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 .
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.