Sampling in Unitary Invariant Subspaces Associated to LCA Groups

Abstract

In this paper a sampling theory for unitary invariant subspaces associated to locally compact abelian (LCA) groups is deduced. Working in the LCA group context allows to obtain, in a unified way, sampling results valid for a wide range of problems which are interesting in practice, avoiding also cumbersome notation. Along with LCA groups theory, the involved mathematical technique is that of frame theory which meets matrix analysis when appropriate dual frames are computed.

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

“…[2,4,10,11,17,20,21], and sampling in U -invariant subspaces Refs. [8,12,13,19]. Besides, as it was showed in Section 4.2, the present approach opens new sampling settings: for instance, those related with crystallographic groups involving examples of practical interest.…”

On Regular Generalized Sampling in T-Invariant Subspaces of a Hilbert Space: An Overview

“…, L ; Sections 4 and 5 are devoted to the finite case where both subspaces A a or A a are finite dimensional; finally, in Section 6 the abstract case associated with an LCA group is exhibited. The mathematical development followed along the paper shares some patterns appearing in the case of an unitary operator U studied in previous works (see [15,16,24,25]); some proofs which mimic similar results will be simply referred. Putting all these cases together can help to survey the intrinsic nature of these sampling problems and their relationships.…”

“…The case where T = U is an unitary operator in H has been studied in Refs. [15,16,24,25,32,35], and it generalizes averaged sampling in shift-invariant subspaces in L 2 (R); whenever U is a shift operator, the samples given in (1) are nothing but samples of a convolution operator, i.e., samples of a filtered version of the function itself.…”

“…, s. As a consequence of expression (31), the recovery of any x ∈ A a depends on the frame property of the sequence G j χ m m∈M; j=1,2,...,s in L 2 ( H). This study has been done in [25,Proposition 2]. In order to state the result we need to introduce some necessary preliminaries.…”

“…We also need to characterize its dual frames having the same structure. This is done in [25,Proposition 4]:…”