Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces

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The Mystery of Gabor Frames (2014, methodology): “Janssen left many white spots on the tie, where he could not determine whether G(χ[0,1], αZ× βZ) is a frame or not; he observed that the frame property seems to depend on number theoretic properties of α, β.”
Sampling Expansions in Reproducing Kernel Hilbert and Banach Spaces (2009, background): “For sampling and reconstruction of signals in a shift-invariant space, the reader may refer to [ 2 , 4 , 21 , 23 ] and references therein.”
Fusion frames and distributed processing (2006, unknown): “A similar idea was also used by Aldroubi and Gr¨oching in a quite diﬀerent context in [2].”
SLANTED MATRICES, BANACH FRAMES, AND SAMPLING (2007, methodology): “Now we can prove a Beurling-Landau type theorem [1, 3 , 4, 5] for shift-invariant spaces generated by piecewise dierentiable functions.”
Sampling-50 years after Shannon (2000, background): “Aldroubi and Gröchenig derived generalized versions of the Beurling–Landau theorems based on an appropriate definition of the sampling density [6].”
Localization of Frames, Banach Frames, and the Invertibility of the Frame Operator (2004, methodology): “are being used [ 5 , 6]. The reconstruction in the associated Banach spaces and the stability of this reconstruction then follow automatically.”