Frames generated by actions of countable discrete groups

TL;DR

This paper explores dual frames generated by countable discrete group actions on Hilbert spaces, revealing new insights into shift-invariant spaces and wavelet theory, especially regarding compactly supported dual and bi-orthogonal generators.

Contribution

It establishes the equivalence between module frames over group algebras and ordinary frames in group representations, with applications to wavelet and shift-invariant spaces.

Findings

Shift-invariant subspaces with compactly supported dual frame generators also have compactly supported bi-orthogonal generators.

Uses Swan's theorem to prove that finitely generated projective modules over Laurent polynomial rings are free.

Provides a link between module frames and classical frame theory in the context of group actions.

Abstract

We consider dual frames generated by actions of countable discrete groups on a Hilbert space. Module frames in a class of modules over a group algebra are shown to coincide with a class of ordinary frames in a representation of the group. This has applications to shift-invariant spaces and wavelet theory. One of the main findings in this paper is that whenever a shift-invariant sub space in L2(Rn) has compactly supported dual frame generators then it also has compactly supported bi-orthogonal generators. The crucial part in the proof is a theorem by Swan that states that every finitely generated projective module over the Laurent polynomials in n variables is free.