The Fourier–Laplace transformation and material law operators

Abstract

In this chapter we introduce the Fourier–Laplace transformation and use it to define operator-valued functions of ∂t,ν; the so-called material law operators. These operators will play a crucial role when we deal with partial differential equations. In the equations of classical mathematical physics, like the heat equation, wave equation or Maxwell’s equation, the involved material parameters, such as heat conductivity or permeability of the underlying medium, are incorporated within these operators. Hence, these operators are also called “material law operators”. We start our chapter by defining the Fourier transformation and proving Plancherel’s theorem in the Hilbert space-valued case, which states that the Fourier transformation defines a unitary operator on L2(ℝ; H).