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

信号的现代处理几乎总是在原始信号的数字（离散）版本上进行，这是通过对离散集上的信号采样获得的。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）的信号需要更多的样本来精确重建。振荡率和所需采样率之间的精确数学关系是惊人的微妙，特别是当样本不均匀时。这个项目的中心问题是准确地理解这种关系，包括其他几个自然产生的数学问题。解析函数空间在信号处理的数学理论中一直扮演着重要的角色。在研究非均匀抽样时特别有用的一类自然空间是de Branges空间。在60年代被引入的de Branges空间理论包含了当时大量的数学分析知识，它继续在现代数学中发挥重要作用，为数学各个领域的相互作用提供了一个背景，包括傅里叶分析、谱理论、算子理论、随机矩阵理论、解析数论和数学物理。本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，并利用由此产生的发现来攻克和解决几个长期存在的开放性问题。这些问题中的许多在本质上更为普遍，远远超出了德布朗日空间的范围。德布朗日空间作为模型的原因在于，这类空间已经展示了信号处理器面临的大多数关键困难。该项目的另一个重要目标是发展一种理论，统一经典功能空间的理论，并使用这种统一作为开发新方法的指导方针，以解决关于这些空间的剩余开放问题。

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.