Universal weighted sampling of positive and normal operator orbits
Abstract
We study common weighted sampling designs for operator systems that are exactly observable at nonnegative integer times. For positive operators with original frame condition number at most κ, we construct a common integer design with arbitrarily small relative Gramian error and O(log R) distinct times up to R. The optimal logarithmic counting coefficient for preserving the frame property satisfies C*(κ)∼π−2logκ as κ→∞. Real and integer times give the same infimum, and we determine its asymptotics as κ↓1. A finite bound on the sampled condition number strictly increases this coefficient. For tight frames, we determine the exact optimum under this constraint and construct an integer design attaining it. Without this constraint, the infimum is never attained. For positive operators, a design valid for every finite κ requires counting faster than log R, with arbitrarily slow divergence of the ratio. For unrestricted normal operators and fixed κ>1, at least linear counting is necessary. Under the logarithmic sector condition |argλ|≤ M(−log|λ|) with fixed M≥0, logarithmic sampling is possible with a sharp leading coefficient. For fixed Borel spectral sets satisfying a radial density condition, we characterize logarithmic sampling and determine the minimum upper counting exponent, attained by an integer design. Counterexamples show that the radial condition cannot be omitted.
Related
- Optimal weighted sampling of positive and normal operator orbits
- Density and Spectral Criteria for Universal Nonuniform Sampling of Positive Operator Orbits
- Sharp weighted norm estimates for the positive Bergman operator: the two-parameter case
- Hyponormal Toeplitz operators with operator valued normal symbols
- Global Central Limit Theorems for Markov chains with normal transition operator
BibTeX
@misc{wu2026universal,
title = {{Universal weighted sampling of positive and normal operator orbits}},
author = {Jian Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11888},
eprint = {2610.11888},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11888},
}
RIS
TY - UNPB
TI - Universal weighted sampling of positive and normal operator orbits
AU - Jian Wu
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11888
UR - https://arxiv.org/abs/2610.11888
AB - We study common weighted sampling designs for operator systems that are exactly observable at nonnegative integer times. For positive operators with original frame condition number at most κ, we construct a common integer design with arbitrarily small relative Gramian error and O(log R) distinct times up to R. The optimal logarithmic counting coefficient for preserving the frame property satisfies C^*(κ)∼π^−2logκ as κ→∞. Real and integer times give the same infimum, and we determine its asymptotics as κ↓1. A finite bound on the sampled condition number strictly increases this coefficient. For tight frames, we determine the exact optimum under this constraint and construct an integer design attaining it. Without this constraint, the infimum is never attained. For positive operators, a design valid for every finite κ requires counting faster than log R, with arbitrarily slow divergence of the ratio. For unrestricted normal operators and fixed κ>1, at least linear counting is necessary. Under the logarithmic sector condition |argλ|≤ M(−log|λ|) with fixed M≥0, logarithmic sampling is possible with a sharp leading coefficient. For fixed Borel spectral sets satisfying a radial density condition, we characterize logarithmic sampling and determine the minimum upper counting exponent, attained by an integer design. Counterexamples show that the radial condition cannot be omitted.
ER -