Balian-Low type theorems on homogeneous groups

Abstract

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 $(\\pi, \\mathcal{H}_{\\pi})$ be an irreducible, square-integrable representation modulo the center $Z(N)$ of $N$ on a Hilbert space $\\mathcal{H}_{\\pi}$ of formal dimension $d_\\pi $. If $g \\in \\mathcal{H}_{\\pi}$ is a phase-space localized vector and the set $\\{ \\pi (\\lambda )g : \\lambda \\in \\Lambda \\}$ for a discrete subset $\\Lambda \\subseteq N / Z(N)$ forms a frame for $\\mathcal{H}_{\\pi}$, then its density satisfies the strict inequality $D^-(\\Lambda )> d_\\pi $, where $D^-(\\Lambda )$ is the lower Beurling density. An analogous density condition $D^+(\\Lambda) < d_{\\pi}$ holds for a Riesz sequence in $\\mathcal{H}_{\\pi}$ contained in the orbit of $(\\pi, \\mathcal{H}_{\\pi})$. The proof is based on a deformation theorem for coherent systems, a universality result for coherent frames and Riesz sequences, some results from Banach space theory, and tools from the analysis on homogeneous groups.