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TurkJMath34(2010),393–415.cT¨UB˙ITAKdoi:10.3906/mat-0809-2SomepropertiesofC-fusionframesMohammadHasanFaroughiandRezaAhmadiAbstractIn[10],wegeneralizedtheconceptoffusionframes,namely,c-fusionframes,whichisacontinuousversionofthefusionframes.Inthisarticlewegivesomeimportantpropertiesaboutthegeneralization,namelyerasuresofsubspaces,theboundofc-erasurereconstructionerrorforParsevalc-fusionframes,perturbationofc-fusionframesandtheframeoperatorforfusionpair.KeyWords:Operator,Hilbertspace,Bessel,Frame,Fusionframe,c-fusionframe1.IntroductionandpreliminariesThroughoutthispaperHwillbeaHilbertspaceandHwillbethecollectionofallclosedsubspaceofH.Also,(X,μ)willbeameasurespace,andv:X→[0,+∞)ameasurablemappingsuchthatv=0almosteverywhere(a.e.).WeshalldenotetheunitclosedballofHbyH1.Frameswasfirstintroducedinthecontextofnon-harmonicFourierseries[9].Outsideofsignalprocessing,framesdidnotseemtogeneratemuchinterestuntilthegroundbreakingworkin[8].Sincethenthetheoryofframesbegantobemorewidelystudied.Duringthelast20yearsthetheoryofframeshasgrownuprapidly,withthedevelopmentofseveralnewapplications.Forexample,besidestraditionalapplicationassignalprocessing,imageprocessing,datacompression,andsamplingtheory,framesarenowusedtomitigatetheeffectoflossesinpocket-basedcommunicationsystemsandhencetoimprovetherobustnessofdatatransmissionon[6],andtodesignhigh-rateconstellationwithfulldiversityinmultiple-antennacodedesign[12].In[2,1,3]someapplicationshavebeendeveloped.ThefusionframeswereconsideredbyCasazza,KutyniokandLiinconnectionwithdistributedprocessingandarerelatedtotheconstructionofglobalframes[4,5].Thefusionframetheoryisinfactmoredelicateduetocomplicatedrelationsbetweenthestructureofthesequenceofweightedsubspacesandthelocalframesinthesubspacesandduetotheextremesensitivitywithrespecttochangesoftheweights.In[10]weextendedfusionframestotheircontinuousversionsinmeasurespacesandinthispaperweshallinvestigatesomepropertiesaboutit.1991AMSMathematicsSubjectClassification:Primary46B25,47A05,94A12,68M10.ThisarticlederivesfromthePh.Dthesisofwhichauthor,completedattheuniversityofTabriz.393FAROUGHI,AHMADIDefinition1.1Let{fi}i∈IbeasequenceofmembersofH.Wesaythat{fi}i∈IisaframeforHifthereexist0A≤B∞suchthatforallh∈HAh2≤i∈I|fi,h|2≤Bh2.(1.1)TheconstantsAandBarecalledtheframebounds.IfA,BcanbechosensothatA=B,wecallthisframeanA-tightframeandifA=B=1,itiscalledaParsevalframe.Ifweonlyhavetheupperbound,wecall{fi}i∈IaBesselsequence.If{fi}i∈IisaBesselsequencethenthefollowingoperatorsarebounded:T:l2(I)→H,T(ci)=i∈Icifi(1.2)T∗:H→l2(I),T∗(f)={f,fi}i∈I(1.3)Sf=TT∗f=i∈If,fifi.(1.4)Theseoperatorsarecalledsynthesisoperator;analysisoperatorandframeoperator,respectively.Definition1.2ForacountableindexsetI,let{Wi}i∈IbeafamilyofclosedsubspaceinH,andlet{vi}i∈Ibeafamilyofrealnumbers,calledweights,i.e.,vi0foralli∈I.Then{(Wi,vi)}i∈IisafusionframeforHifthereexist0C≤D∞suchthatforallh∈HCh2≤i∈Ivi2πWi(f)2≤Dh2,(1.5)whereπWiistheorthogonalprojectionontothesubspaceWi.WecallCandDthefusionframebounds.Thefamily{(Wi,vi)}i∈Iiscalledac-tightfusionframe,ifin1.5theconstantsCandDcanbechosensothatC=D,aParsevalfusionframeprovidedC=D=1andanorthonormalfusionbasisifH=i∈IWi.If{(Wi,vi)}i∈Ipossessesanupperfusionframebound,butnotnecessarilyalowerbound,wecallitisaBesselfusionsequencewithBesselfusionboundD.Therepresentationspaceemployedinthissettingis(i∈I⊕Wi)l2={{fi}i∈I|fi∈Wiand{||fi||}i∈I∈l2(I)}.Let{(Wi,vi)}i∈IbeafusionframeforH.Thesynthesisoperator,analysisoperatorandframeoperatoraredefined,respectively,byTW:(i∈I⊕Wi)l2→HwithTW(f)=i∈Ivifi,T∗W:H→(i∈I⊕Wi)l2withT∗W(f)={viπWi(f)}i∈I,394FAROUGHI,AHMADISW(f)=TWT∗W=i∈Iv2iπWi(f).Byproposition3.7in[5],if{(Wi,vi)}i∈IisafusionframeforHwithfusionframeboundsCandDthenSWisapositiveandinvertibleoperatoronHwithCId≤SW≤DId.Thetheoryofframeshasacontinuousversionasfollows.Definition1.3Let(X,μ)beameasurespace.Letf:X→Hbeweaklymeasurable(i.e.,forallh∈H,themappingx→f(x),hismeasurable).Thenfiscalledacontinuousframeorc-frameforHifthereexist0A≤B∞suchthatforallh∈HAh2≤X|f(x),h|2dμ≤Bh2.(1.6)TherepresentationspaceemployedinthissettingisL2(X,μ)={ϕ:X→H|ϕismeasurableandϕ2∞},inwhichϕ2=(X||ϕ(x)||2dμ)12.Thesynthesisoperator,analysisoperatorandframeoperatoraredefined,respectively,byTf:L2(X,μ)→HTfϕ,h=Xϕ(x)f(x),hdμ(x),(1.7)T∗f:H→L2(X,μ)(T∗fh)(x)=h,f(x),x∈X,(1.8)Sf=TfT∗f.(1.9)AlsobyTheorem2.5.in[14],Sfispositive,self-adjointandinvertible.Weneedthefollowingtheoremsandtheproofscanbefoundin[14].Theorem1.4LetfbeacontinuousframeforHwiththeframeoperatorSfandletV:H→Kbeaboundedandinvertibleoperator.ThenV◦fisacontinuousframeforKwiththeframeoperatorVSfV∗.Theorem1.5LetKbeaclosedsubspaceofHandletP:H→Kbeanorthogonalprojection.Thenthefollowingholds:(i)IffisacontinuousframeforHwithboundsAandB,thenPfisacontinuousframeforKwiththeboundsAandB.(ii)IffisacontinuousframeforKwiththeframeoperatorSf,thenforeachh,k∈HPh,k=Xh,S−1ff(x)f(x),kdμ(x).395FAROUGHI,AHMADIThefollowinglemmasandtheoremscanbefoundinoperatortheorytextbooks[13,16,17,18]whichweshallusethiswork.Lemma1.6Letu: