Chapter 8 - Fourier Analysis and Measures

Summary

In this chapter, we recall some basic facts from Fourier analysis which we need for our approach to diffraction theory in Chapter 9. Since Fourier series and their generalisations to almost periodic functions occur only occasionally below, we keep their exposition brief and informal (mostly without proofs, but with proper references). Subsequently, Fourier transforms of functions and measures are covered in more detail, including a brief introduction to volume-averaged (or Eberlein) convolutions.

Fourier series

A (possibly complex-valued) function f of one real variable is called periodic, if f (x + T) = f(x) holds for some T ≠ 0 and all x ∈ ℝ. Clearly, one then has f(x + nT) = f(x) for all n ∈ ℤ. Assuming that the periodic function f is not a constant function, the smallest T > 0 with this property is the fundamental period of f. It is simply called the period of f when misunderstandings are unlikely. Examples are provided by trigonometric functions, such as sin(x), cos(x) or e^ix, all three with fundamental period T = 2π. A natural way to look at such functions in a more general setting is to consider T-periodic functions that are locally integrable (over any compact set K ⊂ ℝ say), so that one can also view them as elements of the Banach space L¹([0, T]); see [DMcK72, Kat04, Pin02] for general background.