# On Gabor g-frames and Fourier series of operators

## TL;DR

This paper introduces Gabor g-frames, a new class of frame-like structures for $L^2(R^d)$ based on Hilbert-Schmidt operators, extending Gabor frames with properties like Janssen representation and biorthogonality, and connects them to Fourier series of operators.

## Contribution

It defines Gabor g-frames using Hilbert-Schmidt operators, establishing their properties and relation to Fourier series of operators, extending Gabor frame theory.

## Key findings

- Gabor g-frames include multi-window Gabor frames as a special case.
- They satisfy Janssen representation and Wexler-Raz biorthogonality conditions.
- Gabor g-frames provide equivalent norms for modulation spaces using weighted $ell^p$-norms.

## 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 D\"orfler, Feichtinger and Gr\"ochenig.