Fourier and Laplace Transforms. Distributions

Abstract

The problems in this chapter concern Fourier and Laplace transforms with their different applications. The physical meaning of the Fourier transform as “frequency analysis” is carefully presented, also obtaining the classical “uncertainty principle”. Starting from the spaces \(L^1(\mathbf{R})\) and \(L^2(\mathbf{R})\), the Fourier transform is extended to the larger space of distributions \({\mathscr {S}}'\), which include the Dirac delta, the Cauchy principal part, and other related distributions. Applications concern ordinary and partial differential equations (in particular, the heat, d’Alembert, and Laplace equations, including a discussion about the uniqueness of solutions), and general linear systems. The important notion of Green function is considered in many details, together with the notion of causality, also in connection with analyticity property (with an introduction to the dispersion relations of Kramers–Kronig). Various examples and applications of Laplace transform are proposed, also in comparison with Fourier transform.