RUI: Rational Schur Functions and their Applications

该项目位于算子理论和复分析的交叉点，这是纯数学的两个领域。算符理论开始于对矩阵（数组）的研究，到世纪初，它为量子力学提供了大量的数学基础。从那时起，算子理论技术有了一些应用，特别是在构造和研究各种有用的函数，即，编码特定类型信息的规则。例如，在控制工程中，自动驾驶仪系统等接受输入、接收反馈和释放输出的设置可以用一般矩阵（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，算子理论技术也被用于通过称为小波构造的东西将信号分解成简单的片段。在这个项目中，主要研究者将使用运营商来研究与一类特定的功能，称为有理函数，出现在这样的应用程序相关的问题。这些问题中有许多部分是适合本科生研究的，本项目的大部分研究将与不同的本科生研究人员一起探索。更确切地说，本项目涉及在某些域上有界的一个或多个变量的有理函数。这样的有理函数包括有限Blaschke产品（FBPs），它在复分析中的因子分解，插值和逼近问题中发挥了重要作用。最近，FBPs出现在一个著名的开放问题的背景下，在算子理论称为Crouzeix的猜想，这基本上说，数值范围的有界算子的希尔伯特空间是一个2-谱集的操作。这个项目的第一个目标是使用FBPs来调查和解决Crouzeix猜想的各种情况，使用Crouzeix猜想来询问和回答关于FBPs的新问题，并探索有关谱集，压缩移位和截断Toeplitz算子的相关问题。这个项目的第二个主题是关于双圆盘和多圆盘上的多元（有理和非有理）函数。具体来说，第二个主要目标是使用模型/实现理论，系统地描述结构和精细的边界规则性的有界解析函数的双盘和多盘。这一目标的动机是最近的工作都合理的内部功能的结构附近的边界奇点和新的规范实现公式一般有界的解析functions.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.