Transcription of Chapter 1 The Fourier Transform
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Chapter 1 The Fourier Transform
Chapter 1 The Fourier Fourier transforms as integralsThere are several ways to define the Fourier Transform of a functionf:R C. In this section, we define it using an integral representation and statesome basic uniqueness and inversion properties, without proof. Thereafter,we will consider the Transform as being defined as a suitable limit of Fourierseries, and will prove the results stated 1 Letf:R R. The Fourier Transform off L1(R), denotedbyF[f](.), is given by the integral:F[f](x) :=1 2 f(t) exp( ixt)dtforx Rfor which the integral exists. We have theDirichlet conditionfor inversion of Fourier 1 Letf:R R.
discontinuity, just as for Fourier series. 2. A truncated cosine wave. f(t) = 8 <: cos3t if ˇ<t<ˇ 1 2 if t= ˇ 0 otherwise: Then, since the cosine is an even function, we have f^( ) = p 2ˇF[f]( ) = Z 1 1 f(t)e i tdt= Z ˇ ˇ cos(3t)cos( t)dt = 2 sin( ˇ) 9 2: 5
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