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Journal of Fourier Analysis and Applications

, Volume 22, Issue 1, pp 36–70 | Cite as

Co-compact Gabor Systems on Locally Compact Abelian Groups

  • Mads Sielemann Jakobsen
  • Jakob LemvigEmail author
Article

Abstract

In this work we extend classical structure and duality results in Gabor analysis on the euclidean space to the setting of second countable locally compact abelian (LCA) groups. We formulate the concept of rationally oversampling of Gabor systems in an LCA group and prove corresponding characterization results via the Zak transform. From these results we derive non-existence results for critically sampled continuous Gabor frames. We obtain general characterizations in time and in frequency domain of when two Gabor generators yield dual frames. Moreover, we prove the Walnut and Janssen representation of the Gabor frame operator and consider the Wexler–Raz biorthogonality relations for dual generators. Finally, we prove the duality principle for Gabor frames. Unlike most duality results on Gabor systems, we do not rely on the fact that the translation and modulation groups are discrete and co-compact subgroups. Our results only rely on the assumption that either one of the translation and modulation group (in some cases both) are co-compact subgroups of the time and frequency domain. This presentation offers a unified approach to the study of continuous and the discrete Gabor frames.

Keywords

Dual frames Duality principle Fiberization Frame Gabor system Janssen representation LCA group  Walnut representation Wexler–Raz biorthogonality relations  Zak transform 

Mathematics Subject Classification

Primary 42C15 Secondary 43A32 43A70 

Notes

Acknowledgments

The authors thank Ole Christensen for useful discussions and for the proof of Theorem 6.6. The first-named author also thanks Hans G. Feichtinger for discussions and pointing out references concerning \(S_0\).

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkLyngbyDenmark

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