunctions,conditionswere
neededthatallowedtopredictthesmoothnessofφjustfromtheinitialdiscrete-
timefilterg.Thesetechniquesrelyonthespeedofconvergenceofthescheme(gj).
Rioulderivesnecessaryandsufficientconditionsforuniformconvergence(gj)toa
limitfunction,whichtheniscontinuous:
Theorem2.2.8.[26]Thecollectionofsequences(gj)convergesuniformly(inthe
senseof2.2.1)toalimitfunctionφ(x)ifandonlyif
g(n)=√2,(2.2.7)
n(−1)ng(n)=0and(2.2.8)
nnmax2j/2|gj(n+1)−gj(n)|→0asj→∞.(2.2.9)

(2).2.8.2.9)(2

Inaddition,onecancharacterizelimitfunctionspossessingstrongerregularityprop-
ertiesbythebehaviorofthe(gj).
Theorem2.2.9.[26]
•0If<gα≤satisfies1,then(gj)(2.2.7),(2.2.8)jwillcanonverd,geforjunifor≥1mly,gjtoisareα-rgularegularoforlimitderαfunctionforsomeϕ.
•IfthesequenceoftheN-thorderdifferencesDNgjconvergesuniformlyinthe
forsensek=of0,..(2.2.1),.,N,DthenkgjϕcisonverNgestimescuniforontinuoumlytoslythekdiffer-thorentiabderle.Fderivativeurtherofmorϕ.e,
in•(Ifgj)is(2.2.7)regisularvalidoforandderr=n(N−1)+αngfor(n)some=00for<αi≤=10,,...then,Ntheand,limitforjfunction≥1,
r˙ϕasso∈Cciate,drto=ϕNby+α(2.1.20)andpmoreossessesover,thethecsamereontinuous-timgularity.ewaveletfunctionψ

Thereisalsoaconverseresult:discrete-timefinitelysupportedwaveletfamilies
arisingfromfiltersassociatedtoanalogscalingfunctionsandwaveletsby(2.1.14)
and(2.1.20),whichareH¨olderregularofacertainorderN+α,N∈N0,0<α≤1,

2.2Discrete-TimeWaveletBasesfor2(Z)

55

pmeetsossessathenothersameconditioordernofwhichdiscrete-timeRioulcallsregulstabilityarity,as[26lo]:ngφaisssatheidtoscalibengstablefunctioifnφ
φ(n)einω=0forallω∈R.
N∈nAlloftheseobservationscoverinparticulartheDaubechiesfamilies.
TheDaubechiesorthonormalcontinuous-timescalingfunctionsandwaveletsoflength
L=4areregularoforderα≈0.55,and,asthestabilityconditioniseasilychecked,
soarethediscrete-timewaveletsarisingfromtheassociatedfilters.Daubechiesfil-
tersoflengthL=6givediscrete-timeregularityoforderr≈1.08,andwithfurther
increasingfilterlength,regularityincreasesaswell,forlargeL,theregularityis
about0.1L(see[26]).
Tconotinsumituous-timeup,thediscfunctions.rete-timeThenoregtionularitofyprregularopertityyiswillalsoconsistenbetwithcrucialreginularithetynextfor
er.haptc

56

2.2

Discrete-Time

Wavelet

Bases

for

2

(Z)

3hapterC

theirDiscrete-TChariactmeeriBesozativonsSpacesand

InChapter1,weintroducedBesovspacesofcontinuous-timefunctionsanddiscussed
differentcharacterizationsofthesespaces,asviaiterateddifferences,Littlewood-
Paleytheoryandintermsofthedecayofwaveletcoefficients.
Inthischapter,wedealwithanalogousfunctionspacesontheintegers.In[31],
R.H.Torresintroducesdiscrete-timeBesovspacesBαp,q(Z),α∈R,0<p,q<∞,by
adaptingaLittlewood-Paleytypecharacterizationforthecontinuous-time(homoge-
neous)BesovspacesB˙αp,q(R).Roughlysaid,Bαp,q(Z)arespacesofsequences,obtained
bBeysoinvstegerpaces.samplingThisowillfbetheband-limitedtopicofdistSectioributionsn3.1.incorrespondingcontinuous-time
Ourmainresultwillbethatthesespacesadmitacharacterizationintermsof
coefficientsfromdiscrete-timewaveletbasesasdescribedinthepreviouschapter:
AsequencewillbeinaαspaceBαp,q(Z)ifandonlyifthecorrespondingcoefficients
areinanormedspacebp,q(Z),describingthedecayofcoefficients.Notethatthese
bαp,q(Z)-normsarejustthetruncatednormsdefinedby(1.2.1)whicharoseinthe
contextofnonlinearapproximationinSection1.2.1.
Therefore,ourresultgivestheanswertothequestionweposedinthissection:a
sequencecanbeapproximatedwithacertainorderifandonlyifitisamemberin
adiscrete-timeBesovspace.
Thisresulthasbeenpreviouslypublishedin[11].
In[31],Torresalsodevelopsadiscrete-timeϕ-transformdecomposition(seeSection
3.2)forBαp,q(Z),whichwillbethestartingpointforourconsiderations.
WederiveourcharacterizationofBesovspacesonZintermsofdiscrete-timewavelet
systemsinSection3.3:Justasinthecontinuouscase,theϕ-transformisinmany
waysquitesimilartoawavelettransform,andthissimilarityallowsaproofofthe
mainresultbystudyingoff-diagonaldecayofcertaininfinitematrices.Thisdecay

57

58

3.1Littlewood-PaleyTypeDefinitionofBαp,q(Z)

behaviorisderivedundersuitableconditionsregardingsupport,vanishingmoments
andsmoothnessofthediscrete-timewavelets.
Thiswaveletcharacterizationallowstoobtainfurtherdescriptionsofthediscrete-
timespacesasinSection3.4,whichcanbeviewedasmore‘intrinsic’incontrastto
theLittlewood-Paleytypedefinition.Inparticular,theydonotrelyonthechoice
functions.iliaryauxofSubsection3.4.1containstheresultsondiscrete-timeBesovspacesintermsof
discrete-timemoduliofsmoothness(comparetotheanalogcontinuous-timespaces
in1.1.1);anotherequivalentcharacterizationintermsofoscillationoverintervalsis
.2.3.4inengiv

3.1Littlewood-PaleyTypeDefinitionofBαp,q(Z)
ThefollowingdefinitionofBesovspacesonZistheoneemployedin[31],exceptfor
achangeofnotationthatisconvenientforourpurposes.
ForadditionalbackgroundinformationonthecorrespondingspacesonR,seeChap-
ter1ore.g.[16,17,14,23].
First,weneedanalogaofSchwartzfunctionsandtempereddistributionsinterms
ofsequences.
Wereferto[31]forbasicfactsconcerningdistributionsonZ.
Definition3.1.1.Acomplex-valuedsequenceη=(η(n))n∈Zsatisfying
n∈supZ|η(n)|(1+|n|)m<∞(3.1.1)
foreverym>0,willbecalledrapidlydecreasing.
Asequencef=(f(n))n∈Zwillbecalled(tempered)distributiononZif
inf{m∈N:sup|f(n)|(1+|n|)−m<∞}<∞.
Z∈n

WewilldenotethespacesofrapidlydecreasingsequencesbyS(Z):itistopologized
bythesup-normsin(3.1.1).S(Z)denotesthespaceofdistributionsonZ,which
isindeedthedualspaceofS(Z).Inordertobeconsistentwiththeusualinner
productnotation,f,η=nf(n)η(n)willstandforthepairingofadistributionf
andtherapidlydecreasingsequenceη.TheFouriertransformoff∈S(Z)isgiven
byf(ω)=k∈Zf(k)e−ikω,extendedtoS(Z)intheusualway.
Ajustificationforthiscanbefoundagainin[31].
Wenextdefinethenotionofaphi-function,whichisthebasisfortheLittlewood-
Paleydefinitionofdiscrete-timeBesovspaces.

3.2ϕ-transformDecompositionofBαp,q(Z)

95

Definition3.1.2.Aphi-functionisafunctionϕc∈S(R)satisfying
suppϕˆc⊆{ω:π/4<|ω|<π},(3.1.2)
forsomeC,ε>0,
|ϕˆc(ω)|>Con{ω:π/4+ε<|ω|<π−ε},(3.1.3)
ϕˆc≡1inasmallneighborhoodof{−π/2,π/2},(3.1.4)
|ϕˆνc(ω)|2=1forω∈R\{0}.(3.1.5)
Z∈νForν∈Z,setϕνc(x)=2−ν+2ϕc(2−ν+2x).Thesuperscriptcservesasareminder
thatϕcisacontinuous-timefunction.Notethatournotationdiffersfrom[31]:here,
smallscalescorrespondtosmallν.Weusedilationbythefactor2−ν+2insteadof
themoreintuitive2−νinordertoobtainaunifiednotationlateron;thishasthe
slightlyawkwardconsequencethatϕcequalsϕ2c.
Wewillnowobtainafamilyofrapidlydecreasingsequencesbysamplingthefunc-
tions(ϕν)ν≥1.Setϕν:=ϕνc|Zforν>1andϕ1:=((χ[−π,π]ϕˆ1c)∨)|Z,wherethe
differentdefinitionforϕ1isduetotechnicalconsiderations,see[31].
LetP(Z)denotethesetofpolynomialsonR,sampledattheintegers.
Definition3.1.3.Letaphi-functionϕc∈S(R)begiven.Forα∈R,0<p,q<∞,
thediscrete-timeBesovspaceBαp,q(Z)isthecollectionofallf∈S/P(Z)(distribu-
tionsonZmodulopolynomialsP(Z)),suchthat
fBαp,q(Z):=((2−ναf∗ϕνlp(Z))q)1/q<∞.
1≥νThisdefinitionisindependentofthechoiceofϕc.Foradistributionf∈S(Z),we
havefBpα,q(Z)=0ifandonlyiff∗ϕν=0forallν≥1.Bytheconditionson
ϕ,thisisequivalenttosuppfˆ={0},orequivalently,tof∈P(Z).Thisiswhy
theBesovspacesaredefinedasspacesofequivalenceclassesmodulopolynomials:
∙Bαp,q(Z)becomesanormfor1≤p,q<∞andaquasi-normingeneral.
Inanalogytothecontinuous-timecase,wehaveaCalder´ontypeformulaforf∈
Z):(Sf=f∗ϕν∗ϕν∗,(3.1.6)
1≥νwithunconditionalconvergenceinS/P(Z)[31].
3.2ϕ-transformDecompositionofBαp,q(Z)
Aϕ-transformtheoremforBαp,q(Z)wasderivedin[31].Itcanbeunderstoodas
acriticallysampledversionof(3.1.6).Wethushaveathandadecompositionfor

60

3.2ϕ-transformDecompositionofBαp,q(Z)

sequencesinthediscrete-timeBesovspacesimilartoanexpansionintoanortho-
normalbasis,thoughnon-orthogonalandwith‘basiselements’thatarecompactly
supportedinFourierdomain.Theorem3.2.1providesanormequivalencethatcan
bereadasacriticallysampledversionofDefinition3.1.3,andwhichyieldsthechar-
acterizationofBαp,q(Z)intermsofthemembershipofϕ-transformcoefficientsinthe
spacebαp,q(Z)wedefinedin(1.2.1).
Consideragainaphi-functionϕc∈S(R),andforν,k∈Zlet
ϕνc,k(x)=2(−ν+2)/2ϕc(2−ν+2x−k).
Fork∈Z,defineϕν,k=ϕνc,k|Zforν>1andϕ1,k=τkϕ1.
In[31],startingfromtheformula(3.1.6)andfollowingthelinesofFrazierand
Jawerth[15],itisderivedthatforanyf∈S(Z)
f=f,ϕν,kϕν,k,(3.2.1)
ν≥1k∈Z
withconvergenceinS/P(Z).Itisalsowell-knownthatcondition(iv)ofDefinition
3.1.2aloneguaranteesthatforf∈2(Z)thedecomposition(3.2.1)convergesinthe
m.norLettheϕ-transformSϕforf∈S(Z)bedefinedbySϕf=s=(sν,k)ν≥1,k∈Z,where
ϕ-transformbyTϕbyTϕt=ν,ktν,kϕν,k.Theconvergenceofthesumisguaranteed
sν,k=f,ϕν,k,andforacomplex-valuedsequencet=(tν,k)ν≥1,k∈Zdefinetheinverse
bythefollowingresult.
Theorem3.2.1.[[31]]Letα∈R,0<p,q<∞.
BothoftheoperatorsSϕ:Bαp,q(Z)→bαp,q(Z)andTϕ:bαp,q(Z)→Bαp,q(Z)arebounded
withfBαp,q(Z)Sϕfbαp,q(Z)andTϕ◦Sϕ=idBpα,q(Z).
Inotherwords,underthesemaps,Bαp,q(Z)isaretractofbαp,q(Z),andBαp,q(Z)canbe
identifiedwiththeclosedsubspaceSϕ(Bαp,q(Z))ofbαp,q(Z).
Wenextgiveamoreprecisestatementconcerningtheconvergenceoftheϕ-transform
decomposition,ifweknowasequencetobelongtoaBesovspace.LetforK∈N
SK(Z):={η∈S(Z):η(n)nm=0,Nm≤K}
andS∞(Z):={η∈S(Z):η(n)nm=0forallm∈N},S−1(Z):=S(Z).
NotethatthedualspaceofSK(Z)canbeidentifiedwithS/PK(Z),thespace
ofequivalenceclassesofdistributionsmodulopolynomialsofdegree≤K,where
(S−1(Z))∼S(Z)andalso(S∞(Z))∼S/P(Z)([17,14]).

