An iterated quasi-interpolation approach for derivative approximation

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.