Mathematical Sciences: Pointwise Fourier Inversion in Several Variables

摘要提案：DMS-9623082 PI：Pinsky 平斯基计划研究点态傅立叶反演秩一对称空间的紧凑和非紧凑型。与此同时，他计划继续波动方程方法傅立叶反演，特别强调聚焦现象之间的联系和速度收敛的傅立叶变换在二维和三维。 他还将研究厄米多项式展开，这是密切相关的薛定谔方程的量子振荡器，以同样的方式，傅立叶变换是有关（达朗贝尔）波动方程。 逐点傅里叶逆问题是近世纪来数学分析中的一个中心问题。 在单变量的情况下，傅立叶分析对于理解信号处理中的滤波和采样以及电气工程和应用科学中的其他时间序列现象至关重要。在多变量的情况下，傅立叶分析与解决物理学、地质学和机械工程中出现的偏微分方程密切相关。吉布斯现象的详细研究和相关的高维例子的分歧和振荡的傅立叶部分和将作出的主要研究者。将本研究的结果与近年来发现许多应用的当前流行的“小波”变换进行对比将是有趣的。

ABSTRACT Proposal: DMS-9623082 PI: Pinsky Pinsky plans to study pointwise Fourier inversion in rank-one symmetric spaces of the compact and non-compact type. At the same time he plans to continue the wave-equation approach to Fourier inversion, with special emphasis on the connection between focusing phenomena and the speed of convergence of Fourier transforms in two and three dimensions. He will also investigate Hermite polynomial expansions, which are closely related to the Schrodinger equation for the quantum oscillator, in the same fashion that Fourier transforms are related to the (d'Alembert) wave equation. The subject of pointwise Fourier inversion has been a central concern in mathematical analysis for nearly a century. In the case of one variable, Fourier analysis is essential to understanding filtering and sampling in signal processing and other time series phenomena in electrical engineering and the applied sciences. In the case of several variables, Fourier analysis is closely related to solving the partial differential equations that occur in physics, geology, and mechanical engineering. A detailed study of Gibbs' phenomenon and related higher-dimensional examples of divergence and oscillation of Fourier partial sums will be made by the principal investigator. It will be interesting to contrast the results of this study with the currently popular "wavelet" transforms, which have found many applications in recent years.