3.2ϕ-transformDecompositionofBαp,q(Z)

61

Lemma3.2.2.Let(ϕν,k)ν≥1,k∈Zbethefamilyofrapidlydecreasingsequencesdefined
3.2.ctionSeinAnyf∈Bαp,q(Z)canbewrittenas
f=f,ϕν,kϕν,k
ν≥1k∈Z
inthesenseof(SK(Z)),K=max{[α−1/p],−1},andforanyη∈SK(Z),weset
f,η:=f,ϕν,kϕν,k,η.
,kν

ProofFirst,noticethatforη∈S(Z),absoluteconvergenceoftheϕ-transform
3.3.2decompbelowositio(sneeRholdsemainrkap3o.3.3int)wisealloswsense,theescomptimareateto|[1ϕν4].,k,ηNo|w,≤letC2η−ν∈(KS+1K.+1/2)Lemma(1+
|k|)−MforM>0.Forf∈Bαp,q(Z),(f,ϕν,k)ν≥1,k∈Z∈bαp,q(Z)byTheorem3.2.1
andparticularlyforanycoefficient|f,ϕν,k|≤C2ν(α−1/p+1/2).
,Hence∞|f,ϕν,k||ϕν,k,η|≤Cν∞=N+1k2ν(α−1/p+1/2)2−ν(K+1+1/2)(1+|k|)−M
k+1N=ν≤Cν∞=N+1k2−ν(K−α+1/p+1),
whichensuresconvergenceforK=max{[α−1/p],−1}as
NNlim→∞f(n)(η(n)−η,ϕν,kϕν,k(n))=0.
=1νZ∈n

3.2.3.Remark1.Forcontinuous-timefunctions,wedecidedbetweenhomogeneousandinho-
mogeneousspaces.Inthediscrete-timecase,wedefinedasinglescaleofthe
spaces,startingfromhomogeneousBesovspaces.AsTorres[31]notes,indis-
cretetime,thenotionofinhomogeneousspacesmakesnosense:wecutoffhigh
frequenciesandthediscrete-timeBesovnormcontrolsthelargescalebehavior
ofthesequence.Controllingthislargescalebehaviorasin1.1.6infactresults
inanormequivalentto∙p(Z).
2.Thereisasamplingtheoremin[31]:itisshownthatfor1<p<∞,the
spacesBαp,q(Z)correspondexactlytothespacesofsamplesoffunctionsin
B˙αp,q(R)∩Eπ,whereEπisthesetoftempereddistributionswhoseFourier
transformsaresupportedon[−π,π].

62

3.3WaveletCharacterizationofBαp,q(Z)

3.3WaveletCharacterizationofBαp,q(Z)
Inthissection,wepresentthecentralresultofthisthesis.Theresultsofthissection
havebeenpublishedin[11].
Weconsiderbiorthogonalfamilies(hj)j≥1and(h˜j)j≥1in2(Z),andassociatedsys-
tems(hj,l)=(hj(∙−2jl)),h˜j,l=(h˜j(∙−2jl)).Weassumethatallinvolvedhj,h˜j
havefinitesupports.Thusitispossibletodefineforf∈S(Z)theoperator
Shf=(f,hj,l)j≥1,l∈Z,andSh˜likewise.Moreover,forallfinitelysupportedcoef-
ficientsequencesd=(dj,l)j≥1,l∈Z,define
Thd=dj,lhj,l.(3.3.1)
,ljAgain,letT˜hbedefinedinananalogousway.Ouraimisthecharacterizationof
f∈Bαp,q(Z)byuseoftheoperatorsThandSh˜.
Inthefollowingwewanttoprovidecriteriaon(hj)j≥1,(h˜j)j≥1toensureanalogsof
3.2.1,withTh,S˜hreplacingTϕ,Sϕ.OurmainresultwillbeTheorem3.3.8below,
whic˙hαmaybeviewedasanalogyto3.2.1,butalsotothewaveletcharacterization
ofBp,q(R).
Intheproofweexploitthestrongsimilaritiesofthebiorthogonalwaveletandϕ-
transforms:botharebasedonbuildingblocksindexedbydyadicscales2j,which
areshiftedalongthegrid2jZ.
Butourresultisnotincludedintheϕ-transformresult:Recallthattheϕ-transform
sequencesϕν,karisebysamplingband-limitedSchwartzfunctionswithinfinitely
manyvanishingmoments.Bycontrast,thewaveletsystemsneedonlyhavefinitely
manyvanishingmoments,andonlyafinitedegreeofsmoothness.Inaddition,we
assumeacontroloverthesupportsoftheinitialsequences(hj)j≥1and(h˜j)j≥1that
cannotbeobtainedbysamplingbandlimitedfunctions.
Thissetupcoversthediscrete-timebiorthogonalwaveletbasesasdescribedinSec-
tion2.2,butitcouldalsobeappliedtoevenmoregeneralsystems,e.g.arisingfrom
cascadealgorithmswheretheanalysisfilterchangesateachscaleinacontrolled
.yawInanycase,theϕ-transformwillbethestartingpointforourconsiderations.The
blueprintforthegeneralproofstrategyisprovidedbythecontinuoustimethe-
ory,ascontainede.g.in[17]or[14].Howevertheargumentsneedtobeadapted
topropertiesofdiscrete-timewaveletfamilies.Thekeytotheproofisthestudy
oftheoff-diagonalbehaviorofthetransitionmatricesA=(hj,l,ϕν,k)j,l,ν,kand
A˜=(ϕν,k,h˜j,l)ν,k,j,l,andtoconcludeboundednessofcertainassociatedoperators
actingonbαp,q(Z).Forthispurpose,westudyhowpropertiesofhj,linfluencethesize
of|hj,l,ϕν,k|.
First,weemployaninequalitythatwillbeusedrepeatedly:

3.3WaveletCharacterizationofBαp,q(Z)

Lemma3.3.1.Letα≤1andM≥N+1.Then,
(1+α|k|)−M(1+|k|)N≤Cα−(N+1).
Z∈k

foorP

(1+α|k|)−M(1+|k|)N=α−M(α−1+|k|)−M(1+|k|)N
k∈Zk∈Z
≤α−M(α−1+|k|)N−M
Z∈k=α−M(αM−N+2(α−1+k)N−M)
1≥k∞≤α−M(αM−N+2(α−1+x)N−Mdx)
0∞=α−N+2α−MxN−Mdx
1−α=α−N+2α−MαM−N−1
≤Cα−(N+1)

63

Lemma3.3.2.Let(hj)j≥1,(ϕν)ν≥1bediscrete-timefamiliessatisfyingthefollowing
onditions:cThereareN≥1andM>0suchthat
nihj(n)=0fori=0,...,N−1,(3.3.2)
Z∈n|hj(n)|≤C2−j/2(1+2−j|n|)−(M+N+2)forn∈Z.(3.3.3)
Further,assumethatforeachν≥1,n∈Zthereisapolynomialpν,nofdegree≤N,
andafunctionΦν:Z×Z→R+suchthat
i)|ϕν(n−k)−pν,n(k)|≤C2−ν/22−νN(1+|k|)NΦν(n,k),
ii)Φν(n,k)≤C,(3.3.4)
iii)Φν(n,k)≤C(1+2−ν|n|)−Mfor|k|<|n|,
2withCindependentofn,ν,k.
νjforThen,≤|(hj∗ϕν)(n)|≤C2(j−ν)(N+1/2)(1+2−ν|n|)−M,(3.3.5)

).3.5(3

64

3.3WaveletCharacterizationofBαp,q(Z)

whereCisaconstantindependentofν,n,k.

ProofThiscanbeverifiedanalogouslytothesecondpartoftheproofofLemma
3.3in[15].Wegivethisproofforthesakeofcompleteness:
.νjLet≤|(hj∗ϕν)(n)|=|hj(k)ϕν(n−k)|
Z∈k(3.=3.2)|hj(k)(ϕν(n−k)−pν,n(k))|
Z∈k≤C|hj(k)|2−ν/22−νN(1+|k|)NΦν(n,k)
(3.3.4i)
Z∈k=C|hj(k)|2−ν/22−νN(1+|k|)NΦν(n,k)
|k|<|n2|
+|hj(k)|2−ν/22−νN(1+|k|)NΦν(n,k)
|k|≥|2n|
=:C(I+II)
Using(3.3.4iii),thefirstsumcanbeestimatedby

I≤C2−(j+ν)/22−νN(1+2−ν|n|)−M(1+2−j|n|)−(M+N+2)(1+|k|)N
|k|<|n|/2
≤C2−(j+ν)/22−νN2j(N+1)(1+2−ν|n|)−M
=C2(j−ν)(N+1/2)(1+2−ν|n|)−M,
whereinthesecondinequalityweusedLemma3.3.1withα=2−j.
From(3.3.4ii),wegetforthesecondsum
II≤C2−(j+ν)/22−νN(1+2−j|n|)−(M+N+2)(1+|k|)N
|k|≥|n|/2
≤C2−(j+ν)/22−νN(1+2−j−1|n|)−M(1+2−j|n|)−(N+2)(1+|k|)N
|k|>|n|/2
≤C2−(j+ν)/22−νN2j(N+1)(1+2−ν|n|)−M
=C2(j−ν)(N+1/2)(1+2−ν|n|)−M,
again,wemadeuseofLemma3.3.1.

