Mathematical Sciences: Fourier analysis of distributions supported on hypersurfaces

小行星9401208 该RUI奖支持与频谱合成相关的谐波分析问题的数学研究，该研究大致指出，平滑，衰减函数在一组近似傅立叶变换中消失。 本项目主要研究流形上的谱综合、傅里叶变换的限制和常系数偏微分方程。 具体而言，该项目涉及欧几里得空间中超曲面支持的分布，以及流形的曲率如何与分布的傅立叶变换的衰减特性相关。 这项工作借鉴了最近的结果，弱谱合成的渐近估计。 由于存在弱谱合成对于一般流形失败的例子，因此有必要研究适当的条件，例如常数零度，这将提供肯定的答案。 调和分析关注的是通过用更简单的分量表示来分解和重构函数。 调和分析的基本工具之一是傅里叶变换，它在偏微分方程的研究中同样发挥着核心作用，这两个领域自然地结合在一起。 变换的分析，限于流形，是一个相对开放的领域，其中一般的结果通常很难实现。

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.