We study the upper and the lower densities of complete and minimal Gabor Gaussians systems. In contrast to the classical lattice case when they are both equal to 1, we prove that the lower density may reach 0 while the upper density may vary at least from $\frac{1}{\pi }$ to e. In the case when the upper density exceeds 1, we establish a sharp inequality relating the upper and the lower densities.

中文翻译：

---

Gabor高斯系统的上下密度

我们研究完整和最小Gabor高斯系统的上，下密度。与经典格情况相反，当它们都等于1时，我们证明较低的密度可以达到0，而较高的密度至少可以从$\frac{\mathrm{1个}}{\pi }$到e。在上限密度超过1的情况下，我们建立了与上限密度和下限密度相关的尖锐不等式。