The norm of the Fourier transform on finite abelian groups

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...

The Fourier Transform and Equations over Finite Abelian Groups

The Norm of the L'-fourier Transform on Unimodular Groups

We discuss sharpness in the Hausdorff Young theorem for unimodular groups. First the functions on unimodular locally compact groups for which equality holds in the Hausdorff Young theorem are determined. Then it is shown that the Hausdorff Young theorem is not sharp on any unimodular group which contains the real Une as a direct summand, or any unimodular group which contains an Abelian normal ...

Fourier analysis on finite abelian groups

[2] The notation L(G) is meant to suggest the presence of the inner product on this space of functions. On a general space X with an integral, the iner product is 〈f1, f2〉 = ∫ X f1f 2. [3] The term character has different meanings in different contexts. The simplest sense is a group homomorphism to C×. However, an equally important use is for the trace of a group homomorphism ρ : G → GLn(k) fro...

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...