Recent developments of the Sinc numerical methods

This paper gives a survey of recent developments of the Sinc numerical methods. A variety of Sinc numerical methods have been developed by Stenger and his school. For a certain class of problems, the Sinc numerical methods have the convergence rates of O(exp(-kapparootn)) with some kappa>0, where n is the number of nodes or bases used in the methods. Recently it has turned out that the Sinc numerical methods can achieve convergence rates of O(exp(-epsilon' n/log n)) with some kappa'>0 for a smaller but still practically meaningful class of problems, and that these convergence rates are best possible. The present paper demonstrates these facts for two Sinc numerical methods: the Sine approximation and the Sinc-collocation method for two-point boundary value problems. (C) 2003 Elsevier B.V. All rights reserved.