Bandlimited Spaces on Some 2-step Nilpotent Lie Groups With One Parseval Frame Generator

TL;DR

This paper investigates band-limited subspaces of L^2 on certain 2-step nilpotent Lie groups, characterizing when they admit Parseval frames and exploring discretization of wavelet transforms.

Contribution

It provides new criteria for the existence of Parseval frames generated by a single function in band-limited subspaces of L^2 on specific nilpotent Lie groups.

Findings

Characterization of band-limited subspaces with Parseval frames

Conditions for discretization of wavelet transforms

Explicit examples illustrating the theoretical results

Abstract

Let $N$ be a step two connected and simply connected non commutative nilpotent Lie group which is square-integrable modulo the center. Let $Z$ be the center of $N$. Assume that $N=P\rtimes M$ such that $P$, and $M$ are simply connected, connected abelian Lie groups, $M$ acts non-trivially on $P$ by automorphisms and $\dim P/Z=\dim M$. We study band-limited subspaces of $L^2(N)$ which admit Parseval frames generated by discrete translates of a single function. We also find characteristics of band-limited subspaces of $L^2(N)$ which do not admit a single Parseval frame. We also provide some conditions under which continuous wavelets transforms related to the left regular representation admit discretization, by some discrete set $\Gamma\subset N$. Finally, we show some explicit examples in the last section.