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H-splines and Quasi-interpolants on a Three Directional Mesh

  • Paul Sablonnière
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 142)

Abstract

Let τ (resp. τ*) be the uniform three-directional mesh of the plane generated by the vectors \({e_1} = \left( {1,0} \right),{e_2} = \left( {0,1} \right),{e_3} = \left( { - 1, - 1} \right)\;\left( {resp.\;e_1^* = (1,0),e_2^* = ( - \frac{1}{2},\frac{{\sqrt 3 }}{2}),e_3^* = ( - \frac{1}{2},\frac{{\sqrt 3 }}{2})} \right)\). Let \( {\text{P}}_{n{\text{ }}}^s(\tau ) \) and \( {\text{P}}_{n{\text{ }}}^s(\tau *) \) be the spaces of piecewise polynomial functions of degree n and smoothness s on these meshes. There exist two interesting families of B-splines, respectively in the spaces \( P_{3r + 1}^{2r}\left( \tau \right),{\text{ }}r{\text{ }} \geqslant {\text{ }}0 \) and \( P_{3r}^{2r - 1}\left( \tau \right),{\text{ }}r{\text{ }} \geqslant {\text{ }}1 \). In the first space, B-splines with minimal support are simultaneously box-splines and H r+1-splines, i.e., their support is the hexagon H r+1, centered at the origin, whose sides are composed of r + 1 edges of triangles of the mesh. In the second space, there exist three types of box-splines whose supports are non regular hexagons. Generalizing examples given in [18] and [19], we construct H r+1-splines in the space \( P_{3r}^{2r - 1}\left( \tau \right) \) as linear combinations of translates of three box-splines. Then we construct various differential and discrete quasi-interpolants (QI) which have the best possible approximation order, for degrees (resp. smoothness orders) ranging from 3 to 10 (resp. from 1 to 6). Their computation is made easier thanks to the symmetry properties of H-splines. Finally, we give some examples of QI with nearly minimal infinite norms, which we call near-best quasi-interpolants.

Keywords

Approximation Order Minimal Support Regular Hexagon Spline Space Composition Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    B.D. Bojanov, H.A. Hakopian, A.A. Sahakian, Spline functions and multivariate interpolation, Kluwer, Dordrecht, 1993.Google Scholar
  2. [2]
    C. de Boor, K. Höllig, Bivariate box-splines and smooth pp functions on a three direction mesh, J. Comput. Appl. Math. 9 (1983), 13–28.zbMATHGoogle Scholar
  3. [3]
    C. de Boor, K. Höllig, S. Riemenschneider, Box-splines, Springer, New-York, 1993.zbMATHGoogle Scholar
  4. [4]
    C. de Boor, Quasiinterpolants and approximation power of multivariate splines in Computation of curves and surfaces, W. Dahmen, M. Gasca, C.A. Micchelli (eds), Kluwer, Dordrecht (1990), 313–345.Google Scholar
  5. [5]
    P.L. Butzer, M. Schmidt, E.L. Stark, L. Voigt, Central factorial numbers; their main properties and some applications, Numer. Funct.Anal. and Optimiz. 10, 5 and 6 (1989), 419–488.Google Scholar
  6. [6]
    C.K. Chui, Multivariate splines, CBMS-NSF Regional Conference Series in Appl. Math., Vol. 54, SIAM, Philadelphia, 1988.CrossRefGoogle Scholar
  7. [7]
    C.K. Chui, M.J. Lai, Algorithms for generating B-nets and graphically displaying spline surfaces on three and four directional meshes, Comp. Aided Geom. Design 8 (1991), 479–493.MathSciNetzbMATHGoogle Scholar
  8. [8]
    L. Collatz, The numerical treatment of differential equations, Springer, Berlin, 1966.Google Scholar
  9. [9]
    W. Dahmen, C.K. Micchelli, Translates of multivariate splines, Linear Algebra Appl. 52 /53 (1983), 217–234.MathSciNetGoogle Scholar
  10. [10]
    W. Dahmen, C.K. Micchelli, Recent progress in multivariate splines, in Approximation Theory IV, C.K. Chui, L.L. Schumaker, J.D. Ward (eds), Academic Press (1983), 27–121.Google Scholar
  11. [11]
    P.O. Frederickson, Triangular spline interpolation, Report 6–70, Lakehead University, 1970.Google Scholar
  12. [12]
    P.O. Frederickson, Generalized triangular splines, Report 7–71, Lakehead University, 1971.Google Scholar
  13. [13]
    P.E. Gill, W. Murray, M.H. Wright, Practical optimization, Academic Press, London, 1997.Google Scholar
  14. [14]
    F. di Guglielmo, Construction d’approximations des espaces de Sobolev sur des réseaux en simplexes, Calcolo 6 (1969), 279–331.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    F. di Guglielmo, Méthode des éléments finis: une famille d’approximations des espaces de Sobolev par les translatées de p fonctions, Calcolo 7 (1970), 185–233.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    M.J. Lai, Fortran subroutines for B-nets of box-splines on three-and four-directional meshes, Numer. Algorithms 2 (1992), 33–38.zbMATHCrossRefGoogle Scholar
  17. [17]
    M.A. Sabin, The use of piecewise forms for the numerical representation of shape, Thesis, Budapest, 1977.Google Scholar
  18. [18]
    P. Sablonnière, Bases de Bernstein et approximants splines, Thèse de Doctorat, Université de Lille, 1982.Google Scholar
  19. [19]
    P. Sablonnière, A catalog of B-splines of degree 10 on a three direction mesh, Publication ANO 132, Université de Lille, 1984.Google Scholar
  20. [20]
    P. Sablonnière, Bernstein-Bézier methods for the construction of bivariate spline approximants, Comput. Aided Geom. Design 2 (1985), 29–36.zbMATHGoogle Scholar
  21. [21]
    P. Sablonnière, Quasi-interpolants associated with H-splines on a three direction mesh, J. Comput. Appl. Math. 66 (1996), 433–442.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    P. Sablonnière, On some families of B-splines on the uniform four-directional mesh of the plane, Conference on multivariate approximation and interpolation with applications in CAGD, signal and image processing. Eilat, Israel, September 7–11, 1998 (unpublished).Google Scholar
  23. [23]
    P. Sablonnière, Quasi-interpolantes splines sobre particiones uniformes, Meeting on Approximation Theory, Ubeda, Spain, July 2000. Prépublication 00–38, Institut de Recherche Mathématique de Rennes.Google Scholar

Copyright information

© Birkhäuser Verlag Basel 2002

Authors and Affiliations

  • Paul Sablonnière
    • 1
  1. 1.Centre de MathématiquesINSA de RennesRennes cedexFrance

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