Systems of Left Translates and Oblique Duals on the Heisenberg Group

TL;DR

This paper characterizes systems of left translates on the Heisenberg group as frames or Riesz sequences using twisted translates, and explores the structure of oblique duals, providing concrete examples and illustrations.

Contribution

It provides a new characterization of left translate systems as frames or Riesz sequences via twisted translates on the Heisenberg group, and analyzes oblique dual structures.

Findings

Characterization of left translate systems as frames or Riesz sequences

Concrete examples of Riesz sequences

Analysis of 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.