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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

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For finite (or in general locally compact) abelian groups, the Fourier transform involves the dual group, i.e. the (continuous) homomorphisms of the group into the unit circle. The analogue for non-abelian compact groups involves the matrix elements of irreducible unitary representations: see the Peter-Weyl theorem.

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