In this paper, a kind of univariate multiquadric quasi-interpolants with the derivatives of approximated function is proposed by combining a univariate multiquadric quasi-interpolant with Lidstone interpolation polynomials proposed in Lidstone (Proc Edinb Math Soc 2:16–19, 1929), Costabile and Dell’ Accio (App Numer Math 52:339–361, 2005) and Catinas (J Appl Funct Anal 4:425–439, 2006). For practical purposes, another kind of approximation operators without any derivative of the approximated function is given using divided differences to approximate the derivatives. Some error bounds and the convergence rates of new operators are derived, which demonstrates that our operators could provide the desired precision by choosing a suitable shape-preserving parameter c and a non-negative integer n. Finally, we make extensive comparison with the other existing methods and give some numerical examples. Moreover, the associated algorithm is easily implemented.

本文通过将单变量多二次拟插值与在Lidstone（Proc Edinb Math Soc 2：16-19，1929），Costabile中提出的Lidstone插值多项式相结合，提出了一种具有近似函数导数的单变量多二次拟插值。和Dell'Accio（App Numer Math 52：339–361，2005）和Catinas（J Appl Funct Anal 4：425–439，2006）。出于实际目的，使用除数差来近似导数，给出了没有近似函数导数的另一种近似算子。推导了一些误差范围和新算子的收敛速度，这表明我们的算子可以通过选择合适的形状保持参数c和非负整数n来提供所需的精度。。最后，我们与其他现有方法进行了广泛的比较，并给出了一些数值示例。而且，相关算法很容易实现。