Approximation of analytic functions: a method of enhanced convergence

Oscar P. Bruno, Fernando Reitich
1994 Mathematics of Computation  
We deal with a method of enhanced convergence for the approximation of analytic functions. This method introduces conformai 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
more » ... a general discussion of this approach. As is known in the case of the Euler transform, conformai 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 conformai 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.
doi:10.1090/s0025-5718-1994-1240654-9 fatcat:egnobjs5qjbf7hfsfcwpg5vw2y