Compactly supported Gabor orthonormal bases
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866914614287007744 |
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| author | Liehr, Lukas |
| author_facet | Liehr, Lukas |
| contents | We characterize all lattices $Λ\subset \mathbb{R}^2$ and all compactly supported functions $g \in L^2(\mathbb{R})$ for which the Gabor system $\left \{ e^{2πi s x} g(x-t) : (t,s) \in Λ\right \}$ forms an orthonormal basis for $L^2(\mathbb{R})$. The characterization is given in geometric terms through translation tilings and discreteness properties of lattice projections. In particular, this resolves a conjecture of Han and Wang on the non-existence of Gabor bases along specific irrational lattices. Finally, we construct Gabor bases that cannot be realized by any product set, answering a problem of Iosevich and Mayeli. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_29984 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Compactly supported Gabor orthonormal bases Liehr, Lukas Functional Analysis 42B10, 42C15, 46B15 We characterize all lattices $Λ\subset \mathbb{R}^2$ and all compactly supported functions $g \in L^2(\mathbb{R})$ for which the Gabor system $\left \{ e^{2πi s x} g(x-t) : (t,s) \in Λ\right \}$ forms an orthonormal basis for $L^2(\mathbb{R})$. The characterization is given in geometric terms through translation tilings and discreteness properties of lattice projections. In particular, this resolves a conjecture of Han and Wang on the non-existence of Gabor bases along specific irrational lattices. Finally, we construct Gabor bases that cannot be realized by any product set, answering a problem of Iosevich and Mayeli. |
| title | Compactly supported Gabor orthonormal bases |
| topic | Functional Analysis 42B10, 42C15, 46B15 |
| url | https://arxiv.org/abs/2605.29984 |