| We consider the following two problems regarding polynomial approximation and interpolation.; In Chapter 1, we investigate the rate of approximation of the Riemann mapping function by generalized Bieberbach polynomials. As auxiliary results (which are also of independent interest), for a bounded Jordan domain G in the complex plane with quasiconformal boundary L, two-sided estimates are obtained for the error in best L2(G) polynomial approximation to functions of the form (z − τ) β, β > −1, and (z − τ) m logl(z − τ), m > −1, l ≠ 0, where τ ∈ L. Furthermore, Andrievskii's lemma that provides an upper bound for the L∞(G) norm of a polynomial pn in terms of the L 2(G) norm of is extended to the case when a finite linear combination (independent of n) of functions of the above form is added to pn. For the case when the boundary of G is piecewise analytic without cusps, these results are used to analyze the improvement in rate of convergence achieved by using augmented, rather than classical, Bieberbach polynomial approximants of the Riemann mapping function of G onto a disk. Finally, numerical results are presented that illustrate the theoretical results obtained.; In Chapter 2, we consider the problem concerning the behavior of “ good” points for multivariate polynomial interpolation on compact subsets of certain curves in . Let E ⊂ be compact and denote the dimension of the space of polynomials of degree at most n in s variables restricted to E. We introduce the notion of an asymptotic interpolation measure (AIM). Such a measure, if it exists, describes the asymptotic distribution of any scheme τn = , n = 1, 2, …, of nodes for multivariate polynomial interpolation for which the norms of the corresponding interpolation operators do not grow geometrically large with n. We demonstrate the existence of AIM's for the finite union of compact subsets of algebraic curves of genus 0 in . It turns out that the theory of logarithmic potentials with external fields plays a useful role in the investigation. For the sets mentioned above we give a computationally simple construction for “good” interpolation schemes. Furthermore, we computed AIM densities and plotted their graphs for some well-known algebraic curves. A Maple V program that can be used for this purpose is included. |
