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[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.

[note article incomplete] This article is incomplete in obvious ways. [/note]

===Different kinds of Fourier transform===

[add]'''Periodic functions and functions defined on $\Z$.'''{{ Let $f:\R\rightarrow\C$ 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 $\hat{f}:\Z\rightarrow\C$ 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 $\{z\in\C:|z|=1\}$ and have a function $f:\mathbb{T}\rightarrow\C$, then the formula for $\hat{f}(n)$ becomes $\int_0^{2\pi}f(e^{i\theta})e^{-in\theta}d\theta$. <br /><br />
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 $|z|=1$.}}[/add]

[add]'''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 $\hat{f}:\Z/N\Z\rightarrow\C$ given by the formula [maths]\hat{f}(r)=N^{-1}\sum_xf(x)e_N(-rx).[/maths]
(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: $f(x)=\sum_r\hat{f}(r)e_n(rx)$.}}[/add]

[add]'''Functions defined from $\R$ to $\R$.''' {{In this case, the Fourier transform takes a function $f$ defined on $\R$ to another function $\hat{f}$ defined on $\R$ by the formula $\hat{f}(\alpha)=\frac 1{\sqrt{2\pi}}\int_{-\infty}^\infty f(x)e^{-i\alpha x}dx$. (Once again, there are several other conventions for the precise definition, which differ in inessential ways from the one we have chosen here.) For suitable functions $f$ this can be inverted as follows: $f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty\hat{f}(\alpha)e^{i\alpha x}d\alpha$.}}[/add]

[add]'''Functions defined on finite Abelian groups.''' {{The Fourier transform for functions defined on $\Z/N\Z$ is a special case of an important abstract definition of the Fourier function for functions defined on a finite Abelian group $G$. A ''character'' on $G$ is defined to be a homomorphism from $G$ to the multiplicative group of non-zero complex numbers (which has to take values of modulus 1, so it can also be defined as a homomorphism to the unit circle in $\C$). It can be shown that the characters on $G$ form an [[w:orthonormal basis]] for the space $L^2(G)$ of functions from $G$ to $\C$ with the inner product $\langle \phi,\psi\rangle=|G|^{-1}\sum_{g\in G}\phi(g)\overline{\psi(g)}$. The Fourier expansion of a function is just its expansion in this basis. More concretely, if $f:G\rightarrow\C$ and $\gamma$ is a character, then $\hat{f}(\gamma)$ is defined to be $\langle f,\gamma\rangle$, and we then have the inversion formula $f=\sum_\gamma\langle f,\gamma\rangle \gamma$, where the sum is over all characters. This gives us the definition we had earlier for functions defined on $\Z/N\Z$, except that we have to identify the character $\gamma_r(x)=e_N(rx)$ with the element $r\in\Z/N\Z$.}}[/add]

[add]'''Functions defined on locally compact Abelian groups.''' {{As the examples of $\mathbb{T}$ and $\R$ show, an Abelian group $G$ does not have to be finite for it to be possible to define a Fourier transform for functions from $G$ to $\C$. Indeed, we can use more or less the same definition and try to expand a function in terms of characters. However, when $G$ is infinite, one does not normally take ''arbitrary'' characters: rather, the group usually has a topological structure and one asks for characters that are ''continuous''. For instance, the continuous characters defined on $\mathbb{T}$ are precisely the functions $z\mapsto z^n$ with $n\in\Z$, and that explains in an abstract way the definition of the Fourier transform for functions defined on $\mathbb{T}$. <br /> <br />
There is some subtlety about what it means to decompose a function into characters, as the example of functions defined on $\R$ shows. There, the characters are functions of the form $x\mapsto e^{i\alpha x}$ with $\alpha\in\R$. In a sense, we write $f$ as a linear combination of characters, but the "linear combination" is an integral rather than a sum.}} [/add]

[add]'''Fourier transforms of generalized functions.''' {{The [[w:Dirac delta function]] is an example of an object that is not in fact a function, but which has a Fourier transform: $\frac 1{\sqrt{2\pi}}\int_{-\infty}^\infty\delta(x)e^{-i\alpha x}dx=\frac 1{\sqrt{2\pi}}$ for every $\alpha$, so its Fourier transform is a constant function. The inverse Fourier transform of the constant function $1/\sqrt{2\pi}$ is $\int_{-\infty}^\infty e^{-i\alpha x}d\alpha$, which can be interpreted as $0$ when $\alpha\ne 0$ and $\infty$ when $\alpha=0$, so we get something delta-function like. This vague idea can be made rigorous using the theory of [[w:Distribution (mathematics)|distributions]]. In general, there are many objects, such as distributions and measures, for which one can usefully define Fourier transforms.}} [/add]

===Basic facts about the Fourier transform===

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

===Articles about the use of the Fourier transform===

* [[Use Fourier identities|Using Fourier identities to estimate integrals]]
* [[If your problem can be expressed in terms of convolutions and inner products then take the Fourier transform]]
* [[Use Fourier expansion to eliminate nasty cutoffs]]

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