Optimal weighted sampling of positive and normal operator orbits
Abstract
We study weighted sampling of operator orbit frames in infinite dimensional separable Hilbert spaces H, keeping the operator and generating vectors fixed. Let Nμ(R) count the sampling times in [0,R]. For a single orbit of a positive operator, let mA(R) count the eigenvalues in (0,e−1/R]. Every real sampling design that preserves the frame property must satisfy the lower bound Nμ(R)≥ mA(R)−Oμ(1). Conversely, for every η>0, we construct an integer sampling design that preserves the frame property and satisfies Nμ(R)≤(1+η)mA(R)+Oη(1). Thus the coefficient $1$ is optimal. Under additional spectral conditions, we attain Nμ(R)=mA(R)+O(1) and characterize the weights that preserve the frame property on the constructed sets. For finitely or countably many orbits of a positive operator, let Pj, j≥1, be the spectral projection corresponding to 2−j−1<−logλ≤2−j and define KA(R)=#{j≥1:2j≤ R, Pj≠0}. Every real sampling design that preserves the frame property satisfies Nμ(R)≥ cμKA(R)−Cμ for R≥2, with constants cμ>0 and Cμ≥0. Conversely, we construct an integer sampling design that preserves the frame property and satisfies Nμ(R)≤ L KA(R)+L for R≥2, where L depends only on the bounds of the given orbit frame. For normal operators of norm one, we prove that the smallest frame condition number is attained among integer sampling designs that preserve the frame property and satisfy a prescribed counting bound, whenever this class is nonempty. Finally, we construct one normal operator with two families of generating vectors for which the optimal growth orders of Nμ(R) are log R and R, respectively, for both real and integer sampling.
Related
- Universal weighted sampling of positive and normal operator orbits
- Sharp weighted norm estimates for the positive Bergman operator: the two-parameter case
- Density and Spectral Criteria for Universal Nonuniform Sampling of Positive Operator Orbits
- Hyponormal Toeplitz operators with operator valued normal symbols
- Global Central Limit Theorems for Markov chains with normal transition operator
BibTeX
@misc{wu2026optimal,
title = {{Optimal weighted sampling of positive and normal operator orbits}},
author = {Jian Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31278},
eprint = {2610.31278},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31278},
}
RIS
TY - UNPB
TI - Optimal weighted sampling of positive and normal operator orbits
AU - Jian Wu
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31278
UR - https://arxiv.org/abs/2610.31278
AB - We study weighted sampling of operator orbit frames in infinite dimensional separable Hilbert spaces H, keeping the operator and generating vectors fixed. Let N_μ(R) count the sampling times in [0,R]. For a single orbit of a positive operator, let m_A(R) count the eigenvalues in (0,e^−1/R]. Every real sampling design that preserves the frame property must satisfy the lower bound N_μ(R)≥ m_A(R)−O_μ(1). Conversely, for every η>0, we construct an integer sampling design that preserves the frame property and satisfies N_μ(R)≤(1+η)m_A(R)+O_η(1). Thus the coefficient $1$ is optimal. Under additional spectral conditions, we attain N_μ(R)=m_A(R)+O(1) and characterize the weights that preserve the frame property on the constructed sets. For finitely or countably many orbits of a positive operator, let P_j, j≥1, be the spectral projection corresponding to 2^−j−1<−logλ≤2^−j and define K_A(R)=#{j≥1:2^j≤ R, P_j≠0}. Every real sampling design that preserves the frame property satisfies N_μ(R)≥ c_μK_A(R)−C_μ for R≥2, with constants c_μ>0 and C_μ≥0. Conversely, we construct an integer sampling design that preserves the frame property and satisfies N_μ(R)≤ L K_A(R)+L for R≥2, where L depends only on the bounds of the given orbit frame. For normal operators of norm one, we prove that the smallest frame condition number is attained among integer sampling designs that preserve the frame property and satisfy a prescribed counting bound, whenever this class is nonempty. Finally, we construct one normal operator with two families of generating vectors for which the optimal growth orders of N_μ(R) are log R and R, respectively, for both real and integer sampling.
ER -