Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Approximation order from certain spaces of smooth bivariate splines on a three-direction mesh

Author: Rong Qing Jia
Journal: Trans. Amer. Math. Soc. 295 (1986), 199-212
MSC: Primary 41A15; Secondary 41A25, 41A63
MathSciNet review: 831196
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \Delta $ be the mesh in the plane obtained from a uniform square mesh by drawing in the north-east diagonal in each square. Let $ \pi _{k,\Delta }^\rho $ be the space of bivariate piecewise polynomial functions in $ {C^\rho }$, of total degree $ \leq k$, on the mesh $ \Delta $. Let $ m(k,\rho )$ denote the approximation order of $ \pi _{k,\Delta }^\rho $. In this paper, an upper bound for $ m(k,\rho )$ is given. In the space $ 3 \leq 2k - 3\rho \leq 7$, the exact values of $ m(k,\rho )$ are obtained:

\begin{displaymath}\begin{array}{*{20}{c}} {m(k,\rho ) = 2k - 2\rho - 1} \hfill ... ...or}}\;2k - 3\rho = 5,6\;{\text{or}}\;7.} \hfill \\ \end{array} \end{displaymath}

In particular, this result answers negatively a conjecture of de Boor and Höllig.

References [Enhancements On Off] (What's this?)

  • [BD] C. de Boor and R. DeVore, Approximation by smooth multivariate splines, Trans. Amer. Math. Soc. 276 (1983), 775-788. MR 688977 (84j:41015)
  • [BH $ _{\mathbf{1}}$] C. de Boor and K. Höllig, $ B$-splines from parallelepipeds, J. Analyse Math. 42 (1982/83), 99-115. MR 729403 (86d:41008)
  • [BH $ _{\mathbf{2}}$] -, Approximation from piecewise $ {C^1}$-cubics: A counterexample, Proc. Amer. Math. Soc. 87 (1983), 649-655. MR 687634 (84j:41014)
  • [BH $ _{\mathbf{3}}$] -, Bivariate box splines and smooth $ pp$ functions on a three-direction mesh, J. Comput. Appl. Math. 9 (1983), 13-28. MR 702228 (85f:41004)
  • [BZ] J. H. Bramble and M. Zlámel, Triangular elements in the finite element method, Math. Comp. 24 (1970), 809-820. MR 0282540 (43:8250)
  • [DM] W. Dahmen and C. A. Micchelli, On the approximation order from certain multivariate spline spaces, J. Austral. Math. Soc. Ser. B 26 (1984), 233-246. MR 765640 (87j:41032)
  • [J] R. Q. Jia, Approximation by smooth bivariate splines on a three direction mesh, Approximation Theory. IV (Chui, Schumaker and Ward, eds.), Adademic Press, New York, 1983. MR 754389 (85m:41019)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 41A15, 41A25, 41A63

Retrieve articles in all journals with MSC: 41A15, 41A25, 41A63

Additional Information

Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society