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）版本上执行，这是通过对离散集上的信号进行采样而获得的。将模拟（连续）信号转换为数字（离散）信号时的基本问题之一是以下问题：能否从样本中恢复原始信号？如果可以，准确度如何？当然，答案在很大程度上取决于正在处理的信号的性质。例如，本质上更复杂（振荡）的信号需要更多样本才能进行精确重建。振荡速率和所需采样速率之间的精确数学关系非常微妙，特别是当样本不均匀时。该项目的重点是准确理解这种关系，包括由此自然产生的其他几个数学问题。解析函数空间在信号处理的数学理论中一直发挥着重要作用。在研究非均匀采样时特别有用的此类空间的一个自然类是德布兰吉斯空间类。德布兰奇空间理论于六十年代提出，涵盖了当时大量的数学分析知识，并且在现代数学中继续发挥着重要作用，为包括傅里叶分析在内的各个数学领域的相互作用提供了基础、谱理论、算子理论、随机矩阵理论、解析数论和数学物理。该项目的主要研究目标是对 de Branges 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。其中许多问题本质上更为普遍，并且远远超出了德布兰奇空间的设置范围。德布兰吉斯空间作为模型的原因在于，此类空间已经表现出信号处理器面临的大部分关键困难。该项目的另一个重要目标是发展一种统一经典功能空间理论的理论，并使用这种统一作为开发新方法的指南，以解决有关这些空间的剩余开放问题。