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On Gabor g-frames and Fourier series of operators

Volume 259 / 2021

Eirik Skrettingland Studia Mathematica 259 (2021), 25-78 MSC: 42C15, 47B38, 47G30, 47B10, 43A32. DOI: 10.4064/sm191115-24-9 Published online: 18 December 2020


We show that Hilbert–Schmidt operators can be used to define frame-like structures for $L^2(\mathbb{R}^d )$ over lattices in $\mathbb{R}^{2d} $ that include multi-window Gabor frames as a special case. These frame-like structures are called Gabor g-frames, since they are examples of g-frames as introduced by Sun. We show that Gabor g-frames share many properties of Gabor frames, including a Janssen representation and Wexler–Raz biorthogonality conditions. A central part of our analysis is a notion of Fourier series of periodic operators based on earlier work by Feichtinger and Kozek, where we show in particular a Poisson summation formula for trace class operators. By choosing operators from certain Banach subspaces of the Hilbert–Schmidt operators, Gabor g-frames give equivalent norms for modulation spaces in terms of weighted $\ell ^p$-norms of an associated sequence, as previously shown for localization operators by Dörfler, Feichtinger and Gröchenig.


  • Eirik SkrettinglandDepartment of Mathematics
    NTNU Norwegian University of Science and Technology
    NO–7491 Trondheim, Norway

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