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 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 L¹(G), i.e. an absolutely integrable function on G. Then the Fourier transform of f is a function on Γ defined by

\hat{f}(\gamma) = \int_G f(x)\, \gamma(-x) \, d\mu

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).

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