RUI: Rational Schur Functions and their Applications

这个项目位于算符理论和复分析的交叉点上，这是纯数学的两个领域。算符理论始于对矩阵(数组)的研究，到二十世纪初，已经为量子力学提供了大量的数学基础。从那时起，算符理论技术有了几个应用，特别是在构造和研究各种有用的函数方面，即编码特定类型信息的规则。例如，在控制工程中，接受输入、接收反馈和释放输出的自动驾驶系统等设置可以用通用矩阵(或算子)建模，而系统的关键信息通常编码在其传递函数中。最近，算符理论技术也被用来将信号分解成简单的片段，通过一种称为小波构造的东西。在这个项目中，主要研究人员将使用运算符来研究与出现在此类应用程序中的一类特定函数(称为有理函数)相关的问题。这些问题中的许多都有适合本科生研究的部分，这个项目中的许多研究将与不同的本科生研究人员一起探索。更准确地说，这个项目涉及一个和多个变量中的有理函数，这些变量有界于某些领域。这些有理函数包括有限Blaschke积(FBP)，它们在复分析中的因式分解、内插和逼近问题中发挥了重要作用。最近，在算子理论中一个著名的公开问题Crouzeix猜想的背景下出现了FBPs，该猜想基本上是说Hilbert空间上的有界算子的数值值域是该算子的2-谱集。这个项目的第一个目标是利用FBP来研究和解决Crouzeix猜想的各种情况，使用Crouzeix猜想来提出和回答关于FBP的新问题，并探索与谱集、压缩移位和截断Toeplitz算子相关的问题。这个项目的第二个主题涉及双圆盘和多圆盘上的多元(有理和非有理)函数。具体地说，第二个主要目标是利用模型/实现理论系统地刻画双圆盘和多圆盘上有界解析函数的结构和精细边界正则性。这一目标是由最近关于边界奇点附近的有理内部函数的结构和一般有界解析函数的新的规范实现公式的工作所推动的。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project lies at the intersection of operator theory and complex analysis, two areas of pure mathematics. Operator theory began with the study of matrices (arrays of numbers), and by the early twentieth century, had given much of the mathematical foundation for quantum mechanics. Since then, operator-theoretic techniques have had several applications, especially in constructing and studying a variety of useful functions, i.e., rules that encode specific types of information. For example, in control engineering, setups such as autopilot systems that accept inputs, receive feedback, and release outputs can be modeled with general matrices (or operators), while a system’s key information is often encoded in its transfer function. Recently, operator theory techniques have also been used to decompose signals into simple pieces, via something called a wavelet construction. In this project, the principal investigator will use operators to study problems related to a specific class of functions, called rational functions, which appear in such applications. Many of these problems have parts that are amenable to undergraduate research, and much of the research in this project will be explored alongside diverse groups of undergraduate researchers.More precisely, this project concerns rational functions in one and more variables that are bounded on certain domains. Such rational functions include finite Blaschke products (FBPs), which have played significant roles in factorization, interpolation, and approximation problems in complex analysis. Recently, FBPs have arisen in the context of a famous open problem in operator theory called Crouzeix’s conjecture, which basically says that the numerical range of a bounded operator on a Hilbert space is a 2-spectral set for that operator. The first goals of this project are to use FBPs to investigate and solve various cases of Crouzeix’s conjecture, use Crouzeix’s conjecture to ask and answer new questions about FBPs, and to explore related questions about spectral sets, compressed shifts, and truncated Toeplitz operators. The second topic of this project concerns multivariate (rational and non-rational) functions on the bidisk and polydisk. Specifically, the second main goal is to use model/realization theory to systematically characterize the structure and fine boundary regularity of bounded analytic functions on the bidisk and polydisk. This goal is motivated by recent work on both the structure of rational inner functions near boundary singularities and new canonical realization formulas for general bounded analytic functions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.