Sampling and Identification of Operators and Applications

在此，我们描述了最近的结果和未来的研究方向，在发展的采样理论的算子。虽然经典的抽样定理断言，带宽限制在一个区间内的函数可以从一个足够精细的抽样网格上的离散样本中恢复，但算子的抽样结果允许恢复具有带宽限制的伪微分算子，基于它们对离散支持分布的响应。在“运营商和应用的采样和识别”（SamOA）项目的第一阶段，我们在之前的见解的基础上，获得了带宽限制由Lebesgue测度小于1的有界Jordan域描述的运营商的显式重建公式。我们的工作包括随机算子，其中Kohn-Nirenberg符号是带宽限制的条件被一个支持条件所取代，这个支持条件是一个算子的随机扩展函数的协方差。我们的结果允许开发两个具有不相关散射信道的广义平稳估计器（WSSUS，在电气工程中广泛使用的通信信道假设）。其中一个估计量适用于具有任意大但有界的协方差函数支持的信道。此外，作为SamOA的一部分，还获得了近似理论算子抽样结果。这些表明，当我们寻求在相关的、时间和频率有限的传输频带中获得有关受限带宽运营商特征的满意信息时，可以使用Schwartz类函数作为标识符。最后，但并非最不重要的是，我们在一系列的论文中讨论了上述的有限维模拟。有限维识别问题与压缩感知应用的时频结构测量矩阵的构造密切相关。事实上，我们的结果包括基追踪（l^1最小化）的性能保证，以使用时频结构化测量恢复足够稀疏的向量。上述结果主要是基于规则间隔的采样节点（和周期权重），即基于欧几里得空间（或其缓和版本）中晶格上支持的标识符。在本文提出的延拓中，我们主要关注不规则采样集。特别是，我们寻求建立适用于给定带宽算子的通用采样集，例如，使用准晶体作为Meyer和Matei建立的函数的通用采样集，或使用Olevskii和Ulanovskii构建的替代通用采样集。此外，我们计划研究概率抽样，即由随机变量生成的抽样集，并建立与学习理论的联系。

Herein, we describe recent results and future research directions in the development of a sampling theory for operators. While the classical sampling theorem asserts that functions that are bandlimited to an interval can be recovered from discrete samples taken on a sufficiently fine sampling grid, sampling results for operators allow for the recovery of pseudodifferential operators with bandlimited Kohn--Nirenberg symbols based on their response to a discretely supported distribution. In the first phase of the project Sampling and Identification of Operators and Applications (SamOA), we built on previous insights and obtained explicit reconstruction formulas for operators whose bandlimitation is described by a bounded Jordan domain of Lebesgue measure less than one. Our work includes stochastic operators where the condition that the Kohn-Nirenberg symbol is bandlimited is replaced by a support condition on the covariance of the now stochastic spreading function of an operator. Our results allowed for the development of two estimators for Wide Sense Stationary with Uncorrelated Scattering channels (WSSUS, a widely used assumption on communications channels in electrical engineering). One of these estimators is applicable to channels with arbitrarily large, but bounded support of the covariance function.In addition to the above, approximation theoretic operator sampling results were obtained as part of SamOA. These show that Schwartz class functions can be used as identifiers when we seek to obtain satisfactory information on the characteristics of bandlimited operators in relevant, time and frequency limited, transmission bands.Last, but not least, we discussed a finite dimensional analogue of the above in a series of papers. The finite dimensional identification problem is intimately linked to the construction of time-frequency structured measurement matrices for compressive sensing applications. Indeed, our results include performance guarantees for Basis Pursuit (l^1-minimization) to recover sufficiently sparse vectors using time-frequency structured measurements.Predominately, the results described above are based on regularly spaced sampling nodes (and periodic weights), that is, on identifiers supported on a lattice in Euclidean space (or mollified versions thereof). In the herein proposed continuation, we focus on irregular sampling sets. In particular, we seek to establish universal sampling sets that are applicable to operators of a given bandwidth, for example, using quasicrystals as established as universal sampling sets for functions by Meyer and Matei, or alternative universal sampling sets as those constructed by Olevskii and Ulanovskii. Moreover, we plan to examine probabilistic sampling, that is, sampling sets that are generated by random variables, and establish connections to learning theory.