Mathematical Sciences: Pointwise Fourier Inversion in Several Variables

数学科学：多变量的逐点傅立叶反演

• 批准号: 9623082
• 负责人: Mark Pinsky
• 金额: $ 6.96万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-15 至 1999-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623082&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Pointwise; Fourier; Inversion

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.

摘要 提案：DMS-9623082 PI：Pinsky Pinsky 计划研究紧致型和非紧致型的一阶对称空间中的点傅立叶反演。与此同时，他计划继续采用波动方程方法进行傅里叶反演，特别强调聚焦现象与二维和三维傅里叶变换收敛速度之间的联系。 他还将研究埃尔米特多项式展开式，该展开式与量子振荡器的薛定谔方程密切相关，就像傅里叶变换与（达朗贝尔）波动方程相关一样。 近一个世纪以来，逐点傅里叶反演的主题一直是数学分析的中心问题。 对于一个变量，傅里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。在多个变量的情况下，傅立叶分析与求解物理学、地质学和机械工程中出现的偏微分方程密切相关。首席研究员将详细研究吉布斯现象以及傅立叶部分和的发散和振荡的相关高维示例。将这项研究的结果与当前流行的“小波”变换进行对比将会很有趣，“小波”变换近年来已得到许多应用。