Mathematical Sciences: Approximation Theory and Special Functions

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This project focuses mathematical research on problems of approximation and special functions. The latter comprise the basis of much of approximation theory, although there is a considerable body of work concentrating on analysis of the functions themselves. Work to be done includes studies of weighted potential and weighted polynomials in the complex plane, asymptotics for orthogonal polynomials and special functions, roots of polynomials and extremal problems for Blaschke products and rational functions. Work will also be done on the degree of approximation by Bernstein operators and Bergman kernel methods for numerical conformal mapping. Approximation is the defining feature of mathematical analysis. Compact expressions of solutions of problems arising in the physical universe rarely occur. More often, the solutions must be approximated by known quantities which have desirable properties and are easily computed. This research contributes to the dual goals of understanding the special functions of analysis and discovering their approximation properties.