Criteria for the existence of Schwartz Gabor frames over rational lattices

TL;DR

This paper establishes explicit criteria for when rational lattices in the time-frequency plane admit Schwartz-class Gabor frames, linking lattice properties to frame existence and providing bounds for multi-window frames.

Contribution

It introduces a new explicit inequality criterion based on lattice covolume and non-integrality, advancing understanding of Gabor frame existence over rational lattices.

Findings

Derived an explicit criterion for Schwartz Gabor frames over rational lattices.

Provided an upper bound on the number of Schwartz windows needed for multi-window frames.

Linked lattice properties to frame existence through a new inequality.

Abstract

We give an explicit criterion for a rational lattice in the time-frequency plane to admit a Gabor frame with window in the Schwartz class. The criterion is an inequality formulated in terms of the lattice covolume, the dimension of the underlying Euclidean space, and the index of an associated subgroup measuring the degree of non-integrality of the lattice. For arbitrary lattices we also give an upper bound on the number of windows in the Schwartz class needed for a multi-window Gabor frame.