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.
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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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García, A.G., Hernández-Medina, M.A. & Ibort, A. Knit Product of Finite Groups and Sampling. Mediterr. J. Math. 16, 146 (2019). https://doi.org/10.1007/s00009-019-1417-8
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DOI: https://doi.org/10.1007/s00009-019-1417-8
Keywords
- Knit product of groups
- unitary representation of a group
- finite unitary-invariant subspaces
- finite frames
- dual frames
- left-inverses
- sampling expansions