Polynomial reproduction by symmetric subdivision schemes

Nira Dyn, Kai Hormann, Malcolm A. Sabin, Zuowei Shen

Research output: Contribution to journalArticlepeer-review


We first present necessary and sufficient conditions for a linear, binary, uniform, and stationary subdivision scheme to have polynomial reproduction of degree d and thus approximation order d + 1. Our conditions are partly algebraic and easy to check by considering the symbol of a subdivision scheme, but also relate to the parameterization of the scheme. After discussing some special properties that hold for symmetric schemes, we then use our conditions to derive the maximum degree of polynomial reproduction for two families of symmetric schemes, the family of pseudo-splines and a new family of dual pseudo-splines.

Original languageEnglish
Pages (from-to)28-42
Number of pages15
JournalJournal of Approximation Theory
Issue number1
StatePublished - Nov 2008


  • Approximation order
  • Polynomial generation
  • Polynomial reproduction
  • Quasi-interpolation.
  • Subdivision schemes


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