On the transcendentality condition for Gaussian Gabor frames and Hermite super/multiwindow frames

TL;DR

This paper establishes a new density criterion for Gaussian Gabor frames in higher dimensions, extending previous results to irrational lattices, and explores conditions for uniqueness and super/multi-window frames with Hermitian windows.

Contribution

It provides a reformulated criterion for Gaussian Gabor frames, extends results to irrational lattices, and analyzes super/multi-window frames with Hermitian windows in the Bargmann-Fock space.

Findings

Density criterion for Gaussian Gabor frames in higher dimensions.

Extension of results to irrational lattices.

Conditions for uniqueness in the Bargmann-Fock space.

Abstract

We give a criterion for higher-dimensional Gaussian Gabor frames, which is a reformulation of one of the main results in in a previous article by the first and last authors in more explicit terms. We use this formulation in order to extend a result of Romero, Ulanovskii, and Zlotnikov to lattices given by irrational rotations. We also show that this density criterion for Gaussian Gabor frames is generic in a certain sense. In addition, we also extend the methods of the first and last named authors to the pseudoeffective threshold which gives a condition for uniqueness in the Bargmann-Fock space. We also use this viewpoint to study super and multi-window Gabor frames with Hermitian windows. In particular, we find a density criterion for transcendental lattices.