权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Sampling and Identification of Operators and Applications

运营商和应用程序的抽样和识别

基本信息

批准号：
111001434
负责人：
Professor Götz Eduard Pfander, Ph.D.
金额：
--
依托单位：
Lehrstuhl für Wissenschaftliches Rechnen
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2009
资助国家：
德国
起止时间：
2008-12-31 至 2018-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/111001434?language=en
关键词：
Sampling Identification Operators Applications

项目摘要

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.

在这里，我们描述了最近的结果和未来的研究方向在运营商的抽样理论的发展。虽然经典的采样定理断言，功能，带宽限制到一个区间，可以恢复从离散样本上采取足够精细的采样网格，采样结果的运营商允许恢复pseudodiomatic运营商与带宽限制科恩-尼伦伯格符号的基础上，他们的响应离散支持分布。在该项目的第一阶段采样和识别的运营商和应用程序（SamOA），我们建立在以前的见解，并获得明确的重建公式的运营商，其带宽限制是由一个有界的约旦域勒贝格测量小于1。我们的工作包括随机运营商的条件下，科恩-尼伦伯格符号是带限的支持条件的协方差现在随机扩展函数的运营商取代。我们的研究结果允许开发两个估计的广义平稳与不相关的散射信道（WSSUS，在电气工程中广泛使用的通信信道的假设）。其中一个估计器适用于具有任意大但有界协方差函数支持的信道。除此之外，作为SamOA的一部分，还获得了近似理论算子采样结果。这些表明，Schwartz类函数可以用作标识符时，我们寻求获得满意的信息，在相关的，时间和频率有限的，transmission bands.Last，但并非最不重要的是，我们讨论了有限维模拟上述一系列文件。有限维识别问题与用于压缩传感应用的时频结构测量矩阵的构造密切相关。实际上，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来恢复足够稀疏的向量。主要地，上述结果基于规则间隔的采样节点（和周期性权重），即，基于在欧几里得空间（或其缓和版本）中的晶格上支持的标识符。在这里提出的延续，我们专注于不规则的采样集。特别是，我们寻求建立通用的采样集，适用于运营商的一个给定的带宽，例如，使用准晶体作为建立通用采样集的功能由迈耶和马泰，或替代通用采样集的Olevskii和Ulanovskii构建。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。