Systems of Left Translates and Oblique Duals on the Heisenberg Group

Abstract

In this paper, we characterize the system of left translates $\{L_{(2k,l,m)}g:k,l,m\in \mathbb{Z}\}$, $g\in L^2(\mathbb{H})$, to be a frame sequence or a \emph{Riesz} sequence in terms of the twisted translates of the corresponding function $g^{\lambda }$. Here, $(\mathbb{H}$ denotes the Heisenberg group and $g^{\lambda }$ the inverse Fourier transform of $g$ with respect to the central variable. This type of characterization for a \emph{Riesz} sequence allows us to find some concrete examples. We also study the structure of the oblique dual of the system of left translates $\{L_{(2k,l,m)}g:k,l,m\in \mathbb{Z}\}$ on $(\mathbb{H}$. This result is also illustrated with an example.

arXiv admin note: substantial text overlap with arXiv:2105.07707