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SchematicrepresentationoftheDiracdeltafunctionbyalinesurmountedbyanarrow.Theheightofthearrowisusuallyusedtospecifythevalueofanymultiplicativeconstant,whichwillgivetheareaunderthefunction.Theotherconventionistowritetheareanexttothearrowhead.

DiracdeltafunctionFromWikipedia,thefreeencyclopedia

Inmathematics,theDiracdeltafunction,orfunction,isageneralizedfunction,ordistribution,ontherealnumberlinethatiszeroeverywhereexceptatzero,withanintegralofoneovertheentirerealline.[1][2][3]Thedeltafunctionissometimesthoughtofasaninfinitelyhigh,infinitelythinspikeattheorigin,withtotalareaoneunderthespike,andphysicallyrepresentsthedensityofanidealizedpointmassorpointcharge.[4]ItwasintroducedbytheoreticalphysicistPaulDirac.Inthecontextofsignalprocessingitisoftenreferredtoastheunitimpulsesymbol(orfunction).[5]ItsdiscreteanalogistheKroneckerdeltafunction,whichisusuallydefinedonadiscretedomainandtakesvalues0and1.

Fromapurelymathematicalviewpoint,theDiracdeltaisnotstrictlyafunction,becauseanyextendedrealfunctionthatisequaltozeroeverywherebutasinglepointmusthavetotalintegralzero.[6]Thedeltafunctiononlymakessenseasamathematicalobjectwhenitappearsinsideanintegral.WhilefromthisperspectivetheDiracdeltacanusuallybemanipulatedasthoughitwereafunction,formallyitmustbedefinedasadistributionthatisalsoameasure.Inmanyapplications,theDiracdeltaisregardedasakindoflimit(aweaklimit)ofasequenceoffunctionshavingatallspikeattheorigin.Theapproximatingfunctionsofthesequencearethus"approximate"or"nascent"deltafunctions.

Contents

1Overview2History3Definitions

3.1Asameasure3.2Asadistribution3.3Generalizations

4Properties4.1Scalingandsymmetry4.2Algebraicproperties4.3Translation4.4Compositionwithafunction4.5Propertiesinndimensions

5Fouriertransform6Distributionalderivatives

6.1Higherdimensions

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TheDiracdeltafunctionasthelimit(inthesenseofdistributions)ofthesequenceofzerocenterednormaldistributions

as .

7Representationsofthedeltafunction7.1Approximationstotheidentity7.2Probabilisticconsiderations7.3Semigroups7.4Oscillatoryintegrals7.5Planewavedecomposition7.6Fourierkernels7.7Hilbertspacetheory

7.7.1Spacesofholomorphicfunctions7.7.2Resolutionsoftheidentity

7.8Infinitesimaldeltafunctions8Diraccomb9SokhotskiPlemeljtheorem10RelationshiptotheKroneckerdelta11Applications

11.1Probabilitytheory11.2Quantummechanics11.3Structuralmechanics

12Seealso13Notes14References15Externallinks

Overview

Thegraphofthedeltafunctionisusuallythoughtofasfollowingthewholexaxisandthepositiveyaxis.Despiteitsname,thedeltafunctionisnottrulyafunction,atleastnotausualonewithrangeinrealnumbers.Forexample,theobjectsf(x)=(x)andg(x)=0areequaleverywhereexceptatx=0yethaveintegralsthataredifferent.AccordingtoLebesgueintegrationtheory,iffandgarefunctionssuchthatf=galmosteverywhere,thenfisintegrableifandonlyifgisintegrableandtheintegralsoffandgareidentical.RigoroustreatmentoftheDiracdeltarequiresmeasuretheoryorthetheoryofdistributions.

TheDiracdeltaisusedtomodelatallnarrowspikefunction(animpulse),andothersimilarabstractionssuchasapointcharge,pointmassorelectronpoint.Forexample,tocalculatethedynamicsofabaseballbeinghitbyabat,onecanapproximatetheforceofthebathittingthebaseballbyadeltafunction.Indoingso,onenotonlysimplifiestheequations,butonealsoisabletocalculatethemotionofthebaseballbyonlyconsideringthetotalimpulseofthebatagainsttheballratherthanrequiringknowledgeofthedetailsofhowthebattransferredenergytotheball.

Inappliedmathematics,thedeltafunctionisoftenmanipulatedasakindoflimit(aweaklimit)ofasequenceoffunctions,eachmemberofwhichhasatallspikeattheorigin:forexample,asequenceofGaussiandistributionscenteredattheoriginwithvariancetendingtozero.

History

JosephFourierpresentedwhatisnowcalledtheFourierintegraltheoreminhistreatiseThorieanalytiquedelachaleurintheform:[7]

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whichistantamounttotheintroductionofthefunctionintheform:[8]

Later,AugustinCauchyexpressedthetheoremusingexponentials:[9][10]

Cauchypointedoutthatinsomecircumstancestheorderofintegrationinthisresultwassignificant.[11][12]

Asjustifiedusingthetheoryofdistributions,theCauchyequationcanberearrangedtoresembleFourier'soriginalformulationandexposethefunctionas:

wherethefunctionisexpressedas:

Arigorousinterpretationoftheexponentialformandthevariouslimitationsuponthefunctionfnecessaryforitsapplicationextendedoverseveralcenturies.Theproblemswithaclassicalinterpretationareexplainedasfollows:[13]

ThegreatestdrawbackoftheclassicalFouriertransformationisarathernarrowclassoffunctions(originals)forwhichitcanbeeffectivelycomputed.Namely,itisnecessarythatthesefunctionsdecreasesufficientlyrapidlytozero(intheneighborhoodofinfinity)inordertoinsuretheexistenceoftheFourierintegral.Forexample,theFouriertransformofsuchsimplefunctionsaspolynomialsdoesnotexistintheclassicalsense.TheextensionoftheclassicalFouriertransformationtodistributionsconsiderablyenlargedtheclassoffunctionsthatcouldbetransformedandthisremovedmanyobstacles.

FurtherdevelopmentsincludedgeneralizationoftheFourierintegral,"beginningwithPlancherel'spathbreakingL2theory(1910),continuingwithWiener'sandBochner'sworks(around1930)andculminatingwiththeamalgamationintoL.Schwartz'stheoryofdistributions(1945)...",[14]andleadingtotheformaldevelopmentoftheDiracdeltafunction.

