The fourier transform

Abstract

The Fourier transform is one of the main tools for analyzing functions in L2(ℝ). It appears in all contexts where one wants to extract the frequencies appearing in a given signal. The definition and main properties of the Fourier transform of functions in L1(ℝ)are considered in Section 7.1. An extension of the Fourier transform to a unitary operator on L2(ℝ) is discussed in Section 7.2. Convolution and its interplay with the Fourier transform is described in Section 7.3. Section 7.4 introduces the sampling problem and the Paley–Wiener space. In particular, it is shown how to recover arbitrary functions in the Paley–Wiener space based on their function values on the discrete set Z. Finally, we relate the Fourier transform to the discrete Fourier transform in Section 7.5.