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 0400423 R Torres堪萨斯大学傅立叶分析技术和应用摘要该提案致力于奇异积分，乘积运算，函数空间和近似，估计算子和零形式，以及其他相关问题的分析调查。要使用的主要工具是与时间-频率扩展相关的分解技术。奇异多线性伪微分算子将与它们在非线性分析、Sobolev空间估计和偏微分方程中的应用一起研究。该项目的另一部分是关于准有序几何变换的数学建模，通过它们的傅立叶光谱来测量。这是由生物学中正在进行的研究在分析光散射的纳米结构在生物组织中的livingorganism.A在理论和应用问题的共同任务是找到最好的方式来分解一个复杂的系统，有效地量化可观察到的属性和效果，并保持在一个组织和管理的形式可用的信息。傅立叶分析通过将信号或函数分解为不同振幅和频率的波谱来解码这些信息，同时还提供了一种研究空间顺序，结构几何和数据模式的方法。结合这一分析，多线性方法在非线性科学中的复杂现象将提供进一步的理解的问题，简单的线性或一阶近似是不够的。有序结构的分析对生物学和物理科学的其他领域的研究也很重要。例如，它是一个相关方面的问题，研究宏观物理性质的材料方面的微观结构。