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.

## 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** 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 *L*¹(*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(*i*ω*x*).

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(±*i*ω*x*) or exp(±2π*i*ω*x*).