Primitive tilings and coherent frames
Coherent states was first introduced by Schrödinger in the context of the quantum harmonic oscillators, in his investigation of classical to quantum transition. Coherent states brought about the area of study focused on coherent frames. These are sequences of the form \(\{\pi(\lambda) f\}_{\lambda \in \Lambda}\) in a Hilbert space \(\mathcal{H}\) based on a representation \(\pi\) of a group \(G\) acting on \(\mathcal{H}\), some window function \(f \in \mathcal{H}\), and a discrete subset \(\Lambda \subset G\), for which there exist constants \(A, B>0\) such that \[ A\|h\|^{2} \leq \sum_{\lambda \in \Lambda}|\langle h, \pi(\lambda) f\rangle|^{2} \leq B\|h\|^{2} \quad \text { for all } \quad h \in \mathcal{H}. \] Coherent frames are mostly characterized by conditions on the generating discrete subset. For instance, the Balian-Low theorem describes Gabor frames based on periodic lattices and provides a density condition on lattices. The objective of this paper is to characterize primitive tilings of Lie groups generating aperiodic coherent frames.
scientific article; zbMATH DE number 7392461
scientific article