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We study the space spanned by the integer shifts of a bivariate Gaussianfunction and the problem of reconstructing any function in that space fromsamples scattered across the plane. We identify a large class of lattices, ormore generally semi-regular sampling patterns spread along parallel lines, thatlead to stable reconstruction while having densities close to the criticalvalue given by Landau's limit. At the critical density, we construct examplesof sampling patterns for which reconstruction fails.  In the same vein, we also investigate continuous sampling along non-uniformlyscattered families of parallel lines and identify the threshold density of lineconfigurations at which reconstruction is possible. In a remarkable contrastwith Paley-Wiener spaces, the results are completely different for lines withrational or irrational slopes.  Finally, we apply the sampling results to Gabor systems with bivariateGaussian windows. As a main contribution, we provide a large list of newexamples of Gabor frames with non-complex lattices having volume close to 1.

Sampling in the shift-invariant space generated by the bivariate Gaussian function