Defining the Fourier transform on LCA groups

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 I. If G is an LCA group, then so is I.

Integration

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 I1/4, 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 L^1(G), i.e. an absolutely integrable function on G. Then the Fourier transform of f is a function on I 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 I with the character that takes x to the value exp(iIx).

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 I, i.e. whether to divide by or multiply by a(2I), and in the correspondence between frequencies and characters, whether I corresponds to exp(iIx) or exp(2IiIx).