Extension of shift-invariant frames for locally compact abelian groups

Abstract

Let G be a locally compact abelian group with a uniform lattice sub-
group. In this paper, we verify extension of shift-invariant systems in L2(G) to tight frames. We show that any shift-invariant Bessel sequence with an at most countable number of generators in L2(G) can be extended to a tight frame for its closed linear span by adding another shift-invariant system with at most the same number of generators in L2(G). Also, we yield an extension of the given Bessel sequence to a pair of dual frame sequences.