# On the Parity under Metapletic Operators and an Extension of a Result of Lyubarskii and Nes

## TL;DR

This paper investigates how the parity of the window function in Gabor frames influences their properties, showing that the frame property depends only on parity and extending previous results to more general lattices using symplectic and metaplectic group theory.

## Contribution

It demonstrates that the frame property of Gabor frames with windows in Feichtinger's algebra depends solely on window parity and extends non-frame results to non-separable lattices.

## Key findings

- Frame property depends only on window parity.
- Generalization of non-frame results to non-separable lattices.
- Utilization of symplectic and metaplectic group interplay.

## Abstract

In this work we show that if the frame property of a Gabor frame with window in Feichtinger's algebra and a fixed lattice only depends on the parity of the window, then the lattice can be replaced by any other lattice of the same density without losing the frame property. As a byproduct we derive a generalization of a result of Lyubarskii and Nes, who could show that any Gabor system consisting of an odd window function from Feichtinger's algebra and any separable lattice of density $\frac{n+1}{n}$, $n \in \mathbb{N}_+$, cannot be a Gabor frame for the Hilbert space of square-integrable functions on the real line. We extend this result by removing the assumption that the lattice has to be separable. This is achieved by exploiting the interplay between the symplectic and the metaplectic group.