Approximation of analytic functions : a method of enhanced convergence

We present a method of enhanced convergence for the approximation of analytic functions. This method introduces conformal transformations in the approximation problems in order to help extract the values of a given analytic function from its Taylor expansion. We show that conformal transformations can extend the radius of convergence of a power series far into infinity, enhance substantially its convergence rates inside the circle of convergence, and can produce a rather dramatic improvement in the conditioning of Pade approximation. This improvement, which we discuss theoretically for Stieltjes type functions, is most notorious in cases of very poorly conditioned Pade problems. In some instances, an application of enhanced convergence leads to results which are many orders of magnitude more accurate than those obtained by classical approximants. 1991 Mathematics Subject Classifications: 65B10, 41A21, 41A25.