Discrete and Continuous Extremal Problems in Approximation Theory

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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.