Knit Product of Finite Groups and Sampling

Abstract

A finite sampling theory associated with a unitary representation of a finite non-abelian group \({\mathbf {G}}\) on a Hilbert space is established. The non-abelian group \({\mathbf {G}}\) is a knit product \({\mathbf {N}}\bowtie {\mathbf {H}}\) of two finite subgroups \({\mathbf {N}}\) and \({\mathbf {H}}\) where at least \({\mathbf {N}}\) or \({\mathbf {H}}\) is abelian. 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.