Mathematical Sciences: Real Variable Techniques in the Approximation of Functions and Boundary Value Problems in Nonsmooth Domains

数学科学：非光滑域中函数逼近和边值问题的实变量技术

• 批准号: 9623251
• 负责人: Rodolfo Torres
• 金额: $ 1.65万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 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: 1996-07-15 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623251&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Real; Variable; Techniques

ABSTRACT Proposal: DMS-962325 PI: Torres Torres will continue with his work in harmonic analysis and partial differential equations. This work combines wavelet decompositions, singular integrals, and related real variable techniques. He plans to study functions that can be characterized by their values on a discrete set of points. A scale of discrete function spaces introduced by Torres in his previous work quantifies the oscillatory properties of these functions. He will now use these spaces to measure the errors in various linear and nonlinear approximations of band-limited signals. He intends to provide optimal estimates that cannot be attained with more classical techniques which only take into account the size of the functions. Other technical aspects of this part of the research relate to problems about operator theory and Besov spaces. In the second part of his program, Torres will continue his work on transmission problems in nonsmooth domains. For example, he will consider problems arising in electromagnetism. He will use singular integrals and layer potentials techniques. He also wants to combine his work with recent numerical analysis techniques for the study of large systems of linear equations. Band-limited signals are functions obtained as superposition of periodic waves whose frequencies remain bounded. Most signals in applications are band-limited and they can be recovered from the knowledge of their values on a discrete set of instants in time. This is very important for practical purposes because, often, a set of measured samples of a signal is all that is available from an experiment or a detection procedure. Fourier and Harmonic Analysis techniques codify, transform, and interpret the information and properties of a function or signal in a quantified way. In particular, the discrete character of wavelets and the techniques to be used make them amenable to numerical applications. Error estimates in approximations and the processes to minimize them trans late into results about compression of information. Other related problems are found in tomography and medical imaging. Transmission problems arise in physical situations in which an object interacts with the surrounding ambient space or when an object is composed of several different materials. These problems require the simultaneous solution, inside and outside of a region in space, of a systems of differential equations that are coupled by conditions on the boundary of the region. Effective tools for studying such problems in smooth domains are available, but it is very important to further develop the theory so it can adapted to domains that have corners or edges since they model realistic situation in the applied sciences.

摘要提案：DMS-962325 PI：托雷斯 托雷斯将继续他的工作，调和分析和偏微分方程。这项工作结合了小波分解，奇异积分，和相关的真实的变量技术。他计划研究的功能，可以其特点是其价值观的一组离散的点。一个规模的离散函数空间介绍了托雷斯在他以前的工作量化这些功能的振荡特性。他现在将使用这些空间来测量带限信号的各种线性和非线性近似的误差。 他打算提供最佳估计，不能达到更经典的技术，只考虑到规模的职能。这部分研究的其他技术方面涉及算子理论和Besov空间的问题。 在他的计划的第二部分，托雷斯将继续他的工作，在非光滑传输问题， 域.例如，他将考虑电磁学中出现的问题。 他将使用奇异积分和层势技术。 他还想把联合收割机的工作与最近的数值分析技术结合起来，用于研究大型线性方程组。 带限信号是作为其频率保持有界的周期波的叠加而获得的函数。应用中的大多数信号都是带限的，并且它们可以从它们在离散时刻集合上的值的知识中恢复。这对于实际目的是非常重要的，因为通常，信号的一组测量样本是从实验或检测过程中可获得的全部。傅立叶和谐波分析技术以量化的方式编码、转换和解释函数或信号的信息和属性。特别是，小波的离散特性和所使用的技术使它们适合于数值应用。近似中的误差估计和使其最小化的过程转化为关于信息压缩的结果。 在断层摄影和医学成像中发现了其他相关问题。传输问题出现在物体与周围环境空间相互作用或物体由几种不同材料组成的物理情况下。 这些问题需要同时解决方案，内部和外部的空间区域，微分方程的系统，耦合条件的边界上的区域。在光滑域中研究这类问题的有效工具是可用的，但进一步发展该理论是非常重要的，因此它可以适应具有角或边的域，因为它们在应用科学中模拟现实情况。