Extremal Problems, Orthogonal Polynomials and Approximation Theory

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Grant Title: Extremal Problems, Orthogonal Polynomials, and Approximation Theory Principal Investigator: Edward B. Saff Co-Principal Investigator: E. A. Rakhmanov The project involves the use of potential theory and operator theory to investigate three main problem areas: (1) discretization of the surface of the unit sphere in N-dimensions, including asymptotics for minimal energy arrangements and algorithms for constructing near optimal points; (2) weak and strong asymptotics for orthogonal polynomials with respect to continuous and discrete weights; and (3) best rational and meromorphic approximation in function spaces, including generalizations of the theory of Adamyan, Arov, and Krein. The main problem to be investigated concerns finding nearly optimal arrangements of large numbers of points (typically in the hundreds or thousands) on the surface of a sphere so that these points are uniformly dispersed with respect to area on the sphere (i.e., the proportion of points lying in an arbitrary cap is close to the ratio obtained by dividing the area of the cap by the total area of the sphere). For this purpose the investigators will use minimum energy criteria which were introduced in a previous NSF grant. By analyzing the asymptotic behavior (as the number of points grows large), the investigators aim to devise simple schemes that generate configurations with near minimal energy. The importance of this problem is underscored in a recent article by S.Smale, who lists the study of such spherical point schemes as one of the eighteen main mathematical problem areas for the next century. Indeed this study has a wide variety of practical applications, among which are the following: (1) the structural analysis of large, stable, carbon molecules (known as fullerenes); (2) the determination of optimal sampling points on the Earth's surface for data gathered by orbiting satellites; and (3) the testing of radar sensors for aircraft when spherical coverage around the aircraft is essential.