A kind of multiquadric quasi-interpolation operator satisfying any degree polynomial reproduction property to scattered data

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 modify a multiquadric quasi-interpolation with the property of linear reproducing to scattered data on one-dimensional space, such that a kind of quasi-interpolation operator Lr+1f has the property of r+1(r∈Z,r<0) degree polynomial reproducing and converges up to a rate of r+2. There is no demand for the derivatives of f in the proposed quasi-interpolation Lr+1f, so it does not increase the orders of smoothness of f. Finally, some numerical experiments are shown to compare the approximation capacity of our quasi-interpolation operators with that of WuSchaback's quasi-interpolation scheme and FengLi's quasi-interpolation scheme.