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PreprintCC BY 4.0arXiv

The frameset of the second-order B-Spline along ab=1/q

  • Alexander Stangl
  • Christina Frederick
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Abstract

We give a complete characterization of the zero set of the Zak transform of the second-order B-spline Q2. As a consequence, we obtain a new proof of the Lemvig--Nielsen obstructions to the Gabor frame property of Q2, and, along the hyperbola ab=1/q, we show these obstructions are sharp: every point they do not exclude belongs to the frame set. This gives the first complete characterization of ℱ(Q2) along an entire curve in this family, resolving a question left open by Lemvig and Nielsen.

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Cite

BibTeXRIS
BibTeX
@misc{stangl2026frameset,
  title = {{The frameset of the second-order B-Spline along $ab=1/q$}},
  author = {Alexander Stangl and Christina Frederick},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00481},
  eprint = {2609.00481},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00481},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The frameset of the second-order B-Spline along ab=1/q
AU  - Alexander Stangl
AU  - Christina Frederick
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.00481
UR  - https://arxiv.org/abs/2609.00481
AB  - We give a complete characterization of the zero set of the Zak transform of the second-order B-spline Q_2. As a consequence, we obtain a new proof of the Lemvig--Nielsen obstructions to the Gabor frame property of Q_2, and, along the hyperbola ab=1/q, we show these obstructions are sharp: every point they do not exclude belongs to the frame set. This gives the first complete characterization of ℱ(Q_2) along an entire curve in this family, resolving a question left open by Lemvig and Nielsen.
ER  -