In mathematics, the Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups.
Contents
Definitions
The Fourier transform of a function
For each representation
Let
The inverse Fourier transform at an element
Transform of a convolution
The convolution of two functions
The Fourier transform of a convolution at any representation
Plancherel formula
For functions
where
Fourier transform on finite abelian groups
Since the irreducible representations of finite abelian groups are all of degree 1 and hence equal to the irreducible characters of the group, Fourier analysis on finite abelian groups is significantly simplified. For instance, the Fourier transform yields a scalar- and not matrix-valued function.
Furthermore, the irreducible characters of a group may be put in one-to-one correspondence with the elements of the group.
Let
Note that the right-hand side is simply
The inverse Fourier transform is then given by
Since
and the inverse is given by
Note that the inverse fourier transform is technically an element of
A property that is often useful in probability is that the Fourier transform of the uniform distribution is simply
Relationship with representation theory
There is a direct relationship between the Fourier transform on finite groups and the representation theory of finite groups. The set of complex valued functions on a finite group,
Applications
This generalization of the discrete Fourier transform is used in numerical analysis. A circulant matrix is a matrix where every column is a cyclic shift of the previous one. Circulant matrices can be diagonalized quickly using the fast Fourier transform, and this yields a fast method for solving systems of linear equations with circulant matrices. Similarly, the Fourier transform on arbitrary groups can be used to give fast algorithms for matrices with other symmetries (Åhlander & Munthe-Kaas 2005). These algorithms can be used for the construction of numerical methods for solving partial differential equations that preserve the symmetries of the equations (Munthe-Kaas 2006).