Multiplication Operators on Hilbert Spaces of Dirichlet Series

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Multiplication Operators on Hilbert Spaces of Dirichlet Series
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Stetler, Eric B
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Doctorate ( Ph.D.)
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University of Florida
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Mathematics
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MCCULLOUGH,SCOTT A
Committee Co-Chair:
JURY,MICHAEL THOMAS
Committee Members:
SHEN,LI C
ROBINSON,PAUL L
ROSALSKY,ANDREW J

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dirichlet -- hilbert -- multipliers -- multivariate
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In this thesis, certain classes of Hilbert spaces of one-variable and multivariable Dirichlet series will be examined. Their corresponding multiplier algebras will be explored and, in certain cases, classified up to isometric isomorphism. In the one-variable case, Hilbert spaces of Dirichlet series of one variable with weighted norms will be considered. Assuming certain conditions on the weights, upper and lower bounds for the operator norms of the corresponding multipliers will be derived. The multiplier algebras of these spaces will be completely classified in the case that the weights are suitably chosen. Examples will be given of weight sequences for which the conclusions of Hedenmalm, Lindqvist and Seip and McCarthy hold, but under alternate hypotheses on the weights. Moreover, it will be shown that this result is not subsumed under theirs. In the multivariable case, Hilbert spaces of multivariate Dirichlet series with weighted norms will be investigated. Results analogous to the one-variable case will be obtained under similar hypotheses on the weights.
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by Eric B Stetler.
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Thesis (Ph.D.)--University of Florida, 2014.
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Adviser: MCCULLOUGH,SCOTT A.
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Co-adviser: JURY,MICHAEL THOMAS.

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MULTIPLICATIONOPERATORSONHILBERTSPACESOFDIRICHLETSERIESByERICB.STETLERADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2014

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c2014EricB.Stetler 2

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IdedicatethistoeveryonethathashelpedmethroughthepasttwoyearsofworkonthisdissertationandtoalltheprofessorswhohavehelpedmakemeintothemathematicianIamtoday. 3

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ACKNOWLEDGMENTS Iwouldliketothankthemembersofmycommitteeforsupportingmeoverthepasttwoyears.IwouldespeciallyliketothankDr.McCullough,whohasbeenthebestadvisorIcouldhaveeveraskedfor,andwhohasalwaysbeenthereforme.Withouthim,noneofthiswouldhavebeenpossible.Iwouldalsoliketothankmyfamily,whohavehelpedshapemeintothemanIamtoday.Last,butnotleast,IthankEmily,whohasbeenanever-presentrockformeandwhohaslovedmeunconditionallythroughout. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 ABSTRACT ......................................... 7 CHAPTER 1INTRODUCTION ................................... 9 1.1TheRiemannZetaFunction .......................... 9 1.2DirichletSeries ................................. 11 2DEFINITIONSANDPRELIMINARIES ....................... 13 2.1ReproducingKernelHilbertSpaces ..................... 13 2.2Multipliers .................................... 14 2.3GeneralFactsaboutDirichletSeries ..................... 17 2.4TheSpaceHw ................................. 22 2.5ClassicationTheorems ............................ 25 2.6MultiplicativeWeights ............................. 27 2.7TheProjectedSpaceHwN ........................... 29 3THELOWERBOUND ................................ 31 4THEUPPERBOUND ................................ 36 4.1Kernels ..................................... 36 4.2TheCasewn=n)]TJ /F8 7.97 Tf 6.58 0 Td[(2 .............................. 38 4.3AnUpperBoundforM' ............................ 40 4.4THECASE= ................................ 44 5WEIGHTSGIVENBYMEASURES ......................... 48 5.1GoalsforthisChapter ............................. 48 5.2Preliminaries .................................. 48 5.3TheWeak-Topology ............................. 49 5.4TheSpaceofComplexMeasuresonaCompactHausdorffSpace .... 52 5.5WeightsGivenbyConvexCombinationsofPointMasses ......... 53 5.6WeightsGivenbyCompactlySupportedProbabilityMeasures ...... 55 5.7WeightsGivenbyCompactlySupportedFiniteMeasures ......... 57 5.8WeightsGivenbyRegularMeasures ..................... 58 6EXAMPLES ...................................... 61 7MULTIVARIABLEDIRICHLETSERIES ...................... 67 7.1GoalsintheMultivariableSetting ....................... 67 5

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7.2AnalyticFunctionsofMoreThanOneVariable ............... 68 7.3Denitions,NotationsandConventions .................... 69 7.4TheMultivariableSpaceHw .......................... 71 7.5MultipliersonHw ................................ 73 7.6TheMultivariableProjectedSpaceHwN .................... 73 8THEMULTIVARIABLELOWERBOUND ...................... 75 9ANm-DIMENSIONALEXTENSIONOFTHEHLSRESULT ........... 77 10THEMULTIVARIABLEUPPERBOUND ...................... 93 10.1AnUpperBoundforM'intheMultivariableSetting ............. 93 10.2TheMultivariableCase= ......................... 96 11CONCLUSION .................................... 99 REFERENCES ....................................... 100 BIOGRAPHICALSKETCH ................................ 101 6

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMULTIPLICATIONOPERATORSONHILBERTSPACESOFDIRICHLETSERIESByEricB.StetlerMay2014Chair:ScottMcCulloughMajor:MathematicsInthisdissertation,certainclassesofHilbertfunctionspacesofone-variableandmultivariableDirichletserieswillbeexamined.Theircorrespondingmultiplieralgebraswillbeexploredand,incertaincases,classieduptoisometricisomorphism.Intheone-variablecase,HilbertspacesoftheformHw=(1Xn=n0ann)]TJ /F4 7.97 Tf 6.59 0 Td[(s:1Xn=n0janj2wn<1),willbeconsidered,wherefwng1n=n0isasequenceofpositivenumbers.Assumingcertainconditionsontheseweightsequences,upperandlowerboundsfortheoperatornormsofthecorrespondingmultiplicationoperatorswillbederived.Themultiplieralgebrasofthesespaceswillbecompletelyclassiedinthecasethattheweightsaresuitablychosen.ExampleswillbegivenofweightsequencesforwhichconclusionssimilartothoseofHedenmalm,LindqvistandSeipandMcCarthyhold,butunderalternatehypothesesontheweights.Moreover,itwillbeshownthatthisresultisnotsubsumedundertheirs.Inthemultivariablecase,HilbertspacesoftheformHw=nXnn0ann)]TJ /F3 7.97 Tf 6.59 0 Td[(s:Xnn0janj2wn<1o 7

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willbeinvestigated,whereagain,fwngnn0isasequenceofpositivenumbers.Resultsanalogoustotheone-variablecasewillbeobtainedundersimilarhypothesesontheweights. 8

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CHAPTER1INTRODUCTION 1.1TheRiemannZetaFunctionTheRiemannzetafunction(s)isdenedintheopenhalf-planeRe(s)>1by (s)=1Xn=11 ns.(1)ItislearnedinelementarycalculusthatthisseriesconvergesabsolutelywhenRe(s)>1anddivergesotherwise.Itwasin1790thatLeonardEulerrstconsideredthisseriesforpositiveintegervaluesofs.Indoingso,hediscoveredafundamentalrelationbetweenthisseriesandtheprimes:ifk>1isaninteger,then 1Xn=11 nk=Ypprime1 1)]TJ /F5 11.955 Tf 11.95 0 Td[(p)]TJ /F4 7.97 Tf 6.59 0 Td[(k.(1)Onemayintuitthisbyexpandingeach(1)]TJ /F5 11.955 Tf 9.3 0 Td[(p)]TJ /F4 7.97 Tf 6.58 0 Td[(k))]TJ /F8 7.97 Tf 6.59 0 Td[(1asanabsolutelyconvergentgeometricseries,multiplyingtheresultingseriestogether,andcollectingtermsappropriately.Thereistheobviousquestionastowhethertheinniteproductofsuchgeometricseriesconvergesand,ifso,whetheritconvergestotheseriesontheleft-handsideof( 1 ).Euler'soriginalproofrstpresentedinhisthesisVariaeObservationescircaSeriesInnitas(VariousObservationsaboutInniteSeries)in1737startedwiththeseriesandusedasievingmethodtoattaintheproduct:Supposethatk>1isaninteger.Thedifference(k))]TJ /F6 11.955 Tf 15.82 8.09 Td[(1 2k(k)=1)]TJ /F6 11.955 Tf 15.82 8.09 Td[(1 2k(k)=1+1 3k+1 5k+1 7k+1 9k+issimply(k)withthetermshavingafactorof2removed.Ifwenowlookatthedifference1)]TJ /F6 11.955 Tf 15.82 8.09 Td[(1 2k(k))]TJ /F6 11.955 Tf 14.44 8.09 Td[(1 3k1)]TJ /F6 11.955 Tf 15.82 8.09 Td[(1 2k(k)=1)]TJ /F6 11.955 Tf 15.81 8.09 Td[(1 3k1)]TJ /F6 11.955 Tf 15.82 8.09 Td[(1 2k(s)=1+1 5k+1 7k+1 11k+, 9

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weseethatweget(k)withthetermshavingfactorsof2,3,orbothremoved.Repeatingthisprocesswithconsecutiveprimes,weobservethattherightsideisbeingsieved.Takingthelimitwhichisjustiedbytheconvergenceof( 1 )forintegersbiggerthan1weget1)]TJ /F6 11.955 Tf 18.95 8.09 Td[(1 11k1)]TJ /F6 11.955 Tf 15.82 8.09 Td[(1 7k1)]TJ /F6 11.955 Tf 15.81 8.09 Td[(1 5k1)]TJ /F6 11.955 Tf 15.82 8.09 Td[(1 3k1)]TJ /F6 11.955 Tf 15.81 8.09 Td[(1 2k(k)=1.Dividingtheleftsidebyeverythingexcept(k)andrewritinggives(k)=Ypprime1 1)]TJ /F5 11.955 Tf 11.95 0 Td[(p)]TJ /F4 7.97 Tf 6.58 0 Td[(k.Chebychevextendedthedenitionof(s)toallcomplexnumberswithrealpartgreaterthan1.Onenotesthattheproofof( 1 )goesthroughexactlythesameifitissimplyassumedthatRe(s)>1.Thus, 1Xn=11 ns=Ypprime1 1)]TJ /F5 11.955 Tf 11.96 0 Td[(p)]TJ /F4 7.97 Tf 6.59 0 Td[(s(1)wheneverRe(s)>1.Onemaydeducetheinnitudeofprimesfromthisrelation:Weknowthatif2Rapproaches1fromtheright,then()!1.Iftherewereonlynitelymanyprimes,thenwewouldnecessarilyhavelim!1+Ypprime1 1)]TJ /F5 11.955 Tf 11.96 0 Td[(p)]TJ /F9 7.97 Tf 6.59 0 Td[(=Ypprime1 1)]TJ /F5 11.955 Tf 11.96 0 Td[(p<1.Thiscontradictionshowsthattheremustbeinnitelymanyprimes.Riemannshowedthattheseriesin( 1 )couldbeanalyticallycontinuedtoCnf1g.TheRiemannzetafunction(s)isthusmoregenerallydenedtobetheanalyticcontinuationoftheseriesin( 1 )toCnf1g.Itisawell-knowconjecturealsoduetoRiemannthatthenontrivialzerosof(s)allhaverealpartequalto1=2.Wewillnotbeundertakinganyeffortstoprovethis. 10

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Fors=1,theseriesin( 1 )divergesto1.Sincelims!1(s)]TJ /F6 11.955 Tf 11.95 0 Td[(1)(s)=1,(s)ismeromorphiconCandisanalyticeverywhereexceptforasimplepolewhereithasresidue1. 1.2DirichletSeriesADirichletseriesisaseriesoftheform 1Xn=1an ns,(1)wherethean'sandsarecomplexnumbers.Ifan=1foreachn,thenwegettheRiemannzetafunctionfromSection 1.1 .ADirichletseriesmayormaynotconverge,dependingonthean'sandthechoicefors.Forexample,ifan=n!foreachn,thentheseriesfailstoconvergeanywhere.Ifinsteadanisnonzeroforonlynitelymanyn,thentheseriesconvergeseverywhere,andisinfactentire.ItturnsoutthatifaDirichletseriesconvergesforsomecomplexnumbers0,thenitmustconvergeforallcomplexswithrealpartstrictlylargerthans0afactthatwillappearaspartofalargertheoreminSection 2.3 Denition1. Givenarealnumber,weletdenotetheopenhalf-planefz2C:Re(z)>g.Withthisnewnotation,ifaDirichletseriesconvergesats0,thenitconvergesinRe(s0).Foranyrealnumber,therewillbecertainfunctionswhicharerepresentablebyaDirichletseriesin.Forexample,asseenabove,(s)isrepresentedbyanabsolutelyconvergentDirichletseriesinthehalf-plane1.Inthisdissertation,wewillbeexaminingcollectionsoffunctionswhichareallrepresentablebyDirichletseriesinsomecommonhalf-plane. 11

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Denition2. WedeneDtobethespaceofallfunctionsrepresentablebyDirichletseriesinsomerighthalf-plane.NotethatDisnonemptysinceitcontainsalltheniteDirichletseries. 12

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CHAPTER2DEFINITIONSANDPRELIMINARIES 2.1ReproducingKernelHilbertSpacesLetXbeasetandletHbeaHilbertspaceofcomplex-valuedfunctionsonX.Eachx2XdeterminesalinearfunctionalEx:H)165(!Cdenedbyevaluationatx:Ex(f)=f(x).Ifx2XandifExiscontinuous,theRieszrepresentationtheoremguaranteestheexistenceofauniquefunctionkx2HsuchthatEx(f)=f(x)=hf,kxi.IfExiscontinuousforeachx2X,wesaythatHisareproducingkernelHilbertspace(RKHS)thenamecomingfromthefactthatthefunctionskx,whendottedagainstf2H,reproducethevalueoffatx.Thefunctionkxiscalledthereproducingkernelforthepointx.Itisausefulfact(andwillbeexploitedheavilyinthisdissertation)thatthespanofthesetfkx:x2XgisdenseinH.Otherwise,wecouldndanonzerofunctionf2Horthogonaltoeachkx.However,thiswouldimplythatf(x)=hf,kxi=0foreachx2Xacontradiction.ThesubjectofRKHS'swassimultaneouslydevelopedbyStefanBergmanandNachmanAronszajninthemid-twentiethcentury.SinceeachkxhasdomainX,wemaydeneanewfunctionk:XX)166(!Cbyk(x,y)=ky(x)=hky,kxi.WecallthisfunctionthereproducingkernelforH.Notethatkissymmetricinthesensethatk(x,y)= k(y,x).Notealsothat,foranynitesubsetfx1,...,xNgofX,the(symmetric)matrix[k(xi,xj)]1i,jNispositivesemi-denitesince,for1,...,NinC,Xi,j ijk(xi,xj)=hXjjkxj,Xiikxii=kXjjkxjk20.AkernelonasetXisafunctionk:XX)166(!Csuchthat 1. k(x,y)= k(y,x);and 13

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2. foranynitesubsetfx1,...,xNgofX,thesymmetricmatrix[k(xi,xj)]1i,jNispositivesemi-denite.We'veseenthateachRKHSdenesareproducingkernel;theconverseisalsotrue: Theorem2.1. [ 3 ]IfkisakernelonthesetX,thenthereisauniqueHilbertspaceoffunctionsonXforwhichkisareproducingkernel.ThistheoremrstappearedinAronszajn'spaperTheoryofReproducingKernels,butisthereinattributedtoE.H.Moore.IfanfeigisanorthonormalbasisforH,then k(x,y)=ky(x)=hky,kxi=Xihky,eiihei,kxi.(2)Weoftenemploythisformulainndingaclosed-formexpressionfork.Forexample,theHardyspaceH2(D)isaRKHSwithorthonormalbasisen(z)=zn.Inthiscase,k(x,y)=1Xn=1hky,eiihei,kxi=1Xn=1 ynxn=1 1)]TJ /F5 11.955 Tf 11.95 0 Td[(x y.Thefunctionk(x,y)=(1)]TJ /F5 11.955 Tf 11.95 0 Td[(x y))]TJ /F8 7.97 Tf 6.59 0 Td[(1isknownastheSzegokernelonthedisk. 2.2MultipliersLetFbeacollectionofcomplex-valuedfunctionsonasetX.AmultiplieronFisafunction':X)166(!Csuchthat'f2Fforeachf2F.ItiscommonforamultiplieronFto,itself,lieinF.Givenamultiplier'onF,wecandenealinearmapM':F)166(!FbyM'f='f.ThecollectionofallsuchmapswhichwewilldenotebyMformsanalgebrawiththeobviousoperations.IfwegiveadditionalstructuretoF,wecanoftensaymoreaboutbothM'andMandcan,incertaincases,evenclassifythemultipliersonF.Forexample,itiswell-knownthattheHardyspaceH2(D)hasforitsmultiplieralgebrathespaceH1(D)offunctionsboundedandholomorphiconthedisk. Theorem2.2. LetHbeaRKHSonXandlet'beamultiplieronH.ThenM'iscontinuous(bounded). 14

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Proof. Itsufcestoverifythehypothesesoftheclosedgraphtheorem.Accordingly,supposethatfn,g,andhareinH,thatfn!g,andthatM'fn!h,bothinH.ItfollowsfromtheCauchy-Schwarzinequalitythatnormconvergenceimpliespointwiseconvergence.Hence,fn(x)!g(x)and(M'fn)(x)='(x)fn(x)!h(x)foreachx2X.Thus'(x)g(x)=h(x),whenceM'g=h. SinceM'iscontinuous,ithasanadjointM'.TheadjointofamultiplicationoperatorM'onaRKHSHhasaninterestingandusefulrelationto':forx2X, M'kx= '(x)kx.(2)Thatis,eachkxisaneigenfunctionofM'withcorrespondingeigenvalue '(x).Toseethis,letf2Handconsiderhf,M'kxi.Now,hf,M'kxi=h'f,kxi='(x)f(x)='(x)hf,kxi=hf, '(x)kxi.Sincethisholdsforeachf2H,wehaveM'kx= '(x)kx.Itisanimmediatecorollarythatthesupremumofj'jonXisatmostkM'k. Proposition2.1. LetTbeacontinuouslinearoperatoronaRKHSHoverthesetX.IfTkx= '(x)kxforeachx2X,thenT=M'.Thatis,Tisgivenbymultiplicationby'. 15

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Proof. Letf2H.ThenhTf,kxi=hf, '(x)kxi='(x)hf,kxi='(x)f(x)=h'f,kxi.Sincethespanofthesetfkx:x2XgisdenseinH,wehavehTf,gi=h'f,giforeachg2H,sothatT=M'. Theorem2.3. IfHisRKHSonXandifMisthecorrespondingmultiplieralgebra,thenMisclosedintheoperatornormtopology. Proof. IffM'ng1n=1isaCauchysequenceinB(H),thensoisfM'ng1n=1.SinceB(H)iscomplete,thereissomeT2B(H)suchthatM'n!Tinoperatornorm.Itfollowsfrom( 2 )thatkM'm)]TJ /F5 11.955 Tf 11.95 0 Td[(M'nkj'm(x))]TJ /F10 11.955 Tf 11.95 0 Td[('n(x)jforeachx2X,sothatthesequencesf'n(x)g1n=1areCauchy(andhenceconvergent).Foreachx,dene'(x)=limn!1'n(x).ThenTkx=limn!1M'nkx=limn!1 'n(x)kx= '(x)kx. 16

