Frames arising from irreducible solvable actions Part I

TL;DR

This paper constructs explicit tight frames and frames in the context of irreducible representations of solvable Lie groups, without assuming integrability, using discrete subgroups and specific functions.

Contribution

It provides explicit constructions of frames from irreducible representations of solvable Lie groups, expanding the understanding of frame theory in non-integrable cases.

Findings

Explicit tight frames from irreducible representations of solvable Lie groups.

Construction of frames using discrete subgroups and smooth functions.

Frames are achieved without assuming the integrability of the representation.

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 $L^{2}\left( M,d\mu_{M}\right) .$