De Branges Spaces as Models for a General Theory of Function Spaces

德布兰吉斯空间作为函数空间一般理论的模型

• 批准号: 1600874
• 负责人: Mishko Mitkovski
• 金额: $ 13.8万
• 依托单位: Clemson University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600874&HistoricalAwards=false
• 关键词: De; Branges; Spaces; Models; General

Modern processing of signals is almost always performed on the digital (discretized) version of the original signals, which is obtained by sampling the signals on a discrete set. One of the fundamental issues when converting an analog (continuous) signal to a digital (discrete) one is the following question: Can the original signals be recovered from the samples, and if so, how accurately? The answer, of course, depends heavily on the nature of the signals that are being processed. For example, signals that are more complex (oscillatory) in nature require more samples for accurate reconstruction. The precise mathematical relationship between the rate of oscillation and the required rate of sampling is surprisingly delicate, especially when the samples are non-uniform. This project is centered on the problem of understanding precisely this relationship, including several other mathematical questions that arise naturally from it. Analytic function spaces have always played an important role in the mathematical theory of signal processing. One natural class of such spaces that is particularly useful when studying non-uniform sampling is the class of de Branges spaces. Introduced in the sixties, the theory of de Branges spaces encompassed a great deal of mathematical analysis knowledge at that time, and it continues to play an important role in modern mathematics, providing a setting for the interplay of various areas of mathematics, including Fourier analysis, spectral theory, operator theory, random matrix theory, analytic number theory, and mathematical physics. The main research objective of this project is to conduct a deeper analysis of de Branges function spaces, and use the resulting findings to attack and resolve several long-standing open problems. Many of these problems are much more general in nature and go far beyond the setting of de Branges spaces. The reason that de Branges spaces serve as a model rests on the fact that this class of spaces already exhibits most of the key difficulties confronting signal processors. Another important goal of this project is to develop a theory that will unify the theory of classical function spaces, and use this unification as a guideline for developing new methods for resolving the remaining open questions about these spaces.

现代信号处理几乎总是在原始信号的数字（离散化）版本上执行，该数字（离散化）版本通过对离散集上的信号进行采样而获得。将模拟（连续）信号转换为数字（离散）信号时的一个基本问题是以下问题：能否从样本中恢复原始信号，如果可以，精度如何？当然，答案在很大程度上取决于被处理信号的性质。例如，本质上更复杂（振荡）的信号需要更多的样本来进行精确重建。振荡速率和所需采样速率之间的精确数学关系令人惊讶地微妙，特别是当样本不均匀时。这个项目的重点是准确地理解这种关系的问题，包括其他几个自然产生的数学问题。解析函数空间一直在信号处理的数学理论中发挥着重要作用。在研究非均匀采样时特别有用的一类自然空间是de布兰日空间。德布兰日空间理论在60年代提出，包含了当时大量的数学分析知识，它在现代数学中继续发挥着重要作用，为数学的各个领域，包括傅立叶分析，谱理论，算子理论，随机矩阵理论，解析数论和数学物理的相互作用提供了一个环境。该项目的主要研究目标是对de布兰日函数空间进行更深入的分析，并利用所得结果来攻击和解决几个长期存在的开放问题。许多这些问题是更普遍的性质，远远超出了设置德布兰日空间。德布兰日空间作为一个模型的原因在于这样一个事实，即这类空间已经表现出大多数的关键困难所面临的信号处理器。该项目的另一个重要目标是开发一个理论，将统一的理论经典的功能空间，并使用这种统一作为指导方针，开发新的方法来解决剩余的开放问题，这些空间。