Extensions of Daubechies' theorem: Reinhardt domains, Hagedorn wavepackets and mixed-state localization operators

TL;DR

This paper extends Daubechies' theorem to multivariate quantum settings using Hagedorn wavepackets and Reinhardt domains, establishing new orthogonality results and linking to Toeplitz operators in quantum harmonic analysis.

Contribution

It introduces Daubechies-type theorems for mixed-state localization operators in quantum harmonic analysis, utilizing Reinhardt domains and Hagedorn wavepackets as key tools.

Findings

Daubechies-type theorems are established for multivariate localization operators.

Reinhardt domains are identified as the natural class of masks for these theorems.

Double orthogonality results are extended to the quantum setting, connecting to Toeplitz operators.

Abstract

Daubechies-type theorems for localization operators are established in the multi-variate setting, where Hagedorn wavepackets are identified as the proper substitute of the Hermite functions. The class of Reinhardt domains is shown to be the natural class of masks that allow for a Daubechies-type result. Daubechies' classical theorem is a consequence of double orthogonality results for the short-time Fourier transform. We extend double orthogonality to the quantum setting and use it to establish Daubechies-type theorems for mixed-state localization operators, a key notion of quantum harmonic analysis. Lastly, we connect the results to Toeplitz operators on quantum Gabor spaces.