# Quantum harmonic analysis on lattices and Gabor multipliers

## TL;DR

This paper develops a quantum harmonic analysis framework on lattices that generalizes classical harmonic analysis and applies to Gabor multipliers, providing new theoretical tools for time-frequency analysis.

## Contribution

It introduces a novel quantum harmonic analysis on lattices, extending classical results and connecting to Gabor multipliers in time-frequency analysis.

## Key findings

- Established convolution and Fourier transform frameworks for operators on lattices.
- Proved analogues of classical harmonic analysis theorems in the quantum setting.
- Unified Gabor multipliers within this quantum harmonic analysis framework.

## Abstract

We develop a theory of quantum harmonic analysis on lattices in $\mathbb{R}^{2d}$. Convolutions of a sequence with an operator and of two operators are defined over a lattice, and using corresponding Fourier transforms of sequences and operators we develop a version of harmonic analysis for these objects. We prove analogues of results from classical harmonic analysis and the quantum harmonic analysis of Werner, including Tauberian theorems and a Wiener division lemma. Gabor multipliers from time-frequency analysis are described as convolutions in this setting. The quantum harmonic analysis is thus a conceptual framework for the study of Gabor multipliers, and several of the results include results on Gabor multipliers as special cases.