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

Nira Dyn, Ron Goldman*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)79-94
Number of pages16
JournalFoundations of Computational Mathematics
Issue number1
StatePublished - Feb 2011


  • Convergence and smoothness analysis
  • Data refinement
  • Lane-Riesenfeld algorithm
  • Nonlinear subdivision schemes
  • Nonlinear symmetric averages
  • p-averages


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