Compactly supported bounded frames on Lie groups

TL;DR

This paper constructs compactly supported, smooth frames on certain Lie groups using induced representations, extending previous work and enabling well-localized frames relevant to time-frequency analysis and wavelet theories.

Contribution

It proves the existence of compactly supported, smooth frames for induced representations on Lie groups, extending prior results to more general groups and solving an open problem.

Findings

Existence of countable sets and functions forming frames on Lie groups

Construction of continuous and smooth functions for frames

Extension of previous work to general connected Lie groups

Abstract

Let $G=NH$ be a Lie group where $N,H$ are closed connected subgroups of $G,$ and $N$ is an exponential solvable Lie group which is normal in $G.$ Suppose furthermore that $N$ admits a unitary character $\chi_{\lambda}$ corresponding to a linear functional $\lambda$ of its Lie algebra. We assume that the map $h\mapsto Ad\left( h^{-1}\right) ^{\ast}\lambda$ defines an immersion at the identity of $H$. Fixing a Haar measure on $H,$ we consider the unitary representation $\pi$ of $G$ obtained by inducing $\chi_{\lambda}.$ This representation which is realized as acting in $L^{2}\left( H,d\mu_{H}\right) $ is generally not irreducible, and we do not assume that it satisfies any integrability condition. One of our main results establishes the existence of a countable set $\Gamma\subset G$ and a function $\mathbf{f}\in L^{2}\left( H,d\mu_{H}\right) $ which is compactly supported and bounded such that $\left\{ \pi\left( \gamma\right) \mathbf{f}:\gamma\in\Gamma\right\} $ is a frame. Additionally, we prove that $\mathbf{f}$ can be constructed to be continuous. In fact, $\mathbf{f}$ can be taken to be as smooth as desired. Our findings extend the work started in \cite{oussa2018frames} to the more general case where $H$ is any connected Lie group. We also solve a problem left open in \cite{oussa2018frames}. Precisely, we prove that in the case where $H$ is an exponential solvable group, there exist a continuous (or smooth) function $\mathbf{f}$ and a countable set $\Gamma$ such that $\left\{ \pi\left( \gamma\right) \mathbf{f}:\gamma\in\Gamma\right\} $ is a Parseval frame. Since the concept of well-localized frames is central to time-frequency analysis, wavelet, shearlet and generalized shearlet theories, our results are relevant to these topics and our approach leads to new constructions which bear potential for applications.