notes15 Fourier Transforms .pdf

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1、Math362:RealandAbstractAnalysisCollegeoftheHolyCross,Spring2005FourierTransformsGivenafunctionf:Rn!R,itsFouriertransformisthefunctionZfˆ(»)=f(x)e¡ix¢»dxRnanditsinverseFouriertransformisthefunctionZfˇ(x)=1f(x)eix¢»d»(2¼)nRnThoughtofasanoperator,theFouriertransformisdenotedbyFandtheinverse

2、FouriertransformbyF¡1.Thatis,F(f)=fˆandF¡1(f)=fˇ.Itshouldbenotedthatitisnotatallobviousthatthesecondformulareallyistheinverseofthefirst.Beforeprovingthis,wewilllookatsomeofthebasicpropertiesoftheFouriertransform.ItishelpfultofirstworkwithinaspecialclassoffunctionscalledtheSchwartzclass.Sch

3、wartzClassDefinition1.AfunctionfissaidtoberapidlydecreasingifforeveryintegerN¸0thereexistsaconstantCNsuchthatNjxjjf(x)j·CNforallx2Rn.Definition2.TheSchwartzclassSisthesetofallfunctionsf2C1(Rn)suchthatfandallofitsderivativesarerapidlydecreasing.ItiseasytoseethattheSchwartzclassisclosedunder

4、differentiationandmultiplicationbypolynomials.Also,sincefunctionsinSareboundedanddecayfasterthananypolynomialasjxj!1,itfollowsthatSchwartzclassfunctionsareintegrable,andthereforeitmakessensetotaketheirFouriertransform.Example1.Fora>0,f(x)=e¡ajxj2isinS.Indimensionn=1,itsFouriertransformisZ

5、1Z1fˆ(»)=e¡ax2e¡ix»dx=e¡a[(x+i»=2a)2+»2=4a2]dx¡1¡1Z1Z1=e¡»2=4ae¡a(x+i»=2a)2dx=e¡»2=4ae¡ax2dx¡1¡1r¼¡»2=4a=eaThefollowingtheoremisthemostimportantalgebraicpropertyofFouriertransforms.1Theorem1.Iff2Sthenfˆ2Sandfcxk=i»kfˆxdkf=ifˆ»kfor1·k·n.Sodifferentiationoffcorrespondstomultiplicationoffˆby

6、apolynomial,andconverselymultiplicationoffbyapolynomialcorrespondstodifferentiationoffˆ.Proof.BytheintegrabilityofSchwartzclassfunctions,thefollowingcalculationsarejusti-fied.IntegrationbypartsinxkgivesZZfc(»)=f(x)e¡ix¢»dx=i»f(x)e¡ix¢»dx=i»fˆ(»)xkxkkkRnRnwhiledifferentiationwithrespectto»kg

7、ivesZfˆ(»)=¡ixf(x)e¡ix¢»dx=¡ixdf(»)»kkkRnThisprovesthetwoformulas.Toprovethatf2Simpliesfˆ2S,firstnoticethatf2Simpliesfˆisbounded,sincefisintegrable.Next,since»fˆ=¡ifckxkandsincefxk2Sitfollowsthat»kfˆisalsobounded.ByinductionitfollowsthatNj»jjfˆ(»)jisboundedforanypositivein