APPROXIMATION OF ANALYTIC-FUNCTIONS - A METHOD OF ENHANCED CONVERGENCE

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 connectio n with certain problems in wave scattering. In sectional sign 2 we pre sent 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 conv ergence rates inside the circles of convergence. We show that conforma l maps can also produce a rather dramatic improvement in the condition ing of Pade approximation. This improvement, which we discuss theoreti cally for Stieltjes-type functions, is most notorious in cases of very poorly conditioned Pade problems. In many instances, an application o f enhanced convergence in conjunction with Pade approximation leads to results which are many orders of magnitude more accurate than those o btained by either classical Pade approximants or the summation of a tr uncated enhanced series.