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ThatT=limn!1M'nisgivenbymultiplicationby'followsfromProposition 2.1 2.3GeneralFactsaboutDirichletSeriesInthissection,wewillbecollectingsomeelementaryfactsregardingconvergenceofDirichletseries,Dirichletconvolutions,productrepresentationsofDirichletseries,andintegralmeansofDirichletseries.Muchlikehowpowerseriesconvergeindisks,Dirichletseriesturnouttoconvergeinhalf-planes. Theorem2.4. [ 2 ]IfP1n=1ann)]TJ /F4 7.97 Tf 6.58 0 Td[(s0convergesandifRe(s1)>Re(s0),thenP1n=1ann)]TJ /F4 7.97 Tf 6.59 0 Td[(s1converges.Moreover,P1n=1ann)]TJ /F4 7.97 Tf 6.59 0 Td[(sconvergesuniformlyoncompactsubsetsofRe(s0).IfP1n=1ann)]TJ /F4 7.97 Tf 6.59 0 Td[(sconvergesin,butdivergesinanystrictlylargerhalf-plane,thentheline+itformstheboundaryoftheregioninwhichP1n=1ann)]TJ /F4 7.97 Tf 6.59 0 Td[(sconverges.Thepointsalong+itatwhichP1n=1ann)]TJ /F4 7.97 Tf 6.59 0 Td[(sconverges(ifthereareany)is,ingeneral,adelicateissue. Corollary1. ADirichletseriesisanalyticinanyopenhalf-planeinwhichitconverges. Proof. SupposethatP1n=1ann)]TJ /F4 7.97 Tf 6.59 0 Td[(sconvergesin,lets0beapointinandletr>0besmallenoughthat D(s0,r),thecloseddiskofradiusrcenteredats0,liesentirelyin.SinceeachniteDirichletseriesisanalyticin D(s0,r),Theorem 2.4 andanapplicationofMorera'stheoremgivesustheanalyticityofP1n=1ann)]TJ /F4 7.97 Tf 6.59 0 Td[(sats0. Itisalsotrue(thoughunimportantforthepurposesofthisdissertation)thatyoucandifferentiateaDirichletseriesterm-by-terminanyopenhalf-planeinwhichitconverges. Theorem2.5. IfP1n=1ann)]TJ /F4 7.97 Tf 6.59 0 Td[(sconvergesabsolutelyatthepoints0andifRe(s1)>Re(s0),thenP1n=1ann)]TJ /F4 7.97 Tf 6.58 0 Td[(s1convergesabsolutely.Inotherwords,absoluteconvergenceoccursinhalf-planes. Proof. Thisfollowsimmediatelyfromthefactthat1Xn=1jann)]TJ /F4 7.97 Tf 6.59 0 Td[(s1j=1Xn=1janjn)]TJ /F18 7.97 Tf 6.58 0 Td[(Re(s1)<1Xn=1janjn)]TJ /F18 7.97 Tf 6.59 0 Td[(Re(s0)=1Xn=1jann)]TJ /F4 7.97 Tf 6.59 0 Td[(s0j<1. 17

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AssociatedtoanygivenDirichletseriesP1n=1ann)]TJ /F4 7.97 Tf 6.59 0 Td[(sareseveralabscissae: 1. theabscissaofconvergencec=inf(Re(s):1Xn=1an nsconverges), 2. theabscissaofabsoluteconvergencea=inf(Re(s):1Xn=1an nsconvergesabsolutely), 3. theabscissaofboundednessb=inf(:1Xn=1an nsconvergestoaboundedfunctionin),and 4. theabscissaofuniformconvergenceu=inf(:1Xn=1an nsconvergesuniformlyin).Weallowanyofthesevaluestobe1(whichhappenswhentheseriesfailstoconvergeanywhere)or(whichhappens,forexample,whentheseriesisnite).Awordofcautionthough:P1n=1ann)]TJ /F4 7.97 Tf 6.59 0 Td[(sneednotconvergetoaboundedfunctioninbonlyineveryproperhalf-planecontainedinb.Norneeditbetrue,ingeneral,thatP1n=1ann)]TJ /F4 7.97 Tf 6.59 0 Td[(sconvergeuniformlyinuonlyineverystrictlysmalleropenhalf-plane.It'sclearthatac.Wealsohaveac+1sinceP1n=1n)]TJ /F4 7.97 Tf 6.58 0 Td[(sconvergesabsolutelyin1.SinceeachniteDirichletseriesisboundedinanyhalf-planeproperlycontainedinC,itfollowsthatbu.Itissomewhatsurprisingthatthesetwonumbersareinfactequal(i.e.u=b)afactprovedbyH.Bohrin[ 4 ].Inthecasethatthean'sareallpositive,allfourabscissaearethesame.InmuchthesamewaythatthecoefcientsofapowerseriesaregivenbytheCauchyintegralformula,thereisanintegralformulagivingthecoefcientsofaconvergentDirichletseriesaswell: 18

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Proposition2.2. [ 2 ]Iff(s)=P1n=1ann)]TJ /F4 7.97 Tf 6.58 0 Td[(sconvergesabsolutelyalongtheverticalstrip0+it,thenforx>0,wehavelimT!11 2TZT)]TJ /F4 7.97 Tf 6.58 0 Td[(Tf(0+it)x0+itdt=8>><>>:anifx=n0otherwise.TheproofconsistsofabasicFourierseries-styleargument.AnimportantconsequenceofProposition 2.2 isthat,iftwoDirichletseriesagreealongaverticallineonwhichtheybothconvergeabsolutely,thentheymustinfactbethesameDirichletseries. Proposition2.3. Iff(s)=P1n=1ann)]TJ /F4 7.97 Tf 6.58 0 Td[(sconvergesabsolutelyfor>a,then limT!11 2TZT)]TJ /F4 7.97 Tf 6.59 0 Td[(Tjf(+it)j2dt=1Xn=1janj2n)]TJ /F8 7.97 Tf 6.59 0 Td[(2.(2)ItiseasilyshownthattheconclusionsofPropositions 2.2 and 2.3 stillholdifconvergesabsolutelyinthehypothesesisreplacedbyconvergesuniformly.Inthisdissertation,wewillencounterHilbertspacesofDirichletserieswithnormsoftheformfoundin( 2 ).Proposition 2.3 willthenallowustoproduceupperboundsforthemultiplicationoperatorsthatariseinthesespaces.IfonetakestwoDirichletseriesP1m=1amm)]TJ /F4 7.97 Tf 6.59 0 Td[(sandP1n=1bnn)]TJ /F4 7.97 Tf 6.59 0 Td[(s,multipliesthemtogether,andcollectstermsintheobviousway,onegets 1Xm=1amm)]TJ /F4 7.97 Tf 6.58 0 Td[(s! 1Xn=1bnn)]TJ /F4 7.97 Tf 6.58 0 Td[(s!=1Xj=1 Xmn=jambn!j)]TJ /F4 7.97 Tf 6.59 0 Td[(s.(2)ThiscanbetakentobethedenitionoftheformalproductoftwoformalDirichletseries.OnenaturallywonderswhatconditionsmustbemetbyP1m=1amm)]TJ /F4 7.97 Tf 6.59 0 Td[(sandP1n=1bnn)]TJ /F4 7.97 Tf 6.59 0 Td[(sif( 2 )istoholdinanythingmorethanaformalsetting.IfbothP1m=1amm)]TJ /F4 7.97 Tf 6.58 0 Td[(sandP1n=1bnn)]TJ /F4 7.97 Tf 6.58 0 Td[(sconvergeabsolutely,itiswell-known(andeasilyproven)that( 2 )holds,andthattheresultingDirichletseriesconvergesabsolutely.Itturnsout,however,thatif 19

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bothseriesconverge,weonlyneedoneofthemtoconvergeabsolutelyfor( 2 )toholdthoughwedolosetheabsoluteconvergenceoftheresultingseries. Proposition2.4. If 1. A=P1m=1amm)]TJ /F4 7.97 Tf 6.58 0 Td[(wconvergesabsolutely, 2. B=P1n=1bnn)]TJ /F4 7.97 Tf 6.58 0 Td[(wconverges,and 3. C=P1j=1)]TJ 5.47 -.72 Td[(Pmn=jambnj)]TJ /F4 7.97 Tf 6.58 0 Td[(w,thenCconvergesandAB=C. Proof. LetAN,BNandCNbetheNth-partialsumsofA,BandCrespectively.ItiseasytoseethatCN=NXj=1ajj)]TJ /F4 7.97 Tf 6.59 0 Td[(wBbN jc,wherebcdenotestheoorfunction.ThusjCN)]TJ /F5 11.955 Tf 11.95 0 Td[(ANBj=NXj=1ajj)]TJ /F4 7.97 Tf 6.59 0 Td[(wBbN jc)]TJ /F4 7.97 Tf 17.3 14.94 Td[(NXj=1ajj)]TJ /F4 7.97 Tf 6.58 0 Td[(wB=NXj=1ajj)]TJ /F4 7.97 Tf 6.59 0 Td[(wBbN jc)]TJ /F5 11.955 Tf 11.96 0 Td[(B.SinceBN!B,jBN)]TJ /F5 11.955 Tf 12.8 0 Td[(Bjisboundedaboveby(say)K.Let>0,chooseMlargeenoughthat1Xj=M+1jajj)]TJ /F4 7.97 Tf 6.59 0 Td[(wj< 2KandjBN)]TJ /F5 11.955 Tf 11.95 0 Td[(Bj< 2P1j=1jajj)]TJ /F4 7.97 Tf 6.59 0 Td[(wj 20

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wheneverN>M.LetN>M2.Thenfor1jM,bN jc>M,sothatNXj=1ajj)]TJ /F4 7.97 Tf 6.59 0 Td[(wBbN jc)]TJ /F5 11.955 Tf 11.96 0 Td[(B=MXj=1ajj)]TJ /F4 7.97 Tf 6.59 0 Td[(wBbN jc)]TJ /F5 11.955 Tf 11.96 0 Td[(B+NXj=M+1ajj)]TJ /F4 7.97 Tf 6.58 0 Td[(wBbN jc)]TJ /F5 11.955 Tf 11.95 0 Td[(BMXj=1jajj)]TJ /F4 7.97 Tf 6.58 0 Td[(wjBbN jc)]TJ /F5 11.955 Tf 11.96 0 Td[(B+NXj=M+1jajj)]TJ /F4 7.97 Tf 6.59 0 Td[(wjBbN jc)]TJ /F5 11.955 Tf 11.96 0 Td[(B.<1Xj=1jajj)]TJ /F4 7.97 Tf 6.59 0 Td[(wj 2P1j=1jajj)]TJ /F4 7.97 Tf 6.58 0 Td[(wj+ 2KK=HencejCN)]TJ /F5 11.955 Tf 11.96 0 Td[(ABjjCN)]TJ /F5 11.955 Tf 11.96 0 Td[(ANBj+jANB)]TJ /F5 11.955 Tf 11.95 0 Td[(ABj<+jAB)]TJ /F5 11.955 Tf 11.95 0 Td[(ANBj.ItfollowsthatlimsupNjCN)]TJ /F5 11.955 Tf 11.95 0 Td[(ABj,andsincewasarbitrary,wegetlimsupNjCN)]TJ /F5 11.955 Tf 11.96 0 Td[(ABj=0.ThusjCN)]TJ /F5 11.955 Tf 11.96 0 Td[(ABj!0asN!1,whenceC=AB. WeclosethissectionwiththefollowingremarkabletheoremofSchnee,whichgivessufcientconditionsforaDirichletseriestoconvergeintherighthalf-plane0: Theorem2.6. [ 13 ]Iff(s)=P1n=1ann)]TJ /F4 7.97 Tf 6.59 0 Td[(s 21

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1. convergesinsome(possiblyremote)half-plane0, 2. hasananalyticcontinuationFto0,and 3. foreach>0,FsatisesthegrowthconditionjF(s)j=O(jsj)asjsj!1ineveryrighthalf-planecontainedin0,thenP1n=1ann)]TJ /F4 7.97 Tf 6.58 0 Td[(s,infact,convergesonallof0.Inparticular,ifaDirichletseriesP1n=1ann)]TJ /F4 7.97 Tf 6.59 0 Td[(shasaboundedanalyticcontinuationto0,thenP1n=1ann)]TJ /F4 7.97 Tf 6.59 0 Td[(sconvergesin0andbythecorollarytoTheorem 2.4 isequaltoitsanalyticcontinuation. 2.4TheSpaceHwLetfwng1n=n0beasequenceofpositiverealnumbersandsupposethatthereisapositiverealnumber0suchthat 1Xn=n0w)]TJ /F8 7.97 Tf 6.58 0 Td[(1nn)]TJ /F8 7.97 Tf 6.58 0 Td[(2<1(2)whenever>0.LetHwdenotetheHilbertspace (1Xn=n0ann)]TJ /F4 7.97 Tf 6.59 0 Td[(s:1Xn=n0janj2wn<1),(2)withinnerproducth,iwdenedbyh1Xn=n0ann)]TJ /F4 7.97 Tf 6.59 0 Td[(s,1Xn=n0bnn)]TJ /F4 7.97 Tf 6.59 0 Td[(siw=1Xn=n0an bnwn.Forsimplicity,wewilltaken0=1unlessotherwisestated. 22

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TheCauchy-Schwarzinequalitygivesusthecontinuityofpointevaluationsin0:ifs=+it20andiff(s)=P1n=1ann)]TJ /F4 7.97 Tf 6.59 0 Td[(s,thenjf(s)j1Xn=1janjn)]TJ /F9 7.97 Tf 6.58 0 Td[(=1Xn=1janjw1=2nw)]TJ /F8 7.97 Tf 6.59 0 Td[(1=2nn)]TJ /F9 7.97 Tf 6.59 0 Td[(vuut 1Xn=1janj2wnvuut 1Xn=1w)]TJ /F8 7.97 Tf 6.58 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(2=kfkwvuut 1Xn=1w)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(2,sothat,iffn!0innorm,thenfn(s)!0asasequenceinC.ItfollowsthatHwisaRKHS,andthus,hasareproducingkernelk(u,z).Sincefw)]TJ /F8 7.97 Tf 6.59 0 Td[(1=2nn)]TJ /F4 7.97 Tf 6.59 0 Td[(sgformsanorthonormalbasisforHw,wecanuse( 2 )tondtheformofk: k(u,z)=hkz,kuiw=1Xn=1hkz,w)]TJ /F8 7.97 Tf 6.59 0 Td[(1=2nn)]TJ /F4 7.97 Tf 6.59 0 Td[(siwhw)]TJ /F8 7.97 Tf 6.59 0 Td[(1=2nn)]TJ /F4 7.97 Tf 6.58 0 Td[(s,kuiw=1Xn=1w)]TJ /F8 7.97 Tf 6.58 0 Td[(1nn)]TJ /F4 7.97 Tf 6.59 0 Td[(u)]TJ ET q .359 w 400.11 -350.58 m 404.7 -350.58 l S Q BT /F4 7.97 Tf 400.11 -355.6 Td[(z.(2)WewillseeinSection 2.6 thatwhenourweightsequencefwng1n=1hascertainnumbertheoreticproperties,thiskerneltakesonamuchmoreaestheticallypleasingform.Let'beamultiplieronHw.IfHwcontainsaconstantfunction(i.e.n0=1),thenit'sclearthat'2Hw;ifn0>1,thenthisisstilltruethoughtheproofisnotquiteastrivial:Letpandqbedistinctprimeslargerthatn0.Since'(s)p)]TJ /F4 7.97 Tf 6.58 0 Td[(s2Hw,wehave'(s)p)]TJ /F4 7.97 Tf 6.59 0 Td[(s=1Xn=n0ann)]TJ /F4 7.97 Tf 6.59 0 Td[(s,sothat'(s)=1Xn=n0an(n=p))]TJ /F4 7.97 Tf 6.59 0 Td[(s.Ifwecanshowthatan=0wheneverp-n,thenthiswillprovethat'isrepresentablebyaDirichletseriesinHwandisthusinHw.Fix>0.Since'(s)p)]TJ /F4 7.97 Tf 6.58 0 Td[(sand'(s)q)]TJ /F4 7.97 Tf 6.58 0 Td[(sare 23

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givenbyabsolutelyconvergentDirichletseriesin,wecanapplyProposition 2.2 toseethatan=limT!11 2TZT)]TJ /F4 7.97 Tf 6.58 0 Td[(T'(+it)p)]TJ /F9 7.97 Tf 6.59 0 Td[()]TJ /F4 7.97 Tf 6.59 0 Td[(itn+itdt=limT!11 2TZT)]TJ /F4 7.97 Tf 6.58 0 Td[(T'(+it)n p+itdt=limT!11 2TZT)]TJ /F4 7.97 Tf 6.58 0 Td[(T'(+it)q)]TJ /F9 7.97 Tf 6.58 0 Td[()]TJ /F4 7.97 Tf 6.59 0 Td[(itnq p+itdt,whichis0unlessnq pisaninteger.Sincepandqaredistinctprimes,itmustbethecasethatan=0whenpdoesn'tdividen. Theorem2.7. Ifn0>1,thenM,themultiplieralgebraofHw,containsno(nonzero)self-adjointsubalgebra;ifn0=1,thentheonlyself-adjointsubalgebraofMisthetrivialalgebraconsistingofscalarmultiplesoftheidentitymap. Proof. SupposethatM'2MandthatM'isgivenbymultiplicationby (i.e.M'=M ).Fromthediscussionprecedingthistheorem,weknowthatboth'and arerepresentablebyDirichletseriesin0;say'(s)=1Xn=n0ann)]TJ /F4 7.97 Tf 6.59 0 Td[(sand (s)=1Xn=n0bnn)]TJ /F4 7.97 Tf 6.59 0 Td[(s.Leti,jn0andobservethath'(s)i)]TJ /F4 7.97 Tf 6.58 0 Td[(s,(ij))]TJ /F4 7.97 Tf 6.59 0 Td[(siw=h1Xn=n0ann)]TJ /F4 7.97 Tf 6.59 0 Td[(si)]TJ /F4 7.97 Tf 6.58 0 Td[(s,(ij))]TJ /F4 7.97 Tf 6.59 0 Td[(siw=h1Xn=n0an(in))]TJ /F4 7.97 Tf 6.59 0 Td[(s,(ij))]TJ /F4 7.97 Tf 6.59 0 Td[(siw=ajwij. 24

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Ontheotherhandh'(s)i)]TJ /F4 7.97 Tf 6.58 0 Td[(s,(ij))]TJ /F4 7.97 Tf 6.59 0 Td[(siw=hi)]TJ /F4 7.97 Tf 6.58 0 Td[(s, (s)(ij))]TJ /F4 7.97 Tf 6.59 0 Td[(siw=hi)]TJ /F4 7.97 Tf 6.58 0 Td[(s,1Xn=n0bnn)]TJ /F4 7.97 Tf 6.58 0 Td[(s(ij))]TJ /F4 7.97 Tf 6.59 0 Td[(siw=hi)]TJ /F4 7.97 Tf 6.58 0 Td[(s,1Xn=n0bn(ijn))]TJ /F4 7.97 Tf 6.58 0 Td[(siw,whichis0unlessjandn0areboth1.Itfollows,thatifn0>1,thenaj=0foreachj,andifn0=1,thenonlya1canbenonzero.Tonishtheproof,simplynotethatmultiplicationbyaconstantdenesamultiplierinthecasethatn0=1. 2.5ClassicationTheoremsIn[ 9 ],Hedenmalm,LindquistandSeip(HLS)classiedthemultipliersonHwinthecasethatwn1byshowingthatthemultipliersonHwwerepreciselythebounded,analyticfunctionsrepresentablebyDirichletseriesin0.Moreprecisely, Theorem2.8. [ 9 ]Ifwn1foreachn,then MH1(0)\D,(2)whereH1(0)isthespaceofbounded,analyticfunctionson0andDisthespaceoffunctionsrepresentablebyDirichletseriesinsomerighthalf-plane,andH1(0)\Disequippedwiththesupremumnormover0.Here,MH1(0)\DmeansMisisometricallyisomorphictoH1(0)\D.In[ 10 ],McCarthyextendedtheresultsofHLStoweightsgivenbyacertainclassofmeasures: Theorem2.9. [ 10 ]IfisapositiveRadonmeasurewith0initssupportandifwnisdenedby wn=Z10n)]TJ /F8 7.97 Tf 6.59 0 Td[(2d()(2) 25

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fornn0,wheren0isthesmallestnaturalnumberforwhichthisintegralisnite,then MH1(0)\D.(2)Inthisdissertation,estimatesonthenormsofthemultiplicationoperatorsoncertainclassesofweightedHilbertspacesofDirichletseriesareestablishedinthecasesthattheweightssatisfysomebasicconditions.Theweightsherearecomplementarytothosegeneratedbymeasuresasconsideredin[ 10 ],overlappingonlyinthemosttrivialcases.Examplesofweightsmeetingtheseconditionsincludethereciprocalsofthedivisorfunction,thesumofthedivisorsfunction,andEuler'stotientfunction(allofwhichwillbedenedinChapter6).Indeed,ineachofthesecasesthenormofamultiplierisidentiedasthesupremumnormofthesymbol'overarighthalfplane.InChapter4,willbereproducingthetheoremofHLSforweightsoftheformwn=n)]TJ /F8 7.97 Tf 6.59 0 Td[(2,whicharegivenby wn=Z10n)]TJ /F8 7.97 Tf 6.59 0 Td[(2d()(2)inthecasethatisapointmassat.InChapter5,wewillextendthisresulttoconvexcombinationsofpointmasseswhich,afterdevelopingthenecessarytheory,willeventuateinarecoveryofMcCarthy'stheorem,withtheonlyadditionalrequirementthatberegular.InChapters7,8,9,and10,wewillbeexaminingHilbertspacesofDirichletseriesofmorethanonevariable(denedinChapter7)andextendingtheresultsoftheone-variablecasetothesenewspaces.ThiswillincludeanextensionofTheorem 2.8 tospacesofmultivariableDirichletserieswithsquare-summablecoefcients.However,certaindifcultieswillariseinthispursuitwhicharenotanissueintheone-variablecase.Inparticular,thetheoremsofBohrandSchneeusedintheone-variablecasedon'tnecessarilycarryovertothemultivariatecaseandsosomecreativitymustbeemployedtogetaroundthis. 26

