Phase space distribution of Gabor expansions

TL;DR

This paper constructs a minimal and complete Gabor system using Gaussian shifts localized at phase space axes, demonstrating it has the lowest possible density among such systems under regular distribution constraints.

Contribution

It introduces a specific Gabor system with minimal density, localized at phase space axes, and compares its density to the von Neumann lattice.

Findings

Number of shifts in a disk is 2/pi times the von Neumann lattice points.

The system has minimal density among all complete Gaussian shift systems with regular distribution.

The constructed system is both complete and minimal in phase space.

Abstract

We present an example of a complete and minimal Gabor system consisting of time-frequency shifts of a Gaussian, localized at the coordinate axes in the time-frequency plane (phase space). Asymptotically, the number of time-frequency shifts contained in a disk centered at the origin is only 2/pi times the number of points from the von Neumann lattice found in the same disk. Requiring a certain regular distribution in phase space, we show that our system has minimal density among all complete and minimal systems of time-frequency shifts of a Gaussian.