正算子与正规算子轨道的通用加权采样
Universal weighted sampling of positive and normal operator orbits
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中文总结 AI 辅助
该研究针对非负整数时刻完全可观的算子系统，构造正算子与正规算子的通用加权采样设计，确定对数采样的最优计数系数及相关约束下的精确最优值，证明径向条件对对数采样的必要性。
中文摘要 AI 辅助
我们研究在非负整数时刻完全可观的算子系统的常见加权采样设计。对于原始框架条件数至多为κ的正算子，我们构造了一个通用整数设计，其相对Gram误差任意小，且到R为止有O(log R)个不同时刻。为保持框架性质，最优对数计数系数满足当κ→∞时，C*(κ)~π⁻²logκ。实时刻与整数时刻给出相同的下确界，且我们确定了当κ↓1时的渐近行为。采样条件数的有限界会严格增大该系数。对于紧框架，我们确定了该约束下的精确最优值，并构造了一个达到该最优值的整数设计。若无此约束，则下确界永远无法达到。对于正算子，对每个有限κ都有效的设计要求计数速度快于log R，且比值的发散速度可任意慢。对于无约束的正规算子和固定κ>1，至少需要线性计数。在固定M≥0的对数扇区条件|argλ|≤M(-log|λ|)下，对数采样是可行的，且具有尖锐的首项系数。对于满足径向密度条件的固定Borel谱集，我们刻画了对数采样并确定了最小上计数指数，该指数可由整数设计达到。反例表明径向条件不能省略。
英文摘要
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 $C^*(κ)\simπ^{-2}\logκ$ as $κ\to\infty$. Real and integer times give the same infimum, and we determine its asymptotics as $κ\downarrow1$. 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 $|\argλ|\le M(-\log|λ|)$ with fixed $M\ge0$, 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.
