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.

傅里叶分析技术与应用摘要本提案致力于研究奇异积分相关算子的分析、类积运算、函数空间和近似、对易子和零形式的估计以及其他相关问题。要使用的主要工具是与时频展开相关的分解技术。将研究奇异多线性伪微分算子及其在非线性分析、Sobolev空间估计和偏微分方程中的应用。该项目的另一部分涉及准有序几何变换的数学建模，通过它们的傅立叶谱进行测量。这是受到正在进行的生物研究的启发，该研究分析了生物体组织中纳米结构对光的散射。在理论和应用问题中，一个共同的任务是找到分解复杂系统的最佳方法，这种方法可以有效地量化可观察到的属性和效果，并将可用信息保持在有组织和可管理的形式中。傅里叶分析通过将信号或函数分解成不同幅度和频率的波谱来解码这些信息，同时也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，非线性科学中复杂现象的多线性方法将提供对简单线性或一阶近似不够的问题的进一步理解。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，从微观结构的角度研究材料的宏观物理性质是一个相关的方面。