3.3WaveletCharacterizationofBαp,q(Z)

65

(ϕν)Remarkν≥13.3satisfying.3.AstheaconsrequiremenequencetsofofLtheemmaLemma3.3t.2,hatwethereobtaisinMfor>η0∈sucSKh(tZ)hatand

|(η∗ϕν)(n)|≤C2−ν(K+1+1/2)(1+2−ν|n|)−M.(3.3.6)
Wewillneedadiscrete-timeanalogontoTaylor’sformula.Forf=(f(n))n∈Z,
considertheforwardandbackwarddifferenceoperatorsΔf(n)=f(n+1)−f(n)
andf(n)=f(n−1)−f(n).
Lemma3.3.4.Letf=(f(n))n∈Z,k∈Z,N∈N.
i)Fork≥0,k≥N
N−1k−N−1
f(n−k)=kif(n)+k−1−mNf(n−m)+Nf(n−k+N).
i=0im=0N−1
ii)Fork<0,k≤−N
f(n−k)=−kΔif(n)+−k−1−mΔNf(n+m)+ΔNf(n−k−N).
N−1−k−N−1
i=0im=0N−1

ProofFirst,bydirectcalculation,ifk≥0
kf(n−k)=kif(n)
i=0i0<kforand−k−k
f(n−k)=Δif(n).
i=0iIntheliterature,thistypeofexpansionissometimesreferredtoasNewton-Gregory
interpolationformula.
i)ForN=k,weget
f(n−k)=kif(n)=kif(n)+Nf(n).
kk−1
i=0ii=0i

66

3.3WaveletCharacterizationofBαp,q(Z)

1:NN−→f(n−k)
=kif(n)+k−1−mNf(n−m)+Nf(n−k+N)
N−1k−N−1
i=0im=0N−1
N−2
=kif(n)+kN−1f(n)+
i=0iN−1
k−N−1
k−1−mN−1f(n−m−1)−N−1f(n−m)+Nf(n−k+N)
1N−=0m=kif(n)+k−m−k−1−mN−1f(n−m)
N−2k−N
i=0im=0N−1N−1
+N−1f(n−k+N−1)
=kif(n)+k−1−mN−1f(n−m)+N−1f(n−k+N−1),
N−2k−N
i=0im=0N−2
whereinthelastequationweusedthatforl,m∈Nlm+1+lm=lm+1+1.

Theproofofii)canbedoneinthesameway.
ThesedifferenceoperatorsarerelatedtotheoperatorDjdefinedinSection2.2.2:
Lemma3.3.5.TheforwardandbackwarddifferenceoperatorsΔ,,definedabove
satisfyforf=(f(n))n∈Z
i)Δmf(n−m)=2−jmDjmf(n)
ii)mf(n)=(−1)m2−jmDjmf(n),
wherem∈NandDjthedifferenceoperatordefinedinsection2.2.2.
i)foPro1:=m2−jDjf(n)=f(n)−f(n−1)=Δf(n−1),
m→m+1:

2−j(m+1)Djm+1f(n)=2−j(m+1)2j(Djmf(n)−Djmf(n−1))
=Δmf(n−m)−Δmf(n−m−1)
=Δm+1f(n−(m+1)).

3.3WaveletCharacterizationofBαp,q(Z)

ii)m=1:−2−jDjf(n)=−f(n)+f(n−1)=f(n),
m→m+1:
(−1)m+12−j(m+1)Djm+1f(n)=(−1)m+12−j(m+1)2j(Djmf(n)−Djmf(n−1))
m−jmmm
==(−m1)f(n2−1)(D−jf(mnf(−n)1)−Djf(n))
=m+1f(n).

67

Forthefollowinglemma,observethatassumptionsconcerningsupportsizeand
vanishingmomentsof(hj)j≥1carryovertothelargersystem(hj,l)j≥1,l∈Z.Moreover,
forthecaseofdiscretewaveletbasesasconstructedinSection2.2,thesupport
propertiesaretriviallyfulfilled,andthevanishingmomentsareensuredbyhaving
enoughvanishingmomentsintheinitialhigh-passfilterh,conferProposition2.2.5.
Lemma3.3.6.Forthediscrete-timefamilies(ϕν,k)ν≥1,k∈Z,theϕ-transformas
definedin(3.2),and(hj,l)j≥1,l∈Z,satisfying
|supphj,l|≤C2j,hj,l∞≤C2−j/2(3.3.7)
n∈Znihj,l(n)=0fori=0,...,N1−1,(3.3.8)
(hj,l)regularoforderN2+εinthesenseof2.2.7,0<ε≤1,(3.3.9)
thefollowinginequalitiesarevalid:
ThereexistC>0,M1,M2∈N,suchthat
νj−M1
|hj,l,ϕν,k|≤C2(j−ν)(N1+1/2)1+|2kν−2l|forj<ν,(3.3.10)
2νj−M2
|hj,l,ϕν,k|≤C2(ν−j)(N2+1/2)1+|2kj−2l|forj≥ν.(3.3.11)
2

PandrooϕfW=e(ϕfirst(nsho))w,thatsatisfyforantheyj≥conditio1andnsνin>Le1,mmtheas3.3.2.equencesSetthingj=n(=hj,20ν(kn))−n2∈jZl
νν,0n∈Z
in(3.3.5)willgivethefirstoftheaboveinequalities.
tioLetnsj(3≥.3.71.),Byfoarsnsumpt∈suppionh(3j.3.8we),hahvje(1satisfies+2−j|(3n|).3.2)≤C,withwhiNch=gNiv1esandbyassump-
|hj(n)|≤C2−j/2(1+2−j|n|)−MforanyM>0.

68

3.3WaveletCharacterizationofBαp,q(Z)

ChoosingM1>0andsettingM=M1+N1+2gives(3.3.3).
Forν>1,thesequence(ϕν,0(n))n∈Zarisesbysamplingfromacontinuous-time
functionϕνc,0∈S(R),whereϕνc,0(x)=2(−ν+2)/2ϕc(2−ν+2x).Bydefinition,ϕccorre-
spondstoϕ2c,s−oν/w2eccan−νrewritetheaboveequationforconvenience:
ϕν,0(n−k)=2ϕ0(2(n−k)).
LetPxbetheTaylorpolynomialofϕ0cofdegreeN1−1inx∈R:
1−N1Px(∙)=(ϕ0c)(m)(x)(∙−x)m.
!m=0mFory∈R,thereisξbetweenxandy,suchthatϕ0c(y)=Px(y)+(ϕ0c)N(N1!1)(ξ).Setting
pν,n(k)=2−ν/2P2−νn(2−ν(n−k))andΦν(n,k)=sup{(ϕ0c(Nξ))!(N1),ξbetweenxandy},
1wehave(3.3.4i),and(3.3.4ii)as(ϕ0c)(N1)∞<∞.
Let|k|<|2n|.Inthiscase,forξbetween2−νnand2−ν(n−k),|ξ|≥2−νmin(|n|,|n−
k|)≥2−ν|n|/2.Sinceϕ0c∈S(R),weobtain(3.3.4iii),as(ϕ0c)(N1)(ξ)≤C(1+
2−ν−1|n|)−M1≤C(1+2−ν|n|)−M1.
Inordertoprovethesecondinequality,wemakeagainuseofLemma3.3.2,exchang-
ingtherolesofϕνandhj.Wewillthereforeneedtocheckthatthesystem(ϕν)
fulfillstherequirementsimposedon(hj)inLemma3.3.2,andviceversa.
Forν≥1,ϕν∈S(Z),whichgives(3.3.3)foranyN∈N.Sincethemomentsofany
orderofϕνvanish(3.3.2)isvalid.Henceitremainstocheckthecondition(3.3.4)
with(hj)replacing(ϕν).Oncethisisachieved,Lemma3.3.2providestheestimate
|(hj∗∗ϕν)(n)|≤C2(ν−j)(N2+1/2)(1+2−j|n|)−M,
andsettingn=2jl−2νkwillfinishtheproof.
Aswechosethefilters∗tobeoffinitelength,supphj∗isfinite.ByLemma3.3.4,for
nandn−kinsupphj,fork>0
khj∗(n−k)=kihj∗(n)
i=0iandfork<0−k
hj∗(n−k)=−ikΔihj∗(n).
=0iFornoutsidethesupportofhj∗,onehastoexpandtheupperseriesintherightresp.
theleftendpointofthesupportinginterval.
Inthecasek>0,setN2−1
pj1,n(k)=kihj∗(n)
i=0i

3.3WaveletCharacterizationofBαp,q(Z)

andfork<0N2−1
pj2,n(k)=−kΔihj∗(n).
i=0iUsingLemmata3.3.4and3.3.5givesfork≥N2
|hj∗(n−k)−pj1,n(k)|
k−N2−1
=|k−1−mN2hj∗(n−m)+N2hj∗(n−k+N2)|
m=0N2−1
k−N2−1k−1−mN2N2∗
≤m=0N2−1+N2∙m∈{0,...sup,k−N2}|hj(n−m)|
≤k∙sup|(−1)N22−jN2DjN2hj∗(n−m)|
N2m∈{0,...,k}
≤C2−jN2(1+|k|)N2∙sup|DN2hj∗(l)|.
l∈{n−k,...,n}
Theregularityconditiononhj∗gives
sup|DN2hj∗(l)|≤C2−j/2
l∈{n−k,...,n}
andduetothefinitesupportofhj∗,fork<|n|/2,
sup|DN2hj∗(l)|≤C2−j/2(1+2−j|l|)−M
l∈{n−k,...,n}
≤C2−j/2(1+2−j−1|n|)−M
≤C2−j/2(1+2−j|n|)−M

foranyM>0.
Likewise,fork<−N2,
|hj∗(n−k)−pj2,n(k)|≤C(1+|k|)N2∙sup|ΔN2hj∗(n+m)|
m∈{0,...,−k−N2}
≤C2−jN2(1+|k|)N2sup|DN2hj∗(l)|.
l∈{n,...,n−k}

69

70

3.3WaveletCharacterizationofBαp,q(Z)

Lemma3.3.7.Letα∈R,0<p,q<∞andlet(ϕν,k)ν≥1,k∈Zbethefamilyof
sequencesdefinedin(3.2).
If(hj,l)j≥1,l∈ZsatisfiestheconditionsofLemma3.3.6withN1>1/(min(1,p))−1−α,
N2>α,thenthematrixA:=(hj,l,ϕν,k)j,l,ν,k,definesaboundedoperatoron
bαp,q(Z),whereAs=(j≥1l∈Zhj,l,ϕν,ksj,l)ν,kfors∈bαp,q(Z).
Also,for(h˜j,l)j≥1,l∈Z,satisfyingtheconditionsofLemma3.3.6,wherenowN1>
α,N2>1/(min(1˜,p))−1−α,thematrixαA˜:=(ϕν,k,h˜j,l)ν,k,j,l,where˜As=
(ν≥1k∈Zϕν,k,hj,lsν,k)j,l,isboundedonbp,q(Z)aswell.
ProofTheproofisquiteclosetothecontinuouscasetreatedin[15,14].We
reproduceithereforthesakeofconvenience.
First,weshowboundednessofA:

pq/p
Asbqα(Z)=2−ν(α−1/p+1/2)|hj,l,ϕν,ksj,l|
p,qν≥1k∈Zj≥1l∈Z
pq/p
≤2(j−ν)(α−1/p+1/2)|hj,l,ϕν,k|2−j(α−1/p+1/2)|sj,l|
ν≥1k∈Zj≥1l∈Zpq/p
≤C2(j−ν)(α−1/p+1/2)|hj,l,ϕν,k|2−j(α−1/p+1/2)|sj,l|
/pqpν≥1k∈Zj<νl∈Z
+2(j−ν)(α−1/p+1/2)|hj,l,ϕν,k|2−j(α−1/p+1/2)|sj,l|
ν≥1k∈Zj≥νl∈Z
=:CIq+IIq

Inthecase1<p<∞,wehavefromLemma3.3.6andMinkowski’sinequalityfor
thefirstterm

Iq
,lj≤C2(j−ν)(α−1/p+1/2+N1+1/2)1+|2νk−2jl|−M12−j(α−1/p+1/2)|s|pq/p
ν≥1k∈Zj<ν,l∈Z2ν

,lj≤C2(j−ν)(α−1/p+1+N1)(1+|2νk−2jl|−M12−j(α−1/p+1/2)|s|)p1/pq/p
ν≥1j<νk∈Zl∈Z2ν

.p/qp)|,kνs|2)/+1/p1−α(ν−(2Z∈k1≥νC≤p/q/p1p)|,ljs|2)/+1/p1−α(j−(2Z∈l)2N−α)(ν−j(2ν≥j1≥νC≤p/q/p1p)|,ljs|2)/+1/p1−α(j−22M−j2|lj2−kν2|+1Z∈l(Z∈k)2N−/p1−α)(ν−j(2ν≥j1≥νC≤p/qp|l,js|2)/+1/p1−α(j−22M−j2|lj2−kν2|+1)2N−/p1−α)(ν71

