Reproducing pairs and Gabor systems at critical density

TL;DR

This paper extends the theory of Gabor systems at critical density by introducing reproducing pairs, proving the existence of a reproducing partner for Gaussian shifts, and generalizing the Balian-Low theorem within this framework.

Contribution

It presents a novel application of reproducing pairs to Gabor systems at critical density, including the existence of a reproducing partner for Gaussian shifts and a generalized Balian-Low theorem.

Findings

Existence of a reproducing partner for Gaussian Gabor systems.

Generalization of the Balian-Low theorem to reproducing pairs.

Solution to an open problem regarding Gabor expansions at critical density.

Abstract

We use the concept of reproducing pairs to study Gabor systems at critical density. First, we present a generalization of the Balian-Low theorem to the reproducing pairs setting. Then, we prove our main result that there exists a reproducing partner for the Gabor system of integer time-frequency shifts of the Gaussian. In other words, the coefficients for this Gabor expansion of a square integrable function can be calculated using inner products with an unstructured family of vectors in $ L^2(\R)$. This solves the possibly last open question for this system.