Frames generated by actions of countable discrete groups

Røysland, Kjetil

doi:10.1090/S0002-9947-2010-05260-2

Frames generated by actions of countable discrete groups
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Trans. Amer. Math. Soc. 363 (2011), 95-108 Request permission

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 subspace in $L^2(\mathbb {R}^n)$ 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.

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