Abstract
We show that if the Gabor system \(\{ g(x-t) e^{2\pi i s x}\}\), \(t \in T\), \(s \in S\), is an orthonormal basis in \(L^2({\mathbb {R}})\) and if the window function g is compactly supported, then both the time shift set T and the frequency shift set S must be periodic. To prove this we establish a necessary functional tiling type condition for Gabor orthonormal bases which may be of independent interest.
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Communicated by Loukas Grafakos.
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Research supported by ISF Grants Nos. 447/16, 227/17 and 1044/21 and ERC Starting Grant No. 713927. Alberto Debernardi Pinos was also partially supported by Ministry of Education and Science of the Republic of Kazakhstan (AP08053326), and by The Center for Research & Development in Mathematics and Applications, through the Portuguese Foundation for Science and Technology (UIDP/04106/2020).
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Debernardi Pinos, A., Lev, N. Gabor orthonormal bases, tiling and periodicity. Math. Ann. 384, 1461–1467 (2022). https://doi.org/10.1007/s00208-021-02324-1
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DOI: https://doi.org/10.1007/s00208-021-02324-1