DE-Sinc methods have almost the same convergence property as SE-Sinc methods even for a family of functions fitting the SE-Sinc methods Part I: Definite integration and function approximation

Abstract

In this paper, the theoretical convergence rate of the trapezoidal rule combined with the double-exponential (DE) transformation is given for a class of functions for which the single-exponential (SE) transformation is suitable. It is well known that the DE transformation enables the rule to achieve a much higher rate of convergence than the SE transformation, and the convergence rate has been analyzed and justified theoretically under a proper assumption. Here, it should be emphasized that the assumption is more severe than the one for the SE transformation, and there actually exist some examples such that the trapezoidal rule with the SE transformation achieves its usual rate, whereas the rule with DE does not. Such cases have been observed numerically, but no theoretical analysis has been given thus far. This paper reveals the theoretical rate of convergence in such cases, and it turns out that the DE’s rate is almost the same as, but slightly lower than that of the SE. By using the analysis technique developed here, the theoretical convergence rate of the Sinc approximation with the DE transformation is also given for a class of functions for which the SE transformation is suitable. The result is quite similar to above; the convergence rate in the DE case is slightly lower than in the SE case. Numerical examples which support those two theoretical results are also given.