Convergence and Smoothness of Nonlinear Lane-Riesenfeld Algorithms in the Functional Setting

We investigate the Lane-Riesenfeld subdivision algorithm for uniform B-splines, when the arithmetic mean in the various steps of the algorithm is replaced by nonlinear, symmetric, binary averaging rules. The averaging rules may be different in different steps of the algorithm. We review the notion of a symmetric binary averaging rule, and we derive some of its relevant properties. We then provide sufficient conditions on the nonlinear binary averaging rules used in the Lane-Riesenfeld algorithm that ensure the convergence of the algorithm to a continuous function. We also show that, when the averaging rules are C² with uniformly bounded second derivatives, then the limit is a C¹ function. A canonical family of nonlinear, symmetric averaging rules, the p-averages, is presented, and the Lane-Riesenfeld algorithm with these averages is investigated.