Dilation of the Weyl symbol and Balian-Low theorem

Ascensi, Gerard; Feichtinger, Hans; Kaiblinger, Norbert

doi:10.1090/S0002-9947-2013-06074-6

Dilation of the Weyl symbol and Balian-Low theorem
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by Gerard Ascensi, Hans G. Feichtinger and Norbert Kaiblinger PDF

Trans. Amer. Math. Soc. 366 (2014), 3865-3880 Request permission

Abstract:

The key result of this paper describes the fact that for an important class of pseudodifferential operators the property of invertibility is preserved under minor dilations of their Weyl symbols. This observation has two implications in time-frequency analysis. First, it implies the stability of general Gabor frames under small dilations of the time-frequency set, previously known only for the case where the time-frequency set is a lattice. Secondly, it allows us to derive a new Balian-Low theorem (BLT) for Gabor systems with window in the standard window class and with general time-frequency families. In contrast to the classical versions of BLT the new BLT does not only exclude orthonormal bases and Riesz bases at critical density, but indeed it even excludes irregular Gabor frames at critical density.

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