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

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.This work has been supported by the grant MTM2017-84098-P from the Spanish Ministerio de Economía y Competitividad (MINECO).Publicad

Sampling Associated with a Unitary Representation of a Semi-Direct Product of Groups: A Filter Bank Approach

This article belongs to the Special Issue New trends on Symmetry and Topology in Quantum MechanicsAn abstract sampling theory associated with a unitary representation of a countable discrete non abelian group G, which is a semi-direct product of groups, on a separable Hilbert space is studied. A suitable expression of the data samples, the use of a filter bank formalism and the corresponding frame analysis allow for fixing the mathematical problem to be solved: the search of appropriate dual frames for l2(G). An example involving crystallographic groups illustrates the obtained results by using either average or pointwise samples.This research was funded by the grant MTM2017-84098-P from the Spanish Ministerio de Economía y Competitividad (MINECO)