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.

抽象的Greenleaf 9531806先前获得的波动方程解决方案的特征时空估计将被扩展并应用，以产生结果，以近似确定紧凑型，时间无关的，与l^2的l^2的潜在l^2的l^2，从近似对其反向造影的知识（或其他确定的散射数据集合）的扩展。根据Sobolev和Lebesgue规范的估计，将研究与表现出尖端奇异性的规范性关系相关的傅立叶积分运算符，无论是简单还是更高阶段。此类估计值在n维空间中的一般线和曲线上的平均运算符的规律性属性应用。最后，将研究由Goncharov的真实类似物产生的傅里叶积分算子的类别的类别，即在N维投影空间中最小程度的代数栖息地的超平面剖面。 该项目的主要对象将是研究几种类型的单数积分运营商和傅立叶积分运算符。这些操作员将功能在一个空间上转变为另一个空间的功能（可能不同）的功能已成为研究各种物理现象的线性部分偏微分方程的中心工具，例如电磁场和声音传播。在该项目中要研究的特定操作员出现在潜在功能的波浪散射中，在断层扫描中，是各种医学成像系统（例如CAT和MRI扫描仪）的数学基础。因此，该项目中考虑的问题的进展将有助于从无创观测中重建未知数量的身体兴趣的程序的理论基础。尽管事实是在ifferent问题中出现的事实，但要研究的运算符具有几个共同的特征，这些功能涉及比原始版本的单数积分和傅立叶积分运算符中所存在的更复杂的几何形状。希望最终，对这些操作员的理解有所改善，将导致改进的重建技术。