Convolution Systems on Discrete Abelian Groups as a Unifying Strategy in Sampling Theory Articles uri icon

publication date

  • February 2020

start page

  • 1

end page

  • 22

issue

  • 1(40)

volume

  • 75

International Standard Serial Number (ISSN)

  • 1422-6383

Electronic International Standard Serial Number (EISSN)

  • 1420-9012

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.

keywords

  • convolution systems; discrete abelian groups; dual frames; sampling expansion; unitary representation of a group