Mesoscopic redundancy for dynamical frames generated by atomic singular model operators

Abstract

Consider the compressed shift $T=P_{K_{\theta }}{M}_z{|}_{K_{\theta }}$ on $K_{\theta }=H^2\ominus \theta H^2$ associated with the atomic singular inner function $\theta (z)=\exp (-a(\zeta +z)/(\zeta -z))$, $a>0$ and $|\zeta |=1$. Let $k_0=P_{K_{\theta }}1$ and define real powers $T^s$ using a holomorphic logarithm near the singleton spectrum of $T$. For any temporal multisequence $\mathsf{S}=(s_k{)}_{k\in I}\subset [0,\infty )$ with uniformly bounded numbers of samples in unit intervals, we prove that $\{T^{s_k}{k}_0:k\in I\}$ is a frame if and only if $\mathsf{S}$ is Kummer-thick: the frequencies $\omega (s_k)=\sqrt{8s_k+4}$, weighted by $1/\omega (s_k)$, have uniformly positive total mass in every sufficiently distant interval of some fixed length. Equivalently, the occupied unit cells fill a fixed positive proportion of every window $[N-C\sqrt{N},N+C\sqrt{N}]$ for some $C>0$ and all sufficiently large $N$. The proof combines a Volterra transmutation into Bessel waves with the Fourier--Bessel Logvinenko--Sereda theorem.

16 pages, 2 figures