Aninfinitesimalformulaforaninfinitelytall,unitimpulsedeltafunction(infinitesimalversionofCauchydistribution)explicitlyappearsinan1827textofAugustinLouisCauchy.[15]SimonDenisPoissonconsideredtheissueinconnectionwiththestudyofwavepropagationasdidGustavKirchhoffsomewhatlater.KirchhoffandHermannvonHelmholtzalsointroducedtheunitimpulseasalimitofGaussians,whichalsocorresponded

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toLordKelvin'snotionofapointheatsource.Attheendofthe19thcentury,OliverHeavisideusedformalFourierseriestomanipulatetheunitimpulse.[16]TheDiracdeltafunctionassuchwasintroducedasa"convenientnotation"byPaulDiracinhisinfluential1930bookThePrinciplesofQuantumMechanics.[17]Hecalleditthe"deltafunction"sinceheuseditasacontinuousanalogueofthediscreteKroneckerdelta.

Definitions

TheDiracdeltacanbelooselythoughtofasafunctiononthereallinewhichiszeroeverywhereexceptattheorigin,whereitisinfinite,

andwhichisalsoconstrainedtosatisfytheidentity

[18]

Thisismerelyaheuristiccharacterization.TheDiracdeltaisnotafunctioninthetraditionalsenseasnofunctiondefinedontherealnumbershastheseproperties.[17]TheDiracdeltafunctioncanberigorouslydefinedeitherasadistributionorasameasure.

Asameasure

Onewaytorigorouslydefinethedeltafunctionisasameasure,whichacceptsasanargumentasubsetAofthereallineR,andreturns(A)=1if0A,and(A)=0otherwise.[19]Ifthedeltafunctionisconceptualizedasmodelinganidealizedpointmassat0,then(A)representsthemasscontainedinthesetA.Onemaythendefinetheintegralagainstastheintegralofafunctionagainstthismassdistribution.Formally,theLebesgueintegralprovidesthenecessaryanalyticdevice.TheLebesgueintegralwithrespecttothemeasuresatisfies

forallcontinuouscompactlysupportedfunctionsf.ThemeasureisnotabsolutelycontinuouswithrespecttotheLebesguemeasureinfact,itisasingularmeasure.Consequently,thedeltameasurehasnoRadonNikodymderivativenotruefunctionforwhichtheproperty

holds.[20]Asaresult,thelatternotationisaconvenientabuseofnotation,andnotastandard(RiemannorLebesgue)integral.

AsaprobabilitymeasureonR,thedeltameasureischaracterizedbyitscumulativedistributionfunction,whichistheunitstepfunction[21]

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ThismeansthatH(x)istheintegralofthecumulativeindicatorfunction1(,x]withrespecttothemeasuretowit,

ThusinparticulartheintegralofthedeltafunctionagainstacontinuousfunctioncanbeproperlyunderstoodasaStieltjesintegral:[22]

Allhighermomentsofarezero.Inparticular,characteristicfunctionandmomentgeneratingfunctionarebothequaltoone.

Asadistribution

Inthetheoryofdistributionsageneralizedfunctionisthoughtofnotasafunctionitself,butonlyinrelationtohowitaffectsotherfunctionswhenitis"integrated"againstthem.Inkeepingwiththisphilosophy,todefinethedeltafunctionproperly,itisenoughtosaywhatthe"integral"ofthedeltafunctionagainstasufficiently"good"testfunctionis.Ifthedeltafunctionisalreadyunderstoodasameasure,thentheLebesgueintegralofatestfunctionagainstthatmeasuresuppliesthenecessaryintegral.

AtypicalspaceoftestfunctionsconsistsofallsmoothfunctionsonRwithcompactsupport.Asadistribution,theDiracdeltaisalinearfunctionalonthespaceoftestfunctionsandisdefinedby[23]

(1)

foreverytestfunction.

Fortobeproperlyadistribution,itmustbe"continuous"inasuitablesense.Ingeneral,foralinearfunctionalSonthespaceoftestfunctionstodefineadistribution,itisnecessaryandsufficientthat,foreverypositiveintegerNthereisanintegerMNandaconstantCNsuchthatforeverytestfunction,onehastheinequality[24]

Withthedistribution,onehassuchaninequality(withCN=1)withMN=0forallN.Thusisadistributionoforderzero.Itis,furthermore,adistributionwithcompactsupport(thesupportbeing{0}).

Thedeltadistributioncanalsobedefinedinanumberofequivalentways.Forinstance,itisthedistributionalderivativeoftheHeavisidestepfunction.Thismeansthat,foreverytestfunction,onehas

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Intuitively,ifintegrationbypartswerepermitted,thenthelatterintegralshouldsimplifyto

andindeed,aformofintegrationbypartsispermittedfortheStieltjesintegral,andinthatcaseonedoeshave

Inthecontextofmeasuretheory,theDiracmeasuregivesrisetoadistributionbyintegration.Conversely,equation(1)definesaDaniellintegralonthespaceofallcompactlysupportedcontinuousfunctionswhich,bytheRieszrepresentationtheorem,canberepresentedastheLebesgueintegralofwithrespecttosomeRadonmeasure.

Generalizations

ThedeltafunctioncanbedefinedinndimensionalEuclideanspaceRnasthemeasuresuchthat

foreverycompactlysupportedcontinuousfunctionf.Asameasure,thendimensionaldeltafunctionistheproductmeasureofthe1dimensionaldeltafunctionsineachvariableseparately.Thus,formally,withx=(x1,x2,...,xn),onehas[5]

(2)

Thedeltafunctioncanalsobedefinedinthesenseofdistributionsexactlyasaboveintheonedimensionalcase.[25]However,despitewidespreaduseinengineeringcontexts,(2)shouldbemanipulatedwithcare,sincetheproductofdistributionscanonlybedefinedunderquitenarrowcircumstances.[26]

ThenotionofaDiracmeasuremakessenseonanyset.[19]ThusifXisaset,x0Xisamarkedpoint,andisanysigmaalgebraofsubsetsofX,thenthemeasuredefinedonsetsAby

isthedeltameasureorunitmassconcentratedatx0.

Anothercommongeneralizationofthedeltafunctionistoadifferentiablemanifoldwheremostofitspropertiesasadistributioncanalsobeexploitedbecauseofthedifferentiablestructure.ThedeltafunctiononamanifoldMcenteredatthepointx0Misdefinedasthefollowingdistribution:

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(3)

forallcompactlysupportedsmoothrealvaluedfunctionsonM.[27]AcommonspecialcaseofthisconstructioniswhenMisanopensetintheEuclideanspaceRn.

