Approximation and Orthogonality in Sobolev Spaces

该研究项目关注的是在Sobolev空间的设置中由简单的多项式函数（和相关问题）的家庭对多变量函数的近似的研究，Sobolev空间是最初为研究数学物理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往往是棘手的，我们往往需要找到一个更易于管理的近似值。 高维空间中区域上的近似方法是这一原理的很好例子，它们在应用数学中的许多问题中是至关重要的。 与一维相反，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。 主要研究者将研究几个问题的近似和正交性的Sobolev空间，依赖于新的连接和想法，最近才发现。 该项目旨在理论理解和构建新的近似方法，并且该工作有可能影响科学计算，数值分析，统计和地球科学。首席研究员将研究Sobolev空间在规则域上的近似和正交性，例如立方体，球，球体和单形。 该项目结合了几个研究课题：近似理论，傅立叶分析，数值分析和正交多项式。 其中一个主要的问题源于该地区的谱方法的数值解偏微分方程。 通过主要研究者和合作者最近的工作，人们越来越清楚地认识到，理解Sobolev空间中的正交性对于Sobolev空间中的近似和计算至关重要。 这项研究将基于最近的进展，在表征最佳逼近多项式的单位球和单位球，在索伯列夫正交多项式，并在谱近似。 预计该项目将导致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.