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In [[mathematics]], the '''Fourier transform on finite groups''' is a generalization of the [[discrete Fourier transform]] from [[cyclic group|cyclic]] to arbitrary [[finite group]]s.
 
==Definitions==
The '''Fourier transform''' of a function <math>f : G \rightarrow \mathbb{C}\,</math>
at a [[representation of a finite group|representation]] <math>\varrho : G \rightarrow GL(d_\varrho, \mathbb{C})\,</math> of <math>G\,</math> is
 
:<math>
\widehat{f}(\varrho) = \sum_{a \in G} f(a) \varrho(a).
</math>
 
So for each representation <math>\varrho\,</math> of <math>G\,</math>, <math>\widehat{f}(\varrho)\,</math> is a <math>d_\varrho \times d_\varrho\,</math> matrix, where <math>d_\varrho\,</math> is the degree of <math>\varrho\,</math>.
 
Let <math>\varrho_i\,</math> be a complete set of inequivalent [[irreducible representation]]s of <math>G</math>. Then<!--, somehow -->, <math>|G| = \sum_i d_{\varrho_i}^2</math>. Then the '''inverse Fourier transform''' at an element <math>a\,</math> of <math>G\,</math> is given by
 
:<math>
f(a) = \frac{1}{|G|} \sum_i d_{\varrho_i} \text{Tr}\left(\varrho_i(a^{-1})\widehat{f}(\varrho_i)\right),
</math>
 
where <math>d_{\varrho_i}\,</math> is the degree of the representation <math>\varrho_i.\,</math>
 
==Properties==
 
===Transform of a convolution===
The '''convolution''' of two functions <math>f, g : G \rightarrow \mathbb{C}\,</math> is defined as
 
:<math>
(f \ast g)(a) = \sum_{b \in G} f(ab^{-1}) g(b).
</math>
 
The Fourier transform of a convolution at any representation <math>\varrho\,</math> of <math>G\,</math> is given by
 
:<math>
\widehat{f \ast g}(\varrho) = \widehat{f}(\varrho)\widehat{g}(\varrho).
</math>
 
===Plancherel formula===
For functions <math>f, g : G \rightarrow \mathbb{C}\,</math>, the Plancherel formula states
 
:<math>
\sum_{a \in G} f(a^{-1}) g(a) = \frac{1}{|G|} \sum_i d_{\varrho_i} \text{Tr}\left(\widehat{f}(\varrho_i)\widehat{g}(\varrho_i)\right),
</math>
 
where <math>\varrho_i\,</math> are the irreducible representations of <math>G.\,</math>
 
==Fourier transform on finite abelian groups==
Since the irreducible representations of finite [[abelian group]]s 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
 
:<math>
\widehat{f}(s) = \sum_{a \in G} f(a) \bar{\chi_s}(a).
</math>
 
Note that the right-hand side is simply <math>\langle f, \chi_s\rangle</math> for the inner product on the vector space of functions from <math>G\,</math> to <math>\mathbb{C}\,</math> defined by
 
:<math>
\langle f, g \rangle = \sum_{a \in G} f(a) \bar{g}(a).
</math>
 
The inverse Fourier transform is then given by
 
:<math>
f(a) = \frac{1}{|G|} \sum_{s \in G} \widehat{f}(s) \chi_s(a).
</math>
 
A property that is often useful in probability is that the Fourier transform of the uniform distribution is simply <math>\delta_{a,0},\,</math> where 0 is the group identity and <math>\delta_{i,j}\,</math> is the [[Kronecker delta]].
 
==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 [[diagonalization|diagonalized]] quickly using the [[fast Fourier transform]], and this  yields a fast method for solving [[system of linear equations|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 partial differential equations|numerical methods for solving partial differential equations]] that preserve the symmetries of the equations {{harv|Munthe-Kaas|2006}}.
 
==See also==
*[[Fourier transform]]
*[[Discrete Fourier transform]]
*[[Representation theory of finite groups]]
*[[Character theory]]
 
==References==
* {{Citation | last1=Åhlander | first1=Krister | last2=Munthe-Kaas | first2=Hans Z. | title=Applications of the generalized Fourier transform in numerical linear algebra | doi=10.1007/s10543-005-0030-3 | mr=2191479  | year=2005 | journal=BIT | volume=45 | issue=4 | pages=819–850}}.
* Diaconis, P. (1988). [http://projecteuclid.org/euclid.lnms/1215467407 ''Group Representations in Probability and Statistics.''] Lecture Notes &mdash; 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.).
* {{Citation | last1=Munthe-Kaas | first1=Hans Z. | title=On group Fourier analysis and symmetry preserving discretizations of PDEs | doi=10.1088/0305-4470/39/19/S14 | mr=2220776  | year=2006 | journal=[[Journal of Physics A]] | volume=39 | issue=19 | pages=5563–5584}}.
* Terras, A. (1999). ''Fourier Analysis on Finite Groups and Applications''. Cambridge: Cambridge University Press.
 
{{DEFAULTSORT:Fourier Transform On Finite Groups}}
[[Category:Fourier analysis]]
[[Category:Finite groups]]

Latest revision as of 00:22, 13 January 2015

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