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.

Pinsky计划研究秩一对称空间中紧致型和非紧致型的逐点傅里叶反演。同时，他计划继续傅里叶反演的波动方程方法，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，它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的关系一样。近一个世纪以来，逐点傅里叶反演一直是数学分析的中心问题。在单变量的情况下，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，傅里叶分析与解决物理、地质和机械工程中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傅里叶部分和的散度和振荡的例子将由首席研究员。将这项研究的结果与近年来发现了许多应用的当前流行的“小波”变换进行对比将是有趣的。