From exact systems to Riesz bases in the Balian-Low theorem

Details From exact systems to Riesz bases in the Balian-Low theorem From exact systems to Riesz bases in the Balian-Low theorem

2009-06-12

arxiv.org/abs/0906.2302v2

Mathematics

Classical Analysis and ODEs

v1: 16 pages. v2: 35 pages. Reorganized the presentation, corrected some typos, and added some additional results. In particul

Scientific paper

We look at the time-frequency localisation of generators of lattice Gabor systems. For a generator of a Riesz basis, this localisation is described by the classical Balian-Low theorem. We establish Balian-Low type theorems for complete and minimal Gabor systems with a frame-type approximation property. These results describe how the best possible localisation of a generator is limited by the degree of control over the coefficients in approximations given by the system, and provide a continuous transition between the classical Balian-Low conditions and the corresponding conditions for generators of complete and minimal systems. Moreover, this holds for the non-symmetric generalisations of these theorems as well.

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