Polynomial Approximation of Conformal Maps

Abstract.

Let G be a finite domain, bounded by a Jordan curve Γ , and let f ₀ be a conformal map of G onto the unit disk. We are interested in the best rate of uniform convergence of polynomial approximation to f ₀ , in the case that Γ is piecewise-analytic without cusps. In particular, we consider the problem of approximating f ₀ by the Bieberbach polynomials π _n and derive results better than those in [5] and [6] for the case that the corners of Γ have interior angles of the form π/N . In the proof, the Lehman formulas for the asymptotic expansion of mapping functions near analytic corners are used. We study the question when these expansions contain logarithmic terms.