Mathematical Sciences: Singular Integrals and Fourier Integrals

数学科学：奇异积分和傅立叶积分

• 批准号: 9531806
• 负责人: Allan Greenleaf
• 金额: $ 11.76万
• 依托单位: University of Rochester
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9531806&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Singular; Integrals; Fourier

Abstract Greenleaf 9531806 Characteristic space-time estimates, previously obtained, for solutions of wave equations will be extended and applied to yield results on the approximate determination of compactly supported, time-independent potentials in L^2 of 3-space from approximate knowledge of their backscattering (or other determined sets of scattering data.) Extensions of such results to potentials with noncompact support and of N-particle type will be pursued. Estimates, in terms of Sobolev and Lebesgue norms, for Fourier integral operators associated with canonical relations exhibiting cusp singularities, both simple and of higher order, will be investigated. Such estimates have applications to regularity properties of averaging operators associated with integrals over generic families of lines and curves in n-dimensional space. Finally, classes of Fourier integral operators arising from real analogues of Goncharov's complexes of hyperplane sections of algebraic arieties of minimal degree in n-dimensional projective space will be studied, with the goal of obtaining composition calculi for them. The principal object of this project will be the study of several types of singular integral operators and Fourier integral operators. Such operators, which transform functions on one space into functions on another (possibly different) space, have become central tools in the study of linear partial differential equations which govern diverse physical phenomena, such as electromagnetic fields and sound propagation. The particular operators to be studied in this project arise in the scattering of waves by potential functions, and in tomography, the mathematical basis for a variety of medical imaging systems, such as CAT and MRI scanners. Thus, progress on the problems considered in this project will contribute to the theoretical underpinnings of procedures for reconstructing unknown quantities of physical interest from noninvasive observations. Despite the fact that they arise in d ifferent problems, the operators to be studied share several common features, which involve more complicated geometry than is present in the original versions of singular integral and Fourier integral operators. It is hoped that, eventually, improved understanding of these operators will lead to improved reconstruction techniques.

已有的波动方程解的格林利夫9531806特征时空估计，将被推广并应用于由L^2的后向散射近似知识(或其它确定的散射数据集)近似确定三维空间中紧支撑的、与时间无关的势的结果。将这些结果推广到具有非紧支承势和N粒子型势能。我们将研究与表现出单阶和高阶尖点奇异性的正则关系有关的傅立叶积分算子的Soblev范数和Lebesgue范数方面的估计。这种估计可应用于n维空间中一般直线和曲线族上与积分相关的平均算子的正则性。最后，我们将研究n维射影空间中极小次代数簇的Goncharov复的实仿所产生的一类傅里叶积分算子，目的是得到它们的复合演算。本项目的主要目标是研究几种类型的奇异积分算子和傅立叶积分算子。这种将一个空间上的函数转换成另一个(可能不同)空间上的函数的算子，已经成为研究线性偏微分方程组的中心工具，线性偏微分方程控制着不同的物理现象，如电磁场和声传播。本项目中要研究的特殊运算符出现在势函数对波的散射中，以及在层析成像中，这是各种医学成像系统的数学基础，如CAT和MRI扫描仪。因此，在这个项目中考虑的问题的进展将有助于从非侵入性观测中重建未知量的物理兴趣的程序的理论基础。尽管它们出现在不同的问题中，但所研究的算子有几个共同的特征，这些特征涉及比原始版本的奇异积分和傅立叶积分算子更复杂的几何。人们希望，最终，对这些运算符的更好理解将导致重建技术的改进。