Fourier Analysis Techniques and Applications

傅立叶分析技术及应用

• 批准号: 0400423
• 负责人: Rodolfo Torres
• 金额: $ 14万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400423&HistoricalAwards=false
• 关键词: Fourier; Analysis; Techniques; Applications

DMS 0400423R TorresUniversity of KansasFourier Analysis Techniques and ApplicationsABSTRACTThis proposal is devoted to investigations in the analysis of operators associated to singular integrals, product-like operations, function spaces and approximations, estimates for commutators and null forms, and other related problems. The main tools to be used are decomposition techniques related to time-frequency expansions. Singular multilinear pseudodifferential operators will be studied together with their application to nonlinear analysis, estimates in Sobolev spaces, and partial differential equations. Another part of the project is concerned with the mathematical modeling of transformations of quasi-ordered geometries as measured by their Fourier spectra. This is motivated by ongoing research in biology in the analysis of the scattering of light by nanostructures in the tissues of living organisms.A common task in both theoretical and applied problems is to find the best way to decompose a complicated system in a way that efficiently quantifies observable properties and effects and keeps the available information in an organized and manageable form. Fourier analysis decodes such information by resolving a signal or function into a spectrum of waves of different amplitudes and frequencies, while providing also a way to study spatial order, structured geometries, and patterns in data. Combined with this analysis, multilinear approaches to complicated phenomena in nonlinear science will provide further understanding of problems where simple linear or first-order approximations are not sufficient. The analysis of ordered structures is also of importance to research in other areas of biology and physical sciences. For example, it is a relevant aspect in questions about the study of macroscopic physical properties of materials in terms of their microscopic structures.

DMS 0400423R Kansasfourier分析技术和应用程序提案的提案专门研究与单数积分相关的操作员，类似产品式操作，功能空间和近似值，换向器和NULL形式的估计以及其他相关问题的估计。要使用的主要工具是与时频扩展相关的分解技术。将研究奇异的多线性假差异操作员，并将其应用于非线性分析，Sobolev空间中的估计以及部分微分方程。该项目的另一部分与通过其傅立叶光谱衡量的准测量几何形状转换的数学建模有关。这是由生物学研究的持续研究激发的，在分析生物组织组织中纳米结构的光散射。理论和应用问题中的一项常见任务是找到以有效量化可观察到的可观察性质和效果的方式分解复杂系统的最佳方法，并将可用信息保持在有组织的和可管理的形式中。傅立叶分析通过将信号或函数解析为不同幅度和频率波的频谱来解码此类信息，同时还提供了一种研究空间秩序，结构性几何形状和数据模式的方法。结合此分析，非线性科学中复杂现象的多线性方法将进一步了解简单的线性或一阶近似不足的问题。对有序结构的分析对于在生物学和物理科学的其他领域进行研究也很重要。例如，这是关于材料的显微镜结构研究宏观物理特性的问题的一个相关方面。