Fourier Analysis: Space, Frequency, and Direction

傅里叶分析：空间、频率和方向

• 批准号: 0900946
• 负责人: Loukas Grafakos
• 金额: $ 19.79万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0900946&HistoricalAwards=false
• 关键词: Fourier; Analysis; Space; Frequency; Direction

Jean Baptiste Joseph Fourier can rightly claim the title of forefather of frequency decompositions. Having postulated in 1807 the theory that essentially arbitrary functions can be expressed as sums of basic sinusoidal waves of various frequencies, Fourier resisted fervent theoretical objections to paint a landscape full of applications to concrete problems. Since then, the development of analysis has furnished new perspectives that have resulted in extraordinary accomplishments in several fields of mathematics and the sciences. Recent techniques include sensitive decompositions localized in both space and phase, which take into consideration directional aspects of the problems. Such decompositions have led to solutions of long-standing conjectures such as the almost everywhere convergence of Fourier series, and the boundedness of the bilinear Hilbert transform. The principal investigator proposes to embark on an extensive study of problems in Fourier Analysis and applications that have a common feature: they require a delicate balance between space, frequency, and direction. The intellectual merit of this proposal lies in the historical value of the problems studied, many of which have naturally arisen over time and have a reputation for their difficulty. This proposal consists of three parts: positive theory, quest for counterexamples, and concrete applications. Proposed work on the positive theory includes extension of the range of boundedness of the bilinear Hilbert transform and other rough singular integrals and a study of m-linear orthogonality and Littlewood-Paley theory. Geometric aspects of frequency decompositions play a crucial role in this study. Counterexamples are sought for the bilinear disc multiplier in the nonlocal square-integrable case and the Carleson-Hunt operator on certain spaces of functions. Concrete applications focus on directional sensitivity in computerized tomography.Fourier Analysis provides decompositions of functions in parts that have common spatial, frequency, and directional characteristics. Just as symphonic music can be analyzed as a finite union of simple notes, certain complicated operations can be represented by their actions on a spectrum of frequencies. Irregularities of signals and images are better located once they are decomposed into small pieces that can be studied individually. The alteration of frequency via multiplication by a typically nonsmooth function, such as intermittent television transmission, calls for a systematic study of preservation of information contained in a signal. Preservation of integrability under frequency alterations serves as a good model to study preservation of information and is the main focus of the theoretical part of this proposal. The broader impact of this proposal is exactly this point, i.e. to provide a solid theoretical foundation or groundwork for modeling protection against the loss of information contained in a signal or image. Applied problems addressed in this project focus on improved results in computerized tomography based on frequency decompositions sensitive to direction. Such applications may lead to sharper computerized tomography images for moving subjects during standard MRI tests.

让·巴蒂斯特·约瑟夫·傅立叶可以理所当然地称之为频率分解的先驱。假设在1807年的理论，基本上任意功能可以表示为总和的基本正弦波的各种频率，傅立叶抵制激烈的理论反对画一幅风景画的应用程序充分具体的问题。从那时起，分析的发展提供了新的视角，在数学和科学的几个领域取得了非凡的成就。最近的技术包括敏感的分解本地化的空间和相位，考虑到方向方面的问题。这样的分解导致了长期存在的问题的解决方案，如傅立叶级数的几乎处处收敛，以及双线性希尔伯特变换的有界性。主要研究者建议着手对傅立叶分析和应用中的问题进行广泛的研究，这些问题具有一个共同的功能：它们需要空间，频率和方向之间的微妙平衡。这一建议的学术价值在于所研究问题的历史价值，其中许多问题是随着时间的推移自然产生的，并以其困难而闻名。该方案包括三个部分：实证理论、寻找反例和具体应用。提出的工作积极的理论包括扩展的范围有界的双线性希尔伯特变换和其他粗糙奇异积分和研究m-线性正交性和Littlewood-Paley理论。频率分解的几何方面在这项研究中起着至关重要的作用。反例寻求双线性圆盘乘子在非局部平方可积的情况下和Carleson-Hunt算子的某些空间的功能。具体的应用集中在计算机断层扫描的方向灵敏度。傅立叶分析提供了具有共同的空间，频率和方向特性的部分的功能分解。正如交响乐可以被分析为简单音符的有限联合一样，某些复杂的操作可以通过它们在频谱上的作用来表示。一旦将信号和图像分解为可以单独研究的小块，就可以更好地定位信号和图像的干扰。通过乘以一个典型的非平滑函数来改变频率，如间歇性的电视传输，需要系统地研究信号中包含的信息的保存。频率变化下可积性的保持是研究信息保持的一个很好的模型，也是本建议理论部分的主要焦点。这一建议的更广泛影响正是这一点，即为防止信号或图像中包含的信息丢失的建模保护提供坚实的理论基础或基础。应用问题解决在这个项目中的重点是改进的结果，在计算机断层扫描的基础上，频率分解敏感的方向。这样的应用可能会导致在标准MRI测试期间移动对象的更清晰的计算机断层扫描图像。