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.
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.
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