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THEORYOFREPRODUCINGKERNELS(')BYN.ARONSZAJNPrefaceThepresentpapermaybeconsideredasasequeltoourpreviouspaperintheProceedingsoftheCambridgePhilosophicalSociety,Theoriegénéraledenoyauxreproduisants—Premièrepartie(vol.39(1944))whichwaswrittenin1942-1943.Intheintroductiontothispaperweoutlinedtheplanofpaperswhichweretofollow.Inthemeantime,however,thegeneraltheoryhasbeendevelopedinmanydirections,andouroriginalplanshavehadtobechanged.Duetowartimeconditionswewerenotable,atthetimeofwritingthefirstpaper,totakeintoaccountalltheearlierinvestigationswhich,althoughsometimesofquiteadifferentcharacter,were,nevertheless,relatedtooursubject.Ourinvestigationisconcernedwithkernelsofaspecialtypewhichhavebeenusedunderdifferentnamesandindifferentwaysinmanydomainsofmathematicalresearch.Weshallthereforebeginourpresentpaperwithashorthistoricalintroductioninwhichweshallattempttoindicatethedif-ferentmannersinwhichthesekernelshavebeenusedbyvariousinvesti-gators,andtoclarifytheterminology.Weshallalsodiscussthemoreim-portanttrendsoftheapplicationofthesekernelswithoutattempting,how-ever,acompletebibliographyofthesubjectmatter.InPartI,weshalldiscussbrieflytheessentialnotionsandresultsofourpreviouspaperandgiveafurtherdevelopmentofthetheoryinanabstractform.InPartII,weshallillustratetheresultsobtainedinthefirstpartbyaseriesofexampleswhichwillgivenewdevelopmentsofalreadyknownap-plicationsofthetheory,aswellassomenewapplications.TableofContentsHistoricalIntroduction.338PartI.GeneralTheory.3421.Definitionofreproducingkernels.3422.Résuméofbasicpropertiesofreproducingkernels.3433.Reproducingkernelsoffinite-dimensionalclasses.3464.CompletionofincompleteHubertspaces.3475.Therestrictionofareproducingkernel.3506.Sumofreproducingkernels.352PresentedtotheSociety,December28,1948,underthetitleHubertspacesandconformaimappings;receivedbytheeditorsNovember26,1948.(')PaperdoneundercontractwiththeOfficeofNavalResearch,N5ori76-16-NR043-046,DivisionofEngineeringScience,HarvardUniversity.337338N.ARONSZAJN[May7.Differenceofreproducingkernels.3548.Productofreproducingkernels.3579.Limitsofreproducingkernels.36210.Constructionofar.k.byresolutionofidentity.36811.Operatorsinspaceswithreproducingkernels.37112.Thereproducingkernelofasumoftwoclosedsubspaces.37513.Finalremarksinthegeneraltheory.380PartII.Examples.3841.Introductoryremarks.384(1)Bergman'skernels.384(2)Harmonickernels.3862.Comparisondomains.3873.Thedifferenceofkernels.3884.ThesquareofakernelintroducedbySzegö.3915.ThekernelH{z,zi)foranellipse.3936.ConstructionofH(z,z¡)forastrip.3947.Limitsofincreasingsequencesofkernels.3968.Constructionofreproducingkernelsbytheprojection-formulaof§12,I.397Bibliography.401HistoricalIntroductionExamplesofkernelsofthetypeinwhichweareinterestedhavebeenknownforalongtime,sincealltheGreen'sfunctionsofself-adjointordinarydifferentialequations(asalsosomeGreen'sfunctions—theboundedones—ofpartialdifferentialequations)belongtothistype.Butthecharacteristicpropertiesofthesekernelsaswenowunderstandthemhaveonlybeenstressedandappliedsincethebeginningofthecentury.Therehavebeenandcontinuetobetwotrendsintheconsiderationofthesekernels.ToexplainthemweshouldmentionthatsuchakernelK(x,y)maybecharacterizedasafunctionoftwopoints,byapropertydiscoveredbyJ.Mercer[l](2)in1909.TothekernelKtherecorrespondsawelldeter-minedclassFoffunctions/(x),inrespecttowhichKpossessestherepro-ducingproperty(E.H.Moore[2]).Ontheotherhand,toaclassoffunc-tionsF,theremaycorrespondakernelKwithreproducingproperty(N.Aronszajn[4]).ThosefollowingthefirsttrendconsideragivenkernelKandstudyitinitself,oreventuallyapplyitinvariousdomains(asintegralequations,theoryofgroups,generalmetricgeometry,andsoon).TheclassFcorrespondingtoKmaybeusedasatoolofresearch,butisintroducedaposteriori(asintheworkofE.H.Moore[2],andmorerecentlyofA.Weil[l],I.GelfandandD.Raikoff[l],andR.Godement[l,2]).Inthesecondtrend,oneisinter-estedprimarilyinaclassoffunctionsF,andthecorrespondingkernelKisusedessentiallyasatoolinthestudyofthefunctionsofthisclass.OneofthebasicproblemsinthiskindofinvestigationistheexplicitconstructionandcomputationofthekernelforagivenclassF.(2)Numbersinbracketsrefertothebibliographyattheendofthepaper.1950]THEORYOFREPRODUCINGKERNELS339ThefirstofthesetrendsoriginatedinthetheoryofintegralequationsasdevelopedbyHubert.Thekernelsconsideredthenwerecontinuouskernelsofpositivedefiniteintegraloperators.ThistheorywasdevelopedbyJ.Mercer[l,2]underthenameofpositivedefinitekernelsandonoccasionhasbeenusedbymanyothersinterestedinintegralequations,especiallyduringtheseconddecadeofthiscentury.Mercerdiscoveredthepropertyn(1)^K(yi,yi)\&jè0,yianypoints,£,•anycomplexnumbers(3),i,t-lcharacterizinghiskernels,amongallthecontinuouskernelsofintegralequa-tions.TothissametrendbelongtheinvestigationsofE.H.Moore[l,2]who,duringthesecondandthirddecadesofthecentury,introducedthesekernelsinthegeneralanalysisunderthenameofpositivehermitianmat-riceswithaviewtoapplicationsinakindofgeneralizationofintegralequations.MooreconsideredkernelsK(x,y)definedonanabstractsetEandcharacterizedbytheprop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