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2.6MultiplicativeWeights Denition3. Afunctionf:N)166(!Cismultiplicativeiff(mn)=f(m)f(n)whenevergcd(m,n)=1andiscompletelymultiplicativeiff(mn)=f(m)f(n)foreachmandninN.Notethatforamultiplicativefunction,f(1)=f(11)=f(1)f(1),sothatf(1)=0orf(1)=1.Intheformercase,fisidentically0sincef(m)=f(1m)=f(1)f(m)=0foreachm.Accordingly,wewillrelegateourdiscussiontothelattercase.Notealsothatfiscompletelydeterminedbyitsvaluesonpowersofprimes.(Iffiscompletelymultiplicative,thenitisentirelydeterminedbyitsvalueson1andtheprimes.)Somewell-knownexamplesofmultiplicativefunctionsarethedivisorfunctiond(n),thesumofdivisorsfunction(n),andtheEulertotientfunction(n)allofwhichwillbeintegralinproducingweightscomplementarytothoseofHLSandMcCarthyforwhichtheirconclusionsneverthelesshold.Supposenowthatfwng1n=1isamultiplicativesequenceofpositivenumbers(i.e.wmn=wmwnwhengcd(m,n)=1)andletHwdenotetheresultingHilbertspace.TheEulerproductformula( 1 )canbeextendedtomoregeneralDirichletserieswhosecoefcientsaremultiplicative:1Xn=1ann)]TJ /F4 7.97 Tf 6.59 0 Td[(s=Ypprime1Xj=0apjp)]TJ /F4 7.97 Tf 6.59 0 Td[(js,providedthattheseriesconvergesabsolutelyats.Inthecasethatourweightsequenceiscompletelymultiplicative,thesuminsidetheaboveproductfurthersimpliesas1Xj=0apjp)]TJ /F4 7.97 Tf 6.58 0 Td[(js=1 1)]TJ /F5 11.955 Tf 11.95 0 Td[(app)]TJ /F4 7.97 Tf 6.58 0 Td[(s, 27

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sothat1Xn=1ann)]TJ /F4 7.97 Tf 6.59 0 Td[(s=Ypprime1 1)]TJ /F5 11.955 Tf 11.96 0 Td[(app)]TJ /F4 7.97 Tf 6.59 0 Td[(s.TheproofsoftheseproductrepresentationsmimictheproofoftheEulerformulagiveninSection 1.1 .Ifourweightsaremultiplicative,thereproducingkernelforHwcanbewrittenask(u,z)= ku(z)=1Xn=1w)]TJ /F8 7.97 Tf 6.58 0 Td[(1nn)]TJ /F4 7.97 Tf 6.58 0 Td[(u)]TJ ET q .359 w 256.25 -142.98 m 260.85 -142.98 l S Q BT /F4 7.97 Tf 256.25 -147.99 Td[(z=Ypprime1Xj=0w)]TJ /F8 7.97 Tf 6.58 0 Td[(1pjp)]TJ /F4 7.97 Tf 6.59 0 Td[(j,where=u+ z;iftheweightsarecompletelymultiplicative,thenwegetk(u,z)=Ypprime1 1)]TJ /F5 11.955 Tf 11.96 0 Td[(w)]TJ /F8 7.97 Tf 6.59 0 Td[(1pp)]TJ /F9 7.97 Tf 6.59 0 Td[(.InChapter4,wewillprovethefollowing:If0,fwng1n=1isamultiplicativesequenceofpositivenumbers(withcorrespond-ingHilbertspaceHw),if'isamultiplieronHw,and,foreachprimepandpositiveintegerk,ourweightssatisfythegrowthconditionwpk)]TJ /F12 5.978 Tf 5.76 0 Td[(1p)]TJ /F8 7.97 Tf 6.58 0 Td[(2wpk,thenkM'kj'j.(Ananalogoustheoremwillbeprovedinthemultivariatecase.)WewillrecovertheresultsofHLSandMcCarthy(withtheprovisothatthe0intheirconclusionsbereplaced,moregenerallybyforsome0)inthecasethatourweightssatisfytheaboveconditions,alongwithafewmoremildassumptions.Examplesofweightsequencesthatmeettheseconditionsincludethereciprocalsofthedivisorfunction,thesumofdivisorsfunction,andtheEulertotientfunction.Moreover,itwillbeshownthatthesenumbertheoreticweightsequencescan'tcomefrommeasuresasin( 2 ). 28

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Itiseasytoshowthatcompletelymultiplicativeweightscanonlycomefrommeasuresinthecasethatisapointmass:Iffwng1n=1iscompletelymultiplicative,thenw1=([0,1))=1,andwecanapplyJensen'sinequalitytotheconvexfunctionx2toseethatZ10n)]TJ /F8 7.97 Tf 6.58 0 Td[(2d()2=(wn)2=wn2=Z10(n)]TJ /F8 7.97 Tf 6.58 0 Td[(2)2d().Equalityonlyoccurswheneithertheintegrandisconstanta.e.-[]ortheconvexfunctionbeingappliedtotheintegralislinear.Itfollowsthen,inourcase,thatmustbeapointmassatsomepointin[0,1). 2.7TheProjectedSpaceHwNLetPNdenotethesetconsistingoftherstNprimesanddenotebyhPNithecollectionofwordsgeneratedbytheprimesinPN:hPNi=f2132pNN:j2Ng.LetHwNdenotetheclosureofthesubspaceofHwspannedbythevectorsn)]TJ /F4 7.97 Tf 6.59 0 Td[(sforn2hPNi.Thatis, HwN=8<:Xn2hPNiann)]TJ /F4 7.97 Tf 6.59 0 Td[(s:1Xn=1ann)]TJ /F4 7.97 Tf 6.58 0 Td[(s2Hw9=;.(2)Finally,letNdenotetheprojectionofHwontoHwN.ThereareseveralthingsthatcanbesaidaboutHwNandN. Lemma1. Suppose'isamultiplieronHwwith'(s)=1Xn=1ann)]TJ /F4 7.97 Tf 6.58 0 Td[(s.Then 1. N'isamultiplieronHwNwithN'(s)=Xn2hPNiann)]TJ /F4 7.97 Tf 6.59 0 Td[(s; 29

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2. withanabuseofnotation,NM'jHwN=MN';and 3. Nk(u,z)=Xn2hPNiw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F4 7.97 Tf 6.59 0 Td[(u)]TJ ET q .359 w 316.34 -42.85 m 320.93 -42.85 l S Q BT /F4 7.97 Tf 316.34 -47.86 Td[(z(2)isthereproducingkernelforHwN.Notethat,iftheweightsaresuchthatXn2hPNiw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(2<1whenever>forsome0,thenNk(u,z)convergesabsolutelywheneveru,z2.ItfollowsfromtheCauchy-Schwarzinequalitythen,thateachfunctioninHwNconvergesabsolutelyanduniformlyinforeach>.Thereasontoconsiderthisprojectedspaceratherthanthefullspaceisthat,whenourweightsaremultiplicative,ourprojectedkerneltakestheformNk(u,z)=Yp2PN1Xj=0w)]TJ /F8 7.97 Tf 6.59 0 Td[(1pjp)]TJ /F4 7.97 Tf 6.59 0 Td[(j,where=u+ zasbefore,andwheretheproductisnownite.Afterafurtherconditionisimposedontheweights,thenitudeoftheproductwillallowustoconsiderconvergencefarthertotheleftthanwewouldotherwisebeableto.Inthecasethatourweightsarecompletelymultiplicative,wehaveNk(u,z)=Yp2PN1 1)]TJ /F5 11.955 Tf 11.96 0 Td[(w)]TJ /F8 7.97 Tf 6.59 0 Td[(1pp)]TJ /F9 7.97 Tf 6.59 0 Td[(. 30

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CHAPTER3THELOWERBOUNDInthissection,wewillproducealowerboundforM'inthecasethattheweightssatisfycertainconvergenceconditions.Asaconsequence,wewillseethatif0andifforeach>thereisaC>0suchthatw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.58 0 Td[(2Cforeachn,thenj'jkM'k.ThisimpositionontheweightsismotivatedbythegrowthrateimplicitinMcCarthy'sweights:since0isinthesupportof,foreach>0,wn=Z10n)]TJ /F8 7.97 Tf 6.58 0 Td[(2d()n)]TJ /F8 7.97 Tf 6.58 0 Td[(2([0,]))w)]TJ /F8 7.97 Tf 6.58 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(21 ([0,]),where([0,])>0byhypothesis.Wenowproceedtothemaintheoremofthissection. Theorem3.1. Letfwng1n=1beasequenceofpositivenumbers,supposethat'isamultiplierofHw,andlet01berealnumbers.If 1Xn=1w)]TJ /F8 7.97 Tf 6.58 0 Td[(1nn)]TJ /F8 7.97 Tf 6.58 0 Td[(21<1(3)whenever1>1and Xn2hPNiw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.58 0 Td[(2<1(3)foreachNwhenever>,then,for>0, 1. eachf2Hwconvergesabsolutelyuniformlyin1+; 2. eachf2HwNconvergesabsolutelyuniformlyin+; 3. HwandHwNarereproducingkernelHilbertspaces,withpointevaluationsbeingcontinuousin1andrespectively; 4. thesequencefN'g1N=1isuniformlyboundedinsupnormonbykM'k;and 31

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5. 'convergesinwithj'jkM'k. Proof. Let>0.AnapplicationoftheCauchy-Schwarzinequalityshowsthatpointevaluationsarecontinuousin1:Ifs=+it21andiff(s)=P1n=1ann)]TJ /F4 7.97 Tf 6.58 0 Td[(s,thenjf(s)j1Xn=1janjn)]TJ /F9 7.97 Tf 6.58 0 Td[(=1Xn=1janjw1=2nw)]TJ /F8 7.97 Tf 6.59 0 Td[(1=2nn)]TJ /F9 7.97 Tf 6.59 0 Td[(vuut 1Xn=1janj2wnvuut 1Xn=1w)]TJ /F8 7.97 Tf 6.58 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(2=kfkwvuut 1Xn=1w)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(2.Thus,HwisaRKHSinwhicheachelementisrepresentablebyanabsolutelyuniformlyconvergentDirichletseriesin1+.TurningourattentiontotheprojectedsubspaceHwN=8<:Xn2hPNiann)]TJ /F4 7.97 Tf 6.59 0 Td[(s:1Xn=1ann)]TJ /F4 7.97 Tf 6.58 0 Td[(s2Hw9=;,iff(s)=Xn2hPNiann)]TJ /F4 7.97 Tf 6.58 0 Td[(s2HwN,anapplicationoftheCauchy-Schwarzinequalityshowsasabovethatpointevaluationsarecontinuousinandthatfconvergesabsolutelyuniformlyin+.Thisprovesitems( 1 ),( 2 )and( 3 ).Moreover,sincek(u,z)isthereproducingkernelforHw,Nk(u,z)isthereproducingkernelfortheprojectedspaceHwNanditem( 2 )fromaboveandCorollary 1 fromChapter2combinetogiveustheanalyticityofeachN'in. 32

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Fromthestandardkernel/eigenfunctionargument,wehave jN'jkNM'jHwNkkNM'kkM'k.(3)Byanormalfamiliesargument,thesequencefN'g1n=1hasasubsequencefNj'g1j=1whichconvergesuniformlyoncompactsetsintosomefunction whichisanalyticin.Hencej jkM'k.Ontheotherhand,thesequencefNj'g1j=1convergesto'uniformlyoncompactsubsetsof1bytheCauchy-Schwarzinequality:if'(s)=P1n=1ann)]TJ /F4 7.97 Tf 6.59 0 Td[(sandif=Re(s)>1,thenj'(s))]TJ /F10 11.955 Tf 11.96 0 Td[(Nj'(s)js Xn2NnhPNjijanj2wns Xn2NnhPNjiw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.58 0 Td[(2,withtheright-handsidetendingto0asj!1.Itfollowsthat ='on1.Since isaboundedanalyticcontinuationof'into,Schnee'stheorem(aslightvariationactually)tellsusthat'convergeson.Corollary 1 fromChapter2givestheanalyticityof'in,andas'= in1,wemusthave'= in.Itfollowsthenthatj'jkM'k. Somefactsaroseintheaboveproofwhichwillbecollectedinthefollowingtheorem. Theorem3.2. Let'beamultiplieronthespaceHw.IfthesequencefN'g1N=1isuniformlyboundedby(say)Binsupnorminthehalf-plane,then 1. thereissomesubsequencefNj'g1j=1convergingpointwiseto'in; 2. 'convergesin;and 3. j'jB.Wenowmovetoamoreinterestingconditiononourweights,whichwillallowustoobtaintheconvergenceconditionsgiveninTheorem 3.1 andwhichaswewillsee 33

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bemoreusefulinproducingboundsforourmultiplierswhenourweightsareofamorenumbertheoreticvariety.Butrst,alemma: Lemma2. ForeachpositiveintegerN,theseriesgivenbyXn2hPNin)]TJ /F4 7.97 Tf 6.59 0 Td[(sconvergesin0. Proof. WehaveXn2hPNin)]TJ /F4 7.97 Tf 6.58 0 Td[(s=Yp2PN1 1)]TJ /F5 11.955 Tf 11.95 0 Td[(p)]TJ /F4 7.97 Tf 6.59 0 Td[(sfarenoughtotheright.SincePn2hPNin)]TJ /F4 7.97 Tf 6.59 0 Td[(shasananalyticcontinuationtoaboundedfunctioninforeach>0,Schnee'stheorem(aftertheappropriatechangeofvariables)tellsusthatPn2hPNin)]TJ /F4 7.97 Tf 6.59 0 Td[(sinfactconvergesinallof.Sincewasarbitrary,itfollowsthatPn2hPNin)]TJ /F4 7.97 Tf 6.58 0 Td[(sconvergesin0. Remark.Schneeisnotactuallyneededinthisproof,butasitwasneededintheproofofTheorem 3.1 ,itseemssensibletouseithereaswell. Theorem3.3. Let0andasequencefwng1n=1ofpositivenumbersbegiven.Ifforeach>,thereexistsaC>0suchthatw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.58 0 Td[(2Cforeachn,andif'isamultiplierofHw,thentheinequalitiesinTheorem 3.1 aresatisedby+1 2andrespectivelyandj'jkM'k. 34

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Proof. Let>0andlet>+1 2+.Then1Xn=1w)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(21Xn=1w)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(2(+=2)n)]TJ /F8 7.97 Tf 6.59 0 Td[(1)]TJ /F9 7.97 Tf 6.59 0 Td[(C+=21Xn=1n)]TJ /F8 7.97 Tf 6.58 0 Td[(1)]TJ /F9 7.97 Tf 6.58 0 Td[(<1,andweseethatcondition( 3 )ofTheorem 3.1 issatisedby1=+1 2.Now,let>0andlet>+.ObservethatXn2hPNiw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(2Xn2hPNiw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(2(+=2)n)]TJ /F9 7.97 Tf 6.59 0 Td[(C+=2Xn2hPnin)]TJ /F9 7.97 Tf 6.59 0 Td[(.Theright-mostsumconvergesbyLemma 2 ,sothatcondition( 3 )ofTheorem 3.1 issatisedby.ItthenfollowsfromTheorem 3.1 thatj'jkM'k. 35

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CHAPTER4THEUPPERBOUND 4.1KernelsWerecallfromChapter2thatakernelonasetXisafunctionk:XX)166(!Csuchthat 1. k(x,y)= k(y,x);and 2. foranynitesubsetfx1,...,xNgofX,thesymmetricmatrix[k(xi,xj)]1i,jNispositivedenite.EachRKHSdenesareproducingkernel;itisthecontentoftheMoore-Aronszajntheorem(Theorem 2.1 )that,givenakernelk,onecanactuallyconstructaRKHSforwhichkisthereproducingkernel.ItwillbeimportantinSection 4.3 (andlaterinChapter10)thatkernelfunctionsareclosedunder 1. addition; 2. multiplication; 3. scalingbypositivenumbers;and 4. limits(whentheyexist).OurapproachtoobtaininganupperboundforkM'kwillndusisometricallyembeddingourspaceHwinanother,simpler,spaceoneonwhichnormsofmultiplicationoperatorsarewell-understood. Theorem4.1. [ 1 ]SupposekisakernelonXX.ThenthereisaHilbertspaceHandtherearevector-valuedfunctionsf,g:X)166(!Hsuchthat k(x,y)=hf(y),f(x)iH=hg(x),g(y)iH.(4)RecallthatamultiplicationoperatorM'onaRKHShasitsadjointM'satisfyingM'ku= '(u)ku 36

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foreachu2X.ItwasseeninProposition 2.1 thatanyboundedmapTsatisfyingthiskernel/eigenfunctionequationhasitsadjointgivenbymultiplication.ItisessentialthatTbeboundedforthistohold.Ausefulwayofdeterminingwhetherornotagivenfunction'isamultiplierontheRKHSHisgivenbythefollowingtheorem: Theorem4.2. [ 1 ]SupposethatTku= '(u)kuforeachu2X.Thefunction'isamultiplieroftheRKHSHornormatmostifandonlyif(2)]TJ /F10 11.955 Tf 11.95 0 Td[('(u) '(z))k(u,z)0(i.e.(2)]TJ /F10 11.955 Tf 11.95 0 Td[('(u) '(z))k(u,z)isapositivesemi-denitekernel).Thisiseasilyveried:LetTbeasgiven.ThenNXi,j=1j i(2)]TJ /F10 11.955 Tf 11.95 0 Td[('(ui) '(uj))k(ui,uj)=2NXi,j=1j ihkuj,kuii)]TJ /F4 7.97 Tf 25.5 14.94 Td[(NXi,j=1j ih '(uj)kuj, '(ui)kuii=2NXi,j=1j ihkuj,kuii)]TJ /F4 7.97 Tf 25.5 14.95 Td[(NXi,j=1j ihTkuj,Tkuii=2kNXj=1jkujk2)-222(kTNXj=1jkujk2,whichisnonnegativebyhypothesis.HencekTfkkfkforeachf2spanfkug.SincespanfkugisdenseinH,TmaybeextendedtoacontinuousmaponH.That'isamultiplieronHnowfollowsfromProposition 2.1 Asaconsequenceofthis,supposethatH1andH2areRKHS's(oversomesetX)withkernelsk1(u,z)andk2(u,z)respectively.Suppose'isamultiplieronH2(withitscorrespondingmultiplicationoperatorboundedinnormby),k2(u,z)isnonvanishingonXX,andthatK(u,z):=k1(u,z) k2(u,z) 37

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isakernel.Then(2)]TJ /F10 11.955 Tf 11.96 0 Td[('(u) '(z))k2(u,z)0)(2)]TJ /F10 11.955 Tf 11.96 0 Td[('(u) '(z))K(u,z)k2(u,z)0)(2)]TJ /F10 11.955 Tf 11.96 0 Td[('(u) '(z))k1(u,z)0.ItfollowsthatthemultiplieralgebraM2ofH2iscontained(asaset)inthemultiplieralgebraM1ofH1. Lemma3. SupposethatH1andH2areRKHS's(overthesamesetX)withcorrespond-ingkernelsk1(u,z)andk2(u,z)andmultiplieralgebrasM1andM2.IfK(u,z):=k1(u,z) k2(u,z)isakernel,thenM2M1. 4.2TheCasewn=n)]TJ /F8 7.97 Tf 6.59 0 Td[(2Inthissection,wewillrecoverthetheoremsofHLSandMcCarthy(again,withtheprovisothatthe0intheirconclusionsbereplacedbyforamoregeneral0)forweightsdenedbywn=n)]TJ /F8 7.97 Tf 6.59 0 Td[(2,whichaspointedoutinSection 2.5 canbewrittenaswn=Z10n)]TJ /F8 7.97 Tf 6.59 0 Td[(2d()inthecasethatconsistsofapointmassat.WhileonemayobtainthisresultfromeitherTheorem 2.8 ofTheorem 2.9 afteranappropriatechangeofvariables,this(slight)generalizationwillserveasasteppingstonetothemainresultinthischapter,andwillbeincludedforcompleteness. 38

