Gabor System Based on the Unitary Dual of the Heisenberg Group

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 $L^2({\mathbb{R}}^{\ast },{\mathcal{B}}_2;d\kappa )$ using the $Schr\ddot{o}dinger$ 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 ${\mathbb{R}}^{\ast }$. 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.

27 pages

References in corpus (1)

• Shift-invariant Spaces with Countably Many Mutually Orthogonal Generators on the Heisenberg group