Projects per year
Abstract
This thesis consists of four papers. The first one introduces generalized translation invariant systems and considers their frame properties, the second and third paper give new results on the theory of Gabor frames, and the fourth is a review paper with proofs and new results on the Feichtinger algebra.
The generalized translation invariant (GTI) systems provide, for the first time, a framework which can describe frame properties of both discrete and continuous systems. The results yield the wellknown characterizations of dual frame pairs and Parseval frames of Gabor, wavelet, curvelet and shearlettype and for (generalized) shiftinvariant systems and their continuous formulations.
This thesis advances the theory of both separable and nonseparable, discrete, semicontinuous and continuous Gabor systems. In particular, the well established structure theory for separable lattice Gabor frames is extended and generalized significantly to Gabor systems with timefrequency shifts along closed subgroups in the timefrequency plane. This includes density results, the Walnut representation, the WexlerRaz biorthogonality relations, the Bessel duality and the duality principle between Gabor frames and Gabor Riesz bases.
The theory of GTI systems and Gabor frames in this thesis is developed and presented in the setting of locally compact abelian groups, however, even in the euclidean setting the results given here improve the existing theory.
Finally, the thesis contains a review paper with proofs of all the major results on the Banach space of functions known as the Feichtinger algebra. This includes many of its different characterizations and treatment of its many equivalent norms, its minimality among all timefrequency shift invariant Banach spaces and aspects of its dual space, operators on the space and the kernel theorem for the Feichtinger algebra. The work also includes new findings such as a characterization among all Banach spaces, a forgotten theorem by Reiter on Banach space isomorphisms of the Feichtinger algebra, and new useful inequalities.
The generalized translation invariant (GTI) systems provide, for the first time, a framework which can describe frame properties of both discrete and continuous systems. The results yield the wellknown characterizations of dual frame pairs and Parseval frames of Gabor, wavelet, curvelet and shearlettype and for (generalized) shiftinvariant systems and their continuous formulations.
This thesis advances the theory of both separable and nonseparable, discrete, semicontinuous and continuous Gabor systems. In particular, the well established structure theory for separable lattice Gabor frames is extended and generalized significantly to Gabor systems with timefrequency shifts along closed subgroups in the timefrequency plane. This includes density results, the Walnut representation, the WexlerRaz biorthogonality relations, the Bessel duality and the duality principle between Gabor frames and Gabor Riesz bases.
The theory of GTI systems and Gabor frames in this thesis is developed and presented in the setting of locally compact abelian groups, however, even in the euclidean setting the results given here improve the existing theory.
Finally, the thesis contains a review paper with proofs of all the major results on the Banach space of functions known as the Feichtinger algebra. This includes many of its different characterizations and treatment of its many equivalent norms, its minimality among all timefrequency shift invariant Banach spaces and aspects of its dual space, operators on the space and the kernel theorem for the Feichtinger algebra. The work also includes new findings such as a characterization among all Banach spaces, a forgotten theorem by Reiter on Banach space isomorphisms of the Feichtinger algebra, and new useful inequalities.
Translated title of the contribution  Gabor frames på lokalt kompakte abelske grupper og relaterede emner 

Original language  English 
Place of Publication  Kgs. Lyngby 

Publisher  Technical University of Denmark 
Number of pages  26 
Publication status  Published  2017 
Series  DTU Compute PHD2016 

Number  436 
ISSN  09093192 
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Projects
 1 Finished

Gabor Frames, their Duals and Applications in Engineering
Jakobsen, M. S., Christensen, O., Lemvig, J., Knudsen, K., Feichtinger, H. & Hernández, E.
Technical University of Denmark
01/10/2013 → 23/11/2016
Project: PhD