Semi-direct product of groups, filter banks and sampling

TL;DR

This paper develops a sampling theory for signals associated with semi-direct product groups using filter banks, providing methods to find dual frames in non-abelian group settings with applications to crystallographic groups.

Contribution

It introduces a novel sampling framework for semi-direct product groups and employs filter bank formalism to identify dual frames in non-abelian group contexts.

Findings

Derived explicit dual frames for semi-direct product group representations.

Applied the theory to crystallographic groups with average and pointwise sampling.

Established a mathematical foundation for sampling in non-abelian group settings.

Abstract

An abstract sampling theory associated to a unitary representation of a countable discrete non abelian group $G$, which is a semi-direct product of groups, on a separable Hilbert space is studied. A suitable expression of the data samples and the use of a filter bank formalism allows to fix the mathematical problem to be solved: the search of appropriate dual frames for $\ell^2(G)$. An example involving crystallographic groups illustrates the obtained results by using average or pointwise samples.