A family of multivariate multiquadric quasi-interpolation operators with higher degree polynomial reproduction

• A family of multivariate multiquadric quasi-interpolants is proposed.
• There is no demand for derivatives of approximated function in quasi-interpolants.
• The quasi-interpolants satisfy any degree polynomial reproduction property.
• The quasi-interpolants can reach up to a higher approximation order.

In this paper, by using multivariate divided difference (Rabut, 2001) to approximate the partial derivative and the idea of the superposition (Waldron, 2009), we modify a multiquadric quasi-interpolation operator (Ling, 2004) based on a dimension-splitting technique with the property of linear reproducing to gridded data on multi-dimensional spaces, such that a family of proposed multivariate multiquadric quasi-interpolation operators Φr+1Φr+1 has the property of r+1(r∈Z,r⩾0) degree polynomial reproducing and converges up to a rate of r+2r+2. In addition, the proposed quasi-interpolation operator only demands information of location points rather than the derivatives of the function approximated. Moreover, we give the approximation error of our quasi-interpolation operator. Finally, some numerical experiments are shown to confirm the approximation capacity of our quasi-interpolation operator.