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From Exact Systems to Riesz Bases in the Balian–Low Theorem

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Abstract

We look at the time–frequency localization of generators of lattice Gabor systems. For a generator of a Riesz basis, this localization 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 localization 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 generalizations of these theorems as well.

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Author notes

  1. Shahaf Nitzan

    Present address: Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

  2. Jan-Fredrik Olsen

    Present address: Centre for Mathematical Sciences, Lund University, P.O. Box 118, 221 00, Lund, Sweden

Authors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), 7491, Trondheim, Norway

    Shahaf Nitzan & Jan-Fredrik Olsen

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  1. Shahaf Nitzan
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  2. Jan-Fredrik Olsen
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Corresponding author

Correspondence to Shahaf Nitzan.

Additional information

Communicated by Karlheinz Gröchenig.

The first author is supported by the ERCIM “Alain Bensoussan” fellowship nr. 2009-05.

For a part of this work, the second author is supported by the Research Council of Norway grant 160192/V30.

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Nitzan, S., Olsen, JF. From Exact Systems to Riesz Bases in the Balian–Low Theorem. J Fourier Anal Appl 17, 567–603 (2011). https://doi.org/10.1007/s00041-010-9150-5

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  • DOI: https://doi.org/10.1007/s00041-010-9150-5

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