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PreprintCC BY 4.0arXiv

Derivatives of Theta Functions and a Problem of Lyubarskii and Nes

  • Lukas Liehr
  • Irina Shafkulovska
  • Mitchell A. Taylor
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Abstract

We characterize all lattices Λ⊂ ℝ2 of rational density for which the Gabor system generated by the first Hermite function along Λ is a frame for L2(ℝ). We prove that these are precisely the lattices whose density satisfies $D(Λ) = \frac{q}{p} $ where p,q ∈ ℕ are coprime with q ≥ p+2, thereby confirming a conjecture of Lyubarskii and Nes. A formalization of our main result in Lean $4$ is also included.

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Cite

BibTeXRIS
BibTeX
@misc{liehr2026derivatives,
  title = {{Derivatives of Theta Functions and a Problem of Lyubarskii and Nes}},
  author = {Lukas Liehr and Irina Shafkulovska and Mitchell A. Taylor},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27846},
  eprint = {2609.27846},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27846},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Derivatives of Theta Functions and a Problem of Lyubarskii and Nes
AU  - Lukas Liehr
AU  - Irina Shafkulovska
AU  - Mitchell A. Taylor
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.27846
UR  - https://arxiv.org/abs/2609.27846
AB  - We characterize all lattices Λ⊂ ℝ^2 of rational density for which the Gabor system generated by the first Hermite function along Λ is a frame for L^2(ℝ). We prove that these are precisely the lattices whose density satisfies $D(Λ) = \frac{q}{p} $ where p,q ∈ ℕ are coprime with q ≥ p+2, thereby confirming a conjecture of Lyubarskii and Nes. A formalization of our main result in Lean $4$ is also included.
ER  -