Numerical Algorithms

, Volume 64, Issue 1, pp 157–180

Complementary Lidstone interpolation on scattered data sets

Original Paper

Abstract

Recently we have introduced a new technique for combining classical bivariate Shepard operators with three point polynomial interpolation operators (Dell’Accio and Di Tommaso, On the extension of the Shepard-Bernoulli operators to higher dimensions, unpublished). This technique is based on the association, to each sample point, of a triangle with a vertex in it and other ones in its neighborhood to minimize the error of the three point interpolation polynomial. The combination inherits both degree of exactness and interpolation conditions of the interpolation polynomial at each sample point, so that in Caira et al. (J Comput Appl Math 236:1691–1707, 2012) we generalized the notion of Lidstone Interpolation (LI) to scattered data sets by combining Shepard operators with the three point Lidstone interpolation polynomial (Costabile and Dell’Accio, Appl Numer Math 52:339–361, 2005). Complementary Lidstone Interpolation (CLI), which naturally complements Lidstone interpolation, was recently introduced by Costabile et al. (J Comput Appl Math 176:77–90, 2005) and drawn on by Agarwal et al. (2009) and Agarwal and Wong (J Comput Appl Math 234:2543–2561, 2010). In this paper we generalize the notion of CLI to bivariate scattered data sets. Numerical results are provided.

Keywords

Combined Shepard operators Complementary Lidstone interpolation Functional approximation Error analysis 

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References

  1. 1.
    Agarwal, R.P., Pinelas, S., Wong, P.J.Y.: Complementary Lidstone interpolation and boundary value problems. J. Inequal. Appl., Article ID 624631, 30 pp. (2009)Google Scholar
  2. 2.
    Agarwal, R.P., Wong, P.J.Y.: Error Inequalities in Polynomial Interpolation and Their Applications. Kluver Academic Publishers, Dordrecht (1993)MATHCrossRefGoogle Scholar
  3. 3.
    Agarwal, R.P., Wong, P.J.Y.: Piecewise complementary Lidstone interpolation and error inequalities. J. Comput. Appl. Math. 234, 2543–2561 (2010)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Allasia, G., Bracco, C.: Multivariate Hermite-Birkhoff interpolation by a class of cardinal basis functions. Appl. Math. Comput. 218, 9248–9260 (2012)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Barnhill, R.E.: Representation and approximation of surfaces. In: Rice, J.R. (ed.) Mathematical Software III, pp. 68–119. Academic Press, New York (1977)Google Scholar
  6. 6.
    Boas, R.P.: A note on functions of exponential type. Bull. Am. Math. Soc. 47, 750–754 (1941)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Boas, R.P.: Representation of functions by Lidstone series. Duke Math. J. 10, 239–245 (1943)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Buckholtz, J.D., Shaw, J.K.: On functions expandable in Lidstone series. J. Math. Anal. Appl. 47, 626–632 (1974)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Caira, R., Dell’Accio, F.: Shepard–Bernoulli operators. Math. Comput. 76, 299–