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PreprintCC BY 4.0arXiv

The frame set of the first Hermite function

  • Markus Faulhuber
  • Philipp Petersen
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Abstract

We determine the frame set of the first Hermite function, thereby proving a conjecture of Lyubarskii and Nes. A characterization of semi-regular Gabor frames due to Gröchenig, Romero, and Stöckler reduces the problem to a uniqueness question for entire functions in a Gaussian shift-invariant space. We address this question by forming Wronskians of finitely many translates of a Gaussian shift-invariant function. Their common critical points become zeros of increasing multiplicity, while automorphy shows that the Wronskians remain within a Gaussian shift-invariant class. A sharp zero-density theorem then rules out all possible failures of the frame property except at the known rational obstructions. This yields the complete rectangular frame set of the first Hermite function and introduces Wronskian amplification as a new method in Gabor frame theory.

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Cite

BibTeXRIS
BibTeX
@misc{faulhuber2026frame,
  title = {{The frame set of the first Hermite function}},
  author = {Markus Faulhuber and Philipp Petersen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.02610},
  eprint = {2609.02610},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02610},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The frame set of the first Hermite function
AU  - Markus Faulhuber
AU  - Philipp Petersen
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.02610
UR  - https://arxiv.org/abs/2609.02610
AB  - We determine the frame set of the first Hermite function, thereby proving a conjecture of Lyubarskii and Nes. A characterization of semi-regular Gabor frames due to Gröchenig, Romero, and Stöckler reduces the problem to a uniqueness question for entire functions in a Gaussian shift-invariant space. We address this question by forming Wronskians of finitely many translates of a Gaussian shift-invariant function. Their common critical points become zeros of increasing multiplicity, while automorphy shows that the Wronskians remain within a Gaussian shift-invariant class. A sharp zero-density theorem then rules out all possible failures of the frame property except at the known rational obstructions. This yields the complete rectangular frame set of the first Hermite function and introduces Wronskian amplification as a new method in Gabor frame theory.
ER  -