Group Convolutions and Matrix Transforms

Abstract

Given a finite group G (possibly noncommutative) and a field $\mathbb{F}$, group convolutions are constructed based on the group algebra of G over $\mathbb{F}$. Matrices with entries in the group algebra are constructed so that they have a convolution property relative to G. As special cases, the discrete Fourier transform, the discrete Walsh transform and a transform based on the dihedral groups are discussed. The development also shows that higher-dimensional transforms are special cases of the construction where the underlying group is an external direct product of other groups. An illustration of the ideas is given using the dihedral groups and matrix representations. Finally, a generalization of convolution is discussed in terms of group rings.