DE-Sinc methods have almost the same convergence property as SE-Sinc methods even for a family of functions fitting the SE-Sinc methods
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In this paper, the theoretical convergence rate of the Sinc indefinite integration combined with the double-exponential (DE) transformation is given for a class of functions for which the single-exponential (SE) transformation is suitable. Although the DE transformation is considered as an enhanced version of the SE transformation for Sinc-related methods, the function space for which the DE transformation is suitable is smaller than that for SE, and therefore, there exist some examples such that the DE transformation is not better than the SE transformation. Even in such cases, however, some numerical observations in the literature suggest that there is almost no difference in the convergence rates of SE and DE. In fact, recently, the observations have been theoretically explained for two explicit approximation formulas: the Sinc quadrature and the Sinc approximation. The conclusion is that in such cases, the DE’s rate is slightly lower, but almost the same as that of the SE. The contribution of this study is the derivation of the same conclusion for the Sinc indefinite integration. Numerical examples that support the theoretical result are also provided.
Mathematics Subject Classification (2000)41A25 65D30
- 6.Okayama, T., Matsuo, T., Sugihara, M.: Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration. Numer. Math. (2013). doi: 10.1007/s00211-013-0515-y
- 7.Okayama, T., Tanaka, K., Matsuo, T., Sugihara, M.: 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. Numer. Math. (2013). doi: 10.1007/s00211-013-0540-x