−3.3WaveletCharacterizationofBαp,q(Z)

jByH¨older’sinequality,where1/p+1/p=1,theinnerp-sumcanbeestimatedby

(p1/p
(1+|k−2j−νl|)−M12−j(α−1/p+1/2)|sj,l|
k∈Zl∈Zp1/p
=(1+|k−2j−νl|)−M1/p(1+|k−2j−νl|)−M1/p2−j(α−1/p+1/2)|sj,l|
/p1k∈Zl∈Z
≤(1+|k−2j−νl|)−M1∙
k∈Zl∈Z1/pp1/p
(1+|k−2j−νl|)−M12−jp(α−1/p+1/2)|sj,l|p
l∈Z1/p
≤C2(ν−j)/p(2−j(α−1/p+1/2)|sj,l|)p,
Z∈lwherethelastinequalityfollowsfromLemma3.3.1.
InsertingthisestimateintoIqandusingtheconditiononN1,wefind
1/pq/p
Iq≤C2(j−ν)(α+N1)(2−j(α−1/p+1/2)|sj,l|)p
/pqν≥1j<νl∈Z
≤C(2−ν(α−1/p+1/2)|sν,k|)p.
ν≥1k∈Z
Similarly,forthesecondterm

2

Z∈lν≥jZ∈k1≥νC≤qII

72

3.3WaveletCharacterizationofBαp,q(Z)

Nowconsiderthecase0<p≤1:
Iq≤C2(j−ν)(α−1/p+1/2+N1+1/2)1+|2kν−2l|2−j(α−1/p+1/2)|sj,l|
νj−M1pq/p
ν≥1k∈Zj<ν,l∈Z2
νj−M1pq/p
≤C2p(j−ν)(α−1/p+N1+1)1+|2k−2l|2−jp(α−1/p+1/2)|sj,l|p
ν≥1k∈Zj<νl∈Z2ν
q/p
≤C2p(j−ν)(α−1/p+N1+1)2−jp(α−1/p+1/2)|sj,l|p
/pqν≥1j<νl∈Z
≤C(2−ν(α−1/p+1/2)|sν,k|)p.
ν≥1k∈Z
Alongthesamelines,forthesecondtermincase0<p≤1
−M2pq/p
IIq≤C2(j−ν)(α−1/p−N2)1+|2νk−2jl|2−j(α−1/p+1/2)|sj,l|
jν≥1k∈Zj≥νl∈Z2
≤C2p(j−ν)(α−1/p−N2)1+|2kj−2l|2−jp(α−1/p+1/2)|sj,l|p
νj−M2pq/p
ν≥1k∈Zj≥νl∈Z2
/pq≤C2p(j−ν)(α−1/p−N2)2(j−ν)2−jp(α−1/p+1/2)|sj,l|p
ν≥1j≥νl∈Z
q/p
=C2p(j−ν)(α−N2)2−jp(α−1/p+1/2)|sj,l|p
p/qν≥1j≥νl∈Z
≤C(2−ν(α−1/p+1/2)|sν,k|)p.
ν≥1k∈Z
ReversingtherolesofjandνintheaboveproofgivesboundednessofA˜:
pq/p
˜Asbqα(Z)=2−j(α−1/p+1/2)|ϕν,k,h˜j,lsν,k|
p,qj≥1l∈Zν≥1k∈Z
pq/p
≤C2(ν−j)(α−1/p+1/2)|ϕν,k,h˜j,l|2−ν(α−1/p+1/2)|sν,k|
j≥1l∈Zj≥νk∈Z
+2(ν−j)(α−1/p+1/2)|ϕν,k,h˜j,l|2−ν(α−1/p+1/2)|sν,k|.
pq/p
j≥1l∈Zj<νk∈Z
ByLemma3.3.6,thesizeof|h˜j,l,ϕν,k|=|h˜j,l,ϕν,k|=|ϕν,k,h˜j,l|againcan
beestimatedby(3.3.10),(3.3.11)respectively,suchthatfor1<p<∞,wecan

3.3WaveletCharacterizationofBαp,q(Z)

73

estimatethefirsttermintheabovesumby
1/pq/p
C2(ν−j)(α+N2)(2−ν(α−1/p+1/2)|sν,k|)p,
j≥1ν≤jk∈Z
andthesecondoneby
1/pq/p
C2(ν−j)(α−N1)(2−ν(α−1/p+1/2)|sν,k|)p.
j≥1ν>jk∈Z
Asinthiscase,wechoseN1>α,N2>1/(min(1,p))−1−α,A˜isboundedandthe
case0<p≤1followsusingthesameargumentsasabove.
Wearenowreadytostateourmainresult:underappropriatesupport,moment
andregularityconditionsonthebiorthogonalwaveletfamilies,themembershipof
adistributionf=(f(n))n∈Zinadiscrete-timeBesovspaceisfullycharacterized
bythedecayofcoefficients(f,h˜j,l)j≥1,l∈Z.Moreover,theassociatedαanalysisand
synthesisoperatorsareisomorphismsontothefullcoefficientspacesbp,q(Z),notjust
subspaces.closedcertaintoonTheorem3.3.8.Letα∈R,0<p,q<∞,N>α,N˜>1/(min{1,p})−1−α.
Supposethat(hj,l),(h˜j,l),j≥1,l∈Z,arebiorthogonalwaveletbasesfor2(Z),
satisfying|supphj|,|supph˜j|≤C2j,(3.3.12)
hj∞≤C2−j/2,h˜j∞≤C2−j/2(3.3.13)
n∈Znihj(n)=0fori=0,...,N˜−1,(3.3.14)
n∈Znih˜j(n)=0fori=0,...,N−1,(3.3.15)
(hj)j≥1regularoforderN+ε(inthesenseof2.2.7),0<ε≤1,(3.3.16)
(h˜j)j≥1regularoforderN˜+ε,˜0<ε˜≤1.(3.3.17)
Thenthefollowingstatementshold:
(a)TheanalysisoperatorSh˜:Bαp,q(Z)→bαp,q(Z)iswell-definedandcontinuous.
(b)Thesynthesisoperatorextendsuniquelytoaboundedoperatorbαp,q(Z)→Bαp,q(Z),
alsodenotedbyTh.Forarbitrary(dj,l)j≥1,l∈Z∈bαp,q(Z),
Th((dj,l)j≥1,l∈Z)=dj,lhj,l,(3.3.18)
l,jwithunconditionalconvergenceintheBesovspacenorm.

74

3.3WaveletCharacterizationofBαp,q(Z)

(c)Th◦Sh˜=idBαp,q(Z),andSh˜◦Th=idbαp,q(Z).Thus,Bαp,q(Z)canbeidentifiedwith
bαp,q(Z)underthemapsSh˜andTh.
(d)WehavethenormequivalencefBαp,q(Z)Sh˜fbαp,q(Z).Moreover,thewavelet
ansionexpf=f,h˜j,lhj,l
,ljholdswithunconditionalconvergenceintheBesovspacenorm.

αKPro=ofmax{[(Compaα−1re/p]to,−1[14}.,The17].)nKFor+1par≥tN(,a),andletthfus∈Bp,q(Z),j≥1,l∈Zand
f=f,ϕν,kϕν,k
,kνholdsin(SK(Z)),byLemma3.2.2.Buth˜j,l∈SK(Z),andtherefore
f,h˜j,l=f,ϕν,kϕν,k,h˜j,l=(A˜(f,ϕν,k))(j,l).
,kν˜˜S˜Here=Aw˜e◦Sϕ.usedBythetheospuerppatoorrt,sizdefinedeandbymomenthetmatcorixnditioA:ns=on(ϕ(νh˜j,k,l,),hj,l(3)ν.3.12,k,j,l.),(3In.3.13short,),(3.3.15),
hLemma3.3.7yieldsthatA˜isboundedonbαp,q(Z),whereasTheorem3.2.1contributes
boundednessofSϕ.Thisprovespart(a).
j,lν,kj,lν,kν,k
Fhor=theopherato,ϕrTh,ϕweholdsconsiderinl2(Zthe).matThenrixfoAr=finit(elyhj,l,suppϕν,kort)j,led,ν,k.sequenceRecallsd,that
Thd=j,ldj,lhj,l
=dj,lhj,l,ϕν,kϕν,k
j,lν,k
=dj,lhj,l,ϕν,kϕν,k
,lj,kν=(Tϕ◦A)(d).
αByHencetheontheassumptionsfinitelyonsupptheortedsystemco(hefficiej)j≥n1t,thesequenceoperas-torwhicAishbareoundeddenseoninbbp,qα((ZZ).)
p,q-ThcoincideswiththeboundedoperatorTϕ◦A.ButthenThhasabounded
d∈bextensionαp,q(Z):toThethenetwholeofspace,restrictionsando(3.3fd.18to)infinitefactsubconsvetsergesofN×uncoZconnditionavergesllyfoinrathell

3.3WaveletCharacterizationofBαp,q(Z)

75

αinnorm(3.3on.18b)p,qco(nZv).ergeThensalso,bowhicundedneshissofTunconditiohimpliesnalconthavttheergence.netoffinitepartialsums
˜Fortheproofofpart(c),considerf∈Bαp,q(Z).Parts(a)andα(b)implythat
ThremaiSh˜fns=toproj,lvef,thahj,ltfhj,listhewithlimituncondiofthetionaexpalnsconiovn.erForgencethisinBpurpp,qo(sZe);wethereffirstoprreoveit
thatf=j,lf,h˜j,lhj,lholdsin(SK(Z)).
Henceletη∈SK.Then,
f,η=f,ϕν,kϕν,k,η=f,ϕν,kϕν,k,η
ν,kν,k
=f,ϕν,kϕν,k,h˜j,lhj,l,η
ν,kj,l
(=∗)f,ϕν,kϕν,k,h˜j,lhj,l,η
,kν,lj=f,h˜j,lhj,l,η
,lj=f,h˜j,lhj,l,η.
,ljTheorderofsummationin(∗)canbeinterchanged,becausetheseriesconverges
lutely:abso(f,ϕν,k)ν,k∈bαp,q(Z)andthematrixA˜:=(ϕν,k,h˜j,l)ν,k,j,lisboundedonbαp,q(Z)
.3.7.3LemmaybyieldsThis(|f,ϕν,k||ϕν,k,h˜j,l|)j,l∈bαp,q(Z),
,kνandinparticular

|f,ϕν,k||ϕν,k,h˜j,l|≤C2j(α−1/p+1/2).
,kνWenotedintheproofofLemma3.3.6that(hj)fulfillstherequirementsimposed
onthefamily(ϕν)inLemma3.3.2.Hence(3.3.3)implies
|hj,l,η|≤C2−j(K+1+1/2)(1+|l|)−M,
forM>0,asη∈SK(Z).

