Efficient computation of the Fourier transform on finite groups

Efficient Computation of the Fourier Transform on Finite Groups

Let G be a finite group, f: G --+ C a function, and p an ir-reducible representation of G. The Fourier transform is defined as j(p) =E SEG f(s)p(s) . Direct computation for all irreducible representations involvesorder IGI2 operations. We derive fast algorithms and develop them for thesymmetric groupSn. There, (n!)2 is reduced to n(n!)a/2, where a is theconstant for matr...

Remarks on Efficient Computation of the Inverse Fourier Transforms on Finite Non-Abelian Groups

The Fourier transform is a classical method in mathematical modeling of systems. Assuming finite non-Abelian groups as the underlying mathematical structure might bring advantages in modeling certain systems often met in computer science and information technologies. Frequent computing of the inverse Fourier transform is usually required in dealing with such systems. These computations require ...

Matrices of Finite Abelian Groups, Finite Fourier Transform and Codes

Finite (or Discrete) Fourier Transforms (FFT) are essential tools in engineering disciplines based on signal transmission, which is the case in most of them. FFT are related with circulant matrices, which can be viewed as group matrices of cyclic groups. In this regard, we introduce a generalization of the previous investigations to the case of finite groups, abelian or not. We make clear the p...

Efficient Computation of Fourier Transforms on Compact Groups

This paper generalizes the fast Fourier transform algorithm to the computationof Fourier transformson compact Lie groups. The basic technique uses factorization of group elements and Gel'fand-Tsetlin bases to simplify the computations, and may be extended to treat the computation of Fourier transforms of nitely supported distributions on the group. Similar transforms may be deened on homogeneou...

The Fourier Transform and Equations over Finite Abelian Groups