Convolution systems on discrete abelian groups as a unifying strategy in sampling theory

2019/05/08 by García, Antonio G., Hernández-Medina, Miguel A., Pérez-Villalón, Gerardo
#20H15 #22B05 #42C15 #94A20 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1905.02999

Abstract

A regular sampling theory in a multiply generated unitary invariant subspace of a separable Hilbert space H is proposed. This subspace is associated to a unitary representation of a countable discrete abelian group G on H. The samples are defined by means of a filtering process which generalizes the usual sampling settings. The multiply generated setting allows to consider some examples where the group G is non-abelian as, for instance, crystallographic groups. Finally, it is worth to mention that classical average or pointwise sampling in shift-invariant subspaces are particular examples included in the followed approach.

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