$L^2$-Approximations of power and logarithmic functions with applications to numerical conformal mapping

Summary.

For a bounded Jordan domain G with quasiconformal boundary L, two-sided estimates are obtained for the error in best $L^2(G)$ polynomial approximation to functions of the form $(z-\tau)^{\beta}, {\beta}>-1$, and $(z-\tau)^m\log^l(z-\tau), m>-1, l\ne 0$, where $\tau\in L$. Furthermore, Andrievskii's lemma that provides an upper bound for the $L^\infty(G)$ norm of a polynomial $p_n$ in terms of the $L^2(G)$ norm of $p_n^\prime$ is extended to the case when a finite linear combination (independent of n) of functions of the above form is added to $p_n$. For the case when the boundary of G is piecewise analytic without cusps, the 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.