正算子轨道的通用非均匀采样：基于完全Stein--Pick准则与密度
Density and Spectral Criteria for Universal Nonuniform Sampling of Positive Operator Orbits
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中文总结 AI 辅助
本文通过Stein恒等式和双谱表示，利用径向Stein--Pick锥和密度条件，完整刻画了正算子轨道非均匀采样的通用精确可观测性，并揭示了谱相互作用导致的额外障碍及临界尺度。
中文摘要 AI 辅助
我们研究正算子系统的非均匀采样。给定局部有限集\\(T\subset[0,\infty)\\)，我们确定何时\\(\{CA^n\}_{n\in\mathbb N}\\)的精确可观测性蕴含对每个正算子\\(A\\)和每个观测算子\\(C\\)的\\(\{CA^t\}_{t\in T}\\)的精确可观测性。利用Stein恒等式和采样可观测性Gramian的双谱表示，我们证明有限谱模型足以检验任意正系统的通用保持性。这产生了在所有矩阵层级上以径向Stein--Pick锥为特征的表征，且常数与维数无关。该准则的标量部分给出\\(0\in T\\)和\\(N_T(R)\asymp R\\)。这些条件在交换情形\\([A,C^*C]=0\\)下也是充分的，但一般情况下不充分；一个满足\\(N_T(R)\asymp R\\)的聚类反例表明谱相互作用产生了额外障碍。该障碍已出现在单生成正对角Carleson框架中，更一般地，对任意给定的有限个生成轨道也出现。尽管如此，我们获得两个正面结果：若自然密度存在，则\\(T\\)是通用的当且仅当\\(0\in T\\)且\\(0<d(T)<\infty\\)，而\\(0\in T\\)、正下Beurling密度和有限上Beurling密度给出另一个充分条件。我们还发展了相对于正则格点的定量扰动理论。特别地，累积条件\\(\sum_{k=0}^{K}|t_k-Nk|^2=o(K^2)\\)迫使相关的二次比较函数在谱边界消失。这确定了平方根作为与固定参考格点比较的临界尺度，并且对于正则幂扰动，揭示了通用采样阈值。最后，该理论适用于由正算子生成的有限和可数无穷族算子轨道。
英文摘要
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 \(0<d(T)<\infty\). More generally, containing zero and a subset of positive finite natural density suffices under linear upper counting. For sets containing zero, universality is invariant under sublinear differences of counting functions. For general normal operators, universal Bessel preservation is instead equivalent to uniformly bounded counts on unit intervals; under a fixed logarithmic sector condition, it is again characterized by linear upper counting.
