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arXiv · 1610.06697

Reproducing pairs and Gabor systems at critical density

Michael Speckbacher·Peter Balazs
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Abstract

We use the concept of reproducing pairs to study Gabor systems at critical density. First, we present a generalization of the Balian-Low theorem to the reproducing pairs setting. Then, we prove our main result that there exists a reproducing partner for the Gabor system of integer time-frequency shifts of the Gaussian. In other words, the coefficients for this Gabor expansion of a square integrable function can be calculated using inner products with an unstructured family of vectors in $ L^2(\R)$. This solves the possibly last open question for this system.

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Michael Speckbacher, Peter Balazs. 2016-10-21. Reproducing pairs and Gabor systems at critical density. https://doi.org/10.1016/j.jmaa.2017.05.079

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