We propose a simple and effective method for designing approximation formulas for weighted analytic functions. We consider spaces of such functions according to weight functions expressing the decay properties of the functions. Then we adopt the minimum worst error of the |$n$|-point approximation formulas in each space for characterizing the optimal sampling points for the approximation. In order to obtain approximately optimal sampling points we consider minimization of a discrete energy related to the minimum worst error. Consequently, we obtain an approximation formula and its theoretical error estimate in each space. In addition, from some numerical experiments, we observe that the formula generated by the proposed method outperforms the corresponding formula derived with sinc approximation, which is near optimal in each space.

中文翻译：

---

通过离散能量最小化设计加权Hardy空间中逼近函数的精确公式

我们提出了一种简单有效的方法来设计加权解析函数的近似公式。我们根据表示函数衰减特性的权重函数来考虑此类函数的空间。然后，我们采用| $ n $ |的最小最差误差。每个空间中的点近似公式，用于表征近似的最佳采样点。为了获得近似最佳的采样点，我们考虑最小化与最小最差误差有关的离散能量。因此，我们获得了每个空间中的近似公式及其理论误差估计。另外，从一些数值实验中，我们观察到，所提出的方法生成的公式优于通过正弦逼近推导的相应公式，该公式在每个空间中都接近最优。