Transcription of Fourier transform techniques 1 The Fourier transform
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Fourier transform techniques 1 The Fourier transform
Fourier transform techniques1 The Fourier transformRecall for a functionf(x) : [ L,L] C, we have the orthogonal expansionf(x) = n= cnein x/L, cn=12L L Lf(y)e in y/Ldy.(1)We think ofcnas representing the amount of a particular eigenfunction with wavenumberkn=n /Lpresent in the functionf(x). So what ifLgoes to ? Notice that the allowedwavenumbers become more and more dense. Therefore, whenL= , we expectf(x)is a su-perposition of an uncountable number of waves corresponding to every wavenumberk R,which can be accomplished by writingf(x)as a integral overkinstead of a sum let s take the limitL formally. Settingkn=n /Land k= /Land using (1), onecan writef(x) =12 n= ( L Lf(y)e iknydy)eiknx this is a Riemann sum for an integral on the intervalk ( , ). TakingL is thesame as taking k 0, which givesf(x) =12 F(k)eikxdk,(2)whereF(k) = f(x)e ikxdx.(3)The functionF(k)is theFourier transformoff(x).
Most of these result from using elementary calculus techniques for the integrals (3) and (2), although a couple require techniques from complex analysis. 1. A Brief table of Fourier transforms ... This is just an algebraic equation whose solution is u^(k) = f^(k) 1 + k2: (9)
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