Gabor systems in C^N with subgroup modulation or translation sets have frame operators that are unitarily equivalent to block-diagonal matrices, with further diagonal and sparsity results under geometric conditions.
Gr¨ ochenig,Foundations of Time-Frequency Analysis, Birkh¨ auser, Boston
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Compactness is established for pseudo-differential operators whose symbols lie in the refined modulation space M^{sharp,q} (0 < q ≤ 1) when acting on a broad family of modulation spaces.