Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Caragea, Andrei, Pfander, Götz
Institution: Catholic University of Eichstätt-Ingolstadt
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
Award: G529166646
OpenAlex ID: W7212087280