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Journal of Mathematics
Volume 2013, Article ID 638254, 12 pages
http://dx.doi.org/10.1155/2013/638254
Research Article

Local Lagrange Interpolations Using Bivariate Splines of Degree Seven on Triangulated Quadrangulations

School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

Received 6 September 2012; Revised 3 December 2012; Accepted 6 December 2012

Academic Editor: Josefa Linares-Perez

Copyright © 2013 Xuqiong Luo and Qikui Du. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. G. Nürnberger and F. Zeilfelder, “Local Lagrange interpolation by cubic splines on a class of triangulations,” in Trends in Approximation Theory, K. Kopotun, T. Lyche, and M. Neamtu, Eds., Innovations in Applied Mathematics, pp. 341–350, Vanderbilt University Press, Nashville, Tenn, USA, 2001. View at Google Scholar
  2. G. Nürnberger and F. Zeilfelder, “Local Lagrange interpolation on Powell-Sabin triangulations and terrain modelling,” in Recent Progress in Multivariate Approximation, vol. 137 of International Series of Numerical Mathematics, pp. 227–244, Birkhäuser, Basel, Switzerland, 2001. View at Google Scholar
  3. G. Nürnberger and F. Zeilfelder, “Lagrange interpolation by bivariate C1splines with optimal approximation order,” Advances in Computational Mathematics, vol. 21, no. 3-4, pp. 381–419, 2004. View at Publisher · View at Google Scholar
  4. G. Nürnberger and F. Zeilfelder, “Developments in bivariate spline interpolation,” Journal of Computational and Applied Mathematics, vol. 121, no. 1-2, pp. 125–152, 2000, Numerical analysis in the 20th century, Vol. I, Approximation theor. View at Publisher · View at Google Scholar
  5. G. Nürnberger and F. Zeilfelder, “Fundamental splines on triangulations,” in Modern Developments in Multivariate Approximation, vol. 145 of International Series of Numerical Mathematics, pp. 215–233, Birkhäuser, Basel, Switzerland, 2003. View at Google Scholar
  6. G. Nürnberger and F. Zeilfelder, “Lagrange interpolation by splines on triangulations,” in Computational Geometry, vol. 34 of AMS/IP Studies in Advanced Mathematics, pp. 113–132, >American Mathematical Society, Providence, RI, USA, 2003. View at Google Scholar
  7. G. Nürnberger, V. Rayevskaya, L. L. Schumaker, and F. Zeilfelder, “Local Lagrange interpolation with C2 splines of degree seven on triangulations,” in Advances in Constructive Approximation: Vanderbilt 2003, pp. 345–370, Nashboro Press, Brentwood, Tenn, USA, 2004. View at Google Scholar
  8. H.-W. Liu and L.-L. Fan, “Lagrange interpolations using bivariate C2 quintic supersplines on double Clough-Tocher refinements,” Computers & Mathematics with Applications, vol. 58, no. 8, pp. 1636–1644, 2009. View at Publisher · View at Google Scholar
  9. G. Nürnberger, V. Rayevskaya, L. L. Schumaker, and F. Zeilfelder, “Local Lagrange interpolation with bivariate splines of arbitrary smoothness,” Constructive Approximation, vol. 23, no. 1, pp. 33–59, 2006. View at Publisher · View at Google Scholar
  10. G. Nürnberger, L. L. Schumaker, and F. Zeilfelder, “Local Lagrange interpolation by bivariate C1 cubic splines,” in Mathematical Methods for Curves and Surfaces III, Innovations in Applied Mathematics., pp. 393–403, Vanderbilt University Press, Nashville, Tenn, USA, 2001. View at Google Scholar
  11. G. Nürnberger, L. L. Schumaker, and F. Zeilfelder, “Lagrange interpolation by C1 cubic splines on triangulations of separable quadrangulations,” in Approximation Theory, Innovations in Applied Mathematics, pp. 405–424, Vanderbilt University Press, Nashville, Tenn, USA, 2002. View at Google Scholar
  12. G. Nürnberger, L. L. Schumaker, and F. Zeilfelder, “Lagrange interpolation by C1 cubic splines on triangulated quadrangulations,” Advances in Computational Mathematics, vol. 21, no. 3-4, pp. 357–380, 2004. View at Publisher · View at Google Scholar
  13. P. Alfeld and L. L. Schumaker, “The dimension of bivariate spline spaces of smoothness r for degree d4r1,” Constructive Approximation, vol. 3, no. 2, pp. 189–197, 1987. View at Publisher · View at Google Scholar
  14. P. Alfeld and L. L. Schumaker, “Upper and lower bounds on the dimension of superspline spaces,” Constructive Approximation, vol. 19, no. 1, pp. 145–161, 2003. View at Google Scholar
  15. G. Nürnberger, L. L. Schumaker, and F. Zeilfelder, “Local Lagrange interpolation by bivariate C1 cubic splines,” in Mathematical Methods for Curves and Surfaces III, Oslo, 2000, Innovations in Applied Mathematics, pp. 393–404, Vanderbilt University Press, Nashville, Tenn, USA, 2001. View at Google Scholar
  16. M.-J. Lai and L. L. Schumaker, “On the approximation power of bivariate splines,” Advances in Computational Mathematics, vol. 9, no. 3-4, pp. 251–279, 1998. View at Publisher · View at Google Scholar
  17. O. Davydov, G. Nürnberger, and F. Zeilfelder, “Bivariate spline interpolation with optimal approximation order,” Constructive Approximation, vol. 17, no. 2, pp. 181–208, 2001. View at Publisher · View at Google Scholar