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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.

Chapter 1 The Fourier Transform 1.1 Fourier transforms as integrals There are several ways to de ne the Fourier transform of a function f: R ! C. In this section, we de ne it using an integral representation and state

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Transcription of Chapter 1 The Fourier Transform

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