Universal Nonuniform Sampling of Positive Operator Orbits via Complete Stein--Pick Criteria and Density

Abstract

We study nonuniform sampling for positive operator systems. Given a locally finite set \(T\subset[0,\infty)\), we determine when exact observability of \(\{CA^n\}_{n\in\mathbb N}\) implies exact observability of \(\{CA^t\}_{t\in T}\) for every positive operator \(A\) and every observation operator \(C\). Using the Stein identity and a double spectral representation of the sampled observability Gramian, we show that finite spectral models suffice to test universal preservation for arbitrary positive systems. This yields a characterization in terms of radial Stein--Pick cones at all matrix levels, with constants independent of the dimension. The scalar part of this criterion yields \(0\in T\) and \(N_T(R)\asymp R\). These conditions are also sufficient in the commuting case \([A,C^*C]=0\), but not in general; a clustered counterexample with \(N_T(R)\asymp R\) shows that spectral interactions create an additional obstruction. This obstruction already occurs for a singly generated positive diagonal Carleson frame and, more generally, for any prescribed finite number of generating orbits. We nevertheless obtain two positive results: if the natural density exists, then \(T\) is universal exactly when \(0\in T\) and \(0<d(T)<\infty\), while \(0\in T\), positive lower Beurling density, and finite upper Beurling density give another sufficient condition. We also develop a quantitative perturbation theory relative to regular lattices. In particular, the cumulative condition \(\sum_{k=0}^{K}|t_k-Nk|^2=o(K^2)\) forces the associated quadratic comparison function to vanish at the spectral boundary. This identifies the square root as a critical scale for comparison with a fixed reference lattice and, for regular power perturbations, reveals the universal sampling threshold. Finally, the theory applies to finite and countable families of operator orbits generated by positive operators.