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1、TheWaveletDimensionFunctionisTheTraceFunctionofAShift-InvariantSystemAmosRonZuoweiShenComputerSciencesDepartmentDepartmentofMathematicsUniversityofWisconsin-MadisonNationalUniversityofSingapore1210WestDayton10KentRidgeCres.Madison,WI53706Singapore119260amos@cs.wisc.edumatzuows@leonis
2、.nus.edu.sgABSTRACTInthisnote,weobservethatthedimensionfunctionassociatedwithawaveletsystemisthetraceoftheGramian bersoftheshift-invariantsystemgeneratedbythenegativedilationsofthemotherwavelets.Whenthisshift-invariantsystemisatightframe,eachoftheGramian bersisanorthogonalprojector,a
3、nditstrace,then,coincideswithitsrank.Thisconnectionleadstosimpleproofsofseveralresultsconcerningthedimensionfunction,andtheargumentsextendtothebi-framecase.AMS(MOS)SubjectClassi cations:Primary42C15,Secondary42C30KeyWords:Dimensionfunction,frames,multiresolutionanalysis,wavelets.This
4、workwassupportedbytheUSNationalScienceFoundationunderGrantsDMS-9872890,DBI-9983114.andANI-0085984,theU.S.ArmyResearchOceunderContractDAAG55-98-1-0443,andtheStrategicWaveletProgramGrantfromtheNationalUniversityofSingapore.1.IntroductionLet:=fgrbea nitesubsetofL(IRd).Thedyadicwavelets
5、ystemgeneratedbythemotherwaveletsii=12istheunionX()=[j2ZZXj();withX():=DjE():jHere,Djisthedyadicdilationoperator,i.e.,jdjjDf(t):=22f(2t);andE()istheshift-invariant(SI)systemgeneratedby,i.e.,E()=fEk:k2ZZd;2g;Ek:f7!f(+k):Giventheset,ourinteresthereisinthedimensionfunction(Weiss'termin
6、ology)Dassociatedwith,whichisde nedasfollows:XrXX1D():=jb(2j(+k))j2;2IRd:ii=1k22ZZdj=1Thedimensionfunctionwasintroducedin[L1]byLemarie-Rieusset,whousedittoprovethatcompactlysupportedorthonormalwaveletsystemscanbeconstructedviaamultiresolutionanalysis(MRA).Amildsmoothnessassumpti
7、onwasassumedofthemotherwaveletsinthisresult.Hethengeneralizedtheresulttothebiorthogonalcaseaswellastothecaseofanarbitraryintegerdilations,undertheassumptionthatthemotherwaveletsarecompactlysupportedorhaveanexponentialdecay(see[L2]).Auscherin[A1]and[A2]extendedLemarie-Rieusset'sresul
8、ttothecasewh