OnalocallycompactHausdorffspaceX,theDiracdeltameasureconcentratedatapointxistheRadonmeasureassociatedwiththeDaniellintegral(3)oncompactlysupportedcontinuousfunctions.Atthislevelofgenerality,calculusassuchisnolongerpossible,howeveravarietyoftechniquesfromabstractanalysisareavailable.Forinstance,themapping isacontinuousembeddingofXintothespaceoffiniteRadonmeasuresonX,equippedwithitsvaguetopology.Moreover,theconvexhulloftheimageofXunderthisembeddingisdenseinthespaceofprobabilitymeasuresonX.[28]

Properties

Scalingandsymmetry

Thedeltafunctionsatisfiesthefollowingscalingpropertyforanonzeroscalar:[29]

andso

(4)

Inparticular,thedeltafunctionisanevendistribution,inthesensethat

whichishomogeneousofdegree1.

Algebraicproperties

Thedistributionalproductofwithxisequaltozero:

Conversely,ifxf(x)=xg(x),wherefandgaredistributions,then

forsomeconstantc.[30]

Translation

TheintegralofthetimedelayedDiracdeltaisgivenby:

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Thisissometimesreferredtoasthesiftingproperty[31]orthesamplingproperty.Thedeltafunctionissaidto"siftout"thevalueatt=T.

Itfollowsthattheeffectofconvolvingafunctionf(t)withthetimedelayedDiracdeltaistotimedelayf(t)bythesameamount:

(using(4): )

Thisholdsunderthepreciseconditionthatfbeatempereddistribution(seethediscussionoftheFouriertransformbelow).Asaspecialcase,forinstance,wehavetheidentity(understoodinthedistributionsense)

Compositionwithafunction

Moregenerally,thedeltadistributionmaybecomposedwithasmoothfunctiong(x)insuchawaythatthefamiliarchangeofvariablesformulaholds,that

providedthatgisacontinuouslydifferentiablefunctionwithgnowherezero.[32]Thatis,thereisauniquewaytoassignmeaningtothedistribution sothatthisidentityholdsforallcompactlysupportedtestfunctionsf.Therefore,thedomainmustbebrokenuptoexcludetheg'=0point.Thisdistributionsatisfies(g(x))=0ifgisnowherezero,andotherwiseifghasarealrootatx0,then

Itisnaturalthereforetodefinethecomposition(g(x))forcontinuouslydifferentiablefunctionsgby

wherethesumextendsoverallrootsofg(x),whichareassumedtobesimple.[32]Thus,forexample

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Intheintegralformthegeneralizedscalingpropertymaybewrittenas

Propertiesinndimensions

Thedeltadistributioninanndimensionalspacesatisfiesthefollowingscalingpropertyinstead:

sothatisahomogeneousdistributionofdegreen.Underanyreflectionorrotation,thedeltafunctionisinvariant:

Asintheonevariablecase,itispossibletodefinethecompositionofwithabiLipschitzfunction[33]

g:RnRnuniquelysothattheidentity

forallcompactlysupportedfunctionsf.

Usingthecoareaformulafromgeometricmeasuretheory,onecanalsodefinethecompositionofthedeltafunctionwithasubmersionfromoneEuclideanspacetoanotheroneofdifferentdimensiontheresultisatypeofcurrent.Inthespecialcaseofacontinuouslydifferentiablefunctiong:RnRsuchthatthegradientofgisnowherezero,thefollowingidentityholds[34]

wheretheintegralontherightisoverg1(0),the(n1)dimensionalsurfacedefinedbyg(x)=0withrespecttotheMinkowskicontentmeasure.Thisisknownasasimplelayerintegral.

Moregenerally,ifSisasmoothhypersurfaceofRn,thenwecanassociatedtoSthedistributionthatintegratesanycompactlysupportedsmoothfunctiongoverS:

whereisthehypersurfacemeasureassociatedtoS.ThisgeneralizationisassociatedwiththepotentialtheoryofsimplelayerpotentialsonS.IfDisadomaininRnwithsmoothboundaryS,thenSisequaltothenormalderivativeoftheindicatorfunctionofDinthedistributionsense:

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wherenistheoutwardnormal.[35][36]Foraproof,seee.g.thearticleonthesurfacedeltafunction.

Fouriertransform

Thedeltafunctionisatempereddistribution,andthereforeithasawelldefinedFouriertransform.Formally,onefinds[37]

Properlyspeaking,theFouriertransformofadistributionisdefinedbyimposingselfadjointnessoftheFouriertransformunderthedualitypairing oftempereddistributionswithSchwartzfunctions.Thus isdefinedastheuniquetempereddistributionsatisfying

forallSchwartzfunctions.Andindeeditfollowsfromthisthat

Asaresultofthisidentity,theconvolutionofthedeltafunctionwithanyothertempereddistributionSissimplyS:

Thatistosaythatisanidentityelementfortheconvolutionontempereddistributions,andinfactthespaceofcompactlysupporteddistributionsunderconvolutionisanassociativealgebrawithidentitythedeltafunction.Thispropertyisfundamentalinsignalprocessing,asconvolutionwithatempereddistributionisalineartimeinvariantsystem,andapplyingthelineartimeinvariantsystemmeasuresitsimpulseresponse.Theimpulseresponsecanbecomputedtoanydesireddegreeofaccuracybychoosingasuitableapproximationfor,andonceitisknown,itcharacterizesthesystemcompletely.SeeLTIsystemtheory:Impulseresponseandconvolution.

TheinverseFouriertransformofthetempereddistributionf()=1isthedeltafunction.Formally,thisisexpressed

andmorerigorously,itfollowssince

forallSchwartzfunctionsf.

Intheseterms,thedeltafunctionprovidesasuggestivestatementoftheorthogonalitypropertyoftheFourierkernelonR.Formally,onehas

Thisis,ofcourse,shorthandfortheassertionthattheFouriertransformofthetempereddistribution

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is

whichagainfollowsbyimposingselfadjointnessoftheFouriertransform.

ByanalyticcontinuationoftheFouriertransform,theLaplacetransformofthedeltafunctionisfoundtobe[38]

Distributionalderivatives

ThedistributionalderivativeoftheDiracdeltadistributionisthedistributiondefinedoncompactlysupportedsmoothtestfunctionsby[39]

Thefirstequalityhereisakindofintegrationbyparts,forifwereatruefunctionthen

Thekthderivativeofisdefinedsimilarlyasthedistributiongivenontestfunctionsby

Inparticular,isaninfinitelydifferentiabledistribution.

Thefirstderivativeofthedeltafunctionisthedistributionallimitofthedifferencequotients:[40]

Moreproperly,onehas

wherehisthetranslationoperator,definedonfunctionsbyh(x)=(x+h),andonadistributionSby

Inthetheoryofelectromagnetism,thefirstderivativeofthedeltafunctionrepresentsapointmagneticdipolesituatedattheorigin.Accordingly,itisreferredtoasadipoleorthedoubletfunction.[41]

Thederivativeofthedeltafunctionsatisfiesanumberofbasicproperties,including:

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[42]

Furthermore,theconvolutionofwithacompactlysupportedsmoothfunctionfis

whichfollowsfromthepropertiesofthedistributionalderivativeofaconvolution.

