Density and Spectral Criteria for Universal Nonuniform Sampling of Positive Operator Orbits
Year: 2026
Type: preprint
Abstract: We characterize locally finite sets \(T\subset[0,\infty)\) that preserve exact observability from nonnegative integer times for every bounded positive operator \(A\) and bounded observation operator \(C\). The Stein identity reduces universality to uniform upper and lower bounds on radial Stein--Pick cones over all finite spectral models. Writing \(N_T(R)=\#(T\cap[0,R])\), we prove that universal Bessel preservation is equivalent to \(N_T(R)=O(R)\). The conditions \(0\in T\) and \(N_T(R)\asymp R\) characterize universal observability preservation under \([A,C^*C]=0\), as well as uniform bound preservation for each fixed number of spectral values. However, we construct a clustered set satisfying these conditions whose optimal sampled lower bounds tend to zero as the spectral count grows, despite fixed integer observability bounds. The same set destroys the frame property of a singly generated positive diagonal Carleson frame while preserving its Bessel property. When natural density exists, universality is equivalent to \(0\in T\) and \(0
Source: arXiv (Cornell University)
Author: Wu, Jian
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212017890