Improved upper complexity bounds for the discrete fourier transform

Abstract

Abstract. The linear complexity L2(G ) of a finite group G is the minimal number of additions, subtractions and multiplications by complex constants of absolute value <2 sufficient to evaluate a suitable Fourier transform of ~G. Combining and modifying several classical FFT-algorithms, we show that Lz(G ) =< 81G I log21GI for any finite metabelian group G.

The Power of Nonmonotonicity in Geometric Searching

“…The classical FFT algorithm has been generalized to arbitrary abelian groups [3], [8], [11], and it is possible to compute it in O (N log N ) If only additions are allowed, then the naive algorithm, which involves adding the weights one at a time, has a complexity of O(n 3/2 ). This establishes the upper bounds of Theorem 1.1 and Corollary 1.2.…”

“…A multiple point (x, y) of line (3) is such that, modulo m, (x, −y), (−x, y), or (−x, −y) belongs to the line. Note that the multiplicity can be as high as 4: for example, take the point (3,2) on the line 2X + 3Y = 0 (mod 12).…”

“…A multiple point (x, y) of line (3) is such that, modulo m, (x, −y), (−x, y), or (−x, −y) belongs to the line. Note that the multiplicity can be as high as 4: for example, take the point (3,2) on the line 2X + 3Y = 0 (mod 12). We count the number of pairs ( p, ) for all points and lines, where p = (x, y) is a multiple point of The Power of Nonmonotonicity in Geometric Searching 15 line .…”

Generalized FFTs–A survey of some recent results

“…Perhaps in combination with other recurrences that involve the degree parameter, this complexity could be reduced to O(n log n). 10. Minimal sampling.…”

The Efficient Computation of Fourier Transforms on Semisimple Algebras

“…As per the group algebra case, the Fourier transform of a complex semisimple algebra A is a change of basis from some preferred basis to a basis given by irreducible matrix elements. The complexity of A, C(A) is the least upper bound of an algorithm effecting such a map and thus bounded above a priori by dim(A) 2 . The work presented in this paper, aiming to reduce this naive upper bound, is both motivated as a next "natural" step in algebraic FFT work (see also extensions to the semigroup case [29,27,28]) as well as by a particular application: the study of a certain random walk on the Birman-Murakami-Wenzl (BMW) algebra [47].…”