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Gabor Frame Regions and Non-Frame Obstructions for an even Rational Window

  • Riya Ghosh
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Abstract

We study the Gabor frame properties of the rational window g(x)=(x2−1)/((x2+1)(x2+4)(x2+9)), whose poles occur in symmetric pairs. We prove that every lattice satisfying 0<αβ<1/3 generates a frame, and we establish an additional frame region for β≥1 and 1/3≤ αβ<0.47373. Furthermore, we show that the rational hyperbolas αβ=p/(3p−1), for p≥2, are entirely contained within the frame set. In contrast, we construct explicit non-frame lattice points on the hyperbolas αβ∈{1/3,1/2,2/3,3/4}. Finally, for the density family αβ=p/(p+1), we derive a symmetry reduction of the associated Zibulski-Zeevi matrix, providing numerical evidence for a richer structure of non-frame obstructions.

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Cite

BibTeXRIS
BibTeX
@misc{ghosh2026gabor,
  title = {{Gabor Frame Regions and Non-Frame Obstructions for an even Rational Window}},
  author = {Riya Ghosh},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.19232},
  eprint = {2609.19232},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.19232},
}
RIS
TY  - UNPB
TI  - Gabor Frame Regions and Non-Frame Obstructions for an even Rational Window
AU  - Riya Ghosh
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.19232
UR  - https://arxiv.org/abs/2609.19232
AB  - We study the Gabor frame properties of the rational window g(x)=(x^2−1)/((x^2+1)(x^2+4)(x^2+9)), whose poles occur in symmetric pairs. We prove that every lattice satisfying 0<αβ<1/3 generates a frame, and we establish an additional frame region for β≥1 and 1/3≤ αβ<0.47373. Furthermore, we show that the rational hyperbolas αβ=p/(3p−1), for p≥2, are entirely contained within the frame set. In contrast, we construct explicit non-frame lattice points on the hyperbolas αβ∈{1/3,1/2,2/3,3/4}. Finally, for the density family αβ=p/(p+1), we derive a symmetry reduction of the associated Zibulski-Zeevi matrix, providing numerical evidence for a richer structure of non-frame obstructions.
ER  -