Remarks on multivariate Gaussian Gabor frames (Q382220)

Let \(g_d\) be the \(d\)-dimensional normalized Gaussian \(2^{\frac{d}{4}}e^{-\pi \|t\|^2}\), \(t\in\mathbb R^d\); \(\Lambda\subset \mathbb R^{2d}\) be a full rank lattice; \((g,\Lambda)\) be a Gabor system; \(D(\Lambda)\) be a density of \(\Lambda\). In the one-dimension case the Gabor system \((g_1,\Lambda)\) is a frame if and only if \(D(\Lambda)>1\). The authors prove that this does not extend to higher dimensions.