Fourier Analysis: Space, Frequency, and Direction

让·巴蒂斯特·约瑟夫·傅立叶可以被称为频率分解的鼻祖。1807年，傅立叶提出了一种理论，即本质上任意的函数都可以用不同频率的基本正弦波的和来表示，他抵制了激烈的理论反对，为具体问题描绘了一幅充满应用前景的图景。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。最近的技术包括空间和相位的敏感分解，它考虑了问题的方向性方面。这样的分解导致了长期猜想的解决，例如傅里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。首席研究员建议着手对傅里叶分析和应用中的问题进行广泛的研究，这些问题有一个共同的特点：它们需要在空间、频率和方向之间取得微妙的平衡。这一建议的智力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。本文由实证理论、反例探索和具体应用三个部分组成。关于正理论的建议工作包括双线性Hilbert变换和其他粗糙奇异积分的有界范围的扩展以及m-线性正交性和Littlewood-Paley理论的研究。频率分解的几何方面在本研究中起着至关重要的作用。寻求了非局部平方可积情况下双线性圆盘乘子的反例和某些函数空间上的Carleson-Hunt算子。具体应用集中在计算机断层扫描的方向灵敏度上。傅里叶分析提供了具有共同空间、频率和方向特征的部分的函数分解。正如交响乐可以用简单音符的有限组合来分析一样，某些复杂的操作也可以用它们在频谱上的作用来表示。一旦将不规则的信号和图像分解成可以单独研究的小块，就能更好地定位它们。通过一个典型的非平滑函数的乘法来改变频率，比如断断续续的电视传输，要求系统地研究保存信号中包含的信息。频率变化下的可积性保持是研究信息保存的一个很好的模型，也是本文理论部分的重点。这一建议更广泛的影响也正是这一点，即为防止信号或图像中所含信息丢失的建模保护提供了坚实的理论基础或基础。本项目所解决的应用问题集中在改进基于方向敏感的频率分解的计算机断层扫描结果上。这样的应用可以在标准的MRI测试中为移动的受试者提供更清晰的计算机断层扫描图像。

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.