Discrete and Continuous Extremal Problems in Approximation Theory

技术摘要：本研究将主要集中在三个方面：(i)多元多项式插值的渐近插值测度的存在性和确定；（ii）二维拉普拉斯函数的反问题；(3) n维欧几里德空间中球面上最小能量点的渐近性。关于(i)，该研究有望产生用于某些代数曲线离散化的算法。关于（ii），我们将探索一种基于最佳亚纯近似的均匀体故障（裂纹）无损检测新方法。关于（iii），我们将研究由某些最小能量准则生成的球上点集的极限行为，并设计出“近最优”点集的快速算法。在研究的各个阶段，势理论都是一个有用的工具。摘要：研究将主要集中在三个问题领域：(i)均匀介质中故障（裂缝）的无创检测，（ii）球面上大量采样点的近最优配置，以及（iii）确定某些曲线上的近最优采样点。关于(i)，目标是开发一种有效的算法（程序），仅利用边界数据来确定导电材料内部裂纹的存在和位置。关于（ii），我们将研究在大分子结构分析中出现的球体上的点集，这些点集也可以作为很好的分布位置(例如；（在地球上）为全球天气模型采集气候数据。关于（iii），我们将探索可用于几何设计和数据采样目的的某些曲线上的点集。

Technical Abstract:The research will primarily focus on three areas: (i) theexistence and determination of asymptotic interpolation measures for multivariate polynomial interpolation; (ii) inverseproblems for the 2D Laplacian; (iii) asymptotics for minimalenergy points on the sphere in n-dimensional Euclidean space.Regarding (i), the research is expected to result in algorithmsfor discretization of certain algebraic curves. Concerning (ii)we will explore a new method based on best meromorphic approximationfor the nondestructive testing of homogeneous bodies for faults (cracks). Regarding (iii), we shall investigate the limiting behavior of point sets on the sphere generated by certain minimum energy criteria and also devise fast algorithmsfor "near-optimal" point sets. In all phases of the research,potential theory is expected to be a useful tool.Non-technical Abstract:The research will primarily focus on three problem areas:(i) the noninvasive detection of faults (cracks) in homogeneousmedia, (ii) near-optimal configurations of large numbersof sampling points on a sphere, and (iii) the determination of near-optimal sampling points on certain curves. Regarding(i), the goal is to develop an efficient algorithm (program)that utilizes only boundary data to determine the existenceand location of cracks inside a conducting material. Concerning(ii), we shall investigate point sets on the sphere thatarise in the structural analysis of large molecules, and alsoserve as well distributed locations (eg. on Earth) for samplingclimate data for global weather models. Regarding (iii), we shallexplore point sets on certain curves that can be used forgeometric design and data sampling purposes.