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The geometry of fundamental domains of lattices was used by Han and Wang toconstruct multivariate Gabor frames for separable lattices. We build upon theirresults to obtain Gabor frames with smooth and compactly supported windowfunctions. For this purpose we study pairs of lattices which have equal densityand allow for a common compact and star-shaped fundamental domain. The resultsare then extended to a larger class of lattices via symplectic equivalence.

A geometric construction of tight Gabor frames with multivariate compactly supported smooth windows