New exponential variable transform methods for functions with endpoint singularities
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 semiinfinite 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 suboptimal. 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 usercontrolled accuracy specified by the parameter choice.
 Publication:

arXiv eprints
 Pub Date:
 May 2013
 arXiv:
 arXiv:1305.2643
 Bibcode:
 2013arXiv1305.2643A
 Keywords:

 Mathematics  Numerical Analysis