D. Rockmore, "Fast Fourier analysis for abelian group extensions", Adv. in Appl. Math., 11 (1990), 164--204.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....cases, the problem of efficiently computing the sums is solved by the fast Fourier transform algorithm of Cooley Tukey [9] and its many generalizations. The special case when G is a finite nonabelian group was first treated by Willsky [37] and has since been studied by many authors, see e.g. [1, 3, 4, 6, 7, 8, 10, 12, 18, 33, 34]. The techniques developed in the current paper borrow many ideas from the finite group case. Conversely, many of the new results in this paper may also be formulated for finite groups; see [27, 28, 29, 30] The first real breakthrough in treating continuous compact nonabelian groups occurred in ....

D. Rockmore, Fast Fourier analysis for abelian group extensions, Adv. in Appl. Math. 11 (1990), 164--204.

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D. Rockmore, "Fast Fourier analysis for abelian group extensions", Adv. in Appl. Math., 11 (1990), 164--204.

....associated complexities as T (q) and I(q) Here, a natural subgroup for restriction is B = B(q) the subgroup of upper triangular matrices. It is easy to check that B(q) is a metabelian group; that is, it contains an abelian normal subgroup U , such that the quotient B=U is abelian. It is known [Cl1, R1] that for such a group T (B) O(j B j log j B j) As will be explained fully in Section 2, the representations of SL 2 (q) occur as essentially q irreducible representations of degree q. Thus, 1.1) now specializes to T (q) j SL 2 (q) j j B j T (B) j SL 2 (q) j j B j p X i=1 d ff ....

....If so, as discussed above, it would afford further computational savings since only the Fourier transforms at the principal series representations would need to be computed. Lastly, we would like to comment on the complexity results of Section 3. Recent work in the area of DFT s for finite groups [Ba, Cl2, Cl1, DR, R1, R2] has shown that for several classes of groups, the DFT can be computed in O(j G j log j G j) operations. It would be of great interest if for G = SL 2 (q) the results of Section 3 could be improved to an upper bound of O(q 3 log q) Fast Fourier analysis for SL 2 over a finite field 42 ....

D. Rockmore. Fast Fourier analysis for abelian group extensions. Adv. in Appl. Math., 11:164--204, 1990.

....Fellowship. DFT for any finite group [15, 11, 7] with dramatic speed ups possible for several classes of nonabelian groups. In particular, it is now known that many families of nonabelian groups also possess O(jGj log jGj) FFT s. This includes the symmetric groups [11] metabelian groups [10, 33] and more generally, supersolvable groups [5] Baum and Clausen s recent book [9] is a good source for some of the recent work in this area. Other developments in the finite setting include new efficient algorithms for computing on quotients [16] as well as with sets of orthogonal polynomial [18, ....

....] Theta G is adapted. While such bases always exist, their construction may be difficult. Success in their construction would then pave the way to FFT s as obtained in Section 4. Note that in the case in which L is abelian, then this is the situation of an abelian extension and the methods of [33] can be used to obtain adapted bases as well as FFT s. 4 FFT s for wreath products The complexity results presented here depend mainly on careful use of the structure of the matrices of the induced representations in their standard bases (cf. Definition 5) as well as various properties of ....

D. Rockmore. Fast Fourier analysis for abelian group extensions. Adv. in Appl. Math. 11, 164-204 (1990).

....over all choices of bases, and is bounded by jGj 2 . This bound follows from a direct approach to the computation. We conjecture that all finite groups have complexity O(jGj log c jGj) for some universal constant c. This has already been proved for many different classes of nonabelian groups [19, 58, 56, 7]. The first results of this type obtained for nonabelian groups are due to Willsky. In [66] Willsky studies a particular class of finite state Markov processes evolving on metacyclic groups. In so doing he gives an O(jGj log jGj) FFT for G a metacyclic group, designed to give an efficient ....

D. Rockmore, Fast Fourier analysis for abelian group extensions, Adv. in Appl. Math. 11 (1990), 164--204.

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