Abstract
In this work we show that if the frame property of a Gabor frame with window in Feichtinger’s algebra and a fixed lattice only depends on the parity of the window, then the lattice can be replaced by any other lattice of the same density without losing the frame property. As a byproduct we derive a generalization of a result of Lyubarskii and Nes, who could show that any Gabor system consisting of an odd window function from Feichtinger’s algebra and any separable lattice of density tfrac{n+1}{n}, n in mathbb {N}_+, cannot be a Gabor frame for the Hilbert space of square-integrable functions on the real line. We extend this result by removing the assumption that the lattice has to be separable. This is achieved by exploiting the interplay between the symplectic and the metaplectic group.
Highlights
In this article we extend results derived by Lyubarskii and Nes, who proved that for odd functions in Feichtinger’s algebra S0(R) and separable lattices of rational density n+1 n
We show that the assumption of the separability of the lattice is not necessary and that their result holds for arbitrary lattices of density n+1 n
The main tools in this work are metaplectic operators, which are a certain class of unitary operators acting on L2(Rd), and their interplay with the symplectic group Sp(d), a subgroup of SL(R, 2d)
Summary
In this article we extend results derived by Lyubarskii and Nes, who proved that for odd functions in Feichtinger’s algebra S0(R) and separable lattices of rational density n+1 n.
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