76

3.3WaveletCharacterizationofBαp,q(Z)

Overall,thisgives
|f,ϕν,kϕν,k,h˜j,lhj,l,η|≤|f,ϕν,k||ϕν,k,h˜j,l||hj,l,η|
j,lν,kj,lν,k
≤C2j(α−1/p+1/2)2−j(K+1+1/2)(1+|l|)−M
,lj≤C2−j(K+1−α+1/p)<∞.
jHencef=j≥1l∈Zf,h˜j,lhj,lin(SK(Z)).
InordertoproveThSh˜f=fintheBesovnorm,notethatinparticularϕν,k∈SK(Z),
which,togetherwithTheorem3.2.1,leadsto
Jqf−f,h˜j,lhj,lBαp,q(Z)
Z∈l=1jJ≤C(f−f,h˜j,lhj,l,ϕν,k)ν,kbqαp,q(Z)
∞j=1l∈Z
q=C(f,h˜j,lhj,l,ϕν,k)ν,kbαp,q(Z)
p/qpj=J+1l∈Z
≤C2−ν(α−1/p+1/2)|f,h˜j,lhj,l,ϕν,k|
ν≥1k∈Zj>J,j<νl∈Z
pq/p
+2−ν(α−1/p+1/2)|f,h˜j,lhj,l,ϕν,k|
ν≥1k∈Zj>J,j≥νl∈Z
=:CIq+IIq.
For1<p<∞,wecanproceedanalogouslytotheproofofLemma3.3.7and
estimatethefirsttermby
1/pq/p
Iq≤C2(j−ν)(α+N˜)(2−j(α−1/p+1/2)|f,h˜j,l|)p,
ν≥1j>J,j<νl∈Z
and1/pq/p
IIq≤C2(j−ν)(α−N)(2−j(α−1/p+1/2)|f,h˜j,l|)p.
ν≥1j>J,j≥νl∈Z
yieldsThisJJlim→∞f−f,h˜j,lhj,lBαp,q(Z)=0.(3.3.19)
Z∈l=1j

3.4‘Intrinsic’CharacterizationsofBαp,q(Z)

77

Thecase0<p≤1followsalongthesamelines,usingtheestimatesestablishedin
.3.7.3LemmaFurthermore,bythebiorthogonalityofhj,l,h˜j,l,anys=(si,m)i≥1,m∈Z∈bαp,q(Z)
canbewrittenas
(si,m)i,m=(sj,lhj,l,h˜i,m)i,m,
,ljwhichgivesSh˜◦Th=idbαp,q(Z).
Part(d)isimmediatefrom(c).
Remark3.3.9.Torres[31]characterizesthespacesBαp,q(Z)asspacesofsequences
obtainedbysamplingband-limiteddistributionsinB˙αp,q(R).
Theorem3.3.8providesanotherrelationbetweentheBαp,q(Z)andtheB˙αp,q(R)spaces
moreintermsofmultiresolutionanalysis.
LetF∈B˙αp,q(R),F=n∈Zanτnϕ,whereϕascalingfunctionassociatedtoa
multiresolutionanalysis(i.e.F∈V0inMRAlanguage).
RecallingthatwehaveF∈B˙αp,q(R)ifandonlyifthediscretewaveletcoefficients
(dj,l)j≥1,l∈Z∈bαp,q(Z).Byourtheorem,thisisinfactequivalentto(an)n∈Ztobein
Bαp,q(Z).

3.4‘Intrinsic’CharacterizationsofBαp,q(Z)
Thereareother,more‘intrinsic’possibilitiestodescribeBesovspacesindiscrete
timethanusingLittlewood-Paleytheory.E.g.[30]containsadescriptioninterms
ofmeanoscillationpropertiesofsequencesforsomespecialcasesoftheparameters
α,p,q.Usingourwaveletcharacterizationresult,itiseasytoextendthiskindof
descriptiontothewholeparameterfamily.Thiswillbetheissueofparagraph3.4.2.
Beforewegivethisresult,however,wecangiveanothercharacterizationofthe
discrete-timeBesovspacesintermsofiterateddifferences.Thisresultisaconse-
quenceofTheorem3.3.8.

3.4.1Discrete-TimeModuliofSmoothness
Analogouslytothecontinuous-timefunctionspaces,thediscrete-timeBesovspaces
possessadescriptionviadifferences.Weadaptthenotionmodulusofsmoothness
tofunctionsgivenindiscretetime,andshowαthatforacertainrangeofparameters
α,p,q,thearisingspacescoincidewiththeBp,q(Z)-spacesdefinedviaLittlewood-
Paleytheory(3.1.3).
ForthecorrespondingtheoryonR,seeChapter1,or[25]or[34].

78

3.4‘Intrinsic’CharacterizationsofBαp,q(Z)

Letm∈Z.Forasequencef=(f(n))n∈Z,definethe(forward)differenceoperator
ybmstepofΔmf(n)=f(n+m)−f(n),
andforr∈N+,definethedifferenceoperatoroforderr,stepm,inductivelyby
Δrmf(n)=Δm(Δrm−1f(n)).
Notethatther−thdifferenceoperatorinexplicitformisgivenby
rrΔrmf(n)=(−1)r−kf(n+km).
k=0kDefinition3.4.1.For1<p<∞,t∈R+,ther-thordermodulusofsmooth-
nessoffinlp(Z)isdefinedby
ωpr(f,t)=supΔrmf(∙)p.
m∈Z,|m|<t

.4.1)(3

.4.2)(3

Thelp(Z)−moduliofsmoothnesssharepropertiesoftheirLp(R)−analogs,see[25,
34].Inthefollowing,wewilllistsomeofthemwhichwillbeneededfurtheron.
1.ωpr(f,t)isanincreasingfunctionoft.
2.For1≤s≤randeacht∈R+
ωpr(f,t)≤2r−sωps(f,t),(3.4.1)
andmoreover,iff∈lp(Z)
ωpr(f,t)≤2rfp.(3.4.2)
3.Letf,gbedefinedonZ.Then,foreacht∈R+,
ωpr(f+g,t)≤ωpr(f,t)+ωpr(g,t),(3.4.3)
andforfmultipliedbyascalarα,
ωpr(αf,t)≤|α|ωpr(f,t).(3.4.4)
Asωpr(f,t)vanishesforpolynomialsonZofdegree≤r−1,ωpr(∙,t)isasemi-
normonthesetofsequencesforwhichωpr(f,t)<∞forallt∈R+.

.4.3)(3

3.4‘Intrinsic’CharacterizationsofBαp,q(Z)

4.ForM∈N,
ωpr(f,M∙t)≤Mrωpr(f,t).
Soifωpr(f,t)<∞forsomet>0,itisfiniteforallt∈R+.

79

).4.5(3

Definitiαonp3.4.2.Forα>0,1<p,q<∞,r=α+1,thesequencefissaidto
beinBq(l(Z))if
fBqα(lp(Z)):=((2−jαωpr(f,2j))q)1/q<∞.(3.4.6)
1≥j

The∙Bqα(lp(Z))aresemi-normsingeneralbecauseofthepolynomialcancellation
propertiesofthemoduliofsmoothness;theybecomenormsmodulopolynomialson
Zofdegree≤r−1.Furthermore,theBqα(lp(Z))-normsareallequivalentmodulo
polynomialsusingdifferentmoduliofsmoothnessr>αinthedefinition.
OuraimistoshowthattheBqα(lp(Z))-spacescoincidewiththediscrete-timeBesov
spaces,atleastfortherangeofparametersgiveninDefinition3.4.2.
Westartourpreparationsforthisbyconsideringanorthonormaldiscrete-time
waveletbasisfor2(Z),(hj,l)j≥1,l∈Zwithassociatedscalingsequences(gj,l)j≥1,l∈Z,
satisfying

|suppgj,l|,|supphj,l|≤C2j,,(3.4.7)
gj,l∞≤C2−j/2,hj,l∞≤C2−j/2(3.4.8)
.4.9)(3.InRemark2.2.3,wenotedthatthefamilyofprojections(Pj)j≥1,Pjf=l∈Zf,gj,lgj,l
definesadecreasingsequenceVj=Pj(2(Z))ofclosedsubspaceswhichsharemany
propertiesofanMRAinL2(R).LetthespacesWjbedefinedlikewise,usingthe
projectionsQjf=l∈Zf,hj,lhj,l.
Obviously,wehaveforFj∈Vj,Fj=l∈Zaj,lgj,l,
Fj22(Z)=|aj,l|2
Z∈landanalogouslyforGj∈Wj,Gj=l∈Zdj,lhj,l
Gj22(Z)=|dj,l|2.
Z∈lWewillnowinvestigatethebehavioroftheprojectionsinspacesp(Z),forp=2.
ThefollowingLemmarelatesthep-normoffunctionsinVj,Wjtothep-normsof

80

3.4‘Intrinsic’CharacterizationsofBαp,q(Z)

theircoefficients.Thistypeofresultissometimescalled‘p-stability’;foranalogous
resultsincontinuoustime,seee.g[34],Section8.1.
Lemma3.4.3.Let1<p<∞,(gj,l),(hj,l)satisfying(3.4.7),(3.4.8).
Then,forFj∈Vj,Fj=l∈Zaj,lgj,l,j≥1,
Fjp(Z)2−j/22j/p(|aj,l|p)1/p,(3.4.10)
Z∈laswellasforGj∈Wj,Gj=l∈Zdj,lhj,l,j≥1,
Gjp(Z)2−j/22j/p(|dj,l|p)1/p.(3.4.11)
Z∈l

).4.10(3

).4.11(3

ProofAs(gj,l)satisfies(3.4.7),(3.4.8),wehaveespeciallyforanyM>0that
|gj,l(n)|≤C2−j/2(1+2−j|n−2jl|)−MforanyM>0,
seetheargumentintheproofofLemma3.3.6.UsingthistogetherwithH¨older’s
inequalitygivesfor1/p+1/p=1
Fjpp(Z)=aj,lgj,l(∙)pp(Z)
Z∈l≤(|aj,l||gj,l(n)|1/p|gj,l(n)|1/p)p
n∈Zl∈Z
≤C∙(|aj,l|2−j/2p(1+2−j|n−2jl|)−M/p2−j/2p(1+2−j|n−2jl|)−M/p)p
n∈Zl∈Z
≤C∙|aj,l|p2−j/2(1+2−j|n−2jl|)−M(2−j/2(1+2−j|n−2jl|)−M)p/p.
n∈Zl∈Zj∈Z
WiththehelpofLemma3.3.1,weknowthat
(1+2−j|n−2jl|)−M≤C,
Z∈lsaellwas(1+2−j|n−2jl|)−M≤C∙2j,
Z∈nhatthsuc

3.4‘Intrinsic’CharacterizationsofBαp,q(Z)

81

Fjpp(Z)≤C∙2−j/22j2−jp/2p|aj,l|p=C∙2j2−jp/2|aj,l|p.
l∈Zl∈Z
Theconverseinequalityfollowswiththesamearguments,seee.g.Proposition8.1
in[34].(3.4.11)followsimmediatelyfromtheabovediscussion.
Letnow(hj,l),(gj,l)satisfy
|suppgj,l|,|supphj,l|≤C2j,(3.4.12)
gj,l∞≤C2−j/2,hj,l∞≤C2−j/2(3.4.13)
n∈Znihj,l(n)=0fori=0,...,N−1,(3.4.14)
(gj,l),(hj,l)regularoforderN+ε,0<ε≤1.(3.4.15)
Below,itwillbeusefultoexpressTheorem3.3.8intermsoftheprojectionsPj,Qj.
WeobtainbyLemma3.4.3thatforN>α,thefollowingconditionsareequivalent:
f∈Bαp,q(Z)(3.4.16)
(f,hj,l)j≥1,l∈Z∈bαp,q(Z)(3.4.17)
(j≥1(2−jαQjfp)q)1/q<∞,(3.4.18)
(j≥1(2−jαf−Pjfp)q)1/q<∞,(3.4.19)
wheretheequivalenceof(3.4.18)and(3.4.19)easilyfollowsfromthefactthat
jf−Pjfp≤Qjfp
=1iandbysummingupthegeometricseries.
ThenextsteptowardsourintendedresultisthefollowingLemma,whichinliterature
ofteniscalledaninequalityofBernstein-type:
Lemma3.4.4.Let1<p<∞,r∈N+andlet(gj,l)j≥1,l∈Zsatisfy(3.4.12),
(3.4.13)and(3.4.15)withN≥r.
ForFi∈Vi,i≥1,wehaveforanyj≥1
ωpr(Fi,2j)≤Cmin(2(j−i)r,1)Fip.(3.4.20)

ProofLetFi∈Vi∩lp(Z).ByLemma3.4.3,Fi=l∈Zai,lgi,lwithFip
2−i/2∙2i/p(ai,l)l∈Zp.
Inthecasej>i,wehaveimmediatelyωpr(Fi,2j)≤2rFipby(3.4.2),soweonly

82

3.4‘Intrinsic’CharacterizationsofBαp,q(Z)

treatthecasej≤i.
Considerther-thorderdifferenceoperatorofstep1:
Δ1rFi(∙)pp=|ai,lΔ1rgi,l(n)|p
n∈Zl∈Z
≤(|ai,l||Δ1rgi,l(n)|1/p|Δ1rgi,l(n)|1/p)p,
n∈Zl∈Z
where1/p+1/p=1.
AsN≥r,wehavebytheregularityassumption(3.4.15)that|Δ1rgi,l(n)|=2−ir|Dirgi,l(n)|≤
C2−ir2−i/2.TogetherwithH¨older’sinequality,thisgives
Δ1rFi(∙)pp≤(|ai,l|p|Δ1rgi,l(n)|)(|Δ1rgi,l(n)|)p/p
n∈Zl∈Zl∈Z
≤C|ai,l|p2i∙2−i(r+1/2)∙2−i(r+1/2)(p/p)
Z∈l≤C∙2−irp∙2i(1−p/2)|ai,l|p
Z∈l≤C∙2−irpFipp,
wherethelastinequalityfollowsfromLemma3.4.3.
Thisyieldstheresult,asby(3.4.5)and(3.4.2)
ωpr(Fi,2j)≤C∙2jrωpr(Fi,2)≤C∙2(j−i)rFip.

