Mathematical Sciences: Fourier analysis of distributions supported on hypersurfaces

9401208郭本瑞奖支持对调和分析中与谱合成相关的问题的数学研究，该奖大致指出，光滑的衰减函数从一个集合上消失近似傅立叶变换在同一集合上消失。本课题主要研究流形上的谱合成、傅立叶变换的限制和常系数偏微分方程组。具体地说，该项目涉及欧氏空间中超曲面所支持的分布，以及流形的曲率如何与分布的傅里叶变换的衰减性质相关。这项工作利用了最近的结果，给出了弱谱合成的渐近估计。由于对于一般流形存在弱谱合成失败的例子，因此有必要研究适当的条件，例如常零性，这将提供肯定的答案。调和分析涉及用更简单的分量表示函数的分解和重构。调和分析的基本工具之一是傅里叶变换，它在研究偏微分方程式中扮演着同样重要的角色，其中两个领域自然地结合在一起。变换的分析局限于流形，是一个相对开放的领域，一般结果通常很难实现。

9401208 Guo This RUI award supports mathematical research on problems in harmonic analysis related to spectral synthesis, which roughly states that smooth, decaying functions vanishing off a set approximate Fourier transforms vanishing on the same set. This project is mainly concerned with spectral synthesis on manifolds, the restriction of the Fourier transform and partial differential equations with constant coefficients. Specifically, the project concerns distributions supported by hypersurfaces in Euclidean space and how the curvature of the manifold is related to decay properties of the Fourier transform of the distribution. The work draws on recent results giving asymptotic estimates for weak spectral synthesis. Since there are examples where weak spectral synthesis fails for general manifolds, it is necessary to investigate the proper conditions, such as constant nullity, which will provide positive answers. Harmonic analysis concerns the decomposition and reconstruction of functions through representations in terms of simpler components. One of the fundamental tools of harmonic analysis is the Fourier transform which plays equally central role in the study of partial differential equations where the two fields come together naturally. Analysis of the transform, restricted to manifolds, is a relatively open field in which general results are usually hard to achieve.