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Theorem4.3. Let0,letwn=n)]TJ /F8 7.97 Tf 6.59 0 Td[(2andletHdenotetheresultingHilbertspace.ThenMH1()\D.Again,H1()isthespaceofbounded,analyticfunctionson,DisthespaceoffunctionsrepresentablebyaDirichletseriesinsomerighthalf-plane,H1()\Disendowedwiththesupremumnorm,andMH1()\DmeansMisisometricallyisomorphictoH1()\D. Proof. Thatj'jkM'kfollowsimmediatelyfromTheorem 3.3 withC=1.TheupperboundonkM'kwillrequireabitmorework.Notethat'(s)convergesinbyitem 5 ofTheorem 3.1 .LetfbeaniteDirichletseries,whichisipsofactoinHw(andindeedinanysuchHilbertspaceofDirichletseries).Sincefisabsolutelyconvergent,Proposition 2.4 tellsusthat'fisgivenbyaconvergentDirichletseriesin.Moreover,sinceboth'andfareboundedin,soistheirproductandBohr'stheoremthengivestheuniformconvergenceof'fin+foreach>0.Letting('f)(s)=1Xn=1ann)]TJ /F4 7.97 Tf 6.58 0 Td[(s,wecanapplyProposition 2.3 tothefunction'ftogetkM'fk2=lim!01Xn=1janj2n)]TJ /F8 7.97 Tf 6.59 0 Td[(2(+)=lim!0limT!11 2TZT)]TJ /F4 7.97 Tf 6.59 0 Td[(Tj('f)(++it)j2dtj'j2lim!0limT!11 2TZT)]TJ /F4 7.97 Tf 6.58 0 Td[(Tjf(++it)j2dt=j'j2kfk2.SincetheniteDirichletseriesformadensesubspaceofHwandsinceM'iscontinuous,wehave kM'fkj'jkfk(4) 39

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foreachf2Hw,whencekM'kj'j.ThemapM'7!'isthusanisometryfromMtoH1(0)\D.Toseesurjectivity,simplynotethat,if'2H1(0)\DisnotinM,thenforeachK>0,wecanndaniteDirichletseriesfwithkfk<1suchthatKk'fkj'j,contradictingtheboundednessof'in. 4.3AnUpperBoundforM'Inthissection,wewillbeestablishingtheupperboundpromisedinSection 2.6 .TheproofofTheorem 4.4 isofamoreoperator-theoreticavorthananythingseenuptothispoint.Beforegettingintotheproof,however,somepreliminariesareinorder.GiventwoHilbertspacesHandK(withinnerproductsh,i1andh,i2respectively),wemaybuildanewHilbertspacefromtheiralgebraictensorproductHK.Thisnotationisnotstandardforthetensorproductoftwomodules,butwearereservingthemorestandardnotationHKfortheHilbertspacewewillbeconstructing.Deneasesquilinearformh,ionelementarytensorsbyhh1k1,h2k2i=hh1,h2i1hk1,k2i2andextendbylinearitytoHK.ThecompletionofthisspaceisaHilbertspace,whichwewilldenotebyHK.Notethat,khkk=khkkkk.Itisevidentthat,iffeigandffjgareorthonormalbasesforHandKrespectively,thenfeifjgisanorthonormalsubsetofHK.(Infact,feifjgisanorthonormalbasisforHK.) 40

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Supposenowthatwearegiventwoboundedoperators,S2B(H)andT2B(K).WecanconstructannewmapST:HK)166(!HKbydening(ST)(hk)=(Sh)(Tk)onelementarytensorsandextendingbylinearitytonitelinearcombinationsofelementarytensors.Now,anyelementinspanfeifjgcanbewrittenasNXj=1hjfj,wherethehj'sarelinearcombinationsoftheei's.LetIKdenotetheidentitymaponK.Sincek(SIK)(nXj=1hjej)k2=knXj=1(Shj)ejk2=nXj=1kShjk2kSk2nXj=1khjk2=kSk2knXj=1hjejk2,weseethatSIKiscontinuouswithoperatornormatmostkSk.Asimilarargumentshowthat,ifIHistheidentitymaponH,thenIHTiscontinuouswithoperatornormatmostkTk.WritingST=(SIK)(IHT),weseethatkSTkkSkkTk(infact,kSTk=kSkkTk,suchanormbeingcalledacross-norm),thusbothshowingthecontinuityofSTonthedensesubsetofnitelinearcombinationsofelementarytensorsandproducingaboundfortheoperatornormofthistensorproductofoperatorsintermsoftheoperatornormsofitsindividualcoordinateoperators.NowthatweknowthatSTiscontinuousonthedensesubsetofnitelinearcombinationsofelementarytensors,wecanextendSTtoanoperatorstilldenotedbySTonallofHK.WenowhaveeverythingweneedtoproveTheorem 4.4 41

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Theorem4.4. Letfwng1n=1beasequenceofpositivenumbers,let0,andlet'beamultiplierofHw,andsupposethat'hasananalyticcontinuationtogivenby .If 1. fwng1n=1ismultiplicative;and 2. foreachprimepandpositiveintegerk,wehavewpkp)]TJ /F8 7.97 Tf 6.58 0 Td[(2wpk)]TJ /F12 5.978 Tf 5.75 0 Td[(1,thenkM'kj j.Byreplacingwnbywnn)]TJ /F8 7.97 Tf 6.59 0 Td[(2,itsufcestoestablishtheresultfor=0.Inthiscase,item( 2 )says,simply,thatthesequencefwpkg1k=0isdecreasingforeachprimep. Proof. LetH0andk0denotethespaceandkernelcorrespondingtotheweightswn1.Notethatforuandzin1,thekernelk0(u,z)convergesabsolutelyandisgivenby k0(u,z)=1Xn=1n)]TJ /F9 7.97 Tf 6.59 0 Td[(=Ypprime1 1)]TJ /F5 11.955 Tf 11.95 0 Td[(p)]TJ /F9 7.97 Tf 6.58 0 Td[(,(4)where=u+ z.Forthegeneralsequenceofweightsandforuandzin0(with0asin( 2 )),thekernelkw(u,z)convergesabsolutely,andhasitsownproductrepresentationgivenby Ypprime1Xj=0w)]TJ /F8 7.97 Tf 6.59 0 Td[(1pjp)]TJ /F4 7.97 Tf 6.59 0 Td[(j.(4)Letmmaxf1,0g.Thenforu,z2m,wehave K(u,z)=kw(u,z) k0(u,z).(4)(Notethatk0(u,z)hasnozeros11.)Theproductsforthesekernelsconvergegiving,K(u,z)=Ypprime(1)]TJ /F5 11.955 Tf 11.96 0 Td[(p)]TJ /F9 7.97 Tf 6.59 0 Td[()1Xj=0w)]TJ /F8 7.97 Tf 6.58 0 Td[(1pjp)]TJ /F4 7.97 Tf 6.59 0 Td[(j=Ypprime 1+1Xj=1(w)]TJ /F8 7.97 Tf 6.58 0 Td[(1pj)]TJ /F5 11.955 Tf 11.96 0 Td[(w)]TJ /F8 7.97 Tf 6.59 0 Td[(1pj)]TJ /F12 5.978 Tf 5.75 0 Td[(1)p)]TJ /F4 7.97 Tf 6.59 0 Td[(j!. 42

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Byourassumptionontheweights,wehavew)]TJ /F8 7.97 Tf 6.58 0 Td[(1pj)]TJ /F5 11.955 Tf 11.96 0 Td[(w)]TJ /F8 7.97 Tf 6.59 0 Td[(1pj)]TJ /F12 5.978 Tf 5.75 0 Td[(10,anditfollowsthatK(u,z)ispositivesemideniteonmm.FromTheorem 4.1 ,thefactthatKispositivesemideniteimpliestheexistenceanauxiliaryHilbertspaceHQandafunctionQ:m!HQsuchthat K(u,z)=kw(u,z) k0(u,z)=hQ(u),Q(z)iH.(4)LetH0HQdenotetheHilbertspacetensorproductofH0andHQ.Multiplyingthroughandrewriting,wehave hkwu,kwziw=hk0u,k0zi0hQ(u),Q(z)iH=hk0uQ(u),k0zQ(z)i.(4)ThisequalityofinnerproductshintsatanunderlyingisometrybetweenthespacesHwandH0HQ.Accordingly,wedeneanoperatorVfromS=fkwu:u2mgtoH0HQbyVkwu=k0uQ(u)andextendbylinearitytospanS.Sodened,VisanisometryonspanS,andsincespanSisdenseinHw,Vthusextendscontinuouslytoanisometry,stilldenotedbyVfromHwtoH0HQ.If isunboundedin0,thenthere'snothingtodosincethenj j0=1.Otherwise, isboundedon0and,bySchnee'stheorem,equalto'on.ItfollowsfromTheorem 4.3 that'isamultiplierofH0.LetM',wdenotemultiplicationby'inHwandletM',0denotemultiplicationby'inH0.Fork0uQ(z)inH0HQ,wehave (M',0I)(k0uQ(z))= '(u)k0uQ(z).(4) 43

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Itiseasilycheckedthat VM',w=(M',0I)V,(4)sothatthefollowingdiagramcommutes:HwV)305()222()306(!H0HQM',w??y??yM',0IHw)305()222()306(!VH0HQ.Inparticular,kM',wkkVkkM',0kkVkkM',0kj'j0,whichiswhatwastobeshown. Remark.IffwngisaweightsequencesatisfyingthehypothesesofTheorem 4.4 andifHandkarethecorrespondingHilbertspaceandkernel,thenitem 2 guaranteesthatthesymmetricfunctiondenedbyK(u,z)=kw(u,z) k(u,z)ispositiveforuandzfarenoughtotheright. 4.4THECASE=We'veseenconditionsthatcanbeimposedonourweightsthatwillguaranteethatj'jkM'k;we'veseenconditionsthatcanbeimposedonourweightstoguaranteethatkM'kj j( beingtheanalyticcontinuationof'to).NowsupposethatfwngisasequenceofpositivenumberssimultaneouslysatisfyingtheconditionsofTheorems 3.1 and 4.4 with01: 1. P1n=1w)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(21<1whenever1>1; 2. Pn2hPNiw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(2<1foreachNwhenever>; 3. fwng1n=1ismultiplicative;and 4. foreachprimepandeachpositiveintegerk,wehavewpkp)]TJ /F8 7.97 Tf 6.59 0 Td[(2wpk)]TJ /F12 5.978 Tf 5.75 0 Td[(1. 44

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IfHwistheHilbertspaceresultingfromsuchasequenceandif'isamultiplieronHwwithananalyticcontinuation to,thenTheorems 3.1 and 4.4 togetherwithSchnee'stheoremgivej'jkM'kj j.ThesameconclusionholdsifweinsteadconsiderasequencesatisfyingtheconditionsofTheorems 3.3 and 4.4 ,again,with01: 1. foreach>thereisaC>0suchthatw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(2Cforeachn; 2. fwng1n=1ismultiplicative;and 3. foreachprimepandeachpositiveintegerk,wehavewpkp)]TJ /F8 7.97 Tf 6.59 0 Td[(2wpk)]TJ /F12 5.978 Tf 5.75 0 Td[(1,Therearemanysequencessatisfyingtheseconditions.Indeed,anycompletelymultiplicativesequencefwng1n=1withp)]TJ /F8 7.97 Tf 6.59 0 Td[(2wpp)]TJ /F8 7.97 Tf 6.58 0 Td[(2(n=1,2,3,...)does.If=,thenthemapM'7!'isanisometryfromM,thespaceofmultipliersonHwwithoperatornorm,intoH1()\D,thespaceoffunctions,boundedandholomorphicinwhicharerepresentablebyDirichletseriesinsomerighthalf-planethisspacebeingequippedwiththesupremumnorm.Thenexttheoremshowsthatthismapisinfactsurjectivethateach'2H1()\DgivesrisetoamultiplieronHw. Theorem4.5. If'2H1()\D,then'isamultiplieronHw. Proof. FromtheremarkfollowingtheproofofTheorem 4.4 ,K(u,z)=kw(u,z) k(u,z) 45

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isakernel.Since'isamultiplieronH(seeninSection 4.2 ),itfollowsfromLemma 3 that'isalsoamultiplieronHw. Corollary2. If0=1and 1. P1n=1w)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(21<1whenever1>1; 2. Pn2hPNiw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(2<1foreachNwhenever>; 3. fwng1n=1ismultiplicative;and 4. foreachprimepandeachpositiveintegerk,wehavewpkp)]TJ /F8 7.97 Tf 6.59 0 Td[(2wpk)]TJ /F12 5.978 Tf 5.75 0 Td[(1.then MH1()\D.(4) Proof. ThemapM'7!'isanisometrybyTheorems 3.1 and 4.4 andissurjectivebyTheorem 4.5 Corollary3. If0=and 1. ifforeach>thereisaC>0suchthatw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(2Cforeachn; 2. fwng1n=1ismultiplicative;and 3. foreachprimepandeachpositiveintegerk,wehavewpkp)]TJ /F8 7.97 Tf 6.59 0 Td[(2wpk)]TJ /F12 5.978 Tf 5.75 0 Td[(1,thenMH1()\D.InChapter6,wewillconsiderweightsgivenbythereciprocalsofsomewell-knownnumbertheoreticfunctions.TheseweightswillbeshowntosatisfytheconditionsofCorollary 3 ,andwillthusprovideexamplesofweightsequencesresultinginHilbertspaceswhosemultiplieralgebrascanbeisomorphicallyidentiedasthespaceofbounded,analyticfunctionsrepresentablebyDirichletseriesinforsome0. 46

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Furthermore,byconsideringgrowthrates,itwillbeshownthattheseweightscan'tarisefromanymeasureasin( 2 ). 47

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CHAPTER5WEIGHTSGIVENBYMEASURES 5.1GoalsforthisChapterIn[ 10 ],McCarthyprovedthefollowing: Theorem5.1. IfisapositiveRadonmeasurewith0initssupportandifwnisdenedby wn=Z10n)]TJ /F8 7.97 Tf 6.59 0 Td[(2d(),(5)fornn0,wheren0isthesmallestnaturalnumberforwhichthisintegralisnite,then MH1(0)\D.(5)Inthischapter,wewillusetheresultsfromSection 4.2 ,alongwithsomeadditionalresultstobeobtaineddirectly,torecaptureMcCarthy'sresultmutatismutandisinthecasethatisaregularmeasurewith0initssupport.Ourapproachwillinvolveshowingthat( 4 )holdswhenourweightsaregivenbyconvexcombinationsofpointmasses;proving( 4 )forweightsgivenbycompactlysupportedprobabilitymeasuresbyusingthefactthatsuchmeasurescan,insomesense,beapproximatedbyconvexcombinationsofpointmasses;extendingthisresulttogeneralprobabilitymeasures;andnally,makingtheleapinshortordertoniteregularmeasuresandthentoallregularmeasures.Beforewecandothisthough,somepreliminariesareinorder. 5.2PreliminariesAtopologicalvectorspace(TVS)isvectorspaceequippedwithatopologyforwhichtheoperationsofadditionandscalarmultiplicationarecontinuous.IfXisaTVS,x2Xand2F(F=RorF=C),thenitiseasilyseenthatthemapsy7!x+yandy7!y(6=0)denehomeomorphismsfromXtoitself.Thus,theopensetsinXareinvariantundertranslationandscaling.WhilemuchtheoryhasbeendevelopedinthestudyofTVS's,wewillonlybeexploringthetheorynecessarytoobtainthecentralresultsof 48

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thischapter.Rudin[ 12 ]andConway[ 6 ]areserviceableprimersforthecuriousreaderwantingtostudythesestructuresinnerdetail.Webeginwithsomebasicterminology.GivenanysubsetSofaTVSX,theconvexhullofSdenotedbyhullSisintersectionofalltheconvexsubsetsofXcontainingSandisthesmallestconvexsubsetofXcontainingS.TheclosedconvexhullofthesetSistheintersectionofalltheclosedconvexsetscontainingS.Aspaceiscalledlocallyconvexifeverypointhasaconvexneighborhood.AconvexcombinationinXisalinearcombinationofpointsinXwhosecoefcientsareallnon-negativeandsumto1.AnextremepointofthesetSisapointofSnotlyingonanyopenlinesegmentjoiningtwopointsofS.Thatis,ifxisanextremepointofS,ifyandzareinS,andifx=(1)]TJ /F5 11.955 Tf 12.24 0 Td[(t)y+tz(0t1),theneitherx=yorx=z.OnemayintuitivelythinkoftheextremepointsofSasbeingtheverticesofS.Itcanbeshown(see[ 11 ])thattheconvexhullofasetSissimplythesetofallconvexcombinationsofpointsofS.Thatis,hull(S)=(NXj=1jsj:N2N,sj2S,j0,NXj=1j=1).IfXislocallyconvexandSXiscompactandconvex,thenmorecanbesaid: Theorem5.2. (Krein-Milman)IfSisacompactandconvexsubsetofalocallyconvexspaceX,thenSistheclosedconvexhullofitsextremepoints.Inparticular,thesetofextremepointsofSisnonempty. 5.3TheWeak-TopologyGivenanormedspaceX,wedeneitsdualspaceXtobethesetofallcontinuous(withrespecttothenormtopologyonX),linearfunctionalsonX.Ingeneral,Xmayhavefarmoreopensetsthannecessarytoguaranteethecontinuityofeach 2X.WedenetheweaktopologyonXtobethesmallesttopologyforwhicheach 2Xiscontinuous.TheweaktopologyonXisgeneratedbysetsoftheform )]TJ /F8 7.97 Tf 6.59 0 Td[(1(U),whereUisopeninC(orR)and 2X.Theweaktopologyis(generally)weakerthanthenorm 49

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topology,inasmuchasitisobtainedbytakingthenorm-opensetsofXandthrowingouttheonesnotneededtoguaranteethecontinuityofthefunctionalsinX.Itisusefultohaveanideaofwhatitmeansforanetfxgtoconvergeweaklytothepointx:Ifx!xweakly,then (x)! (x)inF(whichisagain,eitherRorC).Toseethis,let 2XandtakeanopenballBcontainingthepoint (x).Then )]TJ /F8 7.97 Tf 6.59 0 Td[(1(B)isaweaklyopenneighborhoodofxandthereisthereforeasuchthat)x2 )]TJ /F8 7.97 Tf 6.59 0 Td[(1(B);thatis) (x)2B,sothatthenet (x)convergesto (x).Conversely,supposethat (x)convergesto0foreach 2X.LetWbeaweaklyopensetcontaining0.Thentherearefunctionals 1,..., nandpositivenumbers1,...,nsuchthat02n\j=1fx2X:j j(x)j
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WemayisometricallyidentifyXwiththesubspaceofevaluationmapsinXviatheembeddingx7!Ex.ThatkExkkxkisclear;thatkExk=kxkfollowsfromtheHahn-Banachtheorem,which,inpart,guaranteestheexistenceofsome 2Xsuchthatk k=1and (x)=kxk.Wedenetheweak-topologyonXtobetheweakesttopologysuchthatpointevaluationsarecontinuous.Ingeneral,Xmayconsistoffunctionalsnotgivenbypointevaluations.If,however,everyelementofXisoftheformExforsomex2X,wesaythatXisreexive.IfXisaHilbertspace,thenitisaconsequenceoftheRieszrepresentationtheoremthatXisreexive,andthatXisisometricallyisomorphictoX.Itcanalsobeshown(Theorem2.4.4in[ 11 ])that,iff2Xisweak-continuous,thenfisgivenbyapointevaluation,sothatthesepointevaluationsare,withrespecttotheweak-topology,theonlycontinuousfunctionalsonX.Asbefore,anetffginXconvergesweak-tofifandonlyifEx(f)!Ex(f)(i.e.f(x)!f(x))foreachx2X.Thus,weak-convergenceinXissimplypointwiseconvergence.Theproofofthischaracterizationofweak-convergenceisalmostidenticaltotheproofofthecharacterizationofweakconvergence,andwillthereforebeomitted.Theclosedunitballofanormedspaceneednotbecompactinthenorm-topology(andwillneverbeininnitedimensionsaconsequenceofRiesz'slemma).However,incertainnormedspaces(eveninnite-dimensionalones),theunitballiscompactwithrespecttoothertopologies. Theorem5.3. (Banach-Alaoglu)ForeachnormedspaceX,theclosedunitballBinXisweak-compact.TheBanach-AlaoglutheoremtellsusthatanygivennetinBhasasubnetf gsuchthatf (x)gconvergesforeachx2X.Moreover,sinceXisthedualspaceofX,wehavetheclosedunitballBofXbeingweak-compact.IfXisreexive,thentheweak-topologyonXinducesaweak-topologyonX,forwhichtheclosedunit 51

