# Frames arising from irreducible solvable actions Part I

Research paper by **Vignon Oussa**

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

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#### 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 $L^{2}\left( M,d\mu
_{M}\right) .$