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Schwartz-Class Gabor Windows and the Balian-Low Classification

  • Andrei Caragea
  • Götz Pfander
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Abstract

A lattice Gabor frame consists of a window function and its time-frequency shifts along a lattice. We classify the lattices that admit Schwartz-class and, equivalently, Feichtinger frame windows. Except for symplectically irrational lattices of critical covolume 1,, we determine the lattice-dependent achievable window smoothness and decay using the scale of modulation spaces. For symplectically rational lattices, including the rational lattices typically used in digital applications, the classification is governed by an integer lattice parameter introduced herein, the symplectic index gap. Our results extend the classical and amalgam Balian-Low theorems. The companion paper "Tilings, Packings, and Smooth and Compactly Supported Gabor Windows" develops the associated tiling-packing theorem and compact-support constructions.

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Cite

BibTeXRIS
BibTeX
@misc{caragea2026schwartzclass,
  title = {{Schwartz-Class Gabor Windows and the Balian-Low Classification}},
  author = {Andrei Caragea and Götz Pfander},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06255},
  eprint = {2609.06255},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06255},
}
RIS
TY  - UNPB
TI  - Schwartz-Class Gabor Windows and the Balian-Low Classification
AU  - Andrei Caragea
AU  - Götz Pfander
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.06255
UR  - https://arxiv.org/abs/2609.06255
AB  - A lattice Gabor frame consists of a window function and its time-frequency shifts along a lattice. We classify the lattices that admit Schwartz-class and, equivalently, Feichtinger frame windows. Except for symplectically irrational lattices of critical covolume 1,, we determine the lattice-dependent achievable window smoothness and decay using the scale of modulation spaces. For symplectically rational lattices, including the rational lattices typically used in digital applications, the classification is governed by an integer lattice parameter introduced herein, the symplectic index gap. Our results extend the classical and amalgam Balian-Low theorems. The companion paper "Tilings, Packings, and Smooth and Compactly Supported Gabor Windows" develops the associated tiling-packing theorem and compact-support constructions.
ER  -