Harmonic analysis and spaces of analytic functions in several variables

算子理论、调和分析和一元解析函数理论形成了一系列相互交织的领域，这些领域是为了解决纯数学和应用数学中一些最重要的问题而创建的。 例如，诺伯特·维纳（Norbert Wiener）在第二次世界大战中发起了时间序列的研究，目的是解决工程“预测”问题。 时间序列是一系列“随机”数据的术语，例如股票市场价格或正在跟踪的喷气式飞机的位置。 谐波分析可以粗略地描述为一组将“信号”（如声音）分解为基本“音调”的技术，为解决时间序列预测问题提供了一组自然的工具。 算子理论和单变量解析函数理论在涉及反馈的相关工程“控制”问题中取得了类似的成功，例如设计恒温器或自动驾驶仪。 这些问题在某种意义上都涉及“一维”数据，算子理论、调和分析和解析函数理论为它们带来了令人印象深刻的相关技术工具箱。 这些理论的多变量版本存在以解决多维问题（例如，图像或视频处理），但不同理论之间的联系较弱，也不太清楚。 这个项目的目标是扩展这些基础学科的一些已知的和特别强的相互作用，以建立一个更强大的工具箱来解决基本的多维问题。这个项目将进一步发展的具体主题是：（1）当一个正三角多项式可以分解为一个平方时的特征，（2）多圆盘上哈代空间的弱分解，（3）有界解析函数的算子理论模型。 题目（1）的成功将加深我们对多元矩问题、多元多项式、平方和分解与行列式表示以及与von Neumann不等式有关的多元算子理论的理解。主题（2）的成功将弥合调和分析与多元函数论之间的差距，并可能导致Dirichlet级数的应用。 本项目将利用第（3）项的最新创新来研究多变量整函数空间及其与傅立叶分析的联系，这可能对傅立叶分析中古老而困难的完备性问题产生深远的影响。

Operator theory, harmonic analysis, and analytic function theory of one variable form a collection of interwoven fields that were created to solve some of the most important problems in pure and applied mathematics. For example, Norbert Wiener initiated the study of time series for the purpose of tackling engineering "prediction" problems in World War II. Time series is a term for a sequence of "random" data, such as stock market prices or the positions of a jet being tracked. Harmonic analysis, which can loosely be described as a set of techniques for breaking up a "signal," such as a sound, into fundamental "tones," provided a natural set of tools for solving time series prediction problems. Operator theory and analytic function theory in one variable have had similar success in related engineering "control" problems involving feedback, for example, designing a thermostat or an automatic pilot. These problems all involve "one-dimensional" data in a certain sense, and operator theory, harmonic analysis, and analytic function theory bring an impressive toolbox of related techniques to bear on them. Multivariate versions of these theories exist to address multidimensional problems (e.g., image or video processing), but the connections between the different theories are weaker and less well understood. It is the goal of this project to expand some known and particularly strong interactions of these fundamental subjects to build a more robust toolbox for addressing fundamentally multidimensional problems.The specific topics this project will develop further are (1) characterizations of when a positive trigonometric polynomial can be factored as a single square, (2) weak factorization of Hardy spaces on the polydisk, and (3) operator theoretic models for bounded analytic functions. Success in topic (1) will deepen our understanding of multivariate moment problems, multivariable polynomials, sums of squares decompositions and determinantal representations, and multivariate operator theory related to von Neumann's inequality. Success in topic (2) will bridge a gap between harmonic analysis and function theory in several variables and could lead to applications in Dirichlet series. This project will use recent innovations in item (3) to study spaces of entire functions in several variables and their connections with Fourier analysis, which could have profound implications in old and difficult completeness problems in Fourier analysis.