- Fourier transform on finite groups
mathematics, the Fourier transform on finite groups is a generalization of the discrete Fourier transformfrom cyclic to arbitrary finite groups.
The Fourier transform of a function at a representation of is
So for each representation of , is a matrix, where is the degree of .
Let be the irreducible representations of . Then the inverse Fourier transform at an element of is 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
This generalization of the discrete Fourier transform is used in
numerical analysis. A circulant matrixis a matrix where every column is a cyclic shiftof 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 harv|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 harv|Munthe-Kaas|2006.
Discrete Fourier transform
Representation theory of finite groups
* | year=2005 | journal=BIT | issn=0006-3835 | volume=45 | issue=4 | pages=819–850.
* Diaconis, P. (1988). "Group Representations in Probability and Statistics." Lecture Notes — Monograph Series, Vol. 11. Hayward, California: Institute of Mathematical Statistics.
* Diaconis, P. (1991). "Finite Fourier Methods: Access to Tools." In "Probabilistic Combinatorics and its Applications," Proceedings of Symposia in Applied Mathematics, Vol. 44. Bollobás, B., and Chung, F. R. K. (ed.).
* | year=2006 | journal=
Journal of Physics A| issn=0305-4470 | volume=39 | issue=19 | pages=5563–5584.
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