Finite-density time sampling of an infinite observation window
Let $\mathcal H$ be a separable complex Hilbert space, let $A$ be a bounded normal operator on $\mathcal H$, and let $(g_j)_{j \in J}$ be a countable Bessel family, $\sum_j |\langle f, g_j\rangle|^2 \le B_0 \|f\|^2$. Define $A^t$, $t \ge 0$, by the spectral calculus with $z^t = |z|^t e^{it\arg z}$, $\arg z \in (-\pi, \pi]$, $0^t = 0$ for $t > 0$, and $A^0 = I$. Suppose that $$m\|f\|^2 \le \sum_j \int_0^\infty |\langle f, A^t g_j\rangle|^2 \, dt \le M\|f\|^2 \qquad (f \in \mathcal H)$$ for some $0 < m \le M < \infty$. Aldroubi, Huang and Petrosyan proved that if such an inequality holds with the integral over a finite window $[0, L]$, then it persists, with other constants, when the integral is replaced by a sum over any finite set of times $0 = t_1 < \dots < t_{n+1} = L$ of sufficiently small mesh. For exponentially stable strongly continuous semigroups the analogous discretization holds on $[0, \infty)$. For a locally finite set $T \subset [0, \infty)$ let $$D^+(T) = \limsup_{L \to \infty} \sup_{a \ge 0} \frac{\#(T \cap [a, a + L])}{L}$$ be its upper Beurling density.
Problem. Must there be a locally finite set $T \subset [0, \infty)$ with $D^+(T) < \infty$ and constants $0 < m_T \le M_T < \infty$ such that $$m_T\|f\|^2 \le \sum_{t \in T} \sum_j |\langle f, A^t g_j\rangle|^2 \le M_T\|f\|^2 \qquad (f \in \mathcal H)?$$ No time-dependent weights may be inserted in the discrete sum.
Source: Aldroubi, Cabrelli, Krishtal and Molter (2026), Open Problem 1 and Section 5 (B), "Discretization on unbounded windows". Stated by AIM — Open Applied Problems for bounded normal operators and Bessel families.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- CA
- Keywords
- AIM Open Applied Problems, Beurling density, dynamical sampling, frames
References
@article{aldroubi2026survey,
author = {Aldroubi, Akram and Cabrelli, Carlos and Krishtal, Ilya and Molter, Ursula},
title = {Dynamical sampling: a survey},
journal = {La Matematica},
volume = {5},
number = {2},
year = {2026},
pages = {37},
doi = {10.1007/s44007-026-00215-y},
eprint = {2511.10769},
archivePrefix = {arXiv},
}
@article{aldroubi2019,
author = {Aldroubi, A. and Huang, L. X. and Petrosyan, A.},
title = {Frames induced by the action of continuous powers of an operator},
journal = {J. Math. Anal. Appl.},
volume = {478},
number = {2},
year = {2019},
pages = {1059--1084},
doi = {10.1016/j.jmaa.2019.05.066},
eprint = {1801.10103},
archivePrefix = {arXiv},
}
@article{diazmartin2021,
author = {D{\'\i}az Mart{\'\i}n, Roc{\'\i}o and Medri, Ivan and Molter, Ursula},
title = {Continuous and discrete dynamical sampling},
journal = {J. Math. Anal. Appl.},
volume = {499},
number = {2},
year = {2021},
pages = {125060},
doi = {10.1016/j.jmaa.2021.125060},
eprint = {2006.08046},
archivePrefix = {arXiv},
}
@misc{aim-infinite-time-dynamical-frame-discretization,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Finite-density time sampling of an infinite observation window}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/research/resolved/657-infinite-time-dynamical-frame-discretization.md},
note = {Entry 657 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
Under the assumptions of AIM problem 457, every sufficiently fine uniform time grid gives an unweighted frame on the infinite observation window. The Bessel hypothesis on the initial sensors and the continuous lower frame bound force the normal evolution operator to be bounded away from zero. Its specified Borel logarithm is consequently bounded. Applying the continuous upper frame bound to …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, siavash-sadeghi-457-558/aim_457_uniform_sampling.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00689, "Finite-density time sampling of an infinite observation window", clopen.solutions.