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.

摘要建议：DMS-962325 PI：Torres Torres将继续他的谐波分析和部分微分方程的工作。这项工作结合了小波分解，奇异积分和相关的真实可变技术。他计划研究可以在离散点上以其价值为特征的功能。托雷斯在其先前的工作中引入的离散功能空间的量表量量化了这些功能的振荡特性。现在，他将使用这些空间来测量带限制信号的各种线性和非线性近似值中的误差。 他打算提供最佳的估计，这些估计无法通过更多的经典技术来实现，这些技术仅考虑了功能的大小。该研究的其他技术方面与操作者理论和BESOV空间有关。 在他计划的第二部分中，托雷斯将继续在非平滑域中的传输问题上继续工作。例如，他将考虑在电磁主义中引起的问题。 他将使用奇异的积分和层电位技术。 他还希望将他的工作与最新的数值分析技术相结合，以研究大型线性方程系统。 带限的信号是作为周期性波的叠加获得的函数，其频率保持界限。应用程序中的大多数信号都是限制的，并且可以从及时的一组瞬间集中从其值中恢复它们。这对于实际目的而言非常重要，因为通常，一组测量的信号样本是从实验或检测过程中获得的。傅立叶和谐波分析技术以量化的方式编纂，转换和解释函数或信号的信息和特性。特别是，小波的离散特征和要使用的技术使它们适合数值应用。近似值和将它们最小化的过程中的错误估计及时延迟到有关信息压缩的结果中。 在断层扫描和医学成像中发现了其他相关问题。在物理情况下出现了传输问题，其中一个物体与周围环境空间相互作用或物体由几种不同的材料组成时。 这些问题需要在空间内部和外部的同时解决方案，即由区域边界上的条件耦合的微分方程系统。可以使用有效研究这些问题的有效工具，但是进一步发展该理论非常重要，因此它可以适应具有角落或边缘的域，因为它们在应用科学中对现实的情况进行了建模。