Analysis of subdivision schemes for nets of functions by proximity and controllability
The authors develop tools for analyzing convergence, smoothness and approximation order of net subdivision schemes, that are schemes which recursively refine nets of bivariate continuous functions defined on grids of lines, and generate denser and denser nets. The two main ingredients of their analysis are (i) control on the size of the Lipschitz constants in grid intervals of the univariate net function and/or their derivatives, (ii) proximity to a point subdivision scheme. They derive sufficient conditions for the convergence of these sequence of refined nets, and for the smoothness of the limit function. They also derive approximation orders of net subdivision schemes which are in proximity with positive schemes refining points. The authors introduce an example of a family of blending spline-type net subdivision schemes and then analyze it with the tools developed in the paper. Although it is expected that the limit functions generated by the schemes analogous to higher order spline schemes are smoother, they can show only that the limit functions are \(C^1\). All the schemes in this family have approximation order 2.
scientific article; zbMATH DE number 5987768