On Gabor g-frames and Fourier series of operators

2019/06/23 by Skrettingland, Eirik · 1 citation
#35S05 #42C15 #43A32 #47B10 #47B38 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1906.09662

Abstract

We show that Hilbert-Schmidt operators can be used to define frame-like structures for L²(ℝ^d) over lattices in 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 ℓ^p-norms of an associated sequence, as previously shown for localization operators by Dörfler, Feichtinger and Gröchenig.

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