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.
Heil, A Basis Theory Primer, Expanded edition, Birkh¨ auser, New York
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Block-equivalent finite Gabor frames
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.