Compactly supported bounded frames on Lie groups

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.