Mesoscopic redundancy for dynamical frames generated by atomic singular model operators
Abstract
Consider the compressed shift T=PKθMz|Kθ on Kθ=H2ominusθH2 associated with the atomic singular inner function θ(z)=exp(−a(ζ+z)/(ζ−z)), a>0 and |ζ|=1. Let k0=PKθ1 and define real powers Ts using a holomorphic logarithm near the singleton spectrum of T. For any temporal multisequence S=(sk)k∈ I⊂[0,∞) with uniformly bounded numbers of samples in unit intervals, we prove that {Tskk0:k∈ I} is a frame if and only if S is Kummer-thick: the frequencies ω(sk)=√(8sk+4), weighted by 1/ω(sk), 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√N,N+C√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.
Related
- The index of mesoscopic Wilson--Dirac operators: I
- Two pointwise estimates for singular integral operators over p-adic groups
- Weak type (1,1) boundedness of noncommutative singular integral operators with rough kernels
- Essentially semi-commuting singular integral operators with Cauchy kernel on L2
- Random dynamical systems generated by rough differential equations driven by controlled rough paths
BibTeX
@misc{krishtal2026mesoscopic,
title = {{Mesoscopic redundancy for dynamical frames generated by atomic singular model operators}},
author = {Ilya Krishtal and Javad Mashreghi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27255},
eprint = {2609.27255},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27255},
}
RIS
TY - UNPB
TI - Mesoscopic redundancy for dynamical frames generated by atomic singular model operators
AU - Ilya Krishtal
AU - Javad Mashreghi
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27255
UR - https://arxiv.org/abs/2609.27255
AB - Consider the compressed shift T=P_K_θM_z|_K_θ on K_θ=H^2ominusθH^2 associated with the atomic singular inner function θ(z)=exp(−a(ζ+z)/(ζ−z)), a>0 and |ζ|=1. Let k_0=P_K_θ1 and define real powers T^s using a holomorphic logarithm near the singleton spectrum of T. For any temporal multisequence S=(s_k)_k∈ I⊂[0,∞) with uniformly bounded numbers of samples in unit intervals, we prove that {T^s_kk_0:k∈ I} is a frame if and only if S is Kummer-thick: the frequencies ω(s_k)=√(8s_k+4), weighted by 1/ω(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√N,N+C√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.
ER -