Interpolatory blending net subdivision schemes of Dubuc–Deslauriers type

Abstract

Net subdivision schemes recursively refine nets of univariate continuous functions defined on the lines of planar grids, and generate as limits bivariate continuous functions. In this paper a family of interpolatory net subdivision schemes related to the family of Dubuc–Deslauriers interpolatory subdivision schemes is constructed and analyzed. The construction is based on Gordon blending interpolants to nets of univariate functions, and on a particular class of blending functions with properties related to the Dubuc–Deslauriers schemes. The general analysis tools for net subdivision schemes, developed in a previous paper by the authors, together with the properties of the blending functions, lead to the proof of the convergence of these schemes to limit functions having the same integer smoothness as the limits of the corresponding Dubuc–Deslauriers schemes. These results are proved for net subdivision schemes corresponding to the first 84 members of the Dubuc–Deslauriers family, and conjectured for the rest. A concrete example of a family of piecewise polynomial blending functions is considered, together with the corresponding family of net subdivision schemes. The performance of the first two net subdivision schemes in this family is demonstrated by two examples.