Approximation to the -th derivatives by multiquadric quasi-interpolation method

Quasi-interpolation is very useful in the study of approximation theory and its applications, since it can yield solutions directly without the need to solve any linear system of equations. Based on the good performance (simple to evaluate) of multiquadric functions, the advantages (approximation order, convexity and monotonlcity preserving) of multiquadric quasi-interpolation have been widely discussed. However, it is usually only for the approximation properties of the function itself and the good properties for derivatives, whereas the high order derivatives have been largely ignored. In this paper, we go further into the approximation properties to the k -th derivatives by using multiquadric quasi-interpolation. Furthermore, we develop two kinds of multiquadric quasi-interpolation schemes on bounded interval [ x 0 , x N ] , whose derivatives converge to the corresponding derivatives of the approximated functions. Finally, the numerical experiments are presented to confirm the accuracy of the presented scheme. Both theoretical results and numerical examples show this scheme provides good accuracy even if the data points in [ x 0 , x N ] are irregularly distributed.