Balian-Low type theorems on homogeneous groups

We prove strict necessary density conditions for coherent frames and Riesz sequences on homogeneous groups. Let $N$ be a connected, simply connected nilpotent Lie group with a dilation structure (a homogeneous group) and let $(π, \mathcal{H}_π)$ be an irreducible, square-integrable representation modulo the center $Z(N)$ of $N$ on a Hilbert space $\mathcal{H}_π$ of formal dimension $d_π$. If $g \in \mathcal{H}_π$ is an integrable vector and the set $\{ π(λ)g : λ\in Λ\}$ for a discrete subset $Λ\subseteq N / Z(N)$ forms a frame for $\mathcal{H}_π$, then its density satisfies the strict inequality $D^-(Λ)> d_π$, where $D^-(Λ)$ is the lower Beurling density. An analogous density condition $D^+(Λ) < d_π$ holds for a Riesz sequence in $\mathcal{H}_π$ contained in the orbit of $(π, \mathcal{H}_π)$. The proof is based on a deformation theorem for coherent systems, a universality result for $p$-frames and $p$-Riesz sequences, some results from Banach space theory, and tools from the analysis on homogeneous groups.

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