JournalsjcaVol. 4, No. 4pp. 359–367

A new bound for smooth spline spaces

  • Hal Schenck

    Auburn University, USA
  • Mike Stillman

    Cornell University, Ithaca, USA
  • Beihui Yuan

    Cornell University, Ithaca, USA
A new bound for smooth spline spaces cover

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Abstract

For a planar simplicial complex ΔR2\Delta \subseteq \mathbb{R}^2, Schumaker proves in [22] that a lower bound on the dimension of the space Ckr(Δ)C^r_k(\Delta) of planar splines of smoothness rr and degree kk on Δ\Delta is given by a polynomial PΔ(r,k)P_\Delta(r,k), and Alfeld–Schumaker show in [2] that PΔ(r,k)P_\Delta(r,k) gives the correct dimension when k4r+1k \ge 4r+1. Examples due to Morgan–Scott, Tohaneanu, and Yuan show that the equality \Crk=PΔ(r,k)\Crk =P_\Delta(r,k) can fail for k{2r,2r+1}k \in \{2r, 2r+1\}. In this note we prove that the equality \Crk=PΔ(r,k)\Crk= P_\Delta(r,k) cannot hold in general for k(22r+7)/10k \le (22r+7)/10.

Cite this article

Hal Schenck, Mike Stillman, Beihui Yuan, A new bound for smooth spline spaces. J. Comb. Algebra 4 (2020), no. 4, pp. 359–367

DOI 10.4171/JCA/43