Approximation of analytic functions: a method of enhanced convergence

Abstract

We deal with 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 around a point. An instance of this method, based on the Euler transform, has long been known; recently we introduced more general versions of it in connection with certain problems in wave scattering. In §2 we present a general discussion of this approach. As is known in the case of the Euler transform, conformal transformations can enlarge the region of convergence of power series and can enhance substantially the convergence rates inside the circles of convergence. We show that conformal maps can also produce a rather dramatic improvement in the conditioning of Padé approximation. This improvement, which we discuss theoretically for Stieltjes-type functions, is most notorious in cases of very poorly conditioned Padé problems. In many instances, an application of enhanced convergence in conjunction with Padé approximation leads to results which are many orders of magnitude more accurate than those obtained by either classical Padé approximants or the summation of a truncated enhanced series.