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1、Math362:RealandAbstractAnalysisCollegeoftheHolyCross,Spring2005FourierTransformsGivenafunctionf:Rn!R,itsFouriertransformisthefunctionZf?(?)=f(x)e?ix￠?dxRnanditsinverseFouriertransformisthefunctionZfˇ(x)=1f(x)eix￠?d?(2?)nRnThoughtofasanoperator,theFouriertransformisdenotedbyFandtheinverse
2、FouriertransformbyF?1.Thatis,F(f)=f?andF?1(f)=fˇ.Itshouldbenotedthatitisnotatallobviousthatthesecondformulareallyistheinverseofthe?rst.Beforeprovingthis,wewilllookatsomeofthebasicpropertiesoftheFouriertransform.Itishelpfulto?rstworkwithinaspecialclassoffunctionscalledtheSchwartzclass.Sch
3、wartzClassDe?nition1.AfunctionfissaidtoberapidlydecreasingifforeveryintegerN?0thereexistsaconstantCNsuchthatNjxjjf(x)j·CNforallx2Rn.De?nition2.TheSchwartzclassSisthesetofallfunctionsf2C1(Rn)suchthatfandallofitsderivativesarerapidlydecreasing.ItiseasytoseethattheSchwartzclassisclosedunder
4、di?erentiationandmultiplicationbypolynomials.Also,sincefunctionsinSareboundedanddecayfasterthananypolynomialasjxj!1,itfollowsthatSchwartzclassfunctionsareintegrable,andthereforeitmakessensetotaketheirFouriertransform.Example1.Fora>0,f(x)=e?ajxj2isinS.Indimensionn=1,itsFouriertransformisZ
5、1Z1f?(?)=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?1r???2=4a=eaThefollowingtheoremisthemostimportantalgebraicpropertyofFouriertransforms.1Theorem1.Iff2Sthenf?2Sandfcxk=i?kf?xdkf=if??kfor1·k·n.Sodi?erentiationoffcorrespondstomultiplicationoff?by
6、apolynomial,andconverselymultiplicationoffbyapolynomialcorrespondstodi?erentiationoff?.Proof.BytheintegrabilityofSchwartzclassfunctions,thefollowingcalculationsarejusti-?ed.IntegrationbypartsinxkgivesZZfc(?)=f(x)e?ix￠?dx=i?f(x)e?ix￠?dx=i?f?(?)xkxkkkRnRnwhiledi?erentiationwithrespectto?kg
7、ivesZf?(?)=?ixf(x)e?ix￠?dx=?ixdf(?)?kkkRnThisprovesthetwoformulas.Toprovethatf2Simpliesf?2S,?rstnoticethatf2Simpliesf?isbounded,sincefisintegrable.Next,since?f?=?ifckxkandsincefxk2Sitfollowsthat?kf?isalsobounded.ByinductionitfollowsthatNj?jjf?(?)jisboundedforanypositivein
