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 $G=P⋊M,$ $M=\exp \mathfrak{m}$ 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 $L^2\left(M,d{\mu }_M\right).$

References in corpus (1)

• Shift-invariant spaces on SI/Z Lie groups