On the convergence rates of subdivision algorithms for box spline surfaces (Q1093826)

Let \({\mathfrak C}\) be the control net for some (parametric) spline surface \({\mathfrak S}\). Subdivision algorithms for generating \({\mathfrak S}\) are based on the idea of generating from \({\mathfrak C}\) the refined control net \({\mathfrak RC}\) such that \({\mathfrak RC}\) tends to \({\mathfrak S}\) as the level of refinement tends to zero. For the general setting of box splines the first author and \textit{C. A. Micchelli} [Comput. Aided Geom. Des. 1, 115- 129 (1984; Zbl 0581.65011)] have shown that \({\mathfrak RC}\) converges to \({\mathfrak S}\) at (at least) the rate of the refinement. In the paper under review some mild additional assumptions guaranteeing quadratic rate of convergence are established. Estimates for the convergence rate provide an adequate a priori choice of the level of the refinement of the control net.