Knit Product of Finite Groups and Sampling

  • Antonio G. García
  • Miguel A. Hernández-MedinaEmail author
  • Alberto Ibort


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.


Knit product of groups unitary representation of a group finite unitary-invariant subspaces finite frames dual frames left-inverses sampling expansions 

Mathematics Subject Classification

20C40 42C15 94A20 



The authors wish to thank the referee for his/her valuable and constructive comments. This work has been supported by the Grant MTM2017-84098-P from the Spanish Ministerio de Economía y Competitividad (MINECO).


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Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Carlos III de MadridLeganés-MadridSpain
  2. 2.Departamento de Matemática Aplicada a las Tecnologías de la Información y las Comunicacion, E.T.S.I.T.Information Processing and Telecommunications Center Universidad Politécnica de MadridMadridSpain

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