Distributions

Abstract

The Fourier transform has been constructed for functions in L¹, L² and for measures. Among these, only the L¹ case was effortless. This begs the question of how to define the Fourier transform for other spaces, such as L^p(Rⁿ) and C(Rⁿ). In this chapter, a workable resolution to this issue will be presented. The space of tempered distributions is a class of objects on which differentiation and the Fourier transform are defined. An additional advantage is that all the various functions spaces Rⁿ mentioned above are easily seen as subsets of the space of tempered distributions. This creates a flexible language for doing calculations without the need for concern about where various intermediate objects lie. An illustration of the power in this point of view is offered in the last section where Sobolev spaces and the regularity of Poisson’s equation are discussed.