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arXiv 2303.08334math.NAcs.NA
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DE-Sinc 近似中网格尺寸与截断数选择公式的改进及其理论误差界

Improvement of selection formulas of mesh size and truncation numbers for the DE-Sinc approximation and its theoretical error bound

Tomoaki Okayama, Shota Ogawa

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Keywords: Sinc approximation, double-exponential transformation, error bound, mesh size, truncation number
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AI总结：

针对DE-Sinc近似，提出两种改进的网格尺寸与截断数选择公式，其中第二种通过引入新参数实现真正最优，并给出显式误差界，数值实验显示其误差界显著优于标准公式。

摘要：

应用于双指数衰减函数的 Sinc 近似被称为 DE-Sinc 近似。由于其高效率，该方法已被用于各种应用。在 Sinc 近似中，应最优地选择网格尺寸和截断数以实现其最佳性能。然而，标准选择公式仅是“近最优”的，因为网格尺寸的最优公式不能用截断数的初等函数表示。在本研究中，我们提出了两个改进的选择公式。第一个基于早期研究的概念，该研究为双指数公式得出了更好的选择公式。该公式的性能略优于标准公式，但仍不是最优的。作为第二个选择公式，我们引入一个新参数以提出真正最优的选择公式。我们为两个选择公式提供了显式误差界。数值比较表明，第一个公式给出的误差界优于标准公式，第二个公式给出的误差界远优于标准和第一个公式。

英文摘要：

The Sinc approximation applied to double-exponentially decaying functions is referred to as the DE-Sinc approximation. Because of its high efficiency, this method has been used in various applications. In the Sinc approximation, the mesh size and truncation numbers should be optimally selected to achieve its best performance. However, the standard selection formula has only been "near-optimally" selected because the optimal formula of the mesh size cannot be expressed in terms of elementary functions of truncation numbers. In this study, we propose two improved selection formulas. The first one is based on the concept by an earlier research that resulted in a better selection formula for the double-exponential formula. The formula performs slightly better than the standard one, but is still not optimal. As a second selection formula, we introduce a new parameter to propose truly optimal selection formula. We provide explicit error bounds for both selection formulas. Numerical comparisons show that the first formula gives a better error bound than the standard formula, and the second formula gives a much better error bound than the standard and first formulas.

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