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ballinXisalsoweak-compact.Inparticular,foraHilbertspaceH,B=Bisweak-compact,sothat,foranynetinB,thereexistsasubnetfxgsuchthathx,yiconvergesforeachy2H. 5.4TheSpaceofComplexMeasuresonaCompactHausdorffSpaceLetXbeacompactHausdorffspace,letC(X)bethespaceofcontinuouscomplex-valuedfunctionsonXwithsupremumnormandletC(X)bethedualofC(X).WedenotebyM(X)thespaceofallcomplexmeasuresonXandbyP(X)thesetofallprobabilitymeasuresonX.Themapkk:M(X))166(!Rdenedbykk=jj(X)(wherejj(X)isnecessarilynitebyTheorem6.4in[ 12 ])makesM(X)intoanormedspace(aBanachspaceeven).EachelementofM(X)denesaboundedlinearfunctionalonC(X)inacanonicalway:if2M(X),thenthefunction:C(X))166(!Cdenedbyf7!ZXfdiscontinuouswithoperatornormatmostkk:ZXfdZXjfjdjjkfk1jj(X).ItisthecontentoftheRieszrepresentationtheoremthatthemap7!denesanisometricisomorphismbetweenM(X)andC(X). Theorem5.4. (Rieszrepresentationtheorem)IfXisalocallycompactHausdorffspace,theneveryboundedlinearfunctionalonC0(X)thespaceofcontinuous,compactlysupportedfunctionsonXisrepresentedbyauniqueregularcomplexBorelmeasureinthesensethat(f)=ZXfdforeveryf2C0(X).Moreover,kk=jj(X).Inourpresentsetting,Xiscompactand,ipsofacto,C0(X)=C(X). 52

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WesawinSection 5.3 thatC(X)isnaturallyimbuedwithaweak-topology.Withtheidentication=asabove,thespaceM(X)inheritsitsownweak-topologyinanaturalway:ifisthemapsendingto,thenAM(X)isweak-openifandonlyif(A)isweak-open.Inthistopology,nwk-*)166(!,ZXfdn!ZXfd8f2C(X).ThefollowingtheoremregardingtheconvexhullofthesetofprobabilitymeasuresinconjunctionwiththeKrein-Milmantheorem(Theorem 5.2 )willplayacrucialroleinallowingustojumpfromweightsgivenbypointmassestoweightsgivenbyprobabilitymeasures. Theorem5.5. P(X)isaconvex,weak-compactsetwhoseextremalpointsarethepointmasses.(Foraproofofthis,seepages72-73in[ 11 ].)ItfollowsfromtheKrein-Milmantheoremthateachprobabilitymeasureistheweak-limitofasequenceofconvexcombinationsofpointmasses. 5.5WeightsGivenbyConvexCombinationsofPointMassesTherststeptowardthe(partial)recoveryofMcCarthy'stheoremistoprovethatMH1(1)\Dinthecasethatourweightsaregivenbyconvexcombinationsofpointmasseswithleftmostsupportat1.Towardthatend,supposethat012kandletf1,...,kgbeacollectionofnonnegativenumberswithPj=1.Foreachn,denewn=1n)]TJ /F8 7.97 Tf 6.59 0 Td[(21++kn)]TJ /F8 7.97 Tf 6.58 0 Td[(2k,sothateachwnisconvexcombinationofpointmasses.LetHwdenotetheresultingHilbertspaceandlet'beamultiplieronHw.Notethat,foreach>1,thereisaconstantCsuchthatw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.58 0 Td[(2Cthisfollowingfromthatfactthatwnisgivenbyameasuresupportedonf1,...kg.Thus,thehypothesisofTheorem 3.3 issatisedby 53

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ourweightsequenceanditfollowsthatj'j1kM'k.Fortheotherdirection(j'j1kM'k),rstnotethat,ifHjistheHilbertspacecorrespondingtotheweightsequencewn=n)]TJ /F8 7.97 Tf 6.58 0 Td[(2jandifkkjisthecorrespondingnorm,thenforf2Hw,wehavekfk2w=1kfk21++kkfk2k,sothatf2Hw,f2Hj(j=1,2,...,k).Inparticular, 'isamultiplieronHw,'isamultiplieronHjforeachj.(5)Supposenowthatf2Hwand('f)(s)=1Xn=1nn)]TJ /F4 7.97 Tf 6.59 0 Td[(s.ThenkM'fk2w=1kM'fk21++kkM'fk2k1j'j21kfk21++kj'j2kkfk2kj'j21(1kfk21++kkfk2k)=j'j21kfk2w,withthejumpfromtherstlinetothesecondlinecomingfromTheorem 3.3 .HencekM'kj'j1. 54

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Lastly,thatthemapM'7!'issurjectivefollowsimmediatelyfromTheorem 4.3 and( 5 ),thusprovingthatMH1(1)\D. 5.6WeightsGivenbyCompactlySupportedProbabilityMeasuresInthissection,wewillbeusingTheorem 5.5 tomaketheleapfromweightsgivenbyconvexcombinationsofpointmasses,asconsideredinSection 5.5 ,toweightsgivenbycompactlysupportedprobabilitymeasures.WewillthereuponextendthisresultinSections 5.7 and 5.8 tocompactlysupportednitemeasuresandnally,toallregularmeasureson[0,1),thusobtainingmutatismutandisMcCarthy'sresultwhentheweightsaregivenbyaregularmeasurewith0initssupport.Letbeaprobabilitymeasuresupportedintheclosedinterval[a,b]witha2supp.Denewn=Z10n)]TJ /F8 7.97 Tf 6.59 0 Td[(2d()andletHw,kkw,andMdenotetheresultingHilbertspace,norm,andmultiplieralgebra.Notethat,sinceisaprobabilitymeasure,wn1foreachn.Asbefore,foreach>a,thereisaC>0(namelyC=1 ([a,]))suchthatw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(2C.Thatj'jakM'kthenfollowsfromTheorem 3.3 .Fortheotherdirection,Theorem 5.5 furnishesasequencefjg1j=1ofnitelysupportedprobabilitymeasuresapproachingweak-.Foreachj,letHjandkkjdenotethecorrespondingHilbertspaceandnorm.AsseenattheendofSection 5.4 ,jwk-*)166(!impliesthatZ10n)]TJ /F8 7.97 Tf 6.59 0 Td[(2dj()!Z10n)]TJ /F8 7.97 Tf 6.58 0 Td[(2d(),sothat,ifwn(j)=Z10n)]TJ /F8 7.97 Tf 6.59 0 Td[(2dj(),thenwn(j)!wnasj!1foreachn. 55

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Sincej'ja<1,'isamultiplieroneachHj,asshownintheprevioussection.LetfbeaniteDirichletseries(whichbyvirtueofbeingnite,liesinHwandineachHj),let('f)(s)=1Xn=1ann)]TJ /F4 7.97 Tf 6.58 0 Td[(s,let>0,andtakeNtobelargeenoughthat1Xn=N+1janj2wn<.ThenkM'fk2w=1Xn=1janj2wn=NXn=1janj2wn+1Xn=N+1janj2wnNXn=1janj2wn+=limj!1NXn=1janj2wn(j)+limsupj!1kM'fk2j+j'j2alimsupj!1kfk2j+=j'j2akfk2w+.(Notethatlimsupj!1kfk2j=limj!1kfk2jsincefisnite.)Since>0wasarbitrary,weseethatkM'fk2wj'j2akfk2w,foreachniteDirichletseries,whencekM'kwj'ja. 56

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Thus,M'7!'denesanisometryfromMintoH1(a)\D.Toseethatthismapisinfactonto,simplynotethatif'2H1(a)\Disnotamultiplier,thenforeachK>0,wecanndaniteDirichletseriesfofnormlessthan1(with('f)(s)=P1n=1ann)]TJ /F4 7.97 Tf 6.59 0 Td[(s)andanNsuchthatKNXn=1janj2wn=limj!1NXn=1janj2wn(j)limsupj!1kM'fk2jj'j2a,whichcontradictstheboundednessof'ina.ThusM'7!'denesanisometricisomorphismfromMontoH1(a)\D. 5.7WeightsGivenbyCompactlySupportedFiniteMeasuresWewillnowextendtheconclusionoftheprecedingsectiontoanycompactlysupportedniteregularmeasure.ThiswillgiveuswhatweneedtonallyrecoverMcCarthy'sresultinSection 5.8 .Letbeaniteregularmeasuresupportedin[a,b]witha2supp.WecantransformintoaprobabilitymeasureandthusputourselvesinthesettingofSection 5.6 bynormalization:if:==([0,1)),thenisaprobabilitymeasuresupportedin[a,b]witha2supp.Deneewn=Z10n)]TJ /F8 7.97 Tf 6.59 0 Td[(2d()andletHewandkkewdenotetheassociatedHilbertspaceandnormrespectively.Notethatewn=wn=([0,1)).Thatj'jakM'kfollowsfromthestandardargument.Sincej'ja<1,'isamultiplieronHew(seeSection 5.6 ).LetfbeaniteDirichletseriesandsupposethat('f)(s)=1Xn=1ann)]TJ /F4 7.97 Tf 6.58 0 Td[(s. 57

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ThenkM'fk2w=1Xn=1janj2wn=([0,1))1Xn=1janj2ewn=([0,1))kM'fk2ew([0,1))j'j2akfk2ew=j'j2akfk2w,sothatkM'kwj'ja.Finally,toseethateach'2H1(a)\DisamultiplieronHw,simplyobservethat,since([0,1))1Xn=1janj2ewn=1Xn=1janj2wn,'isamultiplieronHewifandonlyif'isalsoamultiplieronHw.ItfollowsthatthemapM'7!'isbothisometricandsurjective,thusisometricallyidentifyingMwiththespaceH1(a)\D. 5.8WeightsGivenbyRegularMeasuresWearenowinapositiontoprovethefollowing:ifisaregularmeasureon[0,1)withatheleftmostelementofitssupport,ifourweightsequenceisdenedbywn=Z10n)]TJ /F8 7.97 Tf 6.59 0 Td[(2d(),andifHwisthecorrespondingHilbertspace,thenMH1(a)\D. 58

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WeknowfromSection 5.7 thatthisequivalenceholdsinthecasethatourmeasureisgivenby[0,j]d,where[0,j]istheindicatorfunctionontheclosedinterval[0,j].Wewillexploitthisfact,alongwithanappropriatelimitingargument,toobtainthedesiredconclusion.Letwn(j)=Z10n)]TJ /F8 7.97 Tf 6.59 0 Td[(2[0,j]d()andforeachj,letHjandkkjdenotethecorrespondingHilbertspaceandnorm.Notethatwn(j)!wnforeachnasj!1.Theargumentshowingthatj'jakM'kisthesameasbefore.Sincej'ja<1,'isamultiplieroneachHj(aspertheconclusionofSection 5.7 ).LetfbeaniteDirichletseriesandsupposethat('f)(s)=1Xn=1ann)]TJ /F4 7.97 Tf 6.58 0 Td[(s.Let>0andtakeNlargeenoughthat1Xn=N+1janj2wn<. 59

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WethenhavekM'fk2w=1Xn=1janj2wn=NXn=1janj2wn+1Xn=N+1janj2wnNXn=1janj2wn+=limj!1NXn=1janj2wn(j)+=limsupj!1kM'fk2j+j'j2alimsupj!1kfk2j+=j'j2akfk2w+sothatkM'kj'jaonthedensesubsetofniteDirichletseries.Hencej'j1=kM'k.ThatthemapM'7!'issurjectivefollowsfromanargumentsimilartotheoneusedinSection 5.6 60

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CHAPTER6EXAMPLESRecallthatasequenceofpositivenumbersfwng1n=1ismultiplicativeifwmn=wmwnwhenevermandnarecoprimeandiscompletelymultiplicativeifwmn=wmwnforeverypairofnaturalnumbersmandn.WesawinSection 4.4 ofChapter4howtoconstructcompletelymultiplicativeweightsequencesforwhichtheconclusionofMcCarthy'stheoremholdsmutatismutandisbutwhichonlycomefromameasureinthecaseofapointmass.However,theJensen'sinequalityargumentusedtoshowthisindependencedoesn'tquiteworkifoursequenceismultiplicative,butnotcompletelyso.Ingeneral,thequestionofwhetherornotamultiplicativesequencearisesfromameasureismoredifcult.Inthischapter,wewillbeexaminingthreenumber-theoretic,multiplicativeweightssequences,noneofwhichcomefromameasure,andallthreeofwhich,forsome0,satisfytheconditionsfoundinCorollary 3 ofSection 4.4 ,restatedhereforconvenience:If0=and 1. ifforeach>thereisaC>0suchthatw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(2C(8n2N) 2. fwng1n=1ismultiplicative;and 3. foreachprimepandeachpositiveintegerk,wehavewpkp)]TJ /F8 7.97 Tf 6.59 0 Td[(2wpk)]TJ /F12 5.978 Tf 5.75 0 Td[(1,thenMH1()\D. Denition4. Thedivisorfunctiond(n)givesthenumberofpositivedivisorsofthenaturalnumbern.Thesumofdivisorsfunction(n)givesthesumofthepositivedivisorsofn.TheEulertotientfunction(n)givesthenumberofpositiveintegerslessthanorequaltonthatarecoprimeton.Theproofsthateachofthesethreefunctionsismultiplicativearewell-knownandwon'tbegivenhere. 61

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Inthischapter,wewillbeconsideringtheweightsequencesgivenbythereciprocalsofthesethreefunctions.Wewillsee,ineachcase,thatifMisthemultiplieralgebraofthecorrespondingRKHS,thenMH1()\Dforsome0.Moreover,itwillbeshownthattheseweightsequenceshavegrowthratesthatcan'tbematchedbyanysortofmeasure,thusprovingtheindependenceoftheseresultsfromMcCarthy's.Wenowstatethemaintheoremofthischapter: Theorem6.1. Ifwnisgivenbythereciprocalofd(n),(n),or(n),thenthereisnopositivemeasuresuchthat wn=Z10n)]TJ /F8 7.97 Tf 6.59 0 Td[(2d().(6)Ontheotherhand,ineachcasethereisa0suchthatMisisometricallyisomorphictoH1()\D.Notethatifthereweresuchameasure,thenitwouldnecessarilybeaprobabilitymeasuresince,ineachcase,w1=1.Beforemovingintotheproofofthistheorem,somepreliminariesareinorder. Lemma4. Foreachnaturalnumbern,6 2<(n)(n) n2<1.Wealsohavethefollowinggrowthrates,whichalongwithLemma 4 willbeneededtoshowthatourweightsequences(1)can'tarisefromameasureasinEquation( 2 )and(2)meettheconditionsofCorollary 3 fortheappropriate0:forthedivisorfunctiond(n),wehaved(n)=o(n) 62

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foreach>0;forthesumofdivisorsfunction(n),wehavethefollowinggrowthrates,therstofwhichisknownasGronwall'stheorem: limsupn(n) nlnlnn=e(6)(istheEuler-Mascheroniconstant)and (n)=O(n1+)(6)foreach>0;andnally,fortheEulertotientfunction(n),wehaveboth limsupn(n) n=1(6)and liminfn(n) n=0.(6)Theproofsofthesecanbefoundin[ 8 ].WearenowreadytoproveTheorem 6.1 Proof. Supposerstthatwn=1 d(n)andassumebywayofcontradictionthatthereissomepositivemeasuresuchthatwn=1 d(n)=Z10n)]TJ /F8 7.97 Tf 6.59 0 Td[(2d().Letfp1
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suchthatlimj!1njlnlnnj (nj)=e)]TJ /F9 7.97 Tf 6.58 0 Td[(.Hencee)]TJ /F9 7.97 Tf 6.59 0 Td[(=limj!1njlnlnnj (nj)=limj!1Z10n1)]TJ /F8 7.97 Tf 6.59 0 Td[(2jlnlnnjd()limsupj!1Z[0,1=2]n1)]TJ /F8 7.97 Tf 6.59 0 Td[(2jlnlnnjd()limsupj!1([0,1=2])lnlnnj,fromwhichitfollowsthat([0,1=2])=0.Since6 2<(n)(n) n2<1(byLemma 4 ),( 6 )impliestheexistenceofasequencefnig1i=1suchthat(ni) ni2,foreachi,whence1 2limsupini (ni).Thus1 2limsupini (ni)=limsupiZ10n1)]TJ /F8 7.97 Tf 6.58 0 Td[(2id()=limsupiZ[0,1=2]n1)]TJ /F8 7.97 Tf 6.58 0 Td[(2id()+Z(1=2,1)n1)]TJ /F8 7.97 Tf 6.59 0 Td[(2id()=limsupiZ(1=2,1)n1)]TJ /F8 7.97 Tf 6.59 0 Td[(2id()=0, 64

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withthesecond-to-laststepcomingfromthefactthat([0,1=2])=0andthelaststepcomingfromFatou'slemmajustiedsinceeachn1)]TJ /F8 7.97 Tf 6.59 0 Td[(2iisdominatedon(1=2,1)bytheintegrablefunction1.Thiscontradictionprovesthatwn,sodened,can'tcomefromameasure.Finally,letwn=1 (n)andagain,supposebywayofcontradiction,thatthereisasatisfyingwn=1 (n)=Z10n)]TJ /F8 7.97 Tf 6.59 0 Td[(2d().By( 6 ),thereisasequencefnig1i=1suchthatlimi!1ni (ni)=1.Itthenfollowsthat1=limi!1ni (ni)=limi!1Z10n1)]TJ /F8 7.97 Tf 6.58 0 Td[(2id()=limi!1Z[0,1=2)n1)]TJ /F8 7.97 Tf 6.59 0 Td[(2id()+(f1=2g)+Z(1=2,1)n1)]TJ /F8 7.97 Tf 6.59 0 Td[(2id().Now,therstintegralintheparenthesesmustbe0andthesecondintegralintheparenthesesmusttendto0bythedominatedconvergencetheorem,sothat(f1=2g)=1.However,runningthroughthesameargumentwithasequencefurnishedby( 6 )showsthat(f1=2g)=1acontradiction.Wehavethusshownthattheweightsequencesgivenbythereciprocalsofthedivisorfunction,thesumofdivisorsfunction,ortheEulertotientfunctioncan'tarisefromanysortofmeasure.WenowproceedtothesecondclaimofTheorem 6.1 thatineachcase,thereisa0suchthatMisisometricallyisomorphictoH1()\D.Let'srstexaminetheweightsequence,wn=1 d(n),derivedfromthereciprocalofthedivisorfunction.Inthiscase,d(n)=o(n)foreach>0(see[ 8 ]),sothat,foreach 65

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>0,w)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.58 0 Td[(2!0asn!1.Wecan,therefore,certainlyndaC>0suchthatw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(2Cforalln.TheotherconditionsinCorollary 3 aresatisedaswell:wnisclearlymultiplicativeandwpk=1 d(pk)=1 k+1isdecreasingwithkforeachprimep.Therefore,MH1(0)\D.Ifinsteadwn=1 (n),thentheconditionsinCorollary 3 aresatisedby=1 2:thereissomeK>0suchthat(n) nK(see( 6 )),whichimpliesthat(n)n)]TJ /F8 7.97 Tf 6.59 0 Td[(1=(n)n)]TJ /F8 7.97 Tf 6.59 0 Td[(2(1=2)0,( 6 )impliestheexistenceofsomepositivenumberKsuchthat(n) n1+Kforeachn,sothat(n)n)]TJ /F8 7.97 Tf 6.58 0 Td[(2(1 2(1+))
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CHAPTER7MULTIVARIABLEDIRICHLETSERIES 7.1GoalsintheMultivariableSettingOverthecourseofthenextfourchapters,wewillbeexploringthemultiplieralgebrasofHilbertspacesofmultivariableDirichletseries(tobedenedshortly).Wewillbeshowingthat,undersuitablehypotheses,theresultsfromChapters3and4canbeextendedtoDirichletseriesinmvariables,wheremisapositiveinteger.Somecomplicationswillarise,however,thatdon'temergeintheone-variablecase.Wewillhavetoemploynewtechniquestogetaroundthesedifculties,whenpossible,andwillsimplyhavetoweakenthestatementsofourconclusionswhennecessary.Inthischapter,wewillbeexploringsomeofthebasictheoryofcomplexfunctionsofmorethanonevariableandwillbedevelopingsomenotationandconventionsthatwillaidourprogressionthroughtheremainderofthisdissertation.Wewillalsobeextendingmanyofthedenitionsfrompreviouschapterstothemultivariatesetting.InChapter8,wewillbeobtainingalowerboundforthemultiplicationoperatorsonourRKHS'sinthecasethatsomefamiliarhypothesesaresatised.Chapter9willrequirethemostworkaswewillbeestablishingananalogueoftheHLSresultforHilbertspacesofmultivariableDirichletserieswithsquare-summablecoefcients.Thisiswheremostofthedifcultieswillariseintryingtotransitiontomorethanonevariable:whendealingwithDirichletseriesofmorethanonevariable,welackthetheoremsofSchneeandBohr,whichwereinstrumentalinprovingtheintegralrepresentationofthenorminH0thespaceofDirichletserieswithsquare-summablecoefcients.Thisintegral/normrelationwasinturnusedtoderiveanupperboundforthemultiplicationoperatorsonH0.Wewilluseanideafoundin[ 5 ]togetaroundthisissueandshowthat,eveninthemultivariablecase,westillhaveanintegralrepresentationofthenormofcertainelementsinthemultivariateversionofH0. 67

