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If one takes a cyclic group and a function on this group, and performs harmonic analysis on it (classical Fourier analysis), the result is a set of coefficients, each one of them corresponding to frequencies, ie intuitively particular paths in the cyclic Cayley graphs (in this case, loops with step 1, 2, etc...)

Is this the same for more general groups (dihedral, alternating, symmetric, etc..) ? Does n-dimensional coefficients (or individual elements of these matrices) corresponds to particular paths in a corresponding Cayley graph of the group ?

Thanks

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Everything works out basically the same way for finite abelian groups, but for nonabelian groups the answer is somewhat complicated. One can take a class function on the group and decompose it as a sum of characters corresponding to irreducible representations of the group; for an arbitrary function one instead uses matrix coefficients, which in nice cases turn out to be finite analogues of certain classical special functions.

This issue is discussed in Ceccherini-Silberstein, Scarabotti, and Tolli's Harmonic analysis on finite groups as well as Terras' Fourier analysis on finite groups.

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