Frames arising from irreducible solvable actions Part I

Abstract

Let $G$ be a simply connected, connected completely solvable Lie group with Lie algebra $\mathfrak{g}=\mathfrak{p}+\mathfrak{m}.$ Next, let $\pi$ be an infinite-dimensional unitary irreducible representation of $G$ obtained by inducing a character from a closed normal subgroup $P=\exp\mathfrak{p}$ of $G.$ Additionally, we assume that $G=P\rtimes M,$ $M=\exp\mathfrak{m}$ is a closed subgroup of $G,$ $d\mu_{M}$ is a fixed Haar measure on the solvable Lie group $M$ and there exists a linear functional $\lambda\in\mathfrak{p}^{\ast}$ such that the representation $\pi=\pi_{\lambda}=\mathrm{ind}_{P}^{G}\left( \chi_{\lambda}\right) $ is realized as acting in $L^{2}\left( M,d\mu_{M}\right) .$ Making no assumption on the integrability of $\pi_{\lambda}$, we describe explicitly a discrete subgroup $\Gamma\subset G$ and a vector $\mathbf{f}\in L^{2}\left( M,d\mu_{M}\right) $ such that $\pi_{\lambda }\left( \Gamma\right) \mathbf{f}$ is a tight frame for $L^{2}\left( M,d\mu_{M}\right) .$ We also construct compactly supported smooth functions $\mathbf{s}$ and discrete subsets $\Gamma\subset G$ such that $\pi_{\lambda }\left( \Gamma\right) \mathbf{s}$ is a frame for $L^{2}\left( M,d\mu_{M}\right) .$