Higherdimensions

Moregenerally,onanopensetUinthendimensionalEuclideanspaceRn,theDiracdeltadistributioncenteredatapointaUisdefinedby[43]

forallS(U),thespaceofallsmoothcompactlysupportedfunctionsonU.If=(1,...,n)isanymulti

indexanddenotestheassociatedmixedpartialderivativeoperator,thenthethderivativeaofais

givenby[43]

Thatis,thethderivativeofaisthedistributionwhosevalueonanytestfunctionisthethderivativeofata(withtheappropriatepositiveornegativesign).

Thefirstpartialderivativesofthedeltafunctionarethoughtofasdoublelayersalongthecoordinateplanes.Moregenerally,thenormalderivativeofasimplelayersupportedonasurfaceisadoublelayersupportedonthatsurface,andrepresentsalaminarmagneticmonopole.Higherderivativesofthedeltafunctionareknowninphysicsasmultipoles.

Higherderivativesenterintomathematicsnaturallyasthebuildingblocksforthecompletestructureofdistributionswithpointsupport.IfSisanydistributiononUsupportedontheset{a}consistingofasinglepoint,thenthereisanintegermandcoefficientscsuchthat[44]

Representationsofthedeltafunction

Thedeltafunctioncanbeviewedasthelimitofasequenceoffunctions

where(x)issometimescalledanascentdeltafunction.Thislimitismeantinaweaksense:eitherthat

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(5)

forallcontinuousfunctionsfhavingcompactsupport,orthatthislimitholdsforallsmoothfunctionsfwithcompactsupport.Thedifferencebetweenthesetwoslightlydifferentmodesofweakconvergenceisoftensubtle:theformerisconvergenceinthevaguetopologyofmeasures,andthelatterisconvergenceinthesenseofdistributions.

Approximationstotheidentity

Typicallyanascentdeltafunctioncanbeconstructedinthefollowingmanner.LetbeanabsolutelyintegrablefunctiononRoftotalintegral1,anddefine

Inndimensions,oneusesinsteadthescaling

Thenasimplechangeofvariablesshowsthatalsohasintegral1.[45]Oneshowseasilythat(5)holdsforallcontinuouscompactlysupportedfunctionsf,andsoconvergesweaklytointhesenseofmeasures.

Theconstructedinthiswayareknownasanapproximationtotheidentity.[46]Thisterminologyisbecause

thespaceL1(R)ofabsolutelyintegrablefunctionsisclosedundertheoperationofconvolutionoffunctions:fgL1(R)wheneverfandgareinL1(R).However,thereisnoidentityinL1(R)fortheconvolutionproduct:noelementhsuchthatfh=fforallf.Nevertheless,thesequencedoesapproximatesuchanidentityinthesensethat

Thislimitholdsinthesenseofmeanconvergence(convergenceinL1).Furtherconditionsonthe,for

instancethatitbeamollifierassociatedtoacompactlysupportedfunction,[47]areneededtoensurepointwiseconvergencealmosteverywhere.

Iftheinitial=1isitselfsmoothandcompactlysupportedthenthesequenceiscalledamollifier.Thestandardmollifierisobtainedbychoosingtobeasuitablynormalizedbumpfunction,forinstance

Insomesituationssuchasnumericalanalysis,apiecewiselinearapproximationtotheidentityisdesirable.Thiscanbeobtainedbytaking1tobeahatfunction.Withthischoiceof1,onehas

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whichareallcontinuousandcompactlysupported,althoughnotsmoothandsonotamollifier.

Probabilisticconsiderations

Inthecontextofprobabilitytheory,itisnaturaltoimposetheadditionalconditionthattheinitial1inanapproximationtotheidentityshouldbepositive,assuchafunctionthenrepresentsaprobabilitydistribution.Convolutionwithaprobabilitydistributionissometimesfavorablebecauseitdoesnotresultinovershootorundershoot,astheoutputisaconvexcombinationoftheinputvalues,andthusfallsbetweenthemaximumandminimumoftheinputfunction.Taking1tobeanyprobabilitydistributionatall,andletting(x)=1(x/)/asabovewillgiverisetoanapproximationtotheidentity.Ingeneralthisconvergesmorerapidlytoadeltafunctionif,inaddition,hasmean0andhassmallhighermoments.Forinstance,if1istheuniform

distributionon[1/2,1/2],alsoknownastherectangularfunction,then:[48]

AnotherexampleiswiththeWignersemicircledistribution

Thisiscontinuousandcompactlysupported,butnotamollifierbecauseitisnotsmooth.

Semigroups

Nascentdeltafunctionsoftenariseasconvolutionsemigroups.Thisamountstothefurtherconstraintthattheconvolutionofwithmustsatisfy

forall,>0.ConvolutionsemigroupsinL1thatformanascentdeltafunctionarealwaysanapproximationtotheidentityintheabovesense,howeverthesemigroupconditionisquiteastrongrestriction.

Inpractice,semigroupsapproximatingthedeltafunctionariseasfundamentalsolutionsorGreen'sfunctionstophysicallymotivatedellipticorparabolicpartialdifferentialequations.Inthecontextofappliedmathematics,semigroupsariseastheoutputofalineartimeinvariantsystem.Abstractly,ifAisalinearoperatoractingonfunctionsofx,thenaconvolutionsemigrouparisesbysolvingtheinitialvalueproblem

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inwhichthelimitisasusualunderstoodintheweaksense.Setting(x)=(,x)givestheassociatednascentdeltafunction.

Someexamplesofphysicallyimportantconvolutionsemigroupsarisingfromsuchafundamentalsolutionincludethefollowing.

Theheatkernel

Theheatkernel,definedby

representsthetemperatureinaninfinitewireattimet>0,ifaunitofheatenergyisstoredattheoriginofthewireattimet=0.Thissemigroupevolvesaccordingtotheonedimensionalheatequation:

Inprobabilitytheory,(x)isanormaldistributionofvarianceandmean0.Itrepresentstheprobabilitydensityattimet=ofthepositionofaparticlestartingattheoriginfollowingastandardBrownianmotion.Inthiscontext,thesemigroupconditionisthenanexpressionoftheMarkovpropertyofBrownianmotion.

InhigherdimensionalEuclideanspaceRn,theheatkernelis

andhasthesamephysicalinterpretation,mutatismutandis.Italsorepresentsanascentdeltafunctioninthesensethatinthedistributionsenseas0.

