Gabor System Based on the Unitary Dual of the Heisenberg Group

TL;DR

This paper introduces a Gabor system based on the unitary dual of the Heisenberg group and provides conditions for it to form various functional systems like frames and orthonormal bases in a specific Hilbert space.

Contribution

It develops new Gabor system constructions on the Heisenberg group and establishes criteria for their completeness and stability properties.

Findings

Provided a sufficient condition for the Gabor system to be a Bessel sequence.

Derived necessary and sufficient conditions for the Gabor system to be an orthonormal system.

Characterized when the Gabor system forms a Parseval frame, frame sequence, or Riesz sequence.

Abstract

In this paper Gabor system of certain type based on the unitary dual of the Heisenberg group $\mathbb{H}^n$ is introduced and a sufficient condition is obtained for the Gabor system to be a Bessel sequence for $L^2(\mathbb{R}^*,\mathcal{B}_2;d\kappa)$ using the $Schr\"{o}dinger$ representation of $\mathbb{H}^n$, where $\mathcal{B}_2$ denotes the class of Hilbert-Schmidt operators on $L^2(\mathbb{R}^n)$ and $d\kappa$ denotes the Haar measure on $\mathbb{R}^*$. Further a necessary and sufficient condition is provided for the Gabor system to be an orthonormal system, a Parseval frame sequence, a frame sequence and a Riesz sequence.