### 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.

### Different kinds of Fourier transform

Periodic functions and functions defined on . Let be a function such that for every . Then the th *Fourier coefficient* is given by the formula . The function is called the Fourier transform of . Periodic functions are naturally thought of as functions defined on the circle. If we write for the unit circle and have a function , then the formula for becomes .

In the other direction, let be a function from to . We can create a periodic function by defining it to equal . Under some circumstances, and with suitable notions of convergence, one can show that this inverts the previous operation: that is, the sum converges to the function . If we express as a function defined on , then this says that we can write as a doubly infinite power series , defined when .

Functions defined on the group of integers mod . Let be a function from to . Write for . Then the *discrete Fourier transform* of is the function given by the formula

Functions defined from to .

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.

## Comments

## Fourier transforms of measures

Wed, 22/04/2009 - 21:27 — ioannis.parissisWe 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 . I guess some special section should be devoted to measures supported on sub-manifolds of and there should be a chain that connects to curvature and oscillatory integrals. Also a 'Parent' of this article should be ' estimates' but I don't know if there is such an article yet.

yannis

## I've added something about

Wed, 22/04/2009 - 22:32 — gowersI'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.