Fourier transform on finite groups

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In mathematics, the Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups.

Definitions[edit]

The Fourier transform of a function at a representation of is

For each representation of , is a matrix, where is the degree of .

Let be a complete set of inequivalent irreducible representations of . Then the matrix entries of the are mutually orthogonal functions on .[1] Since the dimension of the transform space is equal to , it follows that .

The inverse Fourier transform at an element of is given by

Properties[edit]

Transform of a convolution[edit]

The convolution of two functions is defined as

The Fourier transform of a convolution at any representation of is given by

Plancherel formula[edit]

For functions , the Plancherel formula states

where are the irreducible representations of

Fourier transform on finite abelian groups[edit]

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.

Therefore, we may define the Fourier transform for finite abelian groups as

Note that the right-hand side is simply for the inner product on the vector space of functions from to defined by

The inverse Fourier transform is then given by

A property that is often useful in probability is that the Fourier transform of the uniform distribution is simply where 0 is the group identity and is the Kronecker delta.

Applications[edit]

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).

See also[edit]

References[edit]

  1. ^ Terras 1999, p. 251