Revision of Fourier transforms front page from Wed, 22/04/2009 - 09:41

Quick description

The Fourier transform is a fundamental tool in many parts of mathematics. This is even more so when one looks at various natural generalizations of it. This article contains brief descriptions of the Fourier transform in various contexts and links to articles about its use.

Prerequisites

Basic analysis, complex numbers.

This article is incomplete. This article is incomplete in obvious ways.

Different kinds of Fourier transform

Periodic functions and functions defined on $\Z$ . Let $\R\rightarrow\C$ be a function such that $f(x+2\pi)=f(x)$ for every $x$ . Then the $n$ th Fourier coefficient $\hat{f}(n)$ is given by the formula $\hat{f}(n)=\int_0^{2\pi}f(t)e^{-int}dt$ . The function $\Z\rightarrow\C$ is called the Fourier transform of $f$ . Periodic functions are naturally thought of as functions defined on the circle. If we write $\mathbb{T}$ for the unit circle $|z|=1\}$ and have a function $\mathbb{T}\rightarrow\C$ , then the formula for $\hat{f}(n)$ becomes $\int_0^{2\pi}f(e^{i\theta})e^{-in\theta}d\theta$ .

In the other direction, let $g$ be a function from $\Z$ to $\C$ . We can create a periodic function $f(x)$ by defining it to equal $\sum_{n=-\infty}^\infty g(n)e^{inx}$ . Under some circumstances, and with suitable notions of convergence, one can show that this inverts the previous operation: that is, the sum $\sum_{n=-\infty}^\infty\hat{f}(n)e^{inx}$ converges to the function $f(x)$ . If we express $f$ as a function defined on $\mathbb{T}$ , then this says that we can write $f$ as a doubly infinite power series $f(z)=\sum_{n=-\infty}^\infty \hat{f}(n)z^n$ , defined when $|z|=1$ .

Functions defined on the group $\Z/N\Z$ of integers mod $N$ . Let $f$ be a function from $\Z/N\Z$ to $\C$ . Write $e_N(x)$ for $e^{2\pi ix/N}$ . Then the discrete Fourier transform of $f$ is the function $\Z/N\Z\rightarrow\C$ given by the formula

$\hat{f}(r)=N^{-1}\sum_xf(x)e_N(-rx).$

(There are various alternative conventions for the precise definition here, but they all have the same important properties.) The discrete Fourier transform can be inverted as follows: $f(x)=\sum_r\hat{f}(r)e_n(rx)$ .

Functions defined from $\R$ to $\R$ .

Functions defined on finite Abelian groups.

Functions defined on locally compact Abelian groups.

Basic facts about the Fourier transform

To be included: Parseval/Plancherel identity, inversion formulae, convolution identities.

Articles about the use of the Fourier transform

Using Fourier identities to estimate integrals

If your problem can be expressed in terms of convolutions and inner products, then take the Fourier transform

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Comments

Fourier transforms of measures

Wed, 22/04/2009 - 22:27 — ioannis.parissis

We need a -Fourier transforms of measures and a -Fourier transforms of distributions or you think it should be under the same title? I believe that Fourier transforms of measures deserve a special heading here. But I don't know where to start really. Trying to define Fourier transforms of measures in full generality might be confusing. I would start on the real line or the circle to make things more concrete and simple. Then I would go on defining Fourier transforms of measures in the Euclidean space $\mathbb R^d$ . I guess some special section should be devoted to measures supported on sub-manifolds of $\mathbb R^d$ and there should be a chain that connects to curvature and oscillatory integrals. Also a 'Parent' of this article should be ' $L^2$ estimates' but I don't know if there is such an article yet.

yannis

I've added something about

Wed, 22/04/2009 - 23:32 — gowers

I've added something about generalized functions, but I'm very far from an expert, so feel free to change it if you don't like it. Actually, now that I've done it I'm starting to think that putting Fourier transforms of hypersurfaces together with Fourier transforms of distributions is not very natural at all, so probably some further work is needed.