New Exponential Variable Transform Methods for Functions with Endpoint Singularities, preprint

Abstract. The focus of this article is the approximation of functions which are analytic on a compact interval except at the endpoints. Existing numerical methods for approximating such functions depend upon the use of particular conformal maps from the original interval to either a semi-infinite or an infinite interval, followed by an appropriate approximation procedure on the new region. We first analyse the convergence of these existing methods and show that, in a precisely defined sense, they are sub-optimal. Specifically, they exhibit poor resolution properties, by which we mean that many more degrees of freedom are required to resolve oscillatory functions than standard approximation schemes for analytic functions such as Chebyshev interpolation. To remedy this situation, we introduce two new transforms; one for each of the above settings. We first provide full convergence results for these new approximations. Second, we demonstrate that, for a particular choice of parameters, these methods lead to substantially better resolution properties. Finally, we show that optimal resolution power can be achieved by an appropriate choice of parameters, provided one forfeits convergence beyond some finite but arbitrarily small tolerance. Key words. Chebyshev interpolation, conformal map, endpoint singularity, resolution analysis AMS subject classifications. 1. Introduction. Th