Fourier Transform On Finite Groups - Definitions

Definitions

The Fourier transform of a function at a representation of is

\widehat{f}(\varrho) = \sum_{a \in G} f(a) \varrho(a).

So for each representation of, is a matrix, where is the degree of .

Let be a complete set of inequivalent irreducible representations of . Then, . Then the inverse Fourier transform at an element of is given by

f(a) = \frac{1}{|G|} \sum_i d_{\varrho_i} \text{Tr}\left(\varrho_i(a^{-1})\widehat{f}(\varrho_i)\right),

where is the degree of the representation

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