The orthonormal dilation property for abstract Parseval wavelet frames

TL;DR

This paper establishes conditions under which Parseval wavelet frames associated with abstract groups can be dilated into orthonormal bases, with specific results for groups including the Heisenberg group.

Contribution

It introduces a dilation condition for Parseval wavelet frames in abstract group settings and proves it always holds for certain classes including the Heisenberg group.

Findings

Parseval frames can be dilated to orthonormal bases under certain conditions.

The dilation condition always holds for groups with Heisenberg subgroups.

Applications include familiar wavelet systems in abstract group contexts.

Abstract

In this work we introduce a class of discrete groups containing subgroups of abstract translations and dilations, respectively. A variety of wavelet systems can appear as $\pi(\G)\psi$, where $\pi$ is a unitary representation of a wavelet group and $\G$ is the abstract pseudo-lattice $\G$. We prove a condition in order that a Parseval frame $\pi(\G)\psi$ can be dilated to an orthonormal basis of the form $\tau(\G)\Psi$ where $\tau$ is a super-representation of $\pi$. For a subclass of groups that includes the case where the translation subgroup is Heisenberg, we show that this condition always holds, and we cite familiar examples as applications.