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

Convergence and Smoothness of Nonlinear Lane–Riesenfeld Algorithms in the Functional Setting: "

Abstract

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 C
² with uniformly bounded second derivatives, then the limit is a C
¹ function. A canonical family of nonlinear, symmetric averaging rules, the p-averages, is presented, and the Lane–Riesenfeld algorithm with these averages is investigated.

• Content Type Journal Article
• DOI 10.1007/s10208-010-9080-2
• Authors
  
  
  • Nira Dyn, School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978 Israel
  • Ron Goldman, Department of Computer Science, Rice University, Houston, TX 77251, USA
  
  

• Journal Foundations of Computational Mathematics
• Online ISSN 1615-3383
• Print ISSN 1615-3375

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