Fourier Transform On Finite Groups - Fourier Transform On Finite Abelian Groups

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

$\widehat{f}(s) = \sum_{a \in G} f(a) \bar{\chi_s}(a).$

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

$\langle f, g \rangle = \sum_{a \in G} f(a) \bar{g}(a).$

The inverse Fourier transform is then given by

$f(a) = \frac{1}{|G|} \sum_{s \in G} \widehat{f}(s) \chi_s(a).$

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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