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InChapter10,wewillbeusingthisversionoftheHLSresulttomimictheproofoftheupperboundforkM'kfoundinChapter4.Finally,inChapter11,wewillraisesomequestionsthatmightserveassteppingstonesforfutureresearch.Theproofsthroughoutthefollowingchapterswhichcanbedoneinwaysentirelyanalogoustotheirone-variablecounterparts,willbeelidedinordertoexpeditethediscussion. 7.2AnalyticFunctionsofMoreThanOneVariableWhilethetheoryofcomplexfunctionsofmorethanonevariableis,ingeneral,averydifferentcreaturefromthetheoryofcomplexfunctionsofonevariable,manyofthebetter-knownone-variableresultsdohavetheirhigher-dimensionalanalogues.WewillberestrictingourattentiontofunctionstakingtheirvaluesinC.Asintheonevariablecase,thereareseveralequivalentwaysofdeninganalyticityinmorethanonevariable.Themostconvenientwayforusistosaythat,afunctionf:Cm)166(!C(m1)isanalyticifitisanalyticineachvariableseparately.Asonemightexpect,thereareequivalentwaysofdeninganalyticityintermsofpowersseriesorintermsofhigher-dimensionalversionsoftheCauchy-Riemannequations,forexample.However,thesealternatewaysofdeninganalyticitywon'tbeneededinthisdissertation,andwillthereforebeavoided.Oftremendousimportancetousarethefollowingtwotheorems,whoseone-variableanaloguesarewell-known: Theorem7.1. OnanopensetinCm,theuniformlimitofasequenceanalyticfunctionsisanalytic. Theorem7.2. LetFbeaboundedfamilyofanalyticfunctionintheopensetCm.TheneverysequenceoffunctionsinFhasasubsequencewhichconvergesuniformlyoncompactsubsetsof. 68

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Theproofsoftheabovetheorems,alongwithamorethoroughtreatmentofthetheoryoffunctionsofmorethanonecomplexvariable,canbefoundin[ 7 ].PuttingTheorems 7.1 and 7.2 together,wegetthefollowing: Corollary4. IfffngisauniformlyboundedsequenceofanalyticfunctionsonCm,thenthereissomesubsequenceffnjgconverginguniformlyoncompactsubsetsoftoafunctionf,whichisalsoanalyticin. 7.3Denitions,NotationsandConventionsInthissection,wewillbegivingsomebasicdenitionsanddevelopingnotationsandconventionsthatwillmakesubsequentsectionsandchaptersmucheasiertofollow.Throughout,mwillbeaxednaturalnumber.LetNmdenotethesetofallm-tuplesofnaturalnumbers.Wewouldliketodeneanm-variableDirichletseriestobesomethinglikeX(n1,...,nm)2Nma(n1,...,nm)n)]TJ /F4 7.97 Tf 6.59 0 Td[(s11n)]TJ /F4 7.97 Tf 6.59 0 Td[(smm,afairlynaturalextensionofthedenitionofaDirichletseriesofonevariable.However,certaindifcultiesarisewhenattemptingsuchadenition.Namely,inwhatorderarewetosumtheelements?Intheone-variablecase,thisisn'tanissuesinceNhasaverynaturalordering.Thisalsoisn'tanissueifourmultivariableDirichletseriesconvergesabsolutely,sincewecanthentaketheordertobewhateverwewish.However,iftheseriesfailstoconvergeabsolutely,thendifferentrearrangementsofthetermscanyieldverydifferentsums,orevencausetheseriestodiverge.Accordingly,wenowgiveaprecisedenitionofaformalmultivariableDirichletseries,alongwithanaturalorderinwhichtosumtheterms.WedeneX(n1,...,nm)2Nma(n1,...,nm)n)]TJ /F4 7.97 Tf 6.59 0 Td[(s11n)]TJ /F4 7.97 Tf 6.59 0 Td[(smm=X1n1++nm=1a(n1,...,nm)n)]TJ /F4 7.97 Tf 6.58 0 Td[(s11n)]TJ /F4 7.97 Tf 6.58 0 Td[(smm;wewillalwayssumbyincreasingdegree.Whethersuchaseriesconvergesordivergeswilldependonourcoefcientsandchoicesofs1,s2,...,sm. 69

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WenowdevelopsomeshorthandthatwillmakeourmultivariableDirichletserieslookabitmorefamiliar.Ifn=(n1,...,nm),ifs=(s1,...,sm),andifweadopttheconventionthatn)]TJ /F3 7.97 Tf 6.59 0 Td[(s=n)]TJ /F4 7.97 Tf 6.59 0 Td[(s11n)]TJ /F4 7.97 Tf 6.59 0 Td[(smm,thenwehaveX(n1,...,nm)2Nma(n1,...,nm)n)]TJ /F4 7.97 Tf 6.59 0 Td[(s11n)]TJ /F4 7.97 Tf 6.59 0 Td[(smm=Xn2Nmann)]TJ /F3 7.97 Tf 6.59 0 Td[(s.Themostnaturalwaytodeneoperationsiscoordinate-wise:ifx=(x1,...,xm)andy=(y1,...,ym)areinCm,dene 1. x+y=(x1+y1,...,xm+ym); 2. xy=(x1y1,...,xmym);and 3. x y=x1 y1,...,xm ymwherewe,ofcourse,assumethatyi6=0foreachi.Ifx=(x1,...,xm)andy=(y1,...,ym)areinRm,wewritex=(x1,...,xm)(y1,...,ym)=yifxiyiforeachi,withequalityifandonlyifxi=yiforeachi.Wewritex=(x1,...,xm)<(y1,...,ym)=yifandonlyifxi
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Lemma5. IfXn2Nmann)]TJ /F20 7.97 Tf 6.58 0 Td[(sconvergesabsolutely(oruniformly)alongtheline(1+it,...,m+it),thenak=mYj=1limTj!11 2TjZTj)]TJ /F4 7.97 Tf 6.59 0 Td[(TjXn2Nmann)]TJ /F20 7.97 Tf 6.58 0 Td[(skj+itjjdtj,wherek=(k1,...,km). 7.4TheMultivariableSpaceHwLetfwngnn0beamulti-indexedsequenceofpositivenumbers.WedenethespaceHwinananalogouswaytotheone-variablecase: Hw=nXnn0ann)]TJ /F3 7.97 Tf 6.59 0 Td[(s:Xnn0janj2wn<1o.(7)Aspresented,thisspaceconsistsofpurelyformalmultivariableDirichletseriessatisfyingthegivennormcondition.Ifwewantthisspacetorepresentactualfunctionswithacommondomain,wemustasintheone-variablecaseimposeagrowthconditiononourweightsequence.Accordingly,supposethat(1,...,m)0=(0,...,0)andXnn0w)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(2<1whenever>(1,...,m).TheneachseriesinHwconvergesinQjj(andindeed,absolutelyuniformlyonanyproductofproperhalf-planes).TheproofthateachseriesinHwconvergesabsolutelyuniformlyinQjj+j(j>0foreachj)isessentiallythesameasintheone-variablecase,andwon'tbegiven.Notethatifourweightsarebounded,theneachf2HwconvergesabsolutelyuniformlyinQj1=2+j.Unlessotherwisestated,wewillforsimplicitytaken0=1.Asonemightexpect,eachfinHwisanalytic(asdenedinSection 7.2 )inQjj.Thiscanbeseenbyxingallbutonevariableandusingthefactthat,inQjj+j,eachf2Hwistheuniformlimitofasequenceofniteseries,eachofwhichisanalytic. 71

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Thegrowthconditionimposedonourweights,turnsHwintoaRKHSwithkernelk:QjjQjj)166(!Cmgivenbykw(u,z)=Xn2Nmw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F3 7.97 Tf 6.59 0 Td[(u)]TJ ET q .359 w 303.2 -61.29 m 307.18 -61.29 l S Q BT /F3 7.97 Tf 303.2 -66.79 Td[(z.Wesaythatourweightsequenceissupermultiplicativeifwab=wawbwheneveraandbarecoprimeandsupercompletelymultiplicativeifwab=wawbforeachaandb.Notethatw(a1,...,am)=w(a1,1,1,...,1)w(1,a2,1,...,1)w(1,1,...,1,am).Notealsothatthesequencesdenedbyfw(n,1,1,...,1)g,fw(1,n,1,...,1)g,...,fw(1,1,...,1,n)gareeachmultiplicative(resp.completelymultiplicative)inthesenseofSection 2.6 .Denotebywj,atheelementoffwngwithainthej-thspotand1'severywhereelse.Inthecasethatourweightsequenceissupermultiplicative,u=(u1,...,um),z=(z1,...,zm)2Qjj,andu+ z==(1,...,m),ourkerneltakestheformkw(u,z)=mYi=1YpprimeXj=0w)]TJ /F8 7.97 Tf 6.58 0 Td[(1i,pjp)]TJ /F4 7.97 Tf 6.58 0 Td[(ji.Ifourweightsareinsteadsupercompletelymultiplicative,thenwegetkw(u,z)=mYi=1Ypprime1 1)]TJ /F5 11.955 Tf 11.95 0 Td[(w)]TJ /F8 7.97 Tf 6.59 0 Td[(1i,pp)]TJ /F9 7.97 Tf 6.59 0 Td[(i.Thevalidityoftheseformulasstemsfromthefactthat,inQjjQjj,ourkernelcanbewrittenaskw(u,z)=mYi=11Xn=1w)]TJ /F8 7.97 Tf 6.58 0 Td[(1i,nn)]TJ /F9 7.97 Tf 6.59 0 Td[(i. 72

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7.5MultipliersonHwAfunction'withdomaincontainingQjjissaidtobeamultiplieronHwif'f2Hwwheneverf2Hw.Eachsuch'determinesaboundedlinearmapM':Hw)166(!Hwdenedbyf7!'f(theboundednessfollowingfromanargumentsimilartotheoneseeninSection 2.2 ).Beingbounded,M'hasanadjointM'which,asusual,satisesthekernel/eigenfunctionequationM'ku= '(u)ku.It'simmediatefromthisthatkM'kj'jQjj.However,aswedidintheone-variablecase,wewillbeabletoimprovethisestimatebyusingcertainprojectedspacestomoveufarthertotheleft.Asintheone-variablecase,anymultiplier'onHwmust,itself,beanelementHw.Ifn0=1,thenthisisimmediate;ifn01,thenoneseesthisbygoingthroughtheproofoftheone-variableanalogue(seeSection 2.4 )withtheprimesp=(p1,...,pm)andq=(q1,...,qm)andtheiteratedintegralsmYj=1limTj!11 2TjZTj)]TJ /F4 7.97 Tf 6.59 0 Td[(Tjxj+itjjdtj. 7.6TheMultivariableProjectedSpaceHwNLetPNdenotetherstNprimes,lethPNidenotethesetofwordsinPN,andlethPNimdenotethem-tuplesofelementsofhPNi.WedenetheprojectedspaceHwNtobethesubspacef2Hw:f(s)=Xn2hPNimann)]TJ /F3 7.97 Tf 6.59 0 Td[(sofHwandwedeneNtobetheprojectionofHwontothisspace.Asintheone-variablecase,wehavethefollowingimportantfactsregardingthespaceHwN: 1. if'isamultiplieronHw,thenN'isamultiplieronHwN; 2. NM'jHwN=MN';and 73

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3. Nk(u,z)isthekernelonHwNandNk(u,z)=Xn2hPNimw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F9 7.97 Tf 6.59 0 Td[(,where=u+ z.Regardingthethirditem,ifourweightsequenceissupermultiplicativeandifliesintheappropriateproductofhalf-planes(whichproductofhalf-planeswilldependontheweights),thenNk(u,z)=mYi=1Yp2PNXj=0w)]TJ /F8 7.97 Tf 6.58 0 Td[(1i,pjp)]TJ /F4 7.97 Tf 6.58 0 Td[(ji.Ifourweightsequenceissupercompletelymultiplicative,thenNk(u,z)=mYi=1Yp2PN1 1)]TJ /F5 11.955 Tf 11.95 0 Td[(w)]TJ /F8 7.97 Tf 6.58 0 Td[(1i,pp)]TJ /F9 7.97 Tf 6.59 0 Td[(i.Ineithercase,theproductisnite. 74

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CHAPTER8THEMULTIVARIABLELOWERBOUNDInthischapter,wewillbeextendingthetwomaintheoremsfromChapter3tothemultivariatesetting.However,wewon'tbeabletogetthefullstrengthoftheresultsobtainedthere.SincewelackahigherdimensionalanalogueofSchnee'stheorem,wewillhavetosettleforshowingthat,if'isamultiplieronHwwithanalyticcontinuation toQjj(thej'stobedenedforthwith),thenkM'kj jQjj.Webeginwiththem-dimensionalanalogueofTheorem 3.1 : Theorem8.1. Letfwngn2Nmbeasequenceofpositivenumbers,supposethat'isamultiplieronHw,andlet00lieinRm.If Xn2Nmw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.58 0 Td[(20<1(8)whenever0>0and,foreachN, Xn2hPNimw)]TJ /F8 7.97 Tf 6.58 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(2<1(8)whenever>,thenfor=(1,...,m)>0, 1. eachf2HwconvergesabsolutelyuniformlyinQj0j+j; 2. eachf2HwNconvergesabsolutelyuniformlyinQjj+j; 3. HwandHwNarereproducingkernelHilbertspaceswithpointevaluationsbeingcontinuousinQj0jandQjjrespectively;and 4. thesequencefN'g1N=1isuniformlyboundedinsupnormonQjjbykM'kandconvergesuniformlyoncompactsubsetsofQjjtoaboundedanalyticfunction withj jQjjkM'k.TheproofofthistheoremisvirtuallyidenticaltotheproofofTheorem 3.1 :Conditions( 8 )and( 8 ),alongwithanapplicationoftheCauchy-Schwarzinequality, 75

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giveitems 1 2 and 3 .OnethenusesthefactthatjN'jQjjkNM'jHwNkkNM'kkM'k,inconjunctionwithCorollary 4 fromChapter7toget,aftertakingtheappropriatelimit,item 4 .Wenowmovetothem-dimensionalversionofTheorem 3.3 Theorem8.2. Let0lieinRmandletfwngn2Nmbeasequenceofpositivenumbers.Ifforeach>,thereexistsaC>0inRmsuchthatw)]TJ /F8 7.97 Tf 6.58 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(2Cforeachn,andif'isamultiplierofHw,thentheinequalitiesinTheorem 8.1 aresatisedby+1 2=)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(1+1 2,...,m+1 2andrespectivelyandj jQjjkM'k.TheproofofTheorem 8.2 mimicstheproofofTheorem 3.3 .OneneedonlyadditionallyknowthatXn2hPNimn)]TJ /F3 7.97 Tf 6.58 0 Td[(sconvergesinQ0(andindeed,convergesabsolutelyuniformlyinQjjforeachm-tuple(1,...,m)ofpositivenumbers).ThisiseasilyseensinceXn2hPNimn)]TJ /F3 7.97 Tf 6.58 0 Td[(s=mYj=1Xnj2hPNin)]TJ /F4 7.97 Tf 6.58 0 Td[(sjjandeachPnj2hPNin)]TJ /F4 7.97 Tf 6.59 0 Td[(sjjconvergesabsolutelyuniformlyinforeach>0byLemma 2 76

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CHAPTER9ANM-DIMENSIONALEXTENSIONOFTHEHLSRESULTTheresultofHLSwasindispensableinderivingtheupperboundforkM'kintheone-variablecase;wewillneeditsm-dimensionalanaloguetoderiveasimilarresultinthecasethatHwconsistsofDirichletseriesinm-variables.Ifwn1,thenthecorrespondingRKHSH0consistsoftheDirichletserieswithsquare-summablecoefcients.If'isamultiplieronH0withanalyticcontinuationtoQ0givenby ,thenthefactthatj jQ0kM'kfollowsimmediatelyfromTheorem 8.2 ofChapter8.Toobtaintheupperbound,wewillneedtoderiveanintegralrepresentationfortheH0normofamultiplieronH0.Thiswillbesubstantiallymoredifcultthanintheone-variablecasesincewelackmultivariateversionsofthetheoremsofSchneeandBohr,whichallowedforafairlystraightforwardproofoftheintegral/normrelationwhendealingwithonlyonevariable.Toaccomplishthisgoal,wewilladaptanideafoundin[ 5 ]tom-dimensionstoshowthat canbeuniformlyapproximatedbyasequenceofsmoothedniteDirichletseries.Oncetheintegral/normrelationisobtained,therestoftheproofwill(largely)gothroughasintheone-variablecase. Theorem9.1. Ifwn1,ifH0isthecorrespondingRKHS,andif'isamultiplieronH0withanalyticcontinuationtoQ0givenby ,thenkM'kj jQ0.Afairbitofworkisneededbeforebeginningtheproof.RecallthateachelementofNH0convergesabsolutelyuniformlyinQii(1,...,m>0),andthus,thateachf2H0NisanalyticinQ0.Suppose'(s)=Xn2Nmann)]TJ /F3 7.97 Tf 6.58 0 Td[(s 77

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isafunction(notnecessarilyamultiplieronanyspace)whosecoefcientsareboundedandsuchthateachN'(s)=Xn2hPNimann)]TJ /F3 7.97 Tf 6.59 0 Td[(sisboundedinmodulusinQ0byK<1(KbeingindependentofN).Notethatforanysuch',Montel'stheoremgivesananalyticcontinuation toQ0withj jQ0K.NotealsothatanymultiplieronH0hasitsprojectionssatisfyingthisboundednessconditionforK=kM'k.Let=(1,...,m)>0andlets2Qii.ThenN'(s)=Xn2hPNimann)]TJ /F3 7.97 Tf 6.59 0 Td[(s=Xn12hPNiXn2hPNim)]TJ /F12 5.978 Tf 5.75 0 Td[(1an1nn)]TJ /F3 7.97 Tf 6.59 0 Td[(s1n)]TJ /F4 7.97 Tf 6.59 0 Td[(s11,wheres1liesinQmi=20.NotethattherearrangementofthetermsofN'isvalidbytheabsolute(andindeed,absolutelyuniform)convergenceofN'inQii:sincethean'sarebounded,Xn2hPNimjann)]TJ /F3 7.97 Tf 6.59 0 Td[(sjmaxnjanjXn2hPNimjn)]TJ /F3 7.97 Tf 6.59 0 Td[(sj=maxnjanjmYj=1Yp2PN1 1)]TJ /F5 11.955 Tf 11.95 0 Td[(pRe(sj).Ifs1=(2+it2,...,m+itm)2Qmi=2i,thenwemaythinkoftheaboveasaDirichletseriesinthevariables1.TheabsoluteconvergenceofXn12hPNiXn2hPNim)]TJ /F12 5.978 Tf 5.76 0 Td[(1an1nn)]TJ /F3 7.97 Tf 6.58 0 Td[(s1n)]TJ /F4 7.97 Tf 6.59 0 Td[(s11meanswecanusetheintegralformulainProposition 2.2 toseethat,if=(1,...,m)2Qii,thenXn2hPNim)]TJ /F12 5.978 Tf 5.75 0 Td[(1an1nn)]TJ /F3 7.97 Tf 6.59 0 Td[(s1=limT1!11 2T1ZT1)]TJ /F4 7.97 Tf 6.59 0 Td[(T1N'(+it)n1+it11dt1Kn11.Letting1!1showsthat,foranyxedn1,thesequenceXn2hPNim)]TJ /F12 5.978 Tf 5.76 0 Td[(1an1nn)]TJ /F3 7.97 Tf 6.58 0 Td[(s1ofanalyticfunctionsinm)]TJ /F6 11.955 Tf 12.16 0 Td[(1variablesiseventuallyboundedbyKn11.Since1>0wasarbitrary,thesequenceis,infact,eventuallyboundedbyK,andthemultivariateversion 78

