On Gabor g-frames and Fourier series of operators
Volume 259 / 2021
Abstract
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.
