Splines on the Alfeld split of a simplex and type A root systems

Hal Schenck

Research output: Contribution to journalArticlepeer-review

Abstract

Alfeld (1984) introduced a tetrahedral analog A S (δ3) of the Clough-Tocher split of a triangle. A formula for the dimension of the spline space Ckr(AS(δn)) is conjectured in Foucart and Sorokina (2013). We prove that the graded module of C r-splines on the cone over A S (δn) is isomorphic to the module D r +1 (A n) of multiderivations on the type A n Coxeter arrangement. A theorem of Terao shows that the module of multiderivations of a Coxeter arrangement is free and gives an explicit basis. As a consequence the conjectured formula holds.

Original languageEnglish (US)
Pages (from-to)1-6
Number of pages6
JournalJournal of Approximation Theory
Volume182
DOIs
StatePublished - Jun 2014

Keywords

  • Dimension formula
  • Hyperplane arrangement
  • Spline

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Mathematics(all)
  • Applied Mathematics

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