Discrete and Continuous Extremal Problems in Approximation Theory

University Of South Florida, Tampa FL

Investigators

Abstract

Technical Abstract: The research will primarily focus on three areas: (i) the existence and determination of asymptotic interpolation measures for multivariate polynomial interpolation; (ii) inverse problems for the 2D Laplacian; (iii) asymptotics for minimal energy points on the sphere in n-dimensional Euclidean space. Regarding (i), the research is expected to result in algorithms for discretization of certain algebraic curves. Concerning (ii) we will explore a new method based on best meromorphic approximation for 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 algorithms for "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 homogeneous media, (ii) near-optimal configurations of large numbers of 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 existence and location of cracks inside a conducting material. Concerning (ii), we shall investigate point sets on the sphere that arise in the structural analysis of large molecules, and also serve as well distributed locations (eg. on Earth) for sampling climate data for global weather models. Regarding (iii), we shall explore point sets on certain curves that can be used for geometric design and data sampling purposes.