Investigations in Harmonic Analysis

谐波分析研究

• 批准号: 0456611
• 负责人: Michael Lacey
• 金额: $ 55.14万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-05-15 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0456611&HistoricalAwards=false
• 关键词: Investigations; Harmonic; Analysis

The PI and Xiaochun Li have recently established the boundedness of a degenerate Radon transform, in which a Hilbert transform is computed on choice of directions in the plane. The critical point is that the choice of directions is only assumed to be smooth, with no geometric condition imposed on the choice of directions. The method of proof involves an intricate set of phase plane methods, together with novel Kakeya maximal function. There are several aspects of this transform that remain poorly understood, and the PI will work to resolve some of these issues. In a second direction, the PI, with Sarah Ferguson and Erin Terwilleger have provided the natural extension of the Nehari theorem to 'little' Hankel operators on product Hardy space. Namely, such Hankel operators are bounded iff their symbol is in product BMO. This is fundamental criteria, which opens up a range of questions in operator theory in several complex variables. This range of questions forms the second avenue of investigation that will be pursued in this project. The range of problems to be pursued in this project will require the creation of new techniques in the broad area of analysis. The objects studies arise naturally in physical processes, such as charge distribution namely the Hilbert transform, medical imaging, namely Radon transforms, and control theory, namely Hankel operators. The questions addressed concern how well behaved these objects are, and the resolution of these questions should yield important insights into deeper aspects of these objects. These insights have in the past lead to important advances in signal processing, imaging, and control theory. Postdoctoral associates and graduate students will also be engaged in this project, enhancing the scientific infrastructure of the country. The PI and Xiaochun Li have recently established the boundedness of a degenerate Radon transform, in which a Hilbert transform is computed on choice of directions in the plane. The critical point is that the choice of directions is only assumed to be smooth, with no geometric condition imposed on the choice of directions. The method of proof involves an intricate set of phase plane methods, together with novel Kakeya maximal function. There are several aspects of this transform that remain poorly understood, and the PI will work to resolve some of these issues. In a second direction, the PI, with Sarah Ferguson and Erin Terwilleger have provided the natural extension of the Nehari theorem to 'little' Hankel operators on product Hardy space. Namely, such Hankel operators are bounded iff their symbol is in product BMO. This is fundamental criteria, which opens up a range of questions in operator theory in several complex variables. This range of questions forms the second avenue of investigation that will be pursued in this project. The range of problems to be pursued in this project will require the creation of new techniques in the broad area of analysis. The objects studies arise naturally in physical processes, such as charge distribution namely the Hilbert transform, medical imaging, namely Radon transforms, and control theory, namely Hankel operators. The questions addressed concern who well behaved these objects are, and the resolution of these questions should yield important insights into deeper aspects of these objects. These insights have in the past lead to important advances in signal processing, imaging, and control theory. Postdoctoral associates and graduate students will also be engaged in this project, enhancing the scientific infrastructure of the country.

PI和Xiaoxun Li最近建立了退化Radon变换的有界性，其中Hilbert变换是在平面上选择方向计算的。 关键点在于，方向的选择仅假设是平滑的，没有对方向的选择施加几何条件。 证明的方法涉及一组复杂的相平面方法，连同新的Kakeya极大函数。这个转换有几个方面仍然不太清楚，PI将努力解决其中的一些问题。 在第二个方向上，PI与Sarah Ferguson和Erin Terwilleger一起将Nehari定理自然地扩展到乘积哈代空间上的“小”汉克尔算子。也就是说，这样的Hankel算子是有界的当且仅当它们的符号是乘积BMO。 这是基本的标准，它开辟了一系列的问题，在算子理论在几个复杂的变量。 这一系列问题构成了本项目将进行的第二种调查途径。本项目所要解决的一系列问题将需要在广泛的分析领域创造新的技术。研究对象自然出现在物理过程中，例如电荷分布即希尔伯特变换，医学成像即Radon变换，以及控制理论即汉克尔算子。所解决的问题涉及这些对象的行为有多好，这些问题的解决应该产生对这些对象的更深层次的重要见解。这些见解在过去导致了信号处理，成像和控制理论的重要进展。博士后和研究生也将参与该项目，加强国家的科学基础设施。 PI和Xiaoxun Li最近建立了退化Radon变换的有界性，其中Hilbert变换是在平面上选择方向计算的。 关键点在于，方向的选择仅假设是平滑的，没有对方向的选择施加几何条件。 证明的方法涉及一组复杂的相平面方法，连同新的Kakeya极大函数。这个转换有几个方面仍然不太清楚，PI将努力解决其中的一些问题。 在第二个方向上，PI与Sarah Ferguson和Erin Terwilleger一起将Nehari定理自然地扩展到乘积哈代空间上的“小”汉克尔算子。也就是说，这样的Hankel算子是有界的当且仅当它们的符号是乘积BMO。 这是基本的标准，它开辟了一系列的问题，在算子理论在几个复杂的变量。 这一系列问题构成了本项目将进行的第二种调查途径。本项目所要解决的一系列问题将需要在广泛的分析领域创造新的技术。研究对象自然出现在物理过程中，例如电荷分布即希尔伯特变换，医学成像即Radon变换，以及控制理论即汉克尔算子。所解决的问题涉及到这些对象是谁表现得很好，这些问题的解决应该产生对这些对象更深层次的重要见解。这些见解在过去导致了信号处理，成像和控制理论的重要进展。博士后和研究生也将参与该项目，加强国家的科学基础设施。