Approximation and Orthogonality in Sobolev Spaces

这项研究项目涉及在Sobolev空间的设置中通过更简单的多项式函数族(以及相关问题)来研究多变量函数的逼近，Sobolev空间是最初为研究数学物理中的问题而开发的抽象数学函数空间，今天在许多科学计算环境中使用。一个真正复杂的系统或问题通常是难以解决的，我们经常需要找到一个更容易管理的近似值。高维空间中区域的逼近方法就是这一原理的一个很好的例子，它们在应用数学中的许多问题中都是至关重要的。与一个维度形成对比的是，许多具有根本重要性的更高维度的挑战性问题没有得到解决。主要研究人员将研究Sobolev空间中的几个关于逼近和正交性的问题，这些问题依赖于最近才揭示的新的联系和概念。该项目的目标是从理论上理解和构建新的逼近方法，这项工作有可能影响科学计算、数值分析、统计学和地学。首席调查者将研究正则区域上Sobolev空间的逼近和正交性，如立方体、球、球和单纯形。该项目结合了几个研究课题：逼近理论、傅立叶分析、数值分析和正交多项式。其中一个主要问题源于偏微分方程组数值解的谱方法领域。通过主要研究者和合作者最近的工作，已经变得越来越清楚，理解Sobolev空间中的正交性对于Sobolev空间中的逼近和计算是至关重要的。这项研究将基于单位球面和单位球上多项式最佳逼近的刻画、Sobolev正交多项式的最佳逼近以及谱逼近的最新进展。该项目有望带来新的科学计算方法和新的算法。

This research project is concerned with the study of approximations of functions of several variables by families of simpler, polynomial functions (and related questions) in the setting of Sobolev spaces, which are abstract mathematical function spaces originally developed to study problems in mathematical physics and which are utilized today in a number of scientific computing settings. A truly complex system or problem is often intractable, and we often need to find an approximation that is more manageable. Approximation methods on domains in higher dimensional spaces are good examples of this principle, and they are crucial in many problems in applied mathematics. In contrast to one dimension, many challenging problems in higher dimensions that are of fundamental importance are not resolved. The Principal Investigator will study several problems on approximation and orthogonality in Sobolev spaces that rely on new connections and ideas revealed only recently. The project aims at both theoretical understanding and construction of new approximation methods, and the work has the potential to impact scientific computing, numerical analysis, statistics, and geoscience.The Principal Investigator will study approximation and orthogonality in Sobolev spaces on regular domains, such as cubes, balls, spheres, and simplexes. The project combines several research topics: approximation theory, Fourier analysis, numerical analysis, and orthogonal polynomials. One of the main problems originates from the area of spectral methods for the numerical solution of partial differential equations. Through recent work of the Principal Investigator and collaborators, it has become increasingly clear that understanding orthogonality in Sobolev spaces is crucial for approximation and computation in Sobolev spaces. This research will be based on recent progress in characterization of best approximation by polynomials on the unit sphere and on the unit ball, in Sobolev orthogonal polynomials, and in spectral approximation. The project is expected to lead to new scientific computational methods and new algorithms.