On Gabor g-frames and Fourier series of operators

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, as 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.

arXiv commentv3: Typos and some technicalities fixed. v2: Smaller changes based on helpful input from anonymous referee. Accepted for publication in Studia Mathematica