A characterization of tight and dual generalized translation invariant frames

Mads Sielemann Jakobsen, Jakob Lemvig

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Abstract

We present results concerning generalized translation invariant (GTI) systems on a second countable locally compact abelian group G. These are systems with a family of generators {gj, P}jεJ, pεPJ ⊂ L2(G), where J is a countable index set, and Pj, j ε J are certain measure spaces. Furthermore, for each j we let Γj, be a closed subgroup of G such that G/Γj is compact. A GTI system is then the collection of functions UjεJ{gj, p(· - γ}γεΓj, pεPj. Many well known systems, such as wavelet, shearlet and Gabor systems, both the discrete and continuous types, are GTI systems. We characterize when such systems form tight frames, and when two GTI Bessel systems form dual frames for L2(G). In particular, this offers a unified approach to the theory of discrete and continuous frames and, e.g., yields well known results for discrete and continuous Gabor and wavelet systems.
Original languageEnglish
Title of host publicationProceedings of the 11th International Conference on Sampling Theory and Applications (SampTA 2015)
PublisherIEEE Press
Publication date2015
Pages96-100
ISBN (Print)978-1-4673-7353-1
DOIs
Publication statusPublished - 2015
Event11th International Conference on Sampling Theory and Applications - Washington, United States
Duration: 25 May 201529 May 2015
Conference number: 11
http://www.american.edu/cas/mathstat/sampta2015/index.cfm

Conference

Conference11th International Conference on Sampling Theory and Applications
Number11
CountryUnited States
CityWashington
Period25/05/201529/05/2015
Internet address

Cite this

Jakobsen, M. S., & Lemvig, J. (2015). A characterization of tight and dual generalized translation invariant frames. In Proceedings of the 11th International Conference on Sampling Theory and Applications (SampTA 2015) (pp. 96-100). IEEE Press. https://doi.org/10.1109/SAMPTA.2015.7148858