Varying the time-frequency lattice of Gabor frames

VARYINGTHETIME-FREQUENCYLATTICEOFGABORFRAMESHANSG.FEICHTINGERANDNORBERTKAIBLINGERAbstrat.AGabororWeyl-Heisenbergframeisgeneratedbytime-frequenyshiftsofasquare-integrablefuntion,theGaboratom,alongatime-frequenylattie.ThedualframeisagainaGaborframe,gen-eratedbythedualatom.Ingeneral,Gaborframesarenotstableunderaperturbationofthelattieonstants.WeshowthatforGaboratomsfromthemodulationspaeM1(Rd),whihisdenseinL2(Rd),theGaborframeisstableunderaperturbationofthelattieonstants.Moreover,inthisasethedualatomdependsontinuouslyonthelattieonstants.1.IntrodutionInwavelettheoryitisnaturaltoworkwithwaveletsystemsobtainedbymeansofthestandardtransformationparameters,i.e.,withshiftsalongtheintegersanddilationsbypowersof2.Inontrast,forGaborsystemsthetime-frequenylattie=aZbZ,a;b0,hastobehosenwithare.First,fortoosparselatties(ab1)noframesexist.AlsothestandardvonNeumannlattie(a=b=1)hastobeexluded:aordingtotheBalian-Lowtheoremagoodtime-frequenyloalizationisimpossibleatritialdensity(ab=1),see[Dau92,Se.4.1℄,[Gro01,Se.8.4℄.Ontheotherhand,Gaborframeswithexellenttime-frequenyloalizationexistforoversampledtime-frequenylatties(ab1).Forexample,takingastheGaboratomgtheGaussianfuntionyieldsaGaborframeforanylattiewithab1.Inpartiular,thesetofpairs(a;b)satisfyingthisonditionisanopensetforthishoieofg.However,thisstabilityoftheframeonditionunderaperturbationof(a;b)doesnotholdforallL2-atoms:Forexample,theharateristifuntionofanintervalleadstoastrange(non-open)setofgoodlattieonstants(a;b),wherethe(ir-)rationalityofabplaysakeyrole;thepatternisknownasJanssen’stie[Jan℄.HeneforpositiveresultsinthisdiretionwewillhavetoonsiderGaboratomsfromanappropriatesubspaeofL2(Rd).Anothermotivationforourresultsisthefollowing:Evenifitisknownthatthesetof\goodlattieonstantsforagivenatomgisanopenset,itisnotlearwhetherthedualatomdependsontinuouslyonthelattieonstants.2000MathematisSubjetClassiation.Primary42C15;Seondary47Lxx,94A12.Keywordsandphrases.Gaborframe,Weyl-Heisenbergframe,dualatom,Rieszbasis,stability,perturbation,time-frequenylattie,modulationspae.TheseondauthorwassupportedbytheAustrianSieneFundFWFgrantP-14485.12HANSG.FEICHTINGERANDNORBERTKAIBLINGERInthepresentpaperweprovethestabilityofGaborframesunderaper-turbationofthelattieparametersforGaboratomsfromthemodulationspaeM1(Rd)(Thm.3.10(i)).Theresultsovertheaseofgeneral(henealsonon-separable)time-frequenylattiesRdandtheyareformulatedinawaysuhthatalsotheatomanbeperturbedatthesametime.ThespaeM1(Rd)(originallydenotedS0(Rd))isdenseinL2(Rd)andplaysanimportantroleinGaboranalysis[FZ98℄,[Gro01,Se.12.1℄.AsanothermainresultweshowthatforatomsginM1(Rd)thedualGaboratomdependsontinuously(intheM1-norm)onthelattieonstants,overtheopensub-setofalllattieonstants(a;b)forwhih(g;a;b)generatesaGaborframe(Thm.3.10(ii)).WealsoinludeanalogousresultsforthestabilityofGaborRieszbasisequenesandtheontinuousdependeneoftheirbiorthogonalatom(Thm.4.2).Moreover,thestatementsareprovedforweightedver-sionsofM1(Rd),whihimpliesorrespondingresultsfortheShwartzspae(Corollary4.3).ThereisextensiveliteratureonerningloalperturbationsofGaborframes[CC97,Chr95,Chr96,Chr98,CH97,CLL00,FG89,FZ95,Jin99,SZ99,SZ01℄.Wenotethattheperturbationofthelattieonstants(a;b)isofadierenttypesineitgeneratesanarbitrarilylargeperturbationoftheindividuallattiepoints(ak;bl)forlargek;l2Z.TheproofsfortheloalperturbationresultsarebasedonaPaley-WienertypeperturbationoftheGaborframeoperatorS=Sg;a;b.Bynature,thisapproahannotbeadaptedtoourproblemsineasamatteroffatSdoesnotdependon-tinuouslyintheL2-operatornormonthelattieonstants(a;b),evenforShwartzatomsg.Thetwopositiveresultsatuallyonerningthepertur-bationof(a;b)obtainedbefore([FZ98,Se.3.6.3℄,[CCL,Thm.3.5℄)oversituationsonlywherethelattieparametersaresuÆientlysmall,andtheNeumannseriesrepresentingtheinverseframeoperatorisabsolutelyon-vergent.(AomplementaryobservationwasthatforgeneralL2-atomstheframeonditionmaydependritiallyonthelattieonstants(a;b)evenforarbitrarilysmalla;b0[FJ00℄.)Thepresentapproahisbasedonnewstrategies:theJanssenrepresen-tationoftheGaborframeoperatorandpropertiesofthetwistedonvo-lution.TheresultsinSe.2arestatedforoperatorsdenedasseriesoftime-frequenyshifts.InSe.3theyareappliedtoGaborframeoperators.ExtensionsaregiveninSe.4and,nally,inSe.5weaddonludingremarks.2.PreliminaryResultsLetL(B)denotethespaeofboundedlinearoperatorsonaBanahspaeB.ThespaeL(B)isaBanahspaewiththenorm(2.1)kTkL(B)=supkfkB=1kTfkB:VARYINGTHELATTICE3Fors2Rlet‘1s(Zn)denotetheBanahspaeofomplexsequenesonZngeneratedbythenorm(2.2)kk‘1s(Zn)=Xk2Znj(k)j(1+kkkRn)s:TheWieneramalgamspaeW(C0;‘1s)(Rn)istheBanahspaeofon-tinuousfuntionsgeneratedbythenorm(2.3)kfkW(C0;‘1s)(Rn)=Xk2Znmax2[0;1℄nnjf(k+)jo(1+kkkRn)s:LetGL(Rn)denotethegroupofinvertiblerealnn-matrieswiththenorm(2.4)kLkGL(Rn)=supkxkRn=1kLxkRn:SineallmatrixnormsareequivalentwehaveL!L0inGL(Rn)ifandonlyifthematriesLonvergetoL0entrybyentry.LetDLdenotethedilationbyL2GL(Rn),(2.5)DLf(x)=f(L1x):WeshowthatontheWieneramalgamspaesdenedabovethedilationisjointlyontinuousinitsargumentandthematrixL.Lemma2.1.Lets0.The