Götz E. Pfander, Peter Rashkov

• Constructive design of Gabor frame windows is rare, and most results come from the one-dimensional case. The connection between the geometry of fundamental domains of lattices and Gabor systems was explored first in a series of papers by Han and Wang [HW01], [HW04]. We build upon these results to construct Gabor frames with smooth and compactly supported window functions in higher dimensions. For this purpose we study pairs of lattices with equal density allowing compact and star-shaped fundamental domains. Concrete examples are provided and the results are extended to other special class of lattices. In addition, we make observations on the intricate behavior of Gabor systems with multivariate Gaussian windows.