The Polynomial Caratheodory--Fejer Approximation Method for Jordan Regions

We propose a method for the approximation of analytic functionson Jordan regions that is based on a Carathéodory—Fejértype of economization of the Faber series. The method turnsout to be very effective if the boundary of the region is analytic.It often still works when the region degenerates to a Jordanarc. We also derive related lower and upper bounds for the errorof the best approximation. ^*Research carried out while this author was at ETH Zurich partiallysupported by a Royal Society European Visiting Fellowship.     

The Caratheodory--Fejer Extension of a Finite Geometric Series

It is shown that the Caratheodory—Fejer extension of afinite geometric series can be given explicitly up to a simplepolynomial equation in an auxiliary variable. This result allowsus to analyse the Caratheodory-Fejer approximation method inthe case where the quotients of successive Maclaurin coefficientsof the given function tend to a limit. ^*Research carried out while this author was at ETH Zurich partiallysupported by a Royal Society European Visiting Fellowship.     

On the Convergence of Some Approximate Methods of Conformal Mapping

A method was given in Ellacott (1978) for determining approximatelythe conformal mapping of a Jordan region on to a disc. Someresults on the convergence of this method are given, which canbe used to prove the result (conjectured in Ellacott, 1978)that if the boundary curve is analytic, then convergence isuniform. The corresponding result is also proved for the Bergmanand Szeg Kernel methods with polynomial basis functions. (Theresult is already known for the Szeg? Kernel method, but a differentproof is given.) Also discussed is the use of rational basisfunctions for the Bergman Kernel method. Currently visiting Forschungsinstitut fr Mathematik, ETH-Zentrum,CH-8092, Zurich.