Next,werelatethesizeofcoefficientsatagivenscaletothemodulusofsmoothness,
derivinganinequalityofJackson-type.
Lemma3.4.5.Let1<p<∞,r∈N+,andlet(hj,l)j≥1,l∈Zbeanorthonormal
waveletbasisfor2(Z),satisfying(3.4.12),(3.4.13)and(3.4.14)forsomeN≥r.
Foranyj≥1,
(f,hj,l)l∈Zp≤C∙2j/22−j/pωpr(f,2j).(3.4.21)

ProofLetChbethesmallestintegersuchthat|supphj,l|≤Ch∙2j.Withoutloss
ofgenerality,weassumehtobecausal,i.e.supphj=supphj,0⊆[0,Ch∙2j[.
LetN∈N,0<−k<N.Then(see3.3.4),
Nf(n−k)=NΔif(n−k−N).
i=0i

3.4‘Intrinsic’CharacterizationsofBαp,q(Z)

83

SetN−1
pn(k)=NΔif(n−k−N).
i=0iriteWf,hj,l=f(n)hj,l(n)=f(n)hj∗(2jl−n)
n∈supphj,ln∈supphj,l
0=hj∗(k)f(2jl−k),
k=−Ch∙2j+1
wherehj∗(n)=hj(−n).
Duetothevanishingmomentcondition(3.4.14)on(hj,l),
0|f,hj,l|p=|hj∗(k)(f(2jl−k)−p2jl(k))|p
k=−Ch∙2j+1
00≤(|hj∗(k)|p)p/p∙|ΔNf(2jl−k−N)|p,
k=−Ch∙2j+1k=−Ch∙2j+1
where1/p+1/p=1.
ByLemma3.4.3,hjp2−j/22j/p=2j/22−j/p.Hence,summingoverl,and
observingthateachm∈ZisinthesupportofatmostChshiftsofhj,weobtain
1/p01/p
|f,hj,l|p≤C∙2j/22−j/p|ΔNf(2jl−k−N)|p
l∈Zl∈Zk=−Ch∙2j+1
/p1≤C∙2j/22−j/p|ΔNf(m)|p
Z∈m≤C∙2j/22−j/pωpN(f,2)
≤C∙2j/22−j/pωpr(f,2j),
whereweusedthemonotonicitypropertiesofωpr,specifically(3.4.1),andN≥r.

84

3.4‘Intrinsic’CharacterizationsofBαp,q(Z)

Wewillmakeuseofbothoftheaboveinequalitiesinordertoshowtheequivalence
ofthesemi-norm(3.4.6)andtheBesovsemi-norm.
Theorem3.4.6.Letα>0,1<p,q<∞.
ThespacesBqα(lp(Z))andBαp,q(Z)coincide(modulopolynomials)andmoreover,
∙Bqα(lp(Z))∙Bαp,q(Z).(3.4.22)

Proof(Compareto[34].)Let(Vj)j≥1aN−regularmultiresolutionanalysiswith
NThen,>αf.=LetfP−jPfjfbe+thei≥orjPithogf−onalPi+1fpro.AsjectionPif−ofPfi+1=f(f∈(nVi))nand∈ZNonto≥rVj=.α+1we
canemploy(3.4.20)which,togetherwith(3.4.2)andthetriangleinequality,gives
ωpr(f,2j)≤ωpr(f−Pjf,2j)+ωpr(Pif−Pi+1f,2j)
j≥i≤2rf−Pjfp+C2j−iPif−Pi+1fp
j≥i≤C2j−if−Pifp.
j≥iSo,usingMinkowski’sinequalityand(3.4.19),weget
fBqα(lp(Z))=((2−jαωpr(f,2j))q)1/q
1≥j≤C((2−jα2j−if−Pifp)q)1/q
j≥1i≥j
≤C((2−jαf−Pjfp)q)1/q≤CfBαp,q(Z).
1≥jTheconverseinequalityeasilyfollowsfrom(3.4.21):
fBpα,q(Z)≤C(((2−j(α+1/2−1/p)|f,hj,l|)p)q/p)1/q
j≥1l∈Z
≤C((2−jαωpr(f,2j))q)1/q=CfBqα(lp(Z)).
1≥jforfTheoremurther3.3a.8isnalysisaoffirstdisexamplecrete-timehowBethesovwavspaecletes.cIhanracterizatparticular,ionthecanbfiniteeemplosuppyorted
ofthewaveletshasgreatlyfacilitatedtheproof.
Anothersuchcharacterization,alsoadirectconsequenceofTheorem3.3.8,isgiven
inthenextsubsection.

3.4‘Intrinsic’CharacterizationsofBαp,q(Z)

85

3.4.2MeanOscillationCharacterizationofBαp,q(Z)
Inthissection,wederiveadescriptionofthediscrete-timeBesovspacesintermsof
oscillations,similarlytothemeanoscillationcharacterizationoftheircontinuous-
timecounterparts,see[8,9].
ForthespecialcaseB1p,p/p(Z),1<p<∞,R.H.Torres[30]showedthatthecorre-
spondingnormisequivalenttotheBp(Z)-norm,definedby
1fBp(Z):=((|Ij,l||f(n)−fIj,l|)p)1/p,(3.4.23)
j≥1l∈Zn∈Ij,l
whereIj,l:=[2jl,2j(l+4)]andfIj,ltheaverageoffonIj,l.
Thisdefinesasemi-normingeneralandanormmoduloconstants.
Here,weextendthisresulttoBesovspaceswhereα>0,1<p,q<∞,using
Theorem3.3.8.Weuseaslightlydifferentnotationcomparedtothearticlescited
above,inparticulartheBp(Z)-spaceswillcorrespondtoMO1p,p/p(Z)definedbelow.
Definition3.4.7.Afamilyofintervals(Ij,l)j≥1,l∈Ziscalledafamilyofadmissible
iferingsvco1.AnyoftheintervalsisoftheformIj,l=[2jl,2jl+Lj[,where
2.forj→∞,2−j∙Lj→C,whereC>1.
satisfyingl∈ZIj,l∩Z=Z.
Foranyj≥1,thefamilyJj:=(Ij,l)l∈Zisthefamilyofenlargeddyadicintervals,
Letf=(f(n))n∈Z,Ianarbitraryintervalandm∈N0.ByfI(m)(n),wedenotethe
(unique)polynomialonZofdegreesmallerorequaltom,suchthat
(f(n)−fI(m)(n))nk=0fork=0,1,...,m.(3.4.24)
I∈nBydefinition,fI(m)isthebestapproximationoffonIbyapolynomialofdegree
m,measuredinthel2(Z)-norm.
Themeanoscillationnormisnowdefinedbythenormsoftheresiduals.
Definition3.4.8.Let1<p,q<∞,m∈N+andlet(Ij,l)j≥1,l∈Zbeafamilyof
admissiblecoverings,whereJj=(Ij,l)l∈Zthecoveringofthelineatscalej.Define
forf=(f(n))n∈Z
oscp,m(f,Jj)=((1|f(n)−fI(j,ml)(n))|)p)1/p.(3.4.25)
l∈Z|Ij,l|n∈Ij,l

86

3.4‘Intrinsic’CharacterizationsofBαp,q(Z)

Letα>0,m=α.AsequencefisinMOαp,q(Z)if
fMOαp,q(Z):=((2−j(α−1/p)oscp,m(f,Jj))q)1/q<∞.(3.4.26)
1≥j

Thesemi-normsin(3.4.26)arenormsmodulodiscrete-timepolynomials.

Usingadifferentfamilyofadmissiblecoveringsoradifferentm>αresultsin
equivalentnorms,seeSection9in[9].
Justlikeintheprevioussection,weuseTheorem3.3.8toshowthattheBesovand
meanoscillationnormsareinfactequivalent.
So,letinthefollowingagain(hj,l)j≥1,l∈Zbeanorthogonalwaveletbasisfor2(Z),
satisfying(3.4.12),(3.4.13),(3.4.14)and(3.4.15)forsomeN>α
Theorem3.4.9.Letα>0,1<p,q<∞.
ThespacesMOαp,q(Z)andBαp,q(Z)coincide(modulopolynomials)andmoreover,
∙MOαp,q(Z)∙Bαp,q(Z).(3.4.27)

hProofFirst,letIj,l:=supphj,l.
Duetothemomentcondition(3.4.14)onhj,landasN>m=α,
)m(|f,hj,l|=|f−fIh,hj,l|
l,j)m(≤hj,l∞|f(n)−fIh(n)|
l,jn∈Ijh,l
≤C2−j/2|f(n)−f(hm)(n)|
Il,jn∈Ijh,l
1≤C2j/2|f(n)−f(hm)(n)|.
|Ijh,l|hIj,l
n∈Ij,l

.8,3.3theoremUsingfBαp,q(Z)≤C(f,hj,l)bαp,q(Z)
=C(((2−j(α+1/2−1/p)|f,hj,l|)p)q/p)1/q
j≥1l∈Z
≤C((2−j(α−1/p)((1|f(n)−f(hm)(n))|)p))q/p)1/q,
Ihj≥1l∈Z|Ij,l|n∈Ihj,l
l,j

3.4‘Intrinsic’CharacterizationsofBαp,q(Z)

87

wherethelasttermisequivalenttotheMOαp,q(Z)-norm,as(Ijh,l)j,lisafamilyof
admissiblecoverings(eventually,theindexinghastobechangedinordertobe
conformwiththefirstpostulationofDefinition3.4.7,asthehj,larenotnecessarily
.causal)Inordertoprovetheconverseinequality,notethatforanypolynomialpofdegree
lessorequaltomandagivenintervalI(see[8],proofofTheorem1)
1|f(n)−fI(m)(n)|≤C1|f(n)−p(n)|.(3.4.28)
|I|n∈I|I|n∈I
Let(Ij,l)beafamilyofadmissiblecoverings.Definepm(n):=im=0Δif(n−(m+1)).
Asf(n)−pm(n)=Δm+1f(n−(m+1))andby(3.4.28),
11|Ij,l||f(n)−fI(j,ml)(n))|≤C|Ij,l||f(n)−pm(n)|
n∈Ij,ln∈Ij,l
≤C1|Δm+1f(n−(m+1))|,
|Ij,l|n∈Ij,l
esgivhwhicfMOαp,q(Z)≤C((2−j(α−1/p)((1|Δm+1f(n−(m+1))|)p))q/p)1/q
j≥1l∈Z|Ij,l|n∈Ij,l
≤C((2−j(α−1/p)2−j/p(|Δm+1f(n−m+1)|p)1/p)q)1/q
j≥1n∈Ij,l
≤C((2−jαωpm+1(f,2j))q)1/q
1≥j≤CfBαp,q(Z),
using(3.4.22)asr=m+1bydefinition.
.104.3.RemarkTheusageofenlargeddyadicintervalsinthedefinitionoftheMOαp,q-spacesiscrucial.
jConsiderjtheusualnon-overlappingdyadicintervalfamily(Ij∗,l)j≥1,l∈Z,whereIj∗,l=
[2l,2(l+1)[.Thisfamilyisnotadmissibleintermsofourdefinition.
DefinethespacesBp∗(Z),1<p<∞asthecollectionofsequences(modulocon-
stants)forwhich
1p1/p
fBp∗(Z):=((|Ij∗,l|∗|f(n)−fIj∗,l|))
j≥1l∈Zn∈Ij,l

.efinitis

88

3.4‘Intrinsic’CharacterizationsofBαp,q(Z)

Let(f(n))n∈Zbet∗hesequence∗definedbyf(n)=0forn<0andf(n)=1,n≥0.
Obviously,f∈Bp(Z),astheBp(Z)-normwillnotfeelthe‘discontinuity’at0,
whereastheBp(Z)-norm(usingadmissiblefamilies)will:
Considertheintervalfamily(Ij,l)j≥1,l∈Z,Ij,l=[2jl,2j(l+2)[,whichisadmissible.

