My previous post said that all the familiar variations on Fourier transforms—Fourier series analysis and synthesis, Fourier transforms on the real line, discrete Fourier transforms, etc.—can be unified into a single theory. They’re all instances of a Fourier transform on a locally compact Abelian (LCA) group. The difference between them is the underlying group.
Given an LCA group G, the Fourier transform takes a function on G and returns a function on the dual group of G. We said this much last time, but we didn’t define the dual group; we just stated examples. We also didn’t say just how you define a Fourier transform in this general setting.
Characters and dual groups
Before we can define a dual group, we have to define group homomorphisms. A homomorphism between two groups G and H is a function h between the groups that preserves the group structure. Suppose the group operation is denoted by addition on G and by multiplication on H (as it will be in our application), saying h preserves the group structure means
h(x + y) = h(x) h(y)
for all x and y in G.
Next, let T be the unit circle, i.e. complex numbers with absolute value 1. T is a group with respect to multiplication. (Why T for circle? This is a common notation, anticipating generalization to toruses in all dimensions. A circle is a one-dimensional torus.)
Now a character on G is a continuous homomorphism from G to T. The set of all characters on G is the dual group of G. Call this group Γ. If G is an LCA group, then so is Γ.
The classical Fourier transform is defined by an integral. To define the Fourier transform on a group we have to have a way to do integration on that group. And there’s a theorem that says we can always do that. For every LCA group, there exists a Haar measure μ, and this measure is nice enough to develop our theory. This measure is essentially unique: Any two Haar measures on the same LCA group must be proportional to each other. In other words, the measure is unique up to multiplying by a constant.
On a discrete group—for our purposes, think of the integers and the integers mod m—Haar measure is just counting; the measure of a set is the number of things in the set. And integration with respect to this measure is summation.
Fourier transform defined
Let f be a function in L¹(G), i.e. an absolutely integrable function on G. Then the Fourier transform of f is a function on Γ defined by
What does this have to do with the classical Fourier transform? The classical Fourier transform takes a function of time and returns a function of frequency. The correspondence between the classical Fourier transform and the abstract Fourier transform is to associate the frequency ω with the character that takes x to the value exp(iωx).
There are multiple slightly different conventions for the classical Fourier transform cataloged here. These correspond to different constant multiples in the choice of measure on G and Γ, i.e. whether to divide by or multiply by √(2π), and in the correspondence between frequencies and characters, whether ω corresponds to exp(±iωx) or exp(±2πiωx).