Asymptotics of Analytic Integrals and the Beurling-Malliavin Theory

解析积分的渐进性和 Beurling-Malliavin 理论

• 批准号: 0500852
• 负责人: Alexei Poltoratski
• 金额: --
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500852&HistoricalAwards=false
• 关键词: Asymptotics; Analytic; Integrals; Beurling; Malliavin

The unifying theme of this proposal is the study of the Hilbert transform and its closest mathematical relatives, the Cauchy transform and the Riesz transform, in non-homogeneous settings appearing in various applications. The applications included in this project were in the center of attention of analysts about 50 years ago -- the completeness and minimality problems, specifically the Beurling-Malliavin theory, the "gap and density" theorems of Beurling-Levinson type, the theory of Cartwright and Paley-Wiener spaces, etc. These areas contain some of the deepest results of linear complex analysis. Even modern expositions require hundreds of pagers with some proofs (like the proof of the Beurling-Malliavin multiplier theorem) still looking completely mysterious. The needs of spectral analysis call for a new approach to these problems and for an extension of the classical results. Powerful techniques of modern complex analysis should give rise to vast generalizations of the classical theory and effective applicationsto spectral problems.This project focuses on complex analysis and its applications. Complex analysis is aclassical area of mathematics that continues to play an important role inboth pure and applied studies. One of the canonical objects of complex analysisis the so-called Hilbert transform. Studies of the Hilbert transform allow one to understand the behavior of complex differentiable functions near the boundary of their domains. Despite being one of the most studied objects in all of mathematics,Hilbert transform is far from being completely understood. Moreover, new developmentsin applications, such as mathematical models of solid state physics and differential equations, require considerable extensions of classical results. The goal of this project is to provide such extensions and to apply new results in several areas of analysis and mathematical physics. Among such applications are spectral problems for the string equation and the Schroedinger equation, which describes wave propagation in quantum mechanics.

这个建议的统一主题是研究希尔伯特变换及其最接近的数学亲属，柯西变换和Riesz变换，在非齐次设置出现在各种应用中。 这个项目中包含的应用在大约50年前是分析师关注的中心--完备性和极小性问题，特别是Beurling-Malliavin理论，Beurling-Levinson类型的“间隙和密度”定理，Cartwright和Paley-Wiener空间理论等。 线性复分析的结果。即使是现代的博览会也需要数百个寻呼机，一些证明（如Beurling-Malliavin乘数定理的证明）仍然看起来完全神秘。谱分析的需要要求一个新的方法来解决这些问题，并为经典结果的扩展。现代复分析的强大技术将使经典理论得到广泛的推广，并有效地应用于谱问题。本项目侧重于复分析及其应用。复分析是数学的一个经典领域，在理论研究和应用研究中都扮演着重要的角色。复分析的标准对象之一是所谓的希尔伯特变换。对希尔伯特变换的研究使人们能够理解复可微函数在其域边界附近的行为。尽管希尔伯特变换是所有数学中研究最多的对象之一，但它远未被完全理解。此外，新的发展在应用中，如数学模型的固态物理和微分方程，需要相当大的扩展的经典结果。该项目的目标是提供这样的扩展，并在分析和数学物理的几个领域应用新的成果。这些应用包括弦方程和薛定谔方程的谱问题，薛定谔方程描述了量子力学中的波传播。