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PreprintCC BY 4.0arXiv

Asymptotic safety regions for Gabor frames generated by Hermite functions

  • Markus Faulhuber
  • Irina Shafkulovska
  • Ilya Zlotnikov
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Abstract

The aim of this paper is to establish new regions in the frame sets of Hermite functions hn. A classical result of Gröchenig and Lyubarskii shows that the Gabor system 𝒢(hn,aℤ× bℤ) forms a frame for L2(ℝ) whenever the lattice density exceeds n+1. We show that, for every η>0 and all sufficiently large n, the same Gabor system forms a frame whenever ab≤ n−2/3−η. Moreover, we obtain an asymptotically sharp result near the coordinate axes, i.e., when one of the parameters a or b is small. Namely, for every δ>0 and ρ∈(0,1/2) and all sufficiently large n we prove that if min{a,b}≤ n−1/2−δ and ab≤ 1/2−ρ then 𝒢(hn,aℤ× bℤ) forms a frame.

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Cite

BibTeXRIS
BibTeX
@misc{faulhuber2026asymptotic,
  title = {{Asymptotic safety regions for Gabor frames generated by Hermite functions}},
  author = {Markus Faulhuber and Irina Shafkulovska and Ilya Zlotnikov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01296},
  eprint = {2609.01296},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01296},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Asymptotic safety regions for Gabor frames generated by Hermite functions
AU  - Markus Faulhuber
AU  - Irina Shafkulovska
AU  - Ilya Zlotnikov
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.01296
UR  - https://arxiv.org/abs/2609.01296
AB  - The aim of this paper is to establish new regions in the frame sets of Hermite functions h_n. A classical result of Gröchenig and Lyubarskii shows that the Gabor system 𝒢(h_n,aℤ× bℤ) forms a frame for L^2(ℝ) whenever the lattice density exceeds n+1. We show that, for every η>0 and all sufficiently large n, the same Gabor system forms a frame whenever ab≤ n^−2/3−η. Moreover, we obtain an asymptotically sharp result near the coordinate axes, i.e., when one of the parameters a or b is small. Namely, for every δ>0 and ρ∈(0,1/2) and all sufficiently large n we prove that if min{a,b}≤ n^−1/2−δ and ab≤ 1/2−ρ then 𝒢(h_n,aℤ× bℤ) forms a frame.
ER  -