Dilation of the Weyl symbol and Balian-Low theorem

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 gen...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2014-07, Vol.366 (7), p.3865-3880
Main Authors: ASCENSI, GERARD, FEICHTINGER, HANS G., KAIBLINGER, NORBERT
Format: Article
Language:English
Subjects:
Algebra
Banach algebra
Banach space
Entire functions
Fourier transformations
Interpolation
Mathematical functions
Mathematical lattices
Mathematical theorems
Tensors
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Summary: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-2013-06074-6