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ofMontel'stheorem(Theorem 7.2 )implies,foreachn1,theexistenceofananalyticcontinuation n1(s1)ofPn2Nm)]TJ /F12 5.978 Tf 5.76 0 Td[(1an1nn)]TJ /F3 7.97 Tf 6.58 0 Td[(s1suchthatj n1(s1)jK.Thesameargumentappliedtothefunction'n1(s1)=Xn2Nm)]TJ /F12 5.978 Tf 5.76 0 Td[(1an1nn)]TJ /F3 7.97 Tf 6.58 0 Td[(s1furnishes,foreachn1andn2,ananalyticcontinuation n1n2(s2)ofPn2Nm)]TJ /F12 5.978 Tf 5.76 0 Td[(2an1n2nn)]TJ /F3 7.97 Tf 6.58 0 Td[(s2(s22Qmi=3i)whichisagainboundedinmodulusbyK.Proceedinginthisway,foreachj=1,...,m)]TJ /F6 11.955 Tf 12.74 0 Td[(1andforeachj-tuple(n1,...,nj)ofnaturalnumbers,wegetananalyticcontinuation n1n2...nj(sj)ofPn2Nm)]TJ /F19 5.978 Tf 5.76 0 Td[(jan1...njn)]TJ /F3 7.97 Tf 6.58 0 Td[(sj(sj2Qmi=j+1i)suchthatj n1n2...njjQmi=j+1jK.Now,fors121and(s2,...,sm)2Qmi=2i,wehave (s1,s2,...,sm)=1Xn1=1 n1(s2,...,sm)n)]TJ /F4 7.97 Tf 6.58 0 Td[(s11:Theright-handsideisanalyticwithrespecttos1sincethecoefcientsarebounded,andisanalyticwithrespecttos2,...,smsinceP1n1=1 n1(s2,...,sm)n)]TJ /F4 7.97 Tf 6.58 0 Td[(s1convergesabsolutelyuniformlyforanyxeds121,independentofs2,...,sm(again,sincethecoefcientsarebounded),sothattheseriesontherightisanalyticwithrespecttoeachofitsvariables.Moreover,inQ1,theleftandrightsidesarebothgivenby'.Itfollowsthat1Xn1=1 n1(s2,...,sm)n)]TJ /F4 7.97 Tf 6.58 0 Td[(s11istheanalyticcontinuationof'to1Qmi=2i.Followingasimilarlineofargument,fors221and(s3,...,sm)2Qmi=3i,wehave n1(s2,...,sm)=1Xn2=1 n1n2(s3,...,sm)n)]TJ /F4 7.97 Tf 6.58 0 Td[(s22. 79

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Continuinginthisway,forsj21and(sj+1,...,sm)2Qmi=j+1i,wehave n1...nj)]TJ /F12 5.978 Tf 5.76 0 Td[(1(sj,...,sm)=1Xnj=1 n1...nj(sj+1,...,sm)n)]TJ /F4 7.97 Tf 6.59 0 Td[(sjj.Following[ 5 ],letu0beaC1functionwithcompactsupportin[,1](>0)suchthatZ10u()d=1anddenev(x)=Z1xu()d.Itfollowsfromthisdenitionthatv(0)=1andv(x)=0whenx1.TheMellintransformeuofu,denedbyeu(s)=Z10s)]TJ /F8 7.97 Tf 6.59 0 Td[(1u()d,isentireanddecreasesfasterthananynegativepowerofs.Notethatthesupportofulyingin[,1]keepstheintegrandfromblowingupneartheorigin.IfwedeneZ(c):=Zc+i1c)]TJ /F4 7.97 Tf 6.59 0 Td[(i1,itiseasilycheckedthatv(x)=1 2iZ(c)eu(w+1)x)]TJ /F4 7.97 Tf 6.59 0 Td[(wdw w,wheneverc,x>0.ForeachN,deneacomplexmeasureNbydN=1 2ieu(w+1)Nwdw w. Lemma6. Ifr>)]TJ /F6 11.955 Tf 9.29 0 Td[(1andr6=0,thenZ(r)dNZ(r)djNjNr 2Z(r)jeu(w+1)jjdwj jwjKNrforsomepositiveconstantK. 80

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Proof. Letw=r+it,wheretvariesoverR,but)]TJ /F6 11.955 Tf 9.3 0 Td[(10,sothat1 jwjisboundedawayfrom0by(say)K1>0.Acoupleapplicationsofintegrationbypartsalongwithourassumptionsonugives jeu(r+1+it)j1 jr+1+itj2Z1jxr+1+it(u0(x)+xu00(x))jdxK2 (1+r)2+t2<1(9)forsomeconstantK2>0,sothatZ(r)jeu(w+1)jjdwjZ(r)K2 (1+r)2+t2dt=K3<1.(Itshouldbepointedoutthat( 9 )holdsforanyraslongastisboundedawayfrom0.)ItfollowsthatZ(r)dN1 2Z(r)jeu(w+1)jNrjdwj jwj1 2K1K3Nr,whencetheclaim. Fix0<=(1,...,m)<1andletc1>1ands1>21,s2>22,...,sm>2m.ThenZ(c1) (s1+w1,s2,...,sm)dN(w1)=Z(c1)1Xn1=1 n1(s2,...,sm)n)]TJ /F4 7.97 Tf 6.58 0 Td[(s1)]TJ /F4 7.97 Tf 6.58 0 Td[(w11dN(w1)=1Xn1=1 n1(s2,...,sm)n)]TJ /F4 7.97 Tf 6.58 0 Td[(s11vn1 N,wherepassingtheintegralinsidethesumisjustiedbytheuniformconvergenceofP1n1=1 n1(s2,...,sm)n)]TJ /F4 7.97 Tf 6.58 0 Td[(s1)]TJ /F4 7.97 Tf 6.59 0 Td[(w11(Re(s1+w1)>1alongtheline(c1)).Notethatthelastsumisactuallynitesincev)]TJ /F4 7.97 Tf 6.68 -4.8 Td[(n1 N=0,whenn1N.Now, (s1+w1,s2,...,sm)isanalyticwithrespecttow1intherectanglewithverticesc1+iR,)]TJ /F10 11.955 Tf 9.3 0 Td[(1+iR,)]TJ /F10 11.955 Tf 9.3 0 Td[(1)]TJ /F5 11.955 Tf 11.96 0 Td[(iRandc1)]TJ /F5 11.955 Tf 11.95 0 Td[(iR(sinceRe(s1+w1)>1)andthefunction (s1+w1,s2,...,sm)eu(w1+1)Nw11 w1isanalyticeverywhereexceptat0,sowemayapplytheresiduetheoremtogetZ(c1) (s1+w1,s2,...,sm)dN(w1)= (s1,...,sm))]TJ /F11 11.955 Tf 11.96 16.27 Td[(Z()]TJ /F9 7.97 Tf 6.59 0 Td[(1) (s1+w1,s2,...,sm)dN(w1). 81

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(Notethattheintegralsalongthetopandbottomportionsoftherectangletendto0asR!1since,if)]TJ /F10 11.955 Tf 9.3 0 Td[(1x1c1,then1 2iZc1)]TJ /F9 7.97 Tf 6.59 0 Td[(1 (s1+x1+iR,s2,...,sm)eu(x1+1+iR)Nx1+iRdx1 p x21+R2K 2Zc1)]TJ /F9 7.97 Tf 6.58 0 Td[(1jeu(x1+1+iR)jNx1dx1 p x21+R2andsincethelastintegraltendsto0asRincreases.)Itfollowsthat (s1,...,sm))]TJ /F11 11.955 Tf 11.95 16.27 Td[(Z()]TJ /F9 7.97 Tf 6.59 0 Td[(1) (s1+w1,s2,...,sm)dN(w1)=1Xn1=1 n1(s2,...,sm)n)]TJ /F4 7.97 Tf 6.59 0 Td[(s11vn1 N.Replacings2inthisequationbys2+w2andrepeatingtheaboveargumentgivesontheleft-handsideZ(c2) (s1,s2+w2,...,sm))]TJ /F11 11.955 Tf 11.95 16.27 Td[(Z()]TJ /F9 7.97 Tf 6.59 0 Td[(1) (s1+w1,s2+w2,...,sm)dN(w1)dN(w2)= (s1,...,sm))]TJ /F11 11.955 Tf 11.96 16.27 Td[(Z()]TJ /F9 7.97 Tf 6.58 0 Td[(1) (s1+w1,s2,...,sm)dN(w1))]TJ /F11 11.955 Tf 11.29 16.86 Td[(Z()]TJ /F9 7.97 Tf 6.59 0 Td[(2) (s1,s2+w2,...,sm)dN(w2))]TJ /F11 11.955 Tf 11.95 16.27 Td[(Z()]TJ /F9 7.97 Tf 6.59 0 Td[(2)Z()]TJ /F9 7.97 Tf 6.59 0 Td[(1) (s1+w1,s2+w2,...,sm)dN(w1)dN(w2),andontheright-handsideZ(c2)1Xn1=1 n1(s2+w2,...,sm)n)]TJ /F4 7.97 Tf 6.59 0 Td[(s11vn1 NdN(w2)=1Xn1=1Z(c2) n1(s2+w2,...,sm)dN(w2)n)]TJ /F4 7.97 Tf 6.59 0 Td[(s11vn1 N=1Xn1=1Z(c2)1Xn2=1 n1n2(s3,...,sm)n)]TJ /F4 7.97 Tf 6.59 0 Td[(s2)]TJ /F4 7.97 Tf 6.58 0 Td[(w22dN(w2)n)]TJ /F4 7.97 Tf 6.59 0 Td[(s11vn1 N=1Xn1=11Xn2=1 n1n2(s3,...,sm)n)]TJ /F4 7.97 Tf 6.59 0 Td[(s22n)]TJ /F4 7.97 Tf 6.59 0 Td[(s11vn1 Nvn2 N.Equatingtheleftandrightsides,weget (s1,...,sm))]TJ /F11 11.955 Tf 11.96 16.27 Td[(Z()]TJ /F9 7.97 Tf 6.58 0 Td[(1) (s1+w1,s2,...,sm)dN(w1) 82

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)]TJ /F11 11.955 Tf 11.29 16.86 Td[(Z()]TJ /F9 7.97 Tf 6.59 0 Td[(2) (s1,s2+w2,...,sm)dN(w2))]TJ /F11 11.955 Tf 11.95 16.28 Td[(Z()]TJ /F9 7.97 Tf 6.59 0 Td[(2)Z()]TJ /F9 7.97 Tf 6.59 0 Td[(1) (s1+w1,s2+w2,...,sm)dN(w1)dN(w2)=1Xn1=11Xn2=1 n1n2(s3,...,sm)n)]TJ /F4 7.97 Tf 6.59 0 Td[(s22n)]TJ /F4 7.97 Tf 6.59 0 Td[(s11vn1 Nvn2 N.Notetheimplicituseoftheresiduetheoremincomputingtheleft-handsidevalidsinceP1n1=1 n1(s2+w2,...,sm)n)]TJ /F4 7.97 Tf 6.59 0 Td[(s11v)]TJ /F4 7.97 Tf 6.68 -4.8 Td[(n1 Nisanalyticwithrespecttow2intherectanglewithverticesc2+iR,)]TJ /F10 11.955 Tf 9.3 0 Td[(2+iR,)]TJ /F10 11.955 Tf 9.29 0 Td[(2)]TJ /F5 11.955 Tf 11.37 0 Td[(iRandc2)]TJ /F5 11.955 Tf 11.37 0 Td[(iR(sinceagain,Re(s2+w2)>2).Notealsothatthelastsuminthecalculationoftheright-handsideisnitesincev)]TJ /F4 7.97 Tf 6.67 -4.8 Td[(n1 Nandv)]TJ /F4 7.97 Tf 6.67 -4.8 Td[(n2 Nare0forn1andn2largeenough.Continuingasaboveandexploitingthefactthat n1...nm)]TJ /F12 5.978 Tf 5.76 0 Td[(1(sm)=1Xnm=1an1...nm)]TJ /F12 5.978 Tf 5.76 0 Td[(1n)]TJ /F4 7.97 Tf 6.59 0 Td[(smm,forsm20,weget,fors2Qii,Xn2Nmann)]TJ /F3 7.97 Tf 6.59 0 Td[(smYj=1vnj N)]TJ /F10 11.955 Tf 11.96 0 Td[( (s)=Xm=1X1j1<0.Thus,Xn2Nmann)]TJ /F3 7.97 Tf 6.58 0 Td[(smYj=1vnj N)]TJ /F10 11.955 Tf 11.95 0 Td[( (s)!0asN!1.Moreover,therateofconvergencedependsonlyonourchoiceof=(1,...,m).Wehavethusshownthefollowing: 83

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Lemma7. Foreach=(1,...,m)>0,andwith'and asabove,wehaveXn2Nmann)]TJ /F20 7.97 Tf 6.59 0 Td[(smYj=1vnj N! (s)uniformlyinQjjasN!1.Withthislemmaunderourbelt,wenowhaveeverythingweneedtoderivetheaforementionedintegral/normrelation. Theorem9.2. IfH0isthespaceoffunctionscorrespondingtotheweightsequencewn1,if istheboundedanalyticcontinuationofthemultiplier'(s)=Xn2Nmann)]TJ /F20 7.97 Tf 6.59 0 Td[(stoQ0,andifwedeneIj:=limTj!11 2TjZTj)]TJ /F4 7.97 Tf 6.59 0 Td[(Tj,thenXn2Nmjanj2=lim!0I1Imj (+it)j2dtmdt1,where+it=(1+it1,...,m+itm). Proof. Lets=+it=(1+it1,...,m+itm)andlet'and beasinthestatementofthetheorem.ThenI1Imj (1+it1,...,m+itm)j2dtmdt1=I1ImjXn2Nmann)]TJ /F3 7.97 Tf 6.58 0 Td[(smYj=1vnj N+N(s)j2dtmdt1withN(s)denotingafunctionofswhichgoesto0uniformlyinQiiasN!1(byLemma 7 ).Continuing,wehave=I1Im Xn2Nmann)]TJ /F9 7.97 Tf 6.58 0 Td[()]TJ /F4 7.97 Tf 6.59 0 Td[(itmYj=1vnj N+N(s)! 84

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Xk2Nm akk)]TJ /F9 7.97 Tf 6.59 0 Td[(+itmYj=1vkj N+N(s)!dtmdt1=I1Im Xn,k2Nman akn)]TJ /F9 7.97 Tf 6.58 0 Td[()]TJ /F4 7.97 Tf 6.59 0 Td[(itk)]TJ /F9 7.97 Tf 6.59 0 Td[(+itmYj=1vnj Nvkj N+eN(s)!dtmdt1.WearelettingeN(s)denotethesumofthecrosstermsandlasttermaftercarryingouttheabovemultiplication.NotethateN(s)!0uniformlyinQiiasN!1sincePn2Nmann)]TJ /F3 7.97 Tf 6.59 0 Td[(sQmj=1v)]TJ /F4 7.97 Tf 6.68 -3.57 Td[(nj Nconvergesuniformlytotheboundedfunction inQii.Passingtheintegralinsidethesummation(recallthatthesumisnite)andrearranging,wegetXn,k2Nman ak(nk))]TJ /F9 7.97 Tf 6.59 0 Td[(mYj=1vnj Nvkj NI1Imk nitdtmdt1+I1ImeN(s)dtmdt1=Xn,k2Nman ak(nk))]TJ /F9 7.97 Tf 6.59 0 Td[(mYj=1vnj Nvkj NIjkj njitjdtj+I1ImeN(s)dtmdt1.RecallthatIjkj njitjdtj=8><>:1ifkj=nj0otherwise,sothattheabovecollapsesdowntoXn2Nmjanj2n)]TJ /F8 7.97 Tf 6.58 0 Td[(2mYj=1hvnj Ni2+I1ImeN(s)dtmdt1.LettingbN=I1ImeN(s)dtmdt1,wegetI1Imj (+it)j2dtmdt1=Xn2Nmjanj2n)]TJ /F8 7.97 Tf 6.58 0 Td[(2mYj=1hvnj Ni2+bN. 85

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SincePn2Nmjanj2convergesandsincev(x)isbounded,foreach>0,wecanndaQ2NmsuchthatXnQjanj2n)]TJ /F8 7.97 Tf 6.59 0 Td[(2mYj=1hvnj Ni2XnQjanj2mYj=1hvnj Ni2<.ThusI1Imj (+it)j2dtmdt1)]TJ /F11 11.955 Tf 11.95 11.36 Td[(XnQjanj2n)]TJ /F8 7.97 Tf 6.59 0 Td[(2mYj=1hvnj Ni2XnQjanj2n)]TJ /F8 7.97 Tf 6.58 0 Td[(2mYj=1hvnj Ni2+jbNj<+jbNj.LettingN!1(andrecallingthatv(0)=1),wegetI1Imj (+it)j2dtmdt1)]TJ /F11 11.955 Tf 11.95 11.36 Td[(XnQjanj2n)]TJ /F8 7.97 Tf 6.59 0 Td[(2,whencelim!0I1Imj (+it)j2dtmdt1)]TJ /F11 11.955 Tf 11.96 11.36 Td[(XnQjanj2.Finally,lettingQ!1giveslim!0I1Imj (+it)j2dtmdt1)]TJ /F11 11.955 Tf 11.95 11.36 Td[(Xn2Nmjanj2,andaswasarbitrary,wehavethedesiredresult. Aftertheappropriatechangeofvariables,wegetthefollowing: Theorem9.3. IfHisthespaceoffunctionscorrespondingtotheweightsequencewn=n)]TJ /F8 7.97 Tf 6.59 0 Td[(2,if istheboundedanalyticcontinuationofthemultiplier'(s)=Xn2Nmann)]TJ /F20 7.97 Tf 6.59 0 Td[(stoQii,andifwedeneIj:=limTj!11 2TjZTj)]TJ /F4 7.97 Tf 6.59 0 Td[(Tj,thenXn2Nmjanj2n)]TJ /F8 7.97 Tf 6.58 0 Td[(2=lim!0I1Imj (++it)j2dtmdt1, 86

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where=(1,...,m)and+it=(1+it1,...,m+itm).Withtheintegral/normrelationnowproven,weproceedtotheproofofTheorem 9.1 ,restatedhereforconvenience:Theorem9.1.Ifwn1,ifH0isthecorrespondingRKHS,andif'isamultiplieronH0withanalyticcontinuationtoQ0givenby ,thenkM'kj jQ0. Proof. Thatj jQ0kM'kwasshownattheoutsetofthischapter.Letf2H0beniteandsupposethat('f)(s)=Xn2Nmann)]TJ /F3 7.97 Tf 6.59 0 Td[(s.Sinceeveryfunctionoftheformn)]TJ /F3 7.97 Tf 6.59 0 Td[(sisamultiplieronH0andsinceMisanalgebra,'fisamultiplieronH0.Moreover, fisthe(bounded)analyticcontinuationof'ftoQ0.ThehypothesesofTheorem 9.2 arethussatisedby'fandwemaythereforecallintoplayourintegral/normrelation:kM'fk2=Xn2Nmjanj2=lim!0I1Imj( f)(+it)j2dtmdt1j j2Q0lim!0I1Imjf(+it)j2dtmdt1=j j2Q0kfk2.Hence,kM'kisboundedbyj jQ0onthedensesubsetofniteDirichletseriesandthereforebyj jQ0onallofH0. ItfollowsfromTheorem 9.1 andthediscussionatthebeginningofthischapterthat,foranygivenmultiplier'onH0withanalyticcontinuationtoQ0givenby ,j jQ0=kM'k.Itisnotclear,however,thatthemap)-322(:M)166(!H1(Q0)\DdenedbyM'7!'issurjective.WerequiredthetheoremsofBohrandSchneetoprovethisintheone-variablecaseanditisunknownifwehavehigher-dimensionalanalogues 87

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ofthesetheorems.However,aslightreimaginingofTheorem 4.3 willyieldacompletedescriptionofMinthecasethatwn1. Denition5. ADirichletpolynomialisaDirichletseriesPn2Nmann)]TJ /F20 7.97 Tf 6.58 0 Td[(sforwhichonlynitelymanyofthean'sarenonzero.Let=(1,...,m)>0.DenotebyUthesetofallfunctionswithdomainQiiwhich,foreach=(1,...,m)>0,canbeuniformlyapproximatedbyDirichletpolynomialsinQii+i. Lemma8. EveryelementofU0\H1(Q0)isrepresentablebyaDirichletseriesinsomeproductofhalf-planes. Proof. Let 2U0\H1(Q0),let=(1,...,m)>0,andsupposethatthesequencen (k)(s)=Xn2Nm (k)nn)]TJ /F3 7.97 Tf 6.59 0 Td[(so1k=1ofDirichletpolynomialsconvergesto uniformlyinQii.It'sclearthatthesequencef (k)gisuniformlyCauchyinQii.ApplyingtheintegralformulafromLemma 5 gives,fors=+it=(1+it1,...,m+itm)andforeachn=(n1,...,nm),j (k1)n)]TJ /F10 11.955 Tf 11.95 0 Td[( (k2)njmYj=1limTj!11 2TjZTj)]TJ /F4 7.97 Tf 6.58 0 Td[(Tjj (k1)(s))]TJ /F10 11.955 Tf 11.95 0 Td[( (k2)(s)jnjjdtjj (k1))]TJ /F10 11.955 Tf 11.95 0 Td[( (k2)jQiin,sothateachsequencef (k)ng1k=1isCauchyinC.Foreachn,let n=limk!1 (k)n.WewillshowthattheseriesXn2Nm nn)]TJ /F3 7.97 Tf 6.59 0 Td[(sconvergesfarenoughtotheright,andindeed,tothecorrectthing(towit ). 88

