# Fourier transform on a general finite group on a set of symbols

I'd like to perform Fourier analysis on a sequence of n symbols or letters, but on a general finite group. For a cyclic group (the usual Fourier transform), I found (J. Math. Biol, 36(1997):pp.64-70) an approach where each symbol is replaced by a vector in $\mathbb{R}^n$ or $\mathbb{R}^{n-1}$.

If the same Fourier transform is considered on a finite group, I don't think one can write :

$F(k)=\sum_{g \in G} f(g) \rho_k(g)$

like it is usually the case, since the product of vector $f(g)$ by the matrix $\rho_k(g)$ might not be defined. How can I adress this problem ? Should I consider representations in the same dimension as $\mathbb{R}^n$ ? Any help appreciated...

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Look at W. Rudin's "Fourier Analysis on Groups" for details. –  JavaMan Oct 27 '11 at 17:39