Fast Fourier Analysis for Abelian Group Extensions

Let G be a nite group and f any complex-valued function deened on G and an irreducible complex matrix representation of G. The Fourier transform of f at is deened to be the matrix P s2G f(s)(s). The Fourier transforms of f at all the irreducible representations of G determine f via the Fourier inversion formula f(s) = 1 j Gj P d trace(b f()(s ?1)): Direct computation of all Fourier transforms of f involves on the order of jGj 2 operations as does direct computation of Fourier inversion. Here fast algorithms are obtained for both operations in the case in which G contains some nontrivial normal subgroup K such that G=K is abelian. Consequently, fast algorithms for computing convolutions on G in this situation are also determined. Under the simpliiying assumption of exponent for matrix multiplication equal to 2 (it is 2.38 as of this writing) it is shown that the number of operations needed to compute all Fourier transforms on G is O(j Gj j Kj T(K)+ jGj log(j Gj j Kj)) where T(K) is the number of operations needed to compute Fourier transforms on K. An analogous result is obtained for Fourier inversion and more careful estimates made for arbitrary exponents of matrix multiplication. In particular, in the case in which G is metabelian (an abelian extension of an abelian subgroup) the assumptions hold and the rst large class of noncom-mutative groups is obtained for which Fourier inversion and the computation of all Fourier transforms may be performed in O(jGj log(jGj)) operations.