Harmonic analysis and spaces of analytic functions in several variables

调和分析和多变量解析函数空间

• 批准号: 1363239
• 负责人: Greg Knese
• 金额: $ 13.8万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1363239&HistoricalAwards=false
• 关键词: Harmonic; analysis; spaces; analytic; functions

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.

算符理论、调和分析和单变量解析函数理论形成了一个相互交织的领域的集合，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）为了解决第二次世界大战中的工程“预测”问题，发起了时间序列的研究。时间序列是“随机”数据序列的术语，例如股票市场价格或被跟踪的喷气机位置。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，它为解决时间序列预测问题提供了一套自然的工具。单一变量的算子理论和解析函数理论在涉及反馈的相关工程“控制”问题上取得了类似的成功，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，而算符理论、调和分析和解析函数理论带来了一个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于解决多维问题（例如，图像或视频处理），但不同理论之间的联系较弱且不太容易理解。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，以构建一个更健壮的工具箱来解决基本的多维问题。本项目将进一步发展的具体主题是：(1)正三角多项式何时可以被分解为单个平方的表征，(2)多盘上Hardy空间的弱分解，以及(3)有界解析函数的算子理论模型。主题(1)的成功将加深我们对多元矩问题、多变量多项式、平方和分解和行列式表示以及与冯·诺伊曼不等式相关的多元算子理论的理解。主题(2)的成功将弥合调和分析和函数理论在几个变量之间的差距，并可能导致Dirichlet级数的应用。本课题将利用第(3)项的最新成果，研究多变量全函数空间及其与傅里叶分析的联系，这将对傅里叶分析中古老而困难的完备性问题产生深远的影响。