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.
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 . In this case, the Fourier transform takes a function defined on to another function defined on by the formula . (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 this can be inverted as follows: .
Functions defined on finite Abelian groups. The Fourier transform for functions defined on is a special case of an important abstract definition of the Fourier function for functions defined on a finite Abelian group . A character on is defined to be a homomorphism from 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 ). It can be shown that the characters on form an orthonormal basis for the space of functions from to with the inner product . The Fourier expansion of a function is just its expansion in this basis. More concretely, if and is a character, then is defined to be , and we then have the inversion formula , where the sum is over all characters. This gives us the definition we had earlier for functions defined on , except that we have to identify the character with the element .
Functions defined on locally compact Abelian groups. As the examples of and show, an Abelian group does not have to be finite for it to be possible to define a Fourier transform for functions from to . Indeed, we can use more or less the same definition and try to expand a function in terms of characters. However, when 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 are precisely the functions with , and that explains in an abstract way the definition of the Fourier transform for functions defined on .
There is some subtlety about what it means to decompose a function into characters, as the example of functions defined on shows. There, the characters are functions of the form with . In a sense, we write as a linear combination of characters, but the "linear combination" is an integral rather than a sum.
Fourier transforms of generalized functions. The Dirac delta function is an example of an object that is not in fact a function, but which has a Fourier transform: for every , so its Fourier transform is a constant function. The inverse Fourier transform of the constant function is , which can be interpreted as when and when , so we get something delta-function like. This vague idea can be made rigorous using the theory of distributions. In general, there are many objects, such as distributions and measures, for which one can usefully define Fourier transforms.
Basic facts about the Fourier transform
To be included: Parseval/Plancherel identity, inversion formulae, convolution identities.