A New Class of Interpolatory L-Splines with Adjoint End Conditions

  • Aurelian BejancuEmail author
  • Reyouf S. Al-Sahli
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9213)


A thin plate spline for interpolation of smooth transfinite data prescribed along concentric circles was recently proposed by Bejancu, using Kounchev’s polyspline method. The construction of the new ‘Beppo Levi polyspline’ surface reduces, via separation of variables, to that of a countable family of univariate L-splines, indexed by the frequency integer k. This paper establishes the existence, uniqueness and variational properties of the ‘Beppo Levi L-spline’ schemes corresponding to non-zero frequencies k. In this case, the resulting L-spline end conditions are formulated in terms of adjoint differential operators, unlike the usual ‘natural’ L-spline end conditions, which employ identical operators at both ends. Our L-spline error analysis leads to an \(L^{2}\)-error bound for transfinite surface interpolation with Beppo Levi polysplines.


Interpolation L-spline Beppo Levi polyspline Approximation order 


  1. 1.
    Ahlberg, J.H., Nilson, E.N., Walsh, J.L.: The Theory of Splines and Their Applications. Academic Press, New York (1967)zbMATHGoogle Scholar
  2. 2.
    Al-Sahli, R.S.: \(L\)-spline Interpolation and Biharmonic Polysplines on Annuli, M.Sc thesis, Kuwait University (2012)Google Scholar
  3. 3.
    Bejancu, A.: Semi-cardinal polyspline interpolation with Beppo Levi boundary conditions. J. Approx. Theory 155, 52–73 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bejancu, A.: Transfinite thin plate spline interpolation. Constr. Approx. 34, 237–256 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bejancu, A.: Thin plate splines for transfinite interpolation at concentric circles. Math. Model. Anal. 18, 446–460 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bejancu, A.: Beppo Levi polyspline surfaces. In: Cripps, R.J., Mullineux, G., Sabin, M.A (eds.), The Mathematics of Surfaces XIV, The Institute of Mathematics and its Applications, UK, (2013) ISBN 978-0-905091-30-3Google Scholar
  7. 7.
    Bejancu, A.: Radially symmetric thin plate splines interpolating a circular contour map. J. Comput. Appl. Math. (2015). doi: 10.1016/
  8. 8.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)zbMATHGoogle Scholar
  9. 9.
    Bozzini, M., Rossini, M., Schaback, R.: Generalized Whittle-Matérn and polyharmonic kernels. Adv. Comput. Math. 39, 129–141 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cavoretto, R., Fasshauer, G.E., McCourt, M.J.: An introduction to the Hilbert-Schmidt SVD using iterated Brownian bridge kernels. Numer. Alg. 68, 393–422 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Duchon, J.: Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces. RAIRO Anal. Numer. 10, 5–12 (1976)MathSciNetGoogle Scholar
  12. 12.
    Jerome, J., Pierce, J.: On spline functions determined by singular self-adjoint differential operators. J. Approx. Theory 5, 15–40 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Karlin, S., Ziegler, Z.: Tchebycheffian spline functions. SIAM J. Numer. Anal. 3, 514–543 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kounchev, O.I.: Multivariate Polysplines: Applications to Numerical and Wavelet Analysis. Academic Press, London (2001)Google Scholar
  15. 15.
    Kounchev, O.I., Render, H.: The approximation order of polysplines. Proc. Am. Math. Soc. 132, 455–461 (2003)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kounchev, O.I., Render, H.: Cardinal interpolation with polysplines on annuli. J. Approx. Theory 137, 89–107 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lucas, T.R.: A generalization of \(L\)-splines. Numer. Math. 15, 359–370 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Rabut, C.: Interpolation with radially symmetric thin plate splines. J. Comput. Appl. Math. 73, 241–256 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Schultz, M.H., Varga, R.S.: \(L\)-splines. Numer. Math. 10, 345–369 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Schumaker, L.L.: Spline Functions: Basic Theory, 3rd edn. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  21. 21.
    Sharon, N., Dyn, N.: Bivariate interpolation based on univariate subdivision schemes. J. Approx. Theory 164, 709–730 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsKuwait UniversityKuwaitKuwait

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