Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Approximation order from bivariate $ C\sp{1}$-cubics: a counterexample

Authors: C. de Boor and K. Höllig
Journal: Proc. Amer. Math. Soc. 87 (1983), 649-655
MSC: Primary 41A15; Secondary 41A25, 41A63
MathSciNet review: 687634
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the space of bivariate $ {C^1}$ piecewise cubic functions on a hexagonal mesh of size $ h$ approximates to certain smooth functions only to within $ O({h^3})$ even though it contains a local partition of every cubic polynomial.

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

  • [BD] C. de Boor and R. DeVore, Approximation by smooth multivariate splines, MRC TSR #2319, 1981; Trans. Amer. Math. Soc. (to appear). MR 688977 (84j:41015)
  • [BH$ _{1}$] C. de Boor and K. Höllig, $ B$-splines from parallelepipeds, MRC TSR #2320, 1982.
  • [BH$ _{2}$] -, Bivariate box splines and smooth $ pp$ functions on a three-direction mesh, MRC TSR #2389; J. Comput. Appl. Math. (to appear).
  • [C] de Casteljau, Courbes et surfaces à pôles, "Enveloppe 40.040", Institut National de la Propriété Industrielle, Paris, 1959.
  • [D] W. Dahmen, Approximation by linear combinations of multivariate $ B$-splines, J. Approximation Theory 31 (1981), 299-324. MR 628515 (82m:41010)
  • [Fa] G. Farin, Subsplines über Dreiecken, Dissertation, Braunschweig, 1979.
  • [Fr] O. Frederickson, Quasi-interpolation, extrapolation, and approximation on the plane, Proc. Manitoba Conf. Numerical Math., Winnipeg, 1971, pp. 159-176. MR 0336170 (49:946)
  • [S] M. A. Sabin, The use of piecewise forms for the numerical representation of shape, Dissertation, dapest, Hungary, 1977. MR 0656152 (58:31723)

Similar Articles

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

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

Additional Information

Keywords: $ {\text{B}}$-splines, multivariate, spline functions, degree of approximation, pp, smooth
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society