Mathematical Sciences: Harmonic Analysis and Hyperbolic Partial Differential Equations

数学科学：调和分析和双曲偏微分方程

• 批准号: 9401855
• 负责人: Hart Smith
• 金额: $ 4万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1996-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401855&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Harmonic; Analysis; Hyperbolic

9401855 Smith This award supports mathematical research focusing on applications of harmonic analysis to problems in the theory of hyperbolic differential equations. Work includes the development of estimates for the wave group with Dirichlet conditions on manifolds similar to that obtained with the additional condition that the boundaries be concave. Estimates fail on manifolds with convex boundaries although in these instances there exist multiply reflected geodesics producing gliding rays which travel along the boundary. It is therefore of interest to find worst case example where the estimates fail and try to obtain newer estimates for this situation. Work will also be done on the Hardy space for Fourier integral operators. Efforts will go into analyzing a new function space on which the algebra of Fourier integral operators of order zero act continuously. Certain operators associated with oscillatory problems are not bounded on the standard Hardy spaces will remain bounded on the new spaces. One of the first tasks is to obtain an alternate characterization of the space. Finally, work will be done constructing frames for square integrable functions which diagonalize the wave group. Harmonic analysis combines those elements of mathematics best exemplifying the ideas of synthesis. One seeks to decompose complex problems into fundamental components. These components are then analyzed for their basic characteristics. Finally, the solution is reconstructed through a recombination of the components. The Fourier series and Fourier transform are examples of tools used in this context; one discrete , the other representing a continuous decomposition. More recently the wavelet theory added new dimensions to some of the more classical approaches to harmonic analysis.***

9401855 Smith 该奖项支持专注于调和分析在双曲微分方程理论问题中的应用的数学研究。 工作包括利用流形上的狄利克雷条件对波群进行估计，类似于通过边界为凹的附加条件获得的估计。 估计在具有凸边界的流形上失败，尽管在这些情况下存在多重反射测地线，产生沿着边界传播的滑动射线。 因此，有必要找到估计失败的最坏情况示例，并尝试针对这种情况获得更新的估计。 傅里叶积分算子的 Hardy 空间也将得到研究。 我们将努力分析零阶傅立叶积分算子代数连续作用的新函数空间。 与振荡问题相关的某些算子在标准 Hardy 空间上不受限制，但在新空间上仍然有界。第一个任务是获得空间的替代特征。 最后，将完成构建对角化波群的平方可积函数的框架的工作。 调和分析结合了最能体现综合思想的数学元素。 人们试图将复杂的问题分解为基本的组成部分。 然后分析这些组件的基本特性。 最后，通过组件的重新组合来重建解决方案。 傅里叶级数和傅里叶变换是在这种情况下使用的工具的示例；一个是离散的，另一个表示连续分解。最近，小波理论为一些更经典的调和分析方法添加了新的维度。***