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 L r + 1 f 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 L r + 1 f , 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 Wu–Schaback’s quasi-interpolation scheme and Feng–Li’s quasi-interpolation scheme.