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(−κn)) with some κ>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(−κ′n/log n)) with some κ′>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 Sinc approximation and the Sinc-collocation method for two-point boundary value problems.