Fourier transforms of integrable functions defined on the whole real line \(\mathbb R\) are studied in Chap. 2. First, in Sect. 2.1, the Fourier transform is defined on the Banach space \(L_1(\mathbb R)\). The main properties of the Fourier transform are handled, such as the Fourier inversion formula and the convolution property. Then, in Sect. 2.2, the Fourier transform is introduced as a bijective mapping of the Hilbert space \(L_2(\mathbb R)\) onto itself by the theorem of Plancherel. The Hermite functions, which form an orthogonal basis of \(L_2(\mathbb R)\), are eigenfunctions of the Fourier transform. In Sect. 2.3, we present the Poisson summation formula and Shannon’s sampling theorem. Finally, two generalizations of the Fourier transform are sketched in Sect. 2.5, namely the windowed Fourier transform and the fractional Fourier transform.
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