demidov,generalized functions in mathematical physics

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1、Generalizedfunctionsinmathematicalphysics.MainideasandconceptsA.S.DemidovContentsPrefaceviiNotationxiChapter1.Introductiontoproblemsofmathematicalphysics11.Temperatureatapoint?No!Involumescontractingtothepoint12.Thenotionofδ-sequenceandδ-function43.Somespacesofsmoothfu

2、nctions.Partitionofunity64.Examplesofδ-sequences105.OntheLaplaceequation116.Ontheheatequation197.TheOstrogradsky–Gaussformula.TheGreenformulaeandtheGreenfunction288.TheLebesgueintegral1)319.ThespacesLpandLp40loc10.FunctionsofL1aslinearfunctionalonC∞45loc011.Simplesthyp

3、erbolicequations.GeneralizedSobolevsolutions46Chapter2.ThespacesD[,D#andD0.Elementsofthedistributiontheory(generalizedfunctioninthesenseofL.Schwartz)5912.ThespaceD[oftheSobolevderivatives5913.ThespaceD#ofgeneralizedfunctions6314.Theproblemofregularization6615.Generaliz

4、edfunctionswithapointsupport.TheBoreltheorem6816.ThespaceD0ofgeneralizedfunctions(distributionsbyL.Schwartz)72vviCONTENTSChapter3.ThespacesHs.Pseudodifferentialoperators8117.TheFourierseriesandtheFouriertransform.ThespacesSandS08118.TheFourier–Laplacetransform.ThePaley–

5、Wienertheorem9419.Fundamentalsolutions.Convolution9920.OnspacesHs10221.Onpseudodifferentialoperators(PDO)10622.Onellipticproblems111Addendum.AddendumAnewapproachtothetheoryofgeneralizedfunctions(Yu.V.Egorov)129Bibliography137Index141PrefacePresently,thenotionoffunctioni

6、snotasfinallycrystallizedanddefinitelyestablishedasitseemedattheendofthe19thcen-tury;onecansaythatatpresentthisnotionisstillinevolution,andthatthedisputeconcerningthevibratingstringisstillgoingononly,ofcourse,indifferentscientificcircumstances,involvingotherpersonalitiesan

7、dusingotherterms.LuzinN.N.(1935)[42]Itissymbolicthatinthatsameyearof1935,S.L.Sobolev,whowas26yearsoldthattime,submittedtotheeditorialboardofthejournal“Matematicheskiysbornik”hisfamouswork[61]andpub-lishedatthesametimeitsbriefversionin“DokladyANSSSR”[60].Thisworklaidfou

8、ndationsofacompletelynewoutlookonthecon-ceptoffunction,unexpectedevenforN.N.Luzin—theconceptofageneralizedfunction(in