ThePoissonkernel

ThePoissonkernel

isthefundamentalsolutionoftheLaplaceequationintheupperhalfplane.[49]Itrepresentstheelectrostaticpotentialinasemiinfiniteplatewhosepotentialalongtheedgeisheldatfixedatthedeltafunction.ThePoissonkernelisalsocloselyrelatedtotheCauchydistribution.Thissemigroupevolvesaccordingtotheequation

wheretheoperatorisrigorouslydefinedastheFouriermultiplier

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Oscillatoryintegrals

Inareasofphysicssuchaswavepropagationandwavemechanics,theequationsinvolvedarehyperbolicandsomayhavemoresingularsolutions.Asaresult,thenascentdeltafunctionsthatariseasfundamentalsolutionsoftheassociatedCauchyproblemsaregenerallyoscillatoryintegrals.Anexample,whichcomesfromasolutionoftheEulerTricomiequationoftransonicgasdynamics,[50]istherescaledAiryfunction

AlthoughusingtheFouriertransform,itiseasytoseethatthisgeneratesasemigroupinsomesense,itisnotabsolutelyintegrableandsocannotdefineasemigroupintheabovestrongsense.Manynascentdeltafunctionsconstructedasoscillatoryintegralsonlyconvergeinthesenseofdistributions(anexampleistheDirichletkernelbelow),ratherthaninthesenseofmeasures.

AnotherexampleistheCauchyproblemforthewaveequationinR1+1:[51]

Thesolutionurepresentsthedisplacementfromequilibriumofaninfiniteelasticstring,withaninitialdisturbanceattheorigin.

Otherapproximationstotheidentityofthiskindincludethesincfunction(usedwidelyinelectronicsandtelecommunications)

andtheBesselfunction

Planewavedecomposition

Oneapproachtothestudyofalinearpartialdifferentialequation

whereLisadifferentialoperatoronRn,istoseekfirstafundamentalsolution,whichisasolutionoftheequation

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WhenLisparticularlysimple,thisproblemcanoftenberesolvedusingtheFouriertransformdirectly(asinthecaseofthePoissonkernelandheatkernelalreadymentioned).Formorecomplicatedoperators,itissometimeseasierfirsttoconsideranequationoftheform

wherehisaplanewavefunction,meaningthatithastheform

forsomevector.Suchanequationcanberesolved(ifthecoefficientsofLareanalyticfunctions)bytheCauchyKovalevskayatheoremor(ifthecoefficientsofLareconstant)byquadrature.So,ifthedeltafunctioncanbedecomposedintoplanewaves,thenonecaninprinciplesolvelinearpartialdifferentialequations.

SuchadecompositionofthedeltafunctionintoplanewaveswaspartofageneraltechniquefirstintroducedessentiallybyJohannRadon,andthendevelopedinthisformbyFritzJohn(1955).[52]Chooseksothatn+kisaneveninteger,andforarealnumbers,put

ThenisobtainedbyapplyingapoweroftheLaplaciantotheintegralwithrespecttotheunitspheremeasuredofg(x)forintheunitsphereSn1:

TheLaplacianhereisinterpretedasaweakderivative,sothatthisequationistakentomeanthat,foranytestfunction,

TheresultfollowsfromtheformulafortheNewtonianpotential(thefundamentalsolutionofPoisson'sequation).ThisisessentiallyaformoftheinversionformulafortheRadontransform,becauseitrecoversthevalueof(x)fromitsintegralsoverhyperplanes.Forinstance,ifnisoddandk=1,thentheintegralontherighthandsideis

whereR(,p)istheRadontransformof:

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Analternativeequivalentexpressionoftheplanewavedecomposition,fromGel'fand&Shilov(19661968,I,3.10),is

forneven,and

fornodd.

Fourierkernels

InthestudyofFourierseries,amajorquestionconsistsofdeterminingwhetherandinwhatsensetheFourierseriesassociatedwithaperiodicfunctionconvergestothefunction.ThenthpartialsumoftheFourierseriesofafunctionfofperiod2isdefinedbyconvolution(ontheinterval[,])withtheDirichletkernel:

Thus,

where

AfundamentalresultofelementaryFourierseriesstatesthattheDirichletkerneltendstotheamultipleofthedeltafunctionasN.Thisisinterpretedinthedistributionsense,that

foreverycompactlysupportedsmoothfunctionf.Thus,formallyonehas

ontheinterval[,].

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Inspiteofthis,theresultdoesnotholdforallcompactlysupportedcontinuousfunctions:thatisDNdoesnotconvergeweaklyinthesenseofmeasures.ThelackofconvergenceoftheFourierserieshasledtotheintroductionofavarietyofsummabilitymethodsinordertoproduceconvergence.ThemethodofCesrosummationleadstotheFejrkernel[53]

TheFejrkernelstendtothedeltafunctioninastrongersensethat[54]

foreverycompactlysupportedcontinuousfunctionf.TheimplicationisthattheFourierseriesofanycontinuousfunctionisCesrosummabletothevalueofthefunctionateverypoint.

Hilbertspacetheory

TheDiracdeltadistributionisadenselydefinedunboundedlinearfunctionalontheHilbertspaceL2ofsquareintegrablefunctions.Indeed,smoothcompactlysupportfunctionsaredenseinL2,andtheactionofthedeltadistributiononsuchfunctionsiswelldefined.Inmanyapplications,itispossibletoidentifysubspacesofL2andtogiveastrongertopologyonwhichthedeltafunctiondefinesaboundedlinearfunctional.

Sobolevspaces

TheSobolevembeddingtheoremforSobolevspacesonthereallineRimpliesthatanysquareintegrablefunctionfsuchthat

isautomaticallycontinuous,andsatisfiesinparticular

ThusisaboundedlinearfunctionalontheSobolevspaceH1.EquivalentlyisanelementofthecontinuousdualspaceH1ofH1.Moregenerally,inndimensions,onehasHs(Rn)provideds>n/2.

Spacesofholomorphicfunctions

Incomplexanalysis,thedeltafunctionentersviaCauchy'sintegralformulawhichassertsthatifDisadomaininthecomplexplanewithsmoothboundary,then

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forallholomorphicfunctionsfinDthatarecontinuousontheclosureofD.Asaresult,thedeltafunctionzisrepresentedonthisclassofholomorphicfunctionsbytheCauchyintegral:

Moregenerally,letH2(D)betheHardyspaceconsistingoftheclosureinL2(D)ofallholomorphicfunctionsinDcontinuousuptotheboundaryofD.ThenfunctionsinH2(D)uniquelyextendtoholomorphicfunctionsinD,andtheCauchyintegralformulacontinuestohold.InparticularforzD,thedeltafunctionzisa

continuouslinearfunctionalonH2(D).Thisisaspecialcaseofthesituationinseveralcomplexvariablesinwhich,forsmoothdomainsD,theSzegkernelplaystheroleoftheCauchyintegral.