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Since 2H1(Q0)andsince (k)convergesto uniformlyinQjithereissomeK0>0suchthatj (k)jQiiK0foreachk.Thus,fors=+it=(1+it1,...,m+itm)andforeachn=(n1,...,nm),wehavej (k)njmYj=1limTj!11 2TjZTj)]TJ /F4 7.97 Tf 6.59 0 Td[(Tjj (k)(s)jnjjdtjK0n)j (k)nn)]TJ /F9 7.97 Tf 6.59 0 Td[(jK0.Lettingk!1givesj nn)]TJ /F9 7.97 Tf 6.59 0 Td[(jK0.IfRe(s)==(1,...,m)>1+,thenXn2Nmj nn)]TJ /F3 7.97 Tf 6.59 0 Td[(sj=Xn2Nmj nn)]TJ /F9 7.97 Tf 6.59 0 Td[(jn)]TJ /F8 7.97 Tf 6.58 0 Td[(()]TJ /F9 7.97 Tf 6.58 0 Td[()<1,whencetheabsoluteconvergenceoftheseriesPn2Nm nn)]TJ /F3 7.97 Tf 6.59 0 Td[(s.Now,let0>0,takeswithRe(s)==(1,...,m)>1+,andtakeNlargeenoughthatXnNj n)]TJ /F10 11.955 Tf 11.95 0 Td[( (k)njn)]TJ /F9 7.97 Tf 6.58 0 Td[(XnN(j nj+j (k)nj)n)]TJ /F9 7.97 Tf 6.59 0 Td[(=XnN(j nn)]TJ /F9 7.97 Tf 6.59 0 Td[(j+j (k)nn)]TJ /F9 7.97 Tf 6.59 0 Td[(j)n)]TJ /F8 7.97 Tf 6.58 0 Td[(()]TJ /F9 7.97 Tf 6.58 0 Td[()2K0XnNn)]TJ /F8 7.97 Tf 6.59 0 Td[(()]TJ /F9 7.97 Tf 6.59 0 Td[()<0.ThenXn2Nm nn)]TJ /F3 7.97 Tf 6.59 0 Td[(s)]TJ /F11 11.955 Tf 11.96 11.36 Td[(Xn2Nm (k)nn)]TJ /F3 7.97 Tf 6.58 0 Td[(sXn2Nmj n)]TJ /F10 11.955 Tf 11.96 0 Td[( (k)njn)]TJ /F9 7.97 Tf 6.59 0 Td[(=XnNj n)]TJ /F10 11.955 Tf 11.96 0 Td[( (k)njn)]TJ /F9 7.97 Tf 6.59 0 Td[(+XnNj n)]TJ /F10 11.955 Tf 11.96 0 Td[( (k)njn)]TJ /F9 7.97 Tf 6.58 0 Td[(XnNj n)]TJ /F10 11.955 Tf 11.96 0 Td[( (k)njn)]TJ /F9 7.97 Tf 6.59 0 Td[(+0,sothatlimsupkXn2Nm nn)]TJ /F3 7.97 Tf 6.59 0 Td[(s)]TJ /F11 11.955 Tf 11.96 11.36 Td[(Xn2Nm (k)nn)]TJ /F3 7.97 Tf 6.59 0 Td[(s0. 89

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Finally,since0wasarbitrary,wehavelimk!1Xn2Nm nn)]TJ /F3 7.97 Tf 6.59 0 Td[(s)]TJ /F11 11.955 Tf 11.95 11.36 Td[(Xn2Nm (k)nn)]TJ /F3 7.97 Tf 6.58 0 Td[(s=0,andtheclaimfollows. We'veseen(Lemma 7 )thateachmultiplieronH0liesinU0.Thisobservationleadsustothefollowing. Theorem9.4. ThemapM'7!'isanisometricisomorphismfromthespaceMontothespaceH1(Q0)\U0. Proof. ThatM'7!'isometricallyembedsMintoH1(Q0)\U0followsfromtheprecedingdiscussions.Now,supposethat 2H1(Q0)\U0isrepresentablebytheDirichletseries'(s)=Xn2Nm'nn)]TJ /F3 7.97 Tf 6.59 0 Td[(sinsomeproductofhalf-planes(theexistenceofsucharepresentationfollowingfromLemma 8 ).Since 2U0,thereissomesequencen'(j)(s)=Xn2Nm'(j)nn)]TJ /F3 7.97 Tf 6.59 0 Td[(so1j=1ofDirichletpolynomialsconvergingto uniformlyineachQii(=(1,...,m)>0).Forsome(1,...,m)>0andforeachs2Qii,wehave (s)=Xn2Nm'nn)]TJ /F3 7.97 Tf 6.59 0 Td[(s,withtheconvergencebeingabsolute.WemaythereforeapplyLemma 5 toobtaintheestimatej'n)]TJ /F10 11.955 Tf 11.95 0 Td[('(j)njmYj=1limTj!11 2TjZTj)]TJ /F4 7.97 Tf 6.59 0 Td[(Tjj'(+it))]TJ /F10 11.955 Tf 11.95 0 Td[('(j)(+it)jnj+itjjdtjj')]TJ /F10 11.955 Tf 11.95 0 Td[('(j)jQkkn, 90

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fromwhichitfollowsthatlimj!1'(j)n='nforeachn.Now,assumebywayofcontradictionthat isnotamultiplieronH0andletK>j j2Q0.ThereissomeniteDirichletseriesf(s)=XnNfnn)]TJ /F3 7.97 Tf 6.59 0 Td[(sinH0ofnormatmost1forwhichXn2Nm'nn)]TJ /F3 7.97 Tf 6.59 0 Td[(sXkNfkk)]TJ /F3 7.97 Tf 6.59 0 Td[(s2=Xi2NmXnk=i'nfk2>2K.TakingMlargeenoughthat,XiMXnk=i'nfk2>K,wegetKj j2Q0.Itfollowsthenthat mustbeamultiplieronH0. 91

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Aftertheappropriatechangeofvariables,wecanextendthisresulttothespacesandmultiplieralgebrascorrespondingtoweightsoftheformwn=n)]TJ /F8 7.97 Tf 6.59 0 Td[(2,where=(1,...,m)0. Theorem9.5. Ifourweightsequenceisdenedbywn=n)]TJ /F8 7.97 Tf 6.58 0 Td[(2,andifHandMarethecorrespondingHilbertspaceandmultiplieralgebra,thenthemapM'7!'isanisometricisomorphismfromthespaceMontothespaceH1(Q)\U. 92

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CHAPTER10THEMULTIVARIABLEUPPERBOUND 10.1AnUpperBoundforM'intheMultivariableSettingInthischapter,wederiveanm-dimensionalanalogueofTheorem 4.4 .Recallthataweightsequenceiscalledsupermultiplicativeifwab=wawbwheneveraandbarecoprimeandiscalledsupercompletelymultiplicativeifwab=wawbforeachaandb. Theorem10.1. Letfwngn2Nmbeamulti-indexedsequenceofpositivenumbers,let'beamultiplierontheresultingHilbertspaceHw,let=(1,...,m)0,andassumethat isananalyticcontinuationof'toQjjwith 2U.If 1. fwngn2Nmissupermultiplicative;and 2. foreachprimep2Nmandeachk2Nm,wehavewpkp)]TJ /F8 7.97 Tf 6.59 0 Td[(2wpk)]TJ /F23 5.978 Tf 5.75 0 Td[(1,thenkM'kj jQjj.Notethat,withtheconventionslaidoutinChapter7,wpkp)]TJ /F8 7.97 Tf 6.58 0 Td[(2wpk)]TJ /F24 5.978 Tf 5.76 0 Td[(1,wpkjjp)]TJ /F8 7.97 Tf 6.59 0 Td[(2jjwpkj)]TJ /F12 5.978 Tf 5.76 0 Td[(1jforeachj. Proof. Asbefore,byreplacingwnbywnn)]TJ /F8 7.97 Tf 6.59 0 Td[(2,wemayassumethat=0.Withthisassumption,item 2 issimplytherequirementthatourweightsequencebedecreasinginkforeachp.Let'and beasinthestatementofthetheoremandletH0and 93

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k0(u,z)denotetheRKHSandkernelcorrespondingtotheweightsequencewn1.Foru,z2Q1,k0(u,z)convergesabsolutely,andisthereforegivenbyitsEulerproduct:k0(u,z)=Xn2Nmn)]TJ /F9 7.97 Tf 6.59 0 Td[(=mYj=1Ypprime1 1)]TJ /F5 11.955 Tf 11.95 0 Td[(p)]TJ /F9 7.97 Tf 6.59 0 Td[(j,where,asbefore,=(1,...,m)=u+ z=(u1+ z1,...,um+ zm).Forthegeneralweightsequenceandforu,z2Qjj(0=(1,...,m)asdenedinSection 7.4 ),kw(u,z)convergesabsolutelyandcanthereforebewrittenaskw(u,z)=mYj=1Ypprime1Xi=0w)]TJ /F8 7.97 Tf 6.59 0 Td[(1j,pip)]TJ /F4 7.97 Tf 6.59 0 Td[(ij.Takermaxf1,max1imfigg.DeneanewfunctionK(u,z)onQrQrbyK(u,z)=kw(u,z) k0(u,z).(thedenominatorhavingnorootsinQrQr).RepresentingourkernelsbytheirrespectiveEulerproducts,wecanwriteK(u,z)asK(u,z)=mYj=1QpprimeP1i=0w)]TJ /F8 7.97 Tf 6.58 0 Td[(1j,pip)]TJ /F4 7.97 Tf 6.58 0 Td[(ij Qpprime1 1)]TJ /F4 7.97 Tf 6.58 0 Td[(p)]TJ /F22 5.978 Tf 5.75 0 Td[(j.ItwasseeninChapter4thateachQpprimeP1i=0w)]TJ /F8 7.97 Tf 6.58 0 Td[(1j,pip)]TJ /F4 7.97 Tf 6.58 0 Td[(ij Qpprime1 1)]TJ /F4 7.97 Tf 6.58 0 Td[(p)]TJ /F22 5.978 Tf 5.75 0 Td[(jdenesapositivekernelonr.Foreachj,thereisthenanauxiliaryHilbertspaceHjandafunctionQj:m)166(!HjsuchthatQpprimeP1i=0w)]TJ /F8 7.97 Tf 6.58 0 Td[(1j,pip)]TJ /F4 7.97 Tf 6.58 0 Td[(ij Qpprime1 1)]TJ /F4 7.97 Tf 6.58 0 Td[(p)]TJ /F22 5.978 Tf 5.75 0 Td[(j=hQj(uj),Qj(zj)ij,fromwhichitfollowsthat K(u,z)=kw(u,z) k0(u,z)=mYj=1hQj(uj),Qj(zj)ij.(10) 94

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LetH0Nmj=1HjdenotetheHilbertspacetensorproductformedbyH0andtheHj'sdenedabove.Multiplyingthroughbyk0(u,z)andrewriting,weget hkwu,kwziw=hk0umOj=1Qj(uj),k0zmOj=1Qj(zj)i.(10)ThisequivalencebetweeninnerproductshintsatanunderlyingisometrybetweenHwandH0Nmj=1Hj.Dene V:S=fkwu:u2Ymg)165(!H0mOj=1Hj(10)bykwu7!k0umOj=1Qj(uj)andextendbylinearitytospanS.It'sclearfrom( 10 )thatV,sodened,isanisometry,whichbythedensityofspanSextendscontinuouslytoanisometryonallofHw.Assumethat isboundedinQ0(otherwisethere'snothingtodo).Since 2U0,thisassumptionontheboundednessof isenoughtoguaranteethat'isamultiplieronH0andthattheoperatornormofthecorrespondingmultiplicationoperatorisatmostj jQ0(Theorem 9.4 ).AsinChapter4,letM',0denotemultiplicationby'inH0andletM',wdenotemultiplicationby'inHw.WewanttoshowthatkM',wkj jQ0.Accordingly,deneamapM',0Nmj=1IjonelementarytensorsinH0Nmj=1HjbyM',0mOj=1Ij:k0umOj=1Qj(zj)7! '(u)k0umOj=1Qj(zj)andextendthismapbylinearityandcontinuitytoallofH0Nmj=1Hj.ItiseasilycheckedthatM',w=V(M',0mOj=1Ij)V, 95

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sothatthefollowingdiagramcommutes:HwV)305()222()306(!H0Nmj=1HjM',w??y??yM',0Nmj=1IjHw)305()222()306(!VH0Nmj=1Hj.Finally,kM',wk=kV(M',0mOj=1Ij)VkkVkkM',0mOj=1IjkkVkkM',0kj jQ0,whichiswhatwastobeshown. Remark.IffwngisaweightsequencesatisfyingthehypothesesofTheorem 10.1 andifHandkaretheHilbertspaceandkernelcorrespondingtotheweightsequencewn=n)]TJ /F8 7.97 Tf 6.59 0 Td[(2(>0),then( 10 )guaranteesthatthekernelK(u,z)=kw(u,z) k(u,z)ispositiveforuandzfarenoughtotheright,sothatthemultiplieralgebraofHiscontainedinthemultiplieralgebraofHw. 10.2TheMultivariableCase=Ifwnisasequenceofpositivenumbers(withcorrespondingRKHSHw)satisfyingtheconditionsofbothTheorems 8.1 and 10.1 : 1. Pn2Nmw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(20<1whenever1>0; 2. Pn2hPNimw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.58 0 Td[(2<1foreachNwhenever>; 3. fwngn2Nmissupermultiplicative;and 4. foreachprimep2Nmandeachk2Nm,wehavewpkp)]TJ /F8 7.97 Tf 6.59 0 Td[(2wpk)]TJ /F24 5.978 Tf 5.75 0 Td[(1, 96

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andif00,thenif'isamultiplieronHwwithanalyticcontinuationtoQjjgivenby 2U,thenj jQjjkM'kj jQjj.Inparticular,if=,thenkM'k=j jQjj,sothatthemapM'7!'isanisometricisomorphismfromthespaceM\UintothespaceH1(Q)\U.Thefollowingtheoremshowsthatthismapisalsosurjective,thusisometricallyidentifyingM\UwiththespaceH1(Q)\U. Theorem10.2. If'2H1(Q)\U,then'isamultiplieronHw. Proof. ThisfollowsimmediatelyfromTheorem 9.5 ,Equation( 10 ),andtheremarkfollowingtheproofofTheorem 10.1 Corollary5. Ifwnisasequenceofpositivenumbers(withcorrespondingRKHSHw),if0=0,andifthefollowingconditionshold: 1. Pn2Nmw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(20<1whenever1>0; 2. Pn2hPNimw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.58 0 Td[(2<1foreachNwhenever>; 3. fwngn2Nmissupermultiplicative;and 4. foreachprimep2Nmandeachk2Nm,wehavewpkp)]TJ /F8 7.97 Tf 6.59 0 Td[(2wpk)]TJ /F23 5.978 Tf 5.75 0 Td[(1,thenM\UH1(Y)\U.RecallthatthersttwoconditionsinCorollary 5 aresatisedby+1 2andwhenthegrowthconditionfoundinTheorem 8.2 ismet.Accordingly, Corollary6. Ifwnisasequenceofpositivenumbers(withcorrespondingRKHSHw),if0=0,andifthefollowingconditionshold: 1. foreach>,thereexistsaC>0inRmsuchthatw)]TJ /F8 7.97 Tf 6.59 0 Td[(1nn)]TJ /F8 7.97 Tf 6.59 0 Td[(2C 97

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foreachn; 2. fwngn2Nmissupermultiplicative;and 3. foreachprimep2Nmandeachk2Nm,wehavewpkp)]TJ /F8 7.97 Tf 6.59 0 Td[(2wpk)]TJ /F23 5.978 Tf 5.75 0 Td[(1,thenM\UH1(Y)\U. 98

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CHAPTER11CONCLUSIONInthisdissertation,we'vebeenabletocompletelyclassifythemultiplieralgebrasofcertainHilbertspacesofDirichletseriesbothintheone-variablecaseandinthemultivariablecase.ItremainsopenwhetheronecangiveacompletedescriptionoftheweightsequencesforwhichMH1()\Dforsome0(orMH1(Y)\Dinthemultivariatecase).Theresultsinthisdissertationindicatethatthismightbeahardquestion,sincethereseemstobelittlerelationbetweentheweightsgivenbymeasuresandtheweightsinChapter6.OnemighttrytoextendtheresultsofBohrandSchneetothemultivariatesettingsothatatrueextensionoftheHLSresultmightbeobtained.OnemightalsoconsiderHilbertspacesofDirichletseriesonCmwhosecoefcientsaregivenbymatrices.IfHweresuchaspace,thenamultiplieronHwouldnecessarilybematrix-valued. 99

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REFERENCES [1] Agler,JimandMcCarthy,JohnE.PickinterpolationandHilbertfunctionspaces,vol.44ofGraduateStudiesinMathematics.Providence,RI:AmericanMathematicalSociety,2002. [2] Apostol,TomM.Introductiontoanalyticnumbertheory.NewYork:Springer-Verlag,1976.UndergraduateTextsinMathematics. [3] Aronszajn,N.Theoryofreproducingkernels.Trans.Amer.Math.Soc.68(1950):337. [4] Bohr,Harald.UberdiegleichmassigeKonvergenzDirichletscherReihen.J.ReineAngew.Math143(1913):203. [5] Bombieri,E.andFriedlander,J.B.Dirichletpolynomialapproximationstozetafunctions.Ann.ScuolaNorm.Sup.PisaCl.Sci.(4)22(1995).3:517. [6] Conway,JohnB.Acourseinfunctionalanalysis,vol.96ofGraduateTextsinMathematics.NewYork:Springer-Verlag,1990,seconded. [7] Gunning,RobertC.andRossi,Hugo.Analyticfunctionsofseveralcomplexvariables.AMSChelseaPublishing,Providence,RI,2009.Reprintofthe1965original. [8] Hardy,G.H.andRiesz,M.ThegeneraltheoryofDirichlet'sseries.CambridgeTractsinMathematicsandMathematicalPhysics,No.18.Stechert-Hafner,Inc.,NewYork,1964. [9] Hedenmalm,Hakan,Lindqvist,Peter,andSeip,Kristian.AHilbertspaceofDirichletseriesandsystemsofdilatedfunctionsinL2(0,1).DukeMath.J.86(1997).1:1. [10] McCarthy,JohnE.HilbertspacesofDirichletseriesandtheirmultipliers.Trans.Amer.Math.Soc.356(2004).3:881(electronic). [11] Pedersen,GertK.Analysisnow,vol.118ofGraduateTextsinMathematics.NewYork:Springer-Verlag,1989. [12] Rudin,Walter.Realandcomplexanalysis.NewYork:McGraw-HillBookCo.,1987,thirded. [13] Schnee,Walter.ZumKonvergenzproblemderDirichletschenReihen.Math.Ann.66(1908).3:337. 100

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BIOGRAPHICALSKETCH EricStetlerwasborninJacksonville,Floridain1982,butgrewupmostofhislifeinasmallmountaintowninsouthwestVirginiacalledGalax.Inthefallof2003,hemovedtoDaytonaBeach,wherehespentthenextfouryearsworkingandgoingtoschoolpart-time.In2007,EricmovedtoGainesvilletostudymathematicsattheUniversityofFlorida.Untilhissenioryear,however,hewasundecidedonwhetherhewantedtogotograduateschoolformathematicsorphilosophy;heoptedfortheformer,graduatingfromtheUniversityofFloridain2009withhisbachelor'sdegreeinmathematicsandenteringthemathematicsgraduateprogramthatfall.Heobtainedhismaster'sdegreetwoyearslaterin2011.InMayof2014,EricgraduatedwithhisdoctorateinmathematicsundertheguidanceofDr.ScottMcCullough. 101