Fourier transform on finite groups
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 . Since the dimension of the transform space is equal to , it follows that .
The inverse Fourier transform at an element of is given by
Transform of a convolution
The convolution of two functions is defined as
The Fourier transform of a convolution at any representation of is given by
For functions , the Plancherel formula states
where are the irreducible representations of
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 be the set of group homomorphisms . This forms a group under point-wise multiplication. We may define the Fourier transform for finite abelian groups for an element as the function given by
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
Since is finite abelian, the irreducible characters of a group may be put into one-to one correspondence with elements of the group. If we fix an isomorphism of for each , we may write for the character corresponding to . Thus the Fourier transform can be rewritten as a function .
and the inverse is given by
Note that the inverse fourier transform is technically an element of , but using the isomorphism from to , we get an element of that equals
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.
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, , together with the operations of pointwise addition and convolution, form a ring that is naturally identified with the group ring of over the complex numbers, . Modules of this ring are the same thing as representations. Maschke's theorem implies that is a semisimple ring, so by the Artin–Wedderburn theorem it decomposes as a direct product of matrix rings. The Fourier transform on finite groups explicitly exhibits this decomposition, with a matrix ring of dimension for each irreducible representation.
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).
- Terras 1999, p. 251
- Åhlander, Krister; Munthe-Kaas, Hans Z. (2005), "Applications of the generalized Fourier transform in numerical linear algebra", BIT, 45 (4): 819–850, doi:10.1007/s10543-005-0030-3, MR 2191479.
- Diaconis, Persi (1988), Group representations in probability and statistics, Lecture Notes—Monograph Series, 11, Institute of Mathematical Statistics, Zbl 0695.60012.
- Diaconis, Persi, "Finite Fourier Methods: Access to Tools", in Bollobás, Béla; Chung, Fan R. K., Probabilistic combinatorics and its applications, Proceedings of Symposia in Applied Mathematics, 44, American Mathematical Society, pp. 171–194, ISBN 978-0-8218-6749-5.
- Munthe-Kaas, Hans Z. (2006), "On group Fourier analysis and symmetry preserving discretizations of PDEs", Journal of Physics A, 39 (19): 5563–84, doi:10.1088/0305-4470/39/19/S14, MR 2220776.
- Terras, Audrey (1999), Fourier Analysis on Finite Groups and Applications, Cambridge University Press, p. 251, ISBN 978-0-521-45718-7, Zbl 0928.43001.