Resolutionsoftheidentity

Givenacompleteorthonormalbasissetoffunctions{n}inaseparableHilbertspace,forexample,thenormalizedeigenvectorsofacompactselfadjointoperator,anyvectorfcanbeexpressedas:

Thecoefficients{n}arefoundas:

whichmayberepresentedbythenotation:

aformofthebraketnotationofDirac.[55]Adoptingthisnotation,theexpansionofftakesthedyadicform:[56]

LettingIdenotetheidentityoperatorontheHilbertspace,theexpression

iscalledaresolutionoftheidentity.WhentheHilbertspaceisthespaceL2(D)ofsquareintegrablefunctionsonadomainD,thequantity:

isanintegraloperator,andtheexpressionforfcanberewrittenas:

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ADiraccombisaninfiniteseriesofDiracdeltafunctionsspacedatintervalsofT

TherighthandsideconvergestofintheL2sense.Itneednotholdinapointwisesense,evenwhenfisacontinuousfunction.Nevertheless,itiscommontoabusenotationandwrite

resultingintherepresentationofthedeltafunction:[57]

WithasuitableriggedHilbertspace(,L2(D),*)whereL2(D)containsallcompactlysupportedsmoothfunctions,thissummationmayconvergein*,dependingonthepropertiesofthebasisn.Inmostcasesofpracticalinterest,theorthonormalbasiscomesfromanintegralordifferentialoperator,inwhichcasetheseriesconvergesinthedistributionsense.[58]

Infinitesimaldeltafunctions

Cauchyusedaninfinitesimaltowritedownaunitimpulse,infinitelytallandnarrowDiractypedeltafunctionsatisfying inanumberofarticlesin1827.[59]Cauchydefinedan

infinitesimalinCoursd'Analyse(1827)intermsofasequencetendingtozero.Namely,suchanullsequencebecomesaninfinitesimalinCauchy'sandLazareCarnot'sterminology.

Nonstandardanalysisallowsonetorigorouslytreatinfinitesimals.ThearticlebyYamashita(2007)containsabibliographyonmodernDiracdeltafunctionsinthecontextofaninfinitesimalenrichedcontinuumprovidedbythehyperreals.HeretheDiracdeltacanbegivenbyanactualfunction,havingthepropertythatforeveryrealfunctionFonehas asanticipatedbyFourierandCauchy.

Diraccomb

Asocalleduniform"pulsetrain"ofDiracdeltameasures,whichisknownasaDiraccomb,orastheShahdistribution,createsasamplingfunction,oftenusedindigitalsignalprocessing(DSP)anddiscretetimesignalanalysis.TheDiraccombisgivenastheinfinitesum,whoselimitisunderstoodinthedistributionsense,

whichisasequenceofpointmassesateachoftheintegers.

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Uptoanoverallnormalizingconstant,theDiraccombisequaltoitsownFouriertransform.ThisissignificantbecauseiffisanySchwartzfunction,thentheperiodizationoffisgivenbytheconvolution

Inparticular,

ispreciselythePoissonsummationformula.[60]

SokhotskiPlemeljtheorem

TheSokhotskiPlemeljtheorem,importantinquantummechanics,relatesthedeltafunctiontothedistributionp.v.1/x,theCauchyprincipalvalueofthefunction1/x,definedby

Sokhotsky'sformulastatesthat[61]

Herethelimitisunderstoodinthedistributionsense,thatforallcompactlysupportedsmoothfunctionsf,

RelationshiptotheKroneckerdelta

TheKroneckerdeltaijisthequantitydefinedby

forallintegersi,j.Thisfunctionthensatisfiesthefollowinganalogofthesiftingproperty:if isanydoublyinfinitesequence,then

Similarly,foranyrealorcomplexvaluedcontinuousfunctionfonR,theDiracdeltasatisfiesthesiftingproperty

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ThisexhibitstheKroneckerdeltafunctionasadiscreteanalogoftheDiracdeltafunction.[62]

Applications

Probabilitytheory

Inprobabilitytheoryandstatistics,theDiracdeltafunctionisoftenusedtorepresentadiscretedistribution,orapartiallydiscrete,partiallycontinuousdistribution,usingaprobabilitydensityfunction(whichisnormallyusedtorepresentfullycontinuousdistributions).Forexample,theprobabilitydensityfunctionf(x)ofadiscretedistributionconsistingofpointsx={x1,...,xn},withcorrespondingprobabilitiesp1,...,pn,canbewrittenas

Asanotherexample,consideradistributionwhich6/10ofthetimereturnsastandardnormaldistribution,and4/10ofthetimereturnsexactlythevalue3.5(i.e.apartlycontinuous,partlydiscretemixturedistribution).Thedensityfunctionofthisdistributioncanbewrittenas

Thedeltafunctionisalsousedinacompletelydifferentwaytorepresentthelocaltimeofadiffusionprocess(likeBrownianmotion).ThelocaltimeofastochasticprocessB(t)isgivenby

andrepresentstheamountoftimethattheprocessspendsatthepointxintherangeoftheprocess.Moreprecisely,inonedimensionthisintegralcanbewritten

where1[x,x+]istheindicatorfunctionoftheinterval[x,x+].

Quantummechanics

Wegiveanexampleofhowthedeltafunctionisexpedientinquantummechanics.Thewavefunctionofaparticlegivestheprobabilityamplitudeoffindingaparticlewithinagivenregionofspace.WavefunctionsareassumedtobeelementsoftheHilbertspaceL2ofsquareintegrablefunctions,andthetotalprobabilityoffindingaparticlewithinagivenintervalistheintegralofthemagnitudeofthewavefunctionsquaredovertheinterval.Aset{n}ofwavefunctionsisorthonormaliftheyarenormalizedby

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whereherereferstotheKroneckerdelta.Asetoforthonormalwavefunctionsiscompleteinthespaceofsquareintegrablefunctionsifanywavefunctioncanbeexpressedasacombinationofthen:

with .CompleteorthonormalsystemsofwavefunctionsappearnaturallyastheeigenfunctionsoftheHamiltonian(ofaboundsystem)inquantummechanicsthatmeasurestheenergylevels,whicharecalledtheeigenvalues.Thesetofeigenvalues,inthiscase,isknownasthespectrumoftheHamiltonian.Inbraketnotation,asabove,thisequalityimpliestheresolutionoftheidentity:

Heretheeigenvaluesareassumedtobediscrete,butthesetofeigenvaluesofanobservablemaybecontinuousratherthandiscrete.Anexampleisthepositionobservable,Q(x)=x(x).Thespectrumoftheposition(inonedimension)istheentirerealline,andiscalledacontinuousspectrum.However,unliketheHamiltonian,thepositionoperatorlackspropereigenfunctions.Theconventionalwaytoovercomethisshortcomingistowidentheclassofavailablefunctionsbyallowingdistributionsaswell:thatis,toreplacetheHilbertspaceofquantummechanicsbyanappropriateriggedHilbertspace.[63]Inthiscontext,thepositionoperatorhasacompletesetofeigendistributions,labeledbythepointsyoftherealline,givenby

Theeigenfunctionsofpositionaredenotedby inDiracnotation,andareknownaspositioneigenstates.

