Mathematical Sciences: Harmonic Analysis and Hyperbolic Partial Differential Equations

小行星9401855 该奖项支持数学研究，重点是调和分析在双曲型微分方程理论中的应用。 工作包括发展的波群的狄利克雷条件流形上类似的额外条件，即边界是凹的。 估计失败的流形凸边界，虽然在这些情况下存在多重反射测地线产生滑行射线沿沿着边界。 因此，有必要找到估计失败的最坏情况示例，并尝试获得这种情况的更新估计。 工作也将在哈代空间的傅立叶积分算子。 努力将去分析一个新的功能空间上的代数傅立叶积分算子的零阶连续作用。 某些与振动问题相关的算子在标准哈代空间上是有界的，但在新的空间上仍是有界的.首要任务之一是获得空间的另一种表征。 最后，将完成工作的建设框架的平方可积函数对角化波群。 调和分析结合了这些数学元素，最好地体现了综合的思想。 一种是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，通过组件的重组来重构解决方案。 傅立叶级数和傅立叶变换是在这种情况下使用的工具的例子;一个离散，另一个表示连续分解。最近，小波理论为一些更经典的谐波分析方法增加了新的维度。

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.***