Numerical Algorithms

, Volume 77, Issue 4, pp 1213–1247 | Cite as

Piecewise Chebyshevian splines: interpolation versus design

Original Paper
  • 66 Downloads

Abstract

We consider the wide class of all piecewise Chebyshevian splines with connection matrices at the knots. We prove that a spline space of this class is “good for interpolation” if and only if the spline space obtained by integration is “good for design”. As a consequence, this provides us with a simple practical description of all such spline spaces which can be used for solving Hermite interpolation problems. These results strongly rely on the properties of blossoms.

Keywords

Piecewise Chebyshevian splines Connection matrices Spline Hermite interpolation Schoenberg-Whitney conditions Total positivity (Piecewise) Generalised derivatives B-spline-type bases Knot insertion Blossoms 

Mathematics Subject Classification (2010)

41A05 65D05 65D07 65D17 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barry, P.J.: de Boor-Fix dual functionals and algorithms for Tchebycheffian B-splines curves. Constr. Approx. 12, 385–408 (1996)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Barsky, B.A.: The β-spline, a local representation based on shape parameters and fundamental geometric measures. Ph.D. dissertation, Dept. of Computer Science, University of Utah, Salt Lake City Utah (1981)Google Scholar
  3. 3.
    Barsky, B.A., Beatty, J.C.: Local control of bias and tension in beta-splines. ACM Trans. Graphics 2, 09–134 (1983)CrossRefMATHGoogle Scholar
  4. 4.
    de Boor, C.: A Practical Guide to Splines, revised version, Applied Math. Sc., Springer 27 (2001)Google Scholar
  5. 5.
    de Boor, C., DeVore, R.: A geometric proof of total positivity for spline interpolation. Math. Comput. 45, 497–504 (1985)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Brilleaud, M., Mazure, M.-L.: Mixed hyperbolic/trigonometric spaces for design. Comp. Math. Appl. 64, 2459–2477 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Brilleaud, M., Mazure, M.-L.: Design with L-splines. Num. Algorithms 65, 91–124 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Carnicer, J.-M., Peña, J.-M.: Total positivity and optimal bases. In: Gasca, M., Micchelli, C.A. (eds.) Total Positivity and its Applications, pp. 133–155. Kluwer Academic Pub. (1996)Google Scholar
  9. 9.
    Dyn, N., Micchelli, C.A.: Piecewise polynomial spaces and geometric continuity of curves. Numer. Math. 54, 319–337 (1988)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Goodman, T.N.T.: Properties of beta-splines. J. Approx. Theory 44, 132–153 (1985)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Goodman, T.N.T.: Total positivity and the shape of curves. In: Gasca, M., Micchelli, C.A. (eds.) Total Positivity and its Applications, pp. 157–186. Kluwer Academic Pub. (1996)Google Scholar
  12. 12.
    Karlin, S.J., Studden, W.J.: Tchebycheff Systems: with Applications in Analysis and Statistics. Wiley Interscience, N.Y. (1966)Google Scholar
  13. 13.
    Kayumov, A., Mazure, M.-L.: Chebyshevian splines: interpolation and blossoms. C. R. Acad. Sci. Paris, Ser. I(344), 65–70 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lee, J.H., Yang, S.N.: Shape preserving and shape control with interpolating bézier curves. J. Comput. Applied Math. 28, 269–280 (1989)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lyche, T., Schumaker, L.L.: Total positivity properties of LB-splines, 35–46 (1996)Google Scholar
  16. 16.
    Mazure, M.-L.: Blossoming: a geometrical approach. Constr. Approx. 15, 33–68 (1999)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Mazure, M.-L.: Chebyshev splines beyond total positivity. Adv. Comp. Math. 14, 129–156 (2001)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Mazure, M.-L.: B-spline bases and osculating flats: one result of H.-P. Seidel revisited. ESAIM Math. Model. Numer. Anal. 36, 1177–1186 (2002)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Mazure, M.-L.: Blossoms and optimal bases. Adv. Comp. Math. 20, 177–203 (2004)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Mazure, M.-L.: On the equivalence between existence of B-spline bases and existence of blossoms. Constr. Approx. 20, 603–624 (2004)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Mazure, M.-L.: Chebyshev spaces and Bernstein bases. Constr. Approx. 22, 347–363 (2005)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Mazure, M.-L.: Towards existence of piecewise Chebyshevian B-spline bases. Numerical Algorithms 39, 399–414 (2005)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Mazure, M.-L.: Ready-to-blossom bases in Chebyshev spaces. In: Jetter, K., Buhmann, M., Hauss-mann, W., Schaback, R., Stoeckler, J. (eds.) Topics in Multivariate Approximation and Interpolation, pp. 109–148. Elsevier (2006)Google Scholar
  24. 24.
    Mazure, M.-L.: Extended Chebyshev piecewise spaces characterised via weight functions. J. Approx. Theory 145, 33–54 (2007)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Mazure, M.-L.: Ready-to-blossom bases and the existence of geometrically continuous piecewise Chebyshevian B-splines. C. R. Acad. Sci. Paris Ser. I(347), 829–834 (2009)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Mazure, M.-L.: Finding all systems of weight functions associated with a given extended Chebyshev space. J. Approx. Theory 163, 363–376 (2011)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Mazure, M.-L.: How to build all Chebyshevian spline spaces good for geometric design? Numer. Math. 119, 517–556 (2011)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Mazure, M.-L.: From Taylor interpolation to Hermite interpolation via duality. Jaén J. Approx. 4, 15–45 (2012)MathSciNetMATHGoogle Scholar
  29. 29.
    Mazure, M.-L.: Polynomial splines as examples of Chebyshevian splines. Num. Algorithms 60, 241–262 (2012)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Mazure, M.-L.: Constructing totally positive piecewise Chebyhevian B-splines, preprintGoogle Scholar
  31. 31.
    Melkman, A.: Another proof of the total positivity of the discrete spline collocation matrix. J. Approx. Theory 84, 265–273 (1996)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Mrken, K.: Total positivity of the discrete spline collocation matrix II. J. Approx. Theory 84, 247–264 (1996)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Mühlbach, G.: ECT-B-splines defined by generalized divided differences. J. Comput. Applied Math. 187, 96–122 (2006)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Mühlbach, G.: One sided Hermite interpolation by piecewise different generalized polynomials. J. Comput. Applied Math. 196, 285–298 (2006)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Pottmann, H.: The geometry of Tchebycheffian splines. Comput. Aided Geom. Design 10, 181–210 (1993)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Ramshaw, L.: Blossoms are polar forms. Comput. Aided Geom. Design 6, 323–358 (1989)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Schoenberg, I.J., Whitney, A.: On pólya frequency functions III. The positivity of translation determinants with applications to the interpolation problem by spline curves. Trans. Amer. Math. Soc. 74, 246–259 (1953)MathSciNetMATHGoogle Scholar
  38. 38.
    Schumaker, L.L.: On tchebycheffian spline functions. J. Approx. Theory 18, 278–303 (1981). N.Y.MathSciNetCrossRefGoogle Scholar
  39. 39.
    Schumaker, L.L.: Spline Functions. Wiley Interscience, N.Y. (1981)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Laboratoire Jean KuntzmannUniversité Grenoble-AlpesGrenobleFrance

Personalised recommendations