History and Evolution of the Density Theorem for Gabor Frames

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Deformation of Gabor systems (2013, methodology): “We refer to Heil’s detailed survey [27] of the numerous contributions to the density theorem for Gabor frames after [31] and to [13] for the Balian-Low theorem.”
A Survey on Multicarrier Communications: Prototype Filters, Lattice Structures, and Implementation Aspects (2012, background): “development of Gabor systems from the mathematical point of view may refer to the surveys in [3], [7]–[10].”
The Mystery of Gabor Frames (2014, background): “For a detailed exposition and complete references we refer to the books and surveys [10,30,31,36, 51,52].”
Sampling and interpolation in Bargmann-Fock spaces of polyanalytic functions (2009, background): “See the survey article [28], [13] and the important special windows studied in [35].”
Gabor Frames and Totally Positive Functions (2011, unknown): “The fundamental density theorem asserts that F(g) is always a subset of {(α, β) ∈ R(2)+ : αβ ≤ 1} [10,14,17].”
Gabor Frames in Finite Dimensions (2013, background): “1 Prior to the work of Gabor, von Neumann postulated that the function family which is now referred to as Gaussian Gabor system is complete [68] (see the respective discussions in [45, 48]).”