A Multiquadric Quasi-interpolation Operator Satisfying Any Degree Polynomial Reproduction Property

In this paper, by virtue of using the linear combinations of the shifts of f(x) to approximate the derivatives of f(x) and Waldron’s superposition idea(2009), we construct a new kind of multiquadric quasi-interpolation operator Lr+1f,bas-ing on the quasi-interpolation operator of Zhang-Wu’s and Feng-Zhou’s construct-ing idea. Lr+1f has the property of r+1 degree polynomial reproducing and converges up to a rate of r+2. There is no demand for the derivatives of f(x), so the smoothness order of f(x) does not increase. At last, some numeri-cal experiments are shown to compare the approximation capacity of our quasi-interpolation operators with that of Zhang-Wu’s quasi-interpolation scheme and Feng-Zhou’s quasi-interpolation scheme.