Fourier Analysis

Abstract

Fourier analysis deals with the representation of functions as superpositions of plane waves, of spatial or spatial-temporal nature. It plays in many respects a decisive role in this work. The Schrödinger equation of a free particle is a wave equation whose solutions are superpositions of such plane waves with a particular dispersion relation. The abstract framework of quantum mechanics is reflected in this picture and can be motivated and derived from it. Fourier analysis plays moreover an extraordinarily important role in the mathematical analysis of partial differential equations like the Schrödinger equation and is basic for our considerations. We begin therefore with an elementary introduction to Fourier analysis. We start as usual from the Fourier transformation of rapidly decreasing functions that is then extended to integrable and square integrable functions. The third section of this chapter is devoted to the concept of weak derivative and its relation to Fourier analysis. We introduce rather general L₂-based spaces of weakly differentiable functions that include the usual isotropic Sobolev spaces but also spaces of functions with L₂-bounded mixed derivatives. Much more information on Fourier analysis can be found in monographs like [70] or [77], and on function spaces in [2, 85, 99].