Analysis of subdivision schemes for nets of functions by proximity and controllability

Costanza Conti*, Nira Dyn

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


In this paper we develop tools for the analysis of net subdivision schemes, schemes which recursively refine nets of bivariate continuous functions defined on grids of lines, and generate denser and denser nets. Sufficient conditions for the convergence of such a sequence of refined nets, and for the smoothness of the limit function, are derived in terms of proximity to a bivariate linear subdivision scheme refining points, under conditions controlling some aspects of the univariate functions of the generated nets. Approximation orders of net subdivision schemes, which are in proximity with positive schemes refining points are also derived. The paper concludes with the construction of a family of blending spline-type net subdivision schemes, and with their analysis by the tools presented in the paper. This family is a new example of net subdivision schemes generating C1 limits with approximation order 2.

Original languageEnglish
Pages (from-to)461-475
Number of pages15
JournalJournal of Computational and Applied Mathematics
Issue number4
StatePublished - 15 Sep 2011


  • Controllability
  • Net subdivision schemes
  • Point subdivision schemes
  • Proximity


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