Abstract: Multiquadric(MQ) is a kind of radial basis function developed by Hardy. So far, it has been used to many realms, such as geodesy and numerical solution of differential equations. Up till now, there are known four kinds of multiquadric quasi-interpolation, namely, LA、LB and LC raised by Beatson and Powell and LD by Wu and Schaback, in which, LB is constant reproducing, LC and LD are linear reproducing. Here, firstly, the properties possessed by the basis functions of a quasi-interpolation in the form of LD are given when this quasi-interpolation is linear reproducing. Then, a multiquadric quasi-interpolation is constructed which has the properties of linear reproducing and preserving monotonicity. Moreover, the theoretical analysis to its approximation error is shown. Lastly, the effect is illustrated by two examples. The numerical results indicate that this quasi-interpolation possesses high approximation precision.

Key words: computational mathematics, numerical approximation, numerical analysis, multiquadric(MQ) quasi-interpolation, linear reproducing, preserving monotonicity