1p1/p
fBp(Z)=fMO1p,p/p(Z)=((|I||f(n)−fIj,l|))
j≥1l∈Zj,ln∈Ij,l
=((2−(j+1)(2j+1∙1))p)1/p=(2−p)1/p,
j≥12j≥1

whichgivesf∈Bp(Z).
ThisizatioanlsofotheindicatspacesesthatMOtheα(Z)regularanditthyusconditioforBαn(isZ):crucialforthewaveletcharacter-
p,qp,qConsiderthediscrete-timeHaarfilters(2.1.5).
Thediscrete-timeHaarwaveletbasesandscalingsequencesreadas
Gj(n)=2−j/2forn=0,...,2j−1;(3.4.29)
etherwiso0and−2−j/2forn=0,...,2j−1−1;
therwiseo0Hj(n)=2−j/2forn=2j−1,...,2j−1;(3.4.30)
withtheusualtranslationGj,l=Gj(∙−2jl),Hj,l=Hj(∙−2jl).
Thediscrete-timeHaarsystem(Hj,l)j≥1,l∈Zisanorthonormalbasisfor2(Z),pos-
sessingonevanishingmomentbutnotbeingregularinthesenseof(2.2.7).Itis
easytoshowthatduetothemomentcondition,forf∈Bp(Z),
(f,Hj,l)b1p,p/p(Z)≤CfBp(Z).
Buttheconverseinequalitycannotbeobtained.
Infact,onecan∗adapttheaboveargumentstoshowthat(f,Hj,l)b1p,p/p(Z)isequiv-
alenttotheBp(Z)-norm,whichisnotthesameasBp(Z).
Inasense,therequiredoverlapofadmissibleintervalsisrelatedtothesupport
size,andthustotheregularityofthediscretetimewavelets(recallthecorrelation
betweensupportsizeandregularityfortheDaubechiesfamily.)

hapterC4

Discrete-TimeTriebel-Lizorkin
Spaces

wFaorvtheelet-basedreadertreafatmenmiliartofwithdiscrete-timefunctionBesspaces,ovitspaceswillcannotbbeeextendedsurprisingtothatdiscrete-the
timeversionsofTriebel-Lizorkinspaces.
atThesomicespacedecompsofositsequencesions.wWithereourstudiednotiobynsQ.atSunhand,inw[29e],canmoregivineatermsdescrofiptionsmootinh
termsofdiscrete-timebases.
Inthefirstsection,wegivethedefinitionoftheTriebel-Lizorkinspacesindiscrete
timeandtheirϕ-transformcharacterization.
InSubsection4.2,weestablisharesultanalogoustoTheorem3.3.8forthistypeof
spaces.ThetechniquesusedwillbemostlythesameasfortheBesovspaces.
ForbackgroundinformationconcerningTriebel-Lizorkinspacesincontinuoustime,
werefertotheusualliteraturease.g.[32],[17]or[16].
4.1Definitionandϕ-transformDecomposition

Letϕcaphi-function(see3.1.2)andsetϕνc(x)=2−ν+2ϕc(2−ν+2x)forν∈Z.
Consideragainthefamily(ϕν)ν≥1,obtainedbyϕν:=ϕνc|Zforν>1andϕ1:=
((χ[−π,π]ϕˆ1c)∨)|Z.Recallthatforf∈S(Z),wehavef=ν≥1f∗ϕν∗ϕν∗with
unconditionalconvergenceinS/P(Z)(3.1.6).
Definitiαon4.1.1.Forα∈R,0<p,q<∞,thediscrete-timeTriebel-Lizorkin
spaceFp,q(Z)isthecollectionofallf∈S/P(Z),suchthat
fFαp,q(Z):=((2−να|f∗ϕν|)q)1/qp<∞.
1≥νThisdefinitionisindependentofthechoiceofϕc[29].

89

90

4.1Definitionandϕ-transformDecomposition

α0De<finip,qtion<4.∞,1.b2.eLtheetctheollespacctioneofofc(truncateomplex-valued)codefficientsequencfamiliesessf=p,q((sjZ,l)),j≥for1,l∈αZ,∈forR,
whichsfαp,q(Z):=((2−jα|sj,l|χ˜j,l)q)1/qp<∞,
j≥1l∈Z
whereχ˜j,l(n)=2−j/2,if2jl≤n<2j(l+1);
otherwise.,0Wenotethatthedefinitionoffαp,q(Z)issimilartothatofbαp,q(Z),butusesadiffer-
entsummationorder(firstoverscales,thenoverpositions)andthecharacteristic
functionsχ˜j,l.Hencethecoefficientspacebαp,q(Z)issomewhateasiertohandleby
rison.compaIn[29],Q.Sunrelatestheaboveαspacesviatheϕ−transform,similarlytotheresult
forthediscrete-timeBesovandbp,qspacesinsection3.2.
Forν,k∈Zletagain
ϕνc,k(x)=2(−ν+2)/2ϕc(2−ν+2x−k),
andfork∈Z,defineϕν,k=ϕνc,k|Zforν>1andϕ1,k=τkϕ1.
Recallthatforanyf∈S(Z)(3.2.1)
f=ν≥1k∈Zf,ϕν,kϕν,k,(4.1.1)
withconvergenceinS/P(Z).
sνLet,k=thef,ϕϕν,k-transfor,andmforSϕaforfcomplex-v∈S(aZ)luedbesequedefinedncebty=S(ϕtfν,k=)ν≥s1,k=∈(Zsν,k)defineν≥1,kthe∈Z,invewhererse
ϕ-transformbyTϕbyTϕt=ν,ktν,kϕν,k.Theconvergenceofthesumisjustified
bythefollowingresult.
Theorem4.1.3.([29])Letα∈R,0<p,q<∞.
BothoftheoperatorsSϕ:Fp,qα(Z)→fαp,q(Z)andTϕ:fαp,q(Z)→Fp,qα(Z)arebounded
withfFαp,q(Z)Sϕffαp,q(Z)andTϕ◦Sϕ=idBαp,q(Z).

.1.1)(4

Remark4.1.4.Consideringthespecialchoiceα=0,1<p<∞,q=2,
fFp,02(Z)=((|f∗ϕν|)2)1/2p,
1≥νonecanseethatthisisexactlytheLittlewood-Paleytypedefinitionofthep(Z)-
norm(seeagain[29]).
Thus,ourresultintheupcomingsectionincludesthediscrete-timewaveletcharac-
terizationofthespacesp(Z),1<p<∞.

4.2WaveletCharacterizationofDiscrete-TimeTriebel-LizorkinSpaces91

4.2WaveletCharacterizationofDiscrete-TimeTriebel-
SpaceskinzorLiForourwaveletdescriptionofthespacesFp,qα(Z),weneedanadditionaltool,and
certaininequalitiesrelatedtoit.
ThiswillbetheissueofthenextdefinitionandtheLemmata4.2.2,4.2.3and4.2.4.
Definition4.2.1.Letf=(f(n))n∈Z.
DefinetheHardy-LittlewoodmaximaloperatoronZby
1Mf(k):=a≤k<b,supa,b∈Zb−a|f(n)|.(4.2.1)
n<b≤a

ThenextLemmaistakenfrom[29]andcanbeviewedasadiscrete-timeversionof
theFefferman-Steinmaximalinequality:
Lemma4.2.2.Let1<p,q<∞.Foranyfamilyofsequences(fi)i∈Z
(|Mfi|q)1/qp≤C(|fi|q)1/qp.(4.2.2)
i∈Zi∈Z

Asimilarresultholdsfor0<p,q≤1.
Inthiscase,replacingthemaximaloperatorMbyMr,0<r<min(p,q),inLemma
(4.2.2),definedby

1/r1rMrf(k):=a≤(k<b,supa,b∈Zb−a|f(n)|),
n<b≤a

yieldsLemma4.2.3.
(|Mrfi|q)1/qp≤C(|fi|q)1/qp.
i∈Zi∈Z

Wewillalsoneedthefollowinginequalities:
Thefirstonecanbefoundin[14]andiseasilyadaptedtothediscretecase.

).2.3(4

.2.4(4)

924.2WaveletCharacterizationofDiscrete-TimeTriebel-LizorkinSpaces

Lemma4.2.4.Let(sj,l)j≥1,l∈Z⊂C.Fixν,j≥1,k∈Zand2νk≤n≤2ν(k+1).
ForM1>1
jν|sj,l|(1+|2kν−2l|)−M1≤C2(ν−j)M(|sj,l|χj,l)(n),ifj<ν,(4.2.5)
2l∈Zνjl∈Z
|sj,l|(1+|2k2−j2l|)−M1≤CM(|sj,l|χj,l)(n),ifj≥ν.(4.2.6)
l∈Zl∈Z
Let0<r<1,M2>1/r.Then,
jν|sj,l|(1+|2kν−2l|)−M2≤C2νr−jMr(|sj,l|χj,l)(n),ifj<ν,(4.2.7)
l∈Z2l∈Z
|sj,l|(1+|2νk−2jl|)−M2≤CMr(|sj,l|χj,l)(n),ifj≥ν.(4.2.8)
jl∈Z2l∈Z

Withtheseresultsathand,weareabletoestablishasimilarboundednessresultas
.7:3.3LemmainLemma4.2.5.Letα∈R,0<p,q<∞andlet(ϕν,k)ν≥1,k∈Zbetheϕ−transform
family.If(hj,l)j≥1,l∈ZsatisfiestheconditionsofLemma3.3.6withN1>1/(min(1,p,q))−
onfαp,q(Z),whereAs=(j≥1l∈Zhj,l,ϕν,ksj,l)ν,kfors∈fαp,q(Z).
1−α,N2>α,thenthematrixA:=(hj,l,ϕν,k)j,l,ν,k,definesaboundedoperator
Also,for(h˜j,l)j≥1,l∈Z,satisfyingtheconditionsofLemma3.3.6,wherenowN1>
(ν≥1k∈Zϕν,k,h˜j,lsν,k)j,l,isboundedonfαp,q(Z)aswell.
α,N2>1/(min(1,p,q))−1−α,thematrixA˜:=(ϕν,k,h˜j,l)ν,k,j,l,where˜As=
Proof
Asfαp,q=((2−να|hj,l,ϕν,ksj,l|χ˜ν,k)q)1/qp
ν≥1k∈Zj≥1l∈Z
≤((2−να|hj,l,ϕν,k||sj,l|χ˜ν,k)q)1/qp
ν≥1k∈Zj≥1l∈Z
≤C((2−να|hj,l,ϕν,k||sj,l|χ˜ν,k)q)1/qp
ν≥1k∈Zj<νl∈Z
+((2−να|hj,l,ϕν,k||sj,l|χ˜ν,k)q)1/qp
ν≥1k∈Zj≥νl∈Z
=:CI+II.

