Fourier-Transform

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1、FouriertransformmiscellanyP.B.Kronheimer,forMath114October6,2010ThesenotescoverthematerialthatwetreatedinclassontheFouriertransform,upuntilthetimethatweembarkedontheresultsaboutFourierinversion.TheycoverthefirstbasicpropertiesoftheFouriertransform,andthefirstexamples.Therewillbeanotherhandout

2、soonontheinversiontheorem.Letf2L1(R)(thespaceofcomplex-valued,integrablefunctionsonR).TheFouriertransformoffisthefunctionfOdefinedbyZfO()De2ixf(x)dx:TheintegrandbelongstoL1intheabovedefinition,becauseitsnormisthesameasthatoff.Sothedefinitionmakessense.Example.IffD[1;1],thenZ1fO()De2ix

3、dx112ix1De2ixD1sin(2)D:Thefunctionsin(2)=()isnotintegrable:theFouriertransformofanL1functionisnotL1ingeneral.21.Firstproperties1.FirstpropertiesInthissection,wewrite(asiscommon)“thefunctionf(x)”whenwemean“thefunctionx7!f(x).WewritefOfortheFouriertransformoff.Throughout,

4、weassumethatf2L1(R).2icfO().Property1.1.TheFouriertransformoff(xCc)iseProof.WritedowntheFouriertransformasZ2ixef(xCc)dxandsubstituteyDxCc.Property1.2.TheFouriertransformoff(Ax)is(1=jAj)fO(A1).Proof.WritedowntheFouriertransformasZ2ixef(Ax)dxandsubstituteyDAx.Property1.3.Forall,we

5、havejfO()jkfkL1.Proof.InthedefinitionoffO(),theabsolutevalueoftheintegrandisjfj,soZjfO()jjfj:1Property1.4.Foranyf2L,theFouriertransformfiscontinuous.OProof.ThisfollowsfromtheDominatedConvergenceTheorem.1Property1.5.IffandgareinL,thentheFouriertransformoftheconvolutionfgisfOg.OProof.Thi

6、sappearedontheproblemsetsasanapplicationofFubini’stheorem.AlthoughfOisnotnecessarilyintegrable,itisalwaysthecasethatfO()!0asjj!1.ThisistheRiemann-Lebesguelemma:1Property1.6.Foranyf2L,theFouriertransformfO()tendstozeroasjj!1.3Proof.DothisfirstbyexplicitcalculationforthecasethatfDIforsome

7、intervalIinR.Deducethatitholdsforanystep-functionf.Forageneralintegrablef,let>0begiven,andleththenbeastepfunctionwithkfhkL1.Sincewealreadyknowaboutstep-functions,weknowthereexistsRsuchthatjjRD)jhO()j:NowuseProperty1.3,appliedtothefunctionfhtose