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

建议：DMS-962325 PI：Torres Torres将继续他在调和分析和偏微分方程方面的工作。这项工作结合了小波分解、奇异积分和相关的实变量技术。他计划研究可以通过一组离散的点上的值来表征的函数。Torres在他以前的工作中引入的离散函数空间的尺度量化了这些函数的振荡性质。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中的误差。他打算提供仅考虑函数大小的更经典技术无法获得的最优估计。这部分研究的其他技术方面涉及到关于算子理论和Besov空间的问题。在他的程序的第二部分，Torres将继续他在非光滑区域上传输问题的工作。例如，他会考虑电磁方面的问题。他将使用奇异积分和层势技术。他还希望将他的工作与最近的数值分析技术结合起来，研究大型线性方程组。带限信号是由频率保持有界的周期波叠加而得到的函数。应用中的大多数信号都是频带受限的，它们可以从时间上离散的一组瞬时上的值的知识中恢复出来。这对于实际目的非常重要，因为通常，一组测量的信号样本是从实验或检测过程中所能获得的全部。傅立叶和调和分析技术以量化的方式对函数或信号的信息和属性进行编码、转换和解释。特别是，小波的离散性质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将其最小化的过程转化为关于信息压缩的结果。在断层扫描和医学成像中也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的物理环境中，或者当物体由几种不同的材料组成时。这些问题需要在空间的区域内外同时求解由区域边界上的条件耦合的微分方程组。在光滑域中研究这类问题的有效工具是可用的，但非常重要的是进一步发展该理论，使其能够适应于有角或边的域，因为它们模拟了应用科学中的现实情况。

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.