Given discrete function values sampled at uniform centers, the iterated quasi-interpolation approach for approximating the m th derivative consists of two steps. The first step adopts m successive applications of the operator DQ (the quasi-interpolation operator Q first, and then the differentiation operator D) to get approximated values of the m th derivative at uniform centers. Then, by one further application of the quasi-interpolation operator Q to corresponding approximated derivative values gives the final approximation of the m th derivative. The most salient feature of the approach is that it approximates all derivatives with the same convergence rate. In addition, it is valid for a general multivariate function, compared with the existing iterated interpolation approaches that are only valid for periodic functions, so far. Numerical examples of approximating high-order derivatives using both the iterated and direct approach based on B-spline quasi-interpolation and multiquadric quasi-interpolation are presented at the end of the paper, which demonstrate that the iterated quasi-interpolation approach provides higher approximation orders than the corresponding direct approach.

给定在统一中心采样的离散函数值，用于逼近第m个导数的迭代拟插值方法包括两个步骤。第一步采用算子DQ的m个连续应用程序（首先是拟插值算子Q，然后是微分算子D），以得到均匀中心处第m个导数的近似值。然后，通过将准插值算子Q进一步应用到相应的近似导数值，得出m的最终近似值。三阶导数 该方法最显着的特征是，它以相同的收敛速度近似所有导数。另外，与目前为止仅对周期函数有效的现有迭代插值方法相比，它对一般的多元函数有效。本文最后给出了基于B样条拟插值和多二次拟插值的迭代和直接方法逼近高阶导数的数值示例，这表明迭代拟插值方法提供了更高的逼近阶比相应的直接方法。