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. Suppose that (1) |f|dtconverges and (2)in any finite interval,f,f are piecewise continuous with at most finitely manymaxima/minima/discontinuities.
1 1 jf^( )j2d : (1.2.3) Expression (1.2.2) is called the Fourier integral or Fourier transform of f. Expression (1.2.1) is called the inverse Fourier integral for f. The Plancherel identity suggests that the Fourier transform is a one-to-one norm preserving map of the Hilbert space L2[1 ;1] onto itself (or to another copy of it-self).
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