New exponential variable transform methods for functions with endpoint singularities

TL;DR

This paper introduces new exponential variable transform methods to improve the approximation of functions with endpoint singularities, achieving better resolution properties at the cost of classical convergence.

Contribution

The authors develop two novel transforms for endpoint singularity approximation, providing convergence analysis and demonstrating enhanced resolution capabilities over existing methods.

Findings

New transforms improve resolution for oscillatory functions.

Optimal resolution achieved with parameter tuning.

Methods allow user-controlled accuracy instead of classical convergence.

Abstract

The focus of this article is the approximation of functions which are analytic on a compact interval except at the endpoints. Typical 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 provide full convergence results for these new approximations and then 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 classical convergence. Instead, the resulting method attains a finite, but user-controlled accuracy specified by the parameter choice.