From exact systems to Riesz bases in the Balian-Low theorem
Abstract
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.
Cite
@article{arxiv.0906.2302,
title = {From exact systems to Riesz bases in the Balian-Low theorem},
author = {Shahaf Nitzan and Jan-Fredrik Olsen},
journal= {arXiv preprint arXiv:0906.2302},
year = {2010}
}
Comments
v1: 16 pages. v2: 35 pages. Reorganized the presentation, corrected some typos, and added some additional results. In particular, added proof that main theorem is sharp