A family of C² four-point stationary subdivision schemes with fourth-order accuracy and shape-preserving properties

The four-point interpolatory scheme and the cubic B-spline are examples of the most well-known stationary subdivision procedures. They are based on the space of cubic polynomials and have their respective strengths and weaknesses. In this regard, the purpose of this study is to introduce a new type of subdivision scheme that integrates the advantages of both the four-point and the cubic B-spline schemes. The proposed scheme achieves approximation order ‘four’ and an improved smoothness C², while keeping the same support of the basic limit function as the four-point scheme. Moreover, under some mild conditions, the new scheme has the properties of monotonicity and convexity preservation. Whereas most high-order shape-preserving schemes are non-linear and rather computationally complicated, the proposed scheme is linear and stationary. Several numerical examples are provided to illustrate the efficiency of our subdivision algorithm.