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

摘要提案：DMS-962325 PI: Torres Torres将继续他在谐波分析和偏微分方程方面的研究。这项工作结合了小波分解、奇异积分和相关的实变量技术。他计划研究可以用离散点集合上的值来表征的函数。托雷斯在他之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各种带限信号的线性和非线性近似的误差。他打算提供仅考虑函数大小的更经典技术无法达到的最佳估计。这部分研究的其他技术方面涉及算子理论和贝索夫空间的问题。在他的计划的第二部分，托雷斯将继续他的工作在非光滑域的传输问题。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。他还想把他的工作与最近研究大型线性方程组的数值分析技术结合起来。带限信号是频率保持有界的周期波的叠加得到的函数。应用中的大多数信号是带限的，它们可以从在时间上离散的一组瞬间的值的知识中恢复。这对于实际目的非常重要，因为通常，一组信号的测量样本是实验或检测过程中可用的全部。傅里叶和谐波分析技术以一种量化的方式编纂、变换和解释函数或信号的信息和性质。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。近似值中的误差估计和最小化误差的过程转化为信息压缩的结果。其他相关的问题也存在于断层扫描和医学成像中。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，会出现传输问题。这些问题需要在空间区域内外的微分方程组的同时解，这些微分方程组是由区域边界上的条件耦合的。在光滑领域研究这类问题已经有了有效的工具，但由于这类问题反映了应用科学中的实际情况，因此进一步发展这一理论，使其适应于有角或有边的领域是非常重要的。

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.