Sectorial Finitely Generated Dynamical Frames
Abstract
We study frames of unilateral iterations generated by finitely many vectors under an invertible bounded operator T. Suppose that the spectrum of T lies in {ρeit:0<ρ≤1, |t|≤ c}, where c<π, and that its essential spectrum lies in {eit:|t|≤ ce}, where 0≤ ce≤ c. Real powers are defined through the principal logarithm. Our structural result identifies fractional-orbit synthesis, up to an invertible map and a compact perturbation, with synthesis of projected vector-valued exponentials on an arc containing the essential spectrum. Semi-Fredholm properties and Fredholm index therefore transfer. For a uniformly separated temporal set Λ, this reduction gives a frame when its one-sided lower uniform density satisfies D+-(Λ)>ce/π and its logarithmic block density satisfies L(Λ)>c/π. Completeness at the latter threshold requires only an invertible operator with spectrum in the corresponding angular sector and a complete integer multiorbit. At L(Λ)>2c/π, the complement property is equivalent to a finite condition in channel-cyclic subspaces, with real phase-retrieval consequences. Generalized divided differences admit the same compact reduction and the analogous frame theorem with multiplicities counted. For arbitrary one-point-per-cell sampling with cell length r, the sufficient raw-sampling conditions are rc<π and 2rce<π. Thus, the full range rc<π holds for raw samples when the essential spectrum is contained in {1}; over the unrestricted class, the sharp universal range is rc<π/2. Divided-difference preconditioning restores rc<π for all operators considered and converts exact collisions into logarithmic-derivative data.
Related
- Mesoscopic redundancy for dynamical frames generated by atomic singular model operators
- Ordering the space of finitely generated groups
- Conjugator length in finitely generated groups
- The topological vertex is finitely generated
- Finitely generated groups of diffeomorphisms with prescribed critical regularity in every dimension
BibTeX
@misc{krishtal2026sectorial,
title = {{Sectorial Finitely Generated Dynamical Frames}},
author = {Ilya Krishtal and Javad Mashreghi and Brendan Miller},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22523},
eprint = {2609.22523},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22523},
}
RIS
TY - UNPB
TI - Sectorial Finitely Generated Dynamical Frames
AU - Ilya Krishtal
AU - Javad Mashreghi
AU - Brendan Miller
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22523
UR - https://arxiv.org/abs/2609.22523
AB - We study frames of unilateral iterations generated by finitely many vectors under an invertible bounded operator T. Suppose that the spectrum of T lies in {ρe^it:0<ρ≤1, |t|≤ c}, where c<π, and that its essential spectrum lies in {e^it:|t|≤ c_e}, where 0≤ c_e≤ c. Real powers are defined through the principal logarithm. Our structural result identifies fractional-orbit synthesis, up to an invertible map and a compact perturbation, with synthesis of projected vector-valued exponentials on an arc containing the essential spectrum. Semi-Fredholm properties and Fredholm index therefore transfer. For a uniformly separated temporal set Λ, this reduction gives a frame when its one-sided lower uniform density satisfies D_+^-(Λ)>c_e/π and its logarithmic block density satisfies L(Λ)>c/π. Completeness at the latter threshold requires only an invertible operator with spectrum in the corresponding angular sector and a complete integer multiorbit. At L(Λ)>2c/π, the complement property is equivalent to a finite condition in channel-cyclic subspaces, with real phase-retrieval consequences. Generalized divided differences admit the same compact reduction and the analogous frame theorem with multiplicities counted. For arbitrary one-point-per-cell sampling with cell length r, the sufficient raw-sampling conditions are rc<π and 2rc_e<π. Thus, the full range rc<π holds for raw samples when the essential spectrum is contained in {1}; over the unrestricted class, the sharp universal range is rc<π/2. Divided-difference preconditioning restores rc<π for all operators considered and converts exact collisions into logarithmic-derivative data.
ER -