Similarconsiderationsapplytotheeigenstatesofthemomentumoperator,orindeedanyotherselfadjointunboundedoperatorPontheHilbertspace,providedthespectrumofPiscontinuousandtherearenodegenerateeigenvalues.Inthatcase,thereisasetofrealnumbers(thespectrum),andacollectionyofdistributionsindexedbytheelementsof,suchthat

Thatis,yaretheeigenvectorsofP.Iftheeigenvectorsarenormalizedsothat

inthedistributionsense,thenforanytestfunction,

where

Thatis,asinthediscretecase,thereisaresolutionoftheidentity

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wheretheoperatorvaluedintegralisagainunderstoodintheweaksense.IfthespectrumofPhasbothcontinuousanddiscreteparts,thentheresolutionoftheidentityinvolvesasummationoverthediscretespectrumandanintegraloverthecontinuousspectrum.

Thedeltafunctionalsohasmanymorespecializedapplicationsinquantummechanics,suchasthedeltapotentialmodelsforasingleanddoublepotentialwell.

Structuralmechanics

Thedeltafunctioncanbeusedinstructuralmechanicstodescribetransientloadsorpointloadsactingonstructures.ThegoverningequationofasimplemassspringsystemexcitedbyasuddenforceimpulseIattimet=0canbewritten

wheremisthemass,thedeflectionandkthespringconstant.

Asanotherexample,theequationgoverningthestaticdeflectionofaslenderbeamis,accordingtoEulerBernoullitheory,

whereEIisthebendingstiffnessofthebeam,wthedeflection,xthespatialcoordinateandq(x)theloaddistribution.IfabeamisloadedbyapointforceFatx=x0,theloaddistributioniswritten

AsintegrationofthedeltafunctionresultsintheHeavisidestepfunction,itfollowsthatthestaticdeflectionofaslenderbeamsubjecttomultiplepointloadsisdescribedbyasetofpiecewisepolynomials.

Alsoapointmomentactingonabeamcanbedescribedbydeltafunctions.ConsidertwoopposingpointforcesFatadistancedapart.TheythenproduceamomentM=Fdactingonthebeam.Now,letthedistancedapproachthelimitzero,whileMiskeptconstant.Theloaddistribution,assumingaclockwisemomentactingatx=0,iswritten

Pointmomentscanthusberepresentedbythederivativeofthedeltafunction.Integrationofthebeamequationagainresultsinpiecewisepolynomialdeflection.

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Seealso

Atom(measuretheory)DeltapotentialDiracmeasureFundamentalsolutionGreen'sfunctionLaplacianoftheindicator

Notes

1. Dirac1958,15Thefunction,p.582. Gel'fand&Shilov1968,VolumeI,1.1,1.33. Schwartz1950,p.34. Arfken&Weber2000,p.845. Bracewell1986,Chapter56. Vladimirov1971,5.17. JBFourier(1822).TheAnalyticalTheoryofHeat(http://books.google.com/books?id=

N8EAAAAYAAJ&pg=PA408&dq=%22when+the+integrals+are+taken+between+infinite+limits%22+%22that+is+to+say,+that+we+have+the+equation%22&hl=en&sa=X&ei=rFeT96cEIzKiQKcyfDtDQ&ved=0CD4Q6AEwAA#v=onepage&q=%22when%20the%20integrals%20are%20taken%20between%20infinite%20limits%22%20%22that%20is%20to%20say%2C%20that%20we%20have%20the%20equation%22&f=false)(EnglishtranslationbyAlexanderFreeman,1878ed.).TheUniversityPress.p.408.,cfp449andpp546551.TheoriginalFrenchtextcanbefoundhere(http://books.google.com/books?id=TDQJAAAAIAAJ&pg=PA525&dq=%22c%27est%C3%A0dire+qu%27on+a+l%27%C3%A9quation%22&hl=en&sa=X&ei=SrC7T9yKBorYiALVnc2oDg&sqi=2&ved=0CEAQ6AEwAg#v=onepage&q=%22c%27est%C3%A0dire%20qu%27on%20a%20l%27%C3%A9quation%22&f=false).

8. HikosaburoKomatsu(2002)."Fourier'shyperfunctionsandHeaviside'spseudodifferentialoperators".InTakahiroKawai,KeikoFujita,eds.MicrolocalAnalysisandComplexFourierAnalysis(http://books.google.com/books?id=8GwKzEemrIcC&pg=PA200&dq=%22Fourier+introduced+the%22+%22+function+much+earlier%22&hl=en&sa=X&ei=oJa6T5L2O6SriQKGloCUBw&ved=0CDQQ6AEwAA#v=onepage&q=%22Fourier%20introduced%20the%22%20%22%20function%20much%20earlier%22&f=false).WorldScientific.p.200.ISBN9812381619.

9. TynMyintU.,LokenathDebnath(2007).LinearPartialDifferentialEquationsforScientistsAndEngineers(http://books.google.com/books?id=Zbz5_UvERIIC&pg=PA4&dq=%22It+was+the+work+of+Augustin+Cauchy%22&hl=en&sa=X&ei=RnW6T52LNovYiQLa9mABw&ved=0CDgQ6AEwAA#v=onepage&q=%22It%20was%20the%20work%20of%20Augustin%20Cauchy%22&f=false)(4thed.).Springer.p.4.ISBN0817643931.

10. LokenathDebnath,DambaruBhatta(2007).IntegralTransformsAndTheirApplications(http://books.google.com/books?id=WbZcqdvCEfwC&pg=PA2&dq=%22It+was+the+work+of+Cauchy+that+contained%22&hl=en&sa=X&ei=Jym9T8LNK6OigKm_GYDg&ved=0CDQQ6AEwAA#v=onepage&q=%22It%20was%20the%20work%20of%20Cauchy%20that%20contained%22&f=false)(2nded.).CRCPress.p.2.ISBN1584885750.

