Stationary subdivision schemes reproducing polynomials

A new class of subdivision schemes is presented. Each scheme in this class is a quasi-interpolatory scheme with a tension parameter, which reproduces polynomials up to a certain degree. We find that these schemes extend and unify not only the well-known Deslauriers–Dubuc interpolatory schemes but the quadratic and cubic B-spline schemes. This paper analyzes their convergence, smoothness and accuracy. It is proved that the proposed schemes provide at least the same or better smoothness and accuracy than the aforementioned schemes, when all the schemes are based on the same polynomial space. We also observe with some numerical examples that, by choosing an appropriate tension parameter, our new scheme can remove undesirable artifacts which usually appear in interpolatory schemes with irregularly distributed control points.