Approximation orders and shape preserving properties of the multiquadric trigonometric B-spline quasi-interpolant

The purpose of the paper is to derive some properties of the multiquadric trigonometric B-spline quasi-interpolant. Firstly, the paper captures its approximation orders for high-order derivatives. Based on the error estimate, one can choose the shape parameter properly such that the quasi-interpolant gives optimal approximations to high-order derivatives. Moreover, the approximation orders also show that (from the theoretical point of view) the considered quasi-interpolant can be applied when the approximation of high-order derivatives is needed, i.e., numerical solution of some PDEs, construction of Lyapunov function, etc. Secondly, the paper derives some shape preserving properties of the quasi-interpolant. These properties suggest that the quasi-interpolant may be used in the geometric modeling (CAD, CAM for instance) that requires shape preservation. Finally, to illustrate the validity of the results, some numerical examples are presented. Both theoretical and numerical results demonstrate that the quasi-interpolant cannot only provide excellent approximations to high-order derivatives, but also preserve the shapes well.