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 II: Indefinite integration
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 II: Indefinite integration
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即使对于拟合 SE-Sinc 方法的一系列函数，DE-Sinc 方法也具有与 SE-Sinc 方法几乎相同的收敛特性。
DOI：
10.1007/s00211-013-0541-9
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发表时间：
2013
影响因子：
2.1
通讯作者：
Ken'ichiro Tanaka
中科院分区：
文献类型：
--
作者：
Kazuaki Inaba;Hiroto Takahashi;Yu Kurokawa;Kikuo Kishimoto;Ken'ichiro Tanaka
本文对一类适用于单指数变换的函数，给出了Sinc不定积分结合双指数变换的理论收敛速度.虽然DE变换被认为是Sinc-related方法的SE变换的增强版本，但DE变换适用的函数空间小于SE的函数空间，因此，存在一些DE变换不比SE变换好的例子。然而，即使在这种情况下，文献中的一些数值观察表明SE和DE的收敛速度几乎没有差异。事实上，最近，已经用两个显式近似公式从理论上解释了观察结果：Sinc求积和Sinc近似。结论是，在这种情况下，DE的比率略低，但与SE的比率几乎相同。本研究的贡献是对Sinc不定积分导出了同样的结论。最后给出了支持理论结果的数值例子.
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.