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Frames arising from irreducible solvable actions Part I

Research paper by Vignon Oussa

Indexed on: 18 Dec '16Published on: 18 Dec '16Published in: arXiv - Mathematics - Functional Analysis



Abstract

Let $G$ be a completely solvable Lie group and let $\pi$ be an infinite-dimensional unitary irreducible representation of $G$ obtained by inducing a character from a closed normal subgroup $P$ of $G.$ Additionally, we assume that $G=P\rtimes M,$ $M$ 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 $\Gamma\subset G$ and $\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 $L^{2}\left( M,d\mu _{M}\right) .$