11. IvorGrattanGuinness(2009).ConvolutionsinFrenchMathematics,18001840:FromtheCalculusandMechanicstoMathematicalAnalysisandMathematicalPhysics,Volume2(http://books.google.com/books?id=_GgioErrbW8C&pg=PA653&dq=%22Further,+in+a+double+integral%22&hl=en&sa=X&ei=4gC9T7KVDvDRiALqdTLDQ&ved=0CDgQ6AEwAA#v=onepage&q=%22Further%2C%20in%20a%20double%20integral%22&f=false).Birkhuser.p.653.ISBN3764322381.

12. See,forexample,Desintgralesdoublesquiseprsententsousuneformeindtermine(http://gallica.bnf.fr/ark:/12148/bpt6k90181x/f387)

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(http://gallica.bnf.fr/ark:/12148/bpt6k90181x/f387)13. DragiaMitrovi,Darkoubrini(1998).FundamentalsofAppliedFunctionalAnalysis:Distributions,Sobolev

Spaces(http://books.google.com/books?id=Od5BxTEN0VsC&pg=PA62&dq=%22greatest+drawback+of+the+classical+Fourier+transformation+is+a+rather+narrow+class+of+functions%22&hl=en&sa=X&ei=IKG6T_niFqWfiQLJoODdBg&ved=0CDQQ6AEwAA#v=onepage&q=%22greatest%20drawback%20of%20the%20classical%20Fourier%20transformation%20is%20a%20rather%20narrow%20class%20of%20functions%22&f=false).CRCPress.p.62.ISBN0582246946.

14. ManfredKracht,ErwinKreyszig(1989)."Onsingularintegraloperatorsandgeneralizations".InThemistoclesM.Rassias,ed.TopicsinMathematicalAnalysis:AVolumeDedicatedtotheMemoryofA.L.Cauchy(http://books.google.com/books?id=xIsPrSiDlZIC&pg=PA553&dq=%22To+this+theory%22+%22and+even+more%22++%22that+one+was+able+to+generalize%22&hl=en&sa=X&ei=RJ66Ty7JOLjiAKuoeSUBw&ved=0CDQQ6AEwAA#v=onepage&q=%22To%20this%20theory%22%20%22and%20even%20more%22%20%20%22that%20one%20was%20able%20to%20generalize%22&f=false).WorldScientific.p.553.ISBN9971506661.

15. Laugwitz1989,p.23016. AmorecompletehistoricalaccountcanbefoundinvanderPol&Bremmer1987,V.4.17. Dirac1958,1518. Gel'fand&Shilov1968,VolumeI,1.1,p.119. Rudin1966,1.2020. Hewitt&Stromberg1963,19.6121. Driggers2003,p.2321.SeealsoBracewell1986,Chapter5foradifferentinterpretation.Otherconventionsforthe

assigningthevalueoftheHeavisidefunctionatzeroexist,andsomeofthesearenotconsistentwithwhatfollows.22. Hewitt&Stromberg1965,9.1923. Strichartz1994,2.224. Hrmander1983,Theorem2.1.525. Hrmander1983,3.126. Strichartz1994,2.3Hrmander1983,8.227. Dieudonn1972,17.3.328. Federer1969,2.5.1929. Strichartz1994,Problem2.6.230. Vladimirov1971,Chapter2,Example3(d)31. Weisstein,EricW.,"SiftingProperty"(http://mathworld.wolfram.com/SiftingProperty.html),MathWorld.32. Gel'fand&Shilov19661968,Vol.1,II.2.533. Furtherrefinementispossible,namelytosubmersions,althoughtheserequireamoreinvolvedchangeofvariables

formula.34. Hrmander1983,6.135. Lange2012,pp.293036. GelfandShilov,p.21237. InsomeconventionsfortheFouriertransform.38. Bracewell198639. Gel'fand&Shilov1966,p.2640. Gel'fand&Shilov1966,2.141. Weisstein,EricW.,"DoubletFunction"(http://mathworld.wolfram.com/DoubletFunction.html),MathWorld.42. Thepropertyfollowsbyapplyingatestfunctionandintegrationbyparts.43. Hrmander1983,p.5644. Hrmander1983,p.56Rudin1991,Theorem6.2545. SteinWeiss,Theorem1.1846. Rudin1991,II.6.3147. Moregenerally,oneonlyneeds=1tohaveanintegrableradiallysymmetricdecreasingrearrangement.48. Saichev&Woyczyski1997,1.1The"deltafunction"asviewedbyaphysicistandanengineer,p.349. Stein&Weiss1971,I.150. Valle&Soares2004,7.251. Hrmander1983,7.852. SeealsoCourant&Hilbert1962,14.53. Lang1997,p.31254. IntheterminologyofLang(1997),theFejrkernelisaDiracsequence,whereastheDirichletkernelisnot.

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54. IntheterminologyofLang(1997),theFejrkernelisaDiracsequence,whereastheDirichletkernelisnot.55. Thedevelopmentofthissectioninbraketnotationisfoundin(Levin2002,Coordinatespacewavefunctionsand

completeness,pp.=109ff)56. Davis&Thomson2000,Perfectoperators,p.34457. Davis&Thomson2000,Equation8.9.11,p.34458. delaMadrid,Bohm&Gadella200259. SeeLaugwitz(1989).60. Crdoba1988Hrmander1983,7.261. Vladimirov1971,5.762. Hartmann1997,pp.15415563. Isham1995,6.2

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Externallinks

Hazewinkel,Michiel,ed.(2001),"Deltafunction"(http://www.encyclopediaofmath.org/index.php?title=p/d030950),EncyclopediaofMathematics,Springer,ISBN9781556080104KhanAcademy.orgvideolesson(http://www.khanacademy.org/video/diracdeltafunction)TheDiracDeltafunction(http://www.physicsforums.com/showthread.php?t=73447),atutorialontheDiracdeltafunction.VideoLecturesLecture23(http://ocw.mit.edu/courses/mathematics/1803differentialequationsspring2010/videolectures/lecture23usewithimpulseinputs),alecturebyArthurMattuck.DiracDeltaFunction(http://planetmath.org/encyclopedia/DiracDeltaFunction.html)onPlanetMathTheDiracdeltameasureisahyperfunction(http://www.osakakyoiku.ac.jp/~ashino/pdf/chinaproceedings.pdf)WeshowtheexistenceofauniquesolutionandanalyzeafiniteelementapproximationwhenthesourcetermisaDiracdeltameasure(http://www.ingmat.udec.cl/~rodolfo/Papers/BGR3.pdf)NonLebesguemeasuresonR.LebesgueStieltjesmeasure,Diracdeltameasure.(http://www.mathematik.unimuenchen.de/~lerdos/WS04/FA/content.html)

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