Knit product of finite groups and sampling

TL;DR

This paper develops a finite sampling theory for non-Abelian groups structured as knit products, providing sampling formulas and dual frames, with an application to dihedral groups.

Contribution

It introduces a novel sampling framework for non-Abelian groups formed as knit products, extending finite sampling theory to this class of groups.

Findings

Sampling formulas indexed by subgroup elements are derived.

Dual frames with unitary invariance are constructed using matrix analysis.

An example with dihedral groups demonstrates the theory's application.

Abstract

A finite sampling theory associated with a unitary representation of a finite non Abelian group $\mathbf{G}$ on a Hilbert space is stablished. The non Abelian group $\mathbf{G}$ is a knit product $\mathbf{N}\bowtie \mathbf{H}$ of two finite subgroups $\mathbf{N}$ and $\mathbf{H}$. Sampling formulas where the samples are indexed by either $\mathbf{N}$ or $\mathbf{H}$ are obtained. Using suitable expressions for the involved samples, the problem is reduced to obtain dual frames in the Hilbert space $\ell^2(\mathbf{G})$ having a unitary invariance property; this is done by using matrix analysis techniques. An example involving dihedral groups illustrates the obtained sampling results.