4.2WaveletCharacterizationofDiscrete-TimeTriebel-LizorkinSpaces93

Considerthecase1<p,q<∞.
UsingLemma3.3.6and(4.2.6),wecanestimatethefirsttermby
I≤C((2(j−ν)(N1+1/2)(1+|2kν−2l|)−M1|sj,l|2−ναχ˜ν,k)q)1/qp
νj
2ν≥1k∈Zj<νl∈Z
≤C(((2(j−ν)(N1+1/2−1)2−ναM(|sj,l|χj,l)(n)χ˜ν,k(n))q)1/q)n∈Zp
ν≥1k∈Zj<νl∈Z
=C(((2(j−ν)(N1+1/2−1+α+1/2)M(2−jα|sj,l|χ˜j,l)(n))q)1/q)n∈Zp
ν≥1j<νl∈Z
≤C((M(2−να|sν,k|χ˜ν,k))q)1/qp,
ν≥1k∈Z
wherewesumupageometricseriestheinlastinequality,asN1>−αbyassumption.
TheinequalityinLemma4.2.2yields
I≤C((2−να|sν,k|χ˜ν,k)q)1/qp≤Csfαp,q(Z).(4.2.9)
ν≥1k∈Z
Alongthesamelines,wecanestimatethesecondtermby
II≤C((2(j−ν)(N1+1/2)(1+|2kj−2l|)−M2|sj,l|2−ναχ˜ν,k)q)1/qp
νj
2ν≥1k∈Zj≥νl∈Z
≤C(((2(ν−j)(N2+1/2−α−1/2)M(2−jα|sj,l|χ˜j,l)(n))q)1/q)n∈Zp
ν≥1j≥νl∈Z
≤Csfαp,q(Z),
asweassumedN2>α.
Nowtothecase0<p,q≤1.Let0<r<min(p,q).ByLemmata3.3.6,4.2.3and
).2.8(4I≤C((2(j−ν)(N1−1/r+α+1)Mr(2−jα|sj,l|χ˜j,l))q)1/qp
ν≥1k∈Zj<νl∈Z
≤Csfαp,q(Z),
asN1>1/r−1−αbyassumption.
Theothertermcanbetreatedanalogously.
Onceweestablishedtheaboveresult,weimmediatelygetananalogontoTheorem
.8:3.3Let(hj,l),(h˜j,l),j≥1,l∈Z,arebiorthogonalwaveletbasesfor2(Z).Recallthe
operatorsSh,Th:forf∈S(Z),letShf=(f,hj,l)j≥1,l∈Z.Forfinitelysupported
coefficientsequencesd=(dj,l)j≥1,l∈Z,define
Thd=dj,lhj,l.(4.2.10)
,lj

944.2WaveletCharacterizationofDiscrete-TimeTriebel-LizorkinSpaces

LetSh˜andTh˜bedefinedlikewise.
Theorem4.2.6.Let˜α∈R,0<p,q<∞,N>α,N˜>1/(min{1,p,q})−12−α.
Supposethat(hj,l),(hj,l),j≥1,l∈Z,arebiorthogonalwaveletbasesfor(Z),
satisfying|supphj|,|supph˜j|≤C2j,
hj∞≤C2−j/2,h˜j∞≤C2−j/2,
n∈Znihj(n)=0fori=0,...,N˜−1,
n∈Znih˜j(n)=0fori=0,...,N−1,
(hj)j≥1regularoforderN+ε(inthesenseof2.2.7),0<ε≤1,
(h˜j)j≥1regularoforderN˜+ε,˜0<ε˜≤1.
Thenthefollowingstatementshold:
(a)TheanalysisoperatorSh˜:Fp,qα(Z)→fαp,q(Z)iswell-definedandcontinuous.
(b)Thesynthesisoperatorextendsuniquelytoaboundedoperatorfαp,q(Z)→
Fp,qα(Z),alsodenotedbyTh.Forarbitrary(dj,l)j≥1,l∈Z∈fαp,q(Z),
Th((dj,l)j≥1,l∈Z)=dj,lhj,l,(4.2.11)
,ljwithunconditionalconvergenceintheTriebel-Lizorkinspacenorm.
(c)Th◦Sh˜=idFαp,q(Z),andSh˜◦Th=idfαp,q(Z).Thus,Fp,qα(Z)canbeidentifiedwith
fαp,q(Z)underthemapsSh˜andTh.
(d)WehavethenormequivalencefFαp,q(Z)Sh˜ffαp,q(Z).Moreover,thewavelet
ansionexpf=f,h˜j,lhj,l
,ljholdswithunconditionalconvergenceintheTriebel-Lizorkinspacenorm.

ProofThestructureoftheproofisthesameasforTheorem3.3.8.
wAsehavLemmaethat3.2fo.2rf∈Fimmediatelyp,qα(Z),cja≥rries1,lov∈erZtoandtheK=discrete-timemax{[α−Tr1ie/pb],−1el-Lizork},inspaces,
f=ν,kf,ϕν,kϕν,k
ofholdsAandin(SA˜K(a(Zs))pro.WvidedehabvyeagLemmaainTh=4.2.T5)ϕ◦Ayieldsand(aS),h=andA˜◦theSϕfir,sthencepartbof(b).oundedness

4.2WaveletCharacterizationofDiscrete-TimeTriebel-LizorkinSpaces95

Forunconditionalconvergence,wenotebyapplyingthedominatedconvergence
theorem,itcanbeshownthatevery(dj,l)∈fαp,q(Z)isthelimitofthenetofits
finiterestrictions.ThisandboundednessofThyieldsunconditionalconvergenceof
1)..2.1(4(3.3.19),thetailj∞=J+1l∈Zf,h˜j,lhj,lconvergesagainstronglyto0asJ→∞:
Again,wehave(Th◦Sh)f=f∈(SK(Z))andalsointhenorm.Inanalogyto
Jf−f,h˜j,lhj,lFαp,q(Z)
Z∈l=1jJ≤C(f−f,h˜j,lhj,l,ϕν,k)ν,kfαp,q(Z)
Z∈l=1j∞=C(f,h˜j,lhj,l,ϕν,k)ν,kfαp,q(Z)
j=J+1l∈Z
≤C((2−να|f,h˜j,lhj,l,ϕν,k|χ˜ν,k)q)1/qp
ν≥1k∈Zj>J,j<νl∈Z
+((2−να|f,h˜j,lhj,l,ϕν,k|χ˜ν,k)q)1/qp
ν≥1k∈Zj>J,j≥νl∈Z
=:CI+II
Let1<p<∞.WecanusetheproofofLemma4.2.5toestimatethetruncated
.seriesForthefirstterm,
I≤C((2(j−ν)(N˜+α)M(2−jα|f,h˜j,l|χ˜j,l))q)1/qp,
ν≥1j>J,j<νl∈Z
andII≤C((2(ν−j)(N−α)M(2−jα|f,h˜j,l|χ˜j,l))q)1/qp
ν≥1j>J,j≥νl∈Z
SummingupgeometricseriesandapplyingLemma4.2.2yields
I≤C((M(2−να|f,h˜ν,k|χ˜ν,k))q)1/qp
ν>Jk∈Z
≤C((2−να|f,h˜ν,k|χ˜ν,k)q)1/qp
Z∈k>Jν=C(f,h˜ν,k)χν>Jfαp,q(Z),

96

4.2WaveletCharacterizationofDiscrete-TimeTriebel-LizorkinSpaces

forwhereJ→χν>J∞.isThethecsaharameacteristicrgumentsfunctionapplyofto{(IνI,k)and:νthe>J}.caseThisp≤s1howsfollothawstIin→the0
samefashionfromtheaccordingestimatesestablishedinLemma4.2.5.
yieldsThis

JJlim→∞f−f,h˜j,lhj,lFαp,q(Z)=0.
Z∈l=1jAgain,biorthogonalityofhj,l,h˜j,lgivesSh˜◦Th=idbαp,q(Z)andpart(d)isimmediate
fromtheabove.

DiscussionAndOutlook

Wefinishthisthesiswithadiscussionofourresultsandanoutlookonpossible
k.orwfurtherThecentralresultinthisthesisisthestudyofnecessaryandsufficientconditions
onwaveletbasesforl2(Z)toconstituteunconditionalbasesfordiscrete-timeBesov
spacesBαp,q(Z)inTheorem3.3.8.
ThDiscrete-us,thetimeheurwaisvticelestcofromefficconienttindecayuous-timecanbectheoryhaaracterreizedsubstanintetiatrmsedofbyomemurbresershipult:
inasuitablesequencespace.
Moreprecisely,thearisingcoefficientsareinbαp,q(Z),ifandonlyifthesequenceis
inthespaceBαp,q(Z).
TheresultsofChapter3haveprovidedavarietyofnewcharacterizationsofBesov
Sevspaceseralinposdiscsibleretewaystime,ofshoexplowingitingthatthesethecseharaspactceesrizaaretionsworthsuggwhileestobjecthemselvtsofes,sostudyme.
ofmentiothemn(rouginspiredhlybinytheexistingorderofresultimpsortforance)thetcohentinfollououswingdolistmain.ofproSpecblems:ifically,we

•Extendingtheresultstohigherdimensions:Inparticularforapplicationsin
imageprocessing,atwo-dimensionalresultwouldbedesirable.Forthecon-
tinuousdomaincase,extensionsofthewaveletcharacterizationtoarbitrary
dimensionshavebeenobtained,andweexpectthatanalogousresultsshould
holdforZd.However,aproofofsucharesultwillhavetodealwitheven
moreinvolvednotation.Also,theprecisechoiceofaLittlewood-Paleytype
characterizationofBesovspacesinhigherdimensioncanbeexpectedtohave
astronginfluence,perhapsnotonthespacesitcharacterizes,butontheeffort
necessarytoprovecharacterizationsintermsofwaveletsystemsobtainedfrom
tensorproductsofone-dimensionalwaveletsandscalingfunctions.

•Developingdiscretetimeheuristicsforsignalprocessingalgorithms:Anobvi-
oustaskinfurtherworkistothrowlightonapplicationssuchascompression
ordenoisingfromafullydiscrete-timeviewpoint,whichourresultprovides.
Forthesakeofconcreteness,letusonlymentiondenoising.Theresultsof
DonohoandJohnstonerelyontheembeddinginthecontinuoustimesetting.

97

984.2WaveletCharacterizationofDiscrete-TimeTriebel-LizorkinSpaces

It3.3sho.8uldinstead,bepossiblewithouttoderivassumingeanaloa‘trguesue’ofthesunderlyingeresultcosntinwhicuohusrely-timeonfuncTheorteion.m

•Extendingthecharacterizationtooperators:Thestudyofcontinuityproper-
tiesofsuitableoperatorswithrespecttovariousBesovnormsisanatural
applicationofthewaveletcharacterization(seee.g.[17],Chapter8).The
operatorscouldbediscrete-timeCalder´on-Zygmundoperators,ornon-linear
operatorsdescribinghistogramequalization.

•StudyingfurtherrelationsbetweendiscretetimeandcontinuoustimeBesov
spthaactes:aInBesov[30],TfunctionorresinprovesdiscreteasatimemplingcanthebeoremunderstoforoBesodavsspacerestrictiosshonwofinga
dingbandliofmiteddiscreteBesovtimeinfunctiotoconnintincouonustinuotimeusBtime.esovspaThiscegivviaestherisestincoanemfunction.bed-
Onspacetheintootheritshaconnd,tinwuoeushavetimetheaonaloftgenviacitedtheembscalingeddingfunctioofnodiscretefantimeMRABesowithv
bysuitablethesincsmofunctioothnessn.Thandusvatherenishingexistmotwomentfundapropmenerties,tallywhicdifferenharteemnotbeddingfulfilles.d
Atimethedescriptionory(inofathesuitconabletinuoussense)timecouldtheocloseryasthisagsymptotap,aicndcaproseovideftheadditdiscrionalete
insightintothediscreteandcontinuous-timespaces.Notethatalsotherela-
tiolimits,nbetasweendescribdiscrete-edintimeSectionandc2.2on.3,tinseuous-emsttoimewaindicateveletsuchsystemsaviaconnectiolargen.scale
Theseresultscouldalsoshedadditionallightontherelationshipbetweenal-
gotheorrithmy,respecheuristicstively.derivedfromdiscretetimeandcontinuoustimeBesovspace

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