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Transactions of the American Mathematical Society

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Reproducing formulas for generalized translation invariant systems on locally compact abelian groups


Authors: Mads Sielemann Jakobsen and Jakob Lemvig
Journal: Trans. Amer. Math. Soc. 368 (2016), 8447-8480
MSC (2010): Primary 42C15, 43A32, 43A70; Secondary 43A60, 46C05
DOI: https://doi.org/10.1090/tran/6594
Published electronically: February 10, 2016
MathSciNet review: 3551577
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Abstract: In this paper we connect the well-established discrete frame theory of generalized shift invariant systems to a continuous frame theory. To do so, we let $ \Gamma _{\!j}$, $ j \in J$, be a countable family of closed, co-compact subgroups of a second countable locally compact abelian group $ G$ and study systems of the form $ \bigcup _{j \in J}\{ g_{j,p}(\cdot - \gamma )\}_{\gamma \in \Gamma _{\!j}, p \in P_j}$ with generators $ g_{j,p}$ in $ L^2(G)$ and with each $ P_j$ being a countable or an uncountable index set. We refer to systems of this form as generalized translation invariant (GTI) systems. Many of the familiar transforms, e.g., the wavelet, shearlet and Gabor transform, both their discrete and continuous variants, are GTI systems. Under a technical $ \alpha $ local integrability condition ($ \alpha $-LIC) we characterize when GTI systems constitute tight and dual frames that yield reproducing formulas for $ L^2(G)$. This generalizes results on generalized shift invariant systems, where each $ P_j$ is assumed to be countable and each $ \Gamma _{\!j}$ is a uniform lattice in $ G$, to the case of uncountably many generators and (not necessarily discrete) closed, co-compact subgroups. Furthermore, even in the case of uniform lattices $ \Gamma _{\! j}$, our characterizations improve known results since the class of GTI systems satisfying the $ \alpha $-LIC is strictly larger than the class of GTI systems satisfying the previously used local integrability condition. As an application of our characterization results, we obtain new characterizations of translation invariant continuous frames and Gabor frames for $ L^2(G)$. In addition, we will see that the admissibility conditions for the continuous and discrete wavelet and Gabor transform in $ L^2(\mathbb{R}^n)$ are special cases of the same general characterizing equations.


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Additional Information

Mads Sielemann Jakobsen
Affiliation: Department of Applied Mathematics and Computer Science, Technical University of Denmark, Matematiktorvet 303B, 2800 Kgs. Lyngby, Denmark
Email: msja@dtu.dk

Jakob Lemvig
Affiliation: Department of Applied Mathematics and Computer Science, Technical University of Denmark, Matematiktorvet 303B, 2800 Kgs. Lyngby, Denmark
Email: jakle@dtu.dk

DOI: https://doi.org/10.1090/tran/6594
Keywords: Continuous frame, dual frames, dual generators, g-frame, Gabor frame, generalized shift invariant system, generalized translation invariant system, LCA group, Parseval frame, wavelet frame
Received by editor(s): May 19, 2014
Received by editor(s) in revised form: October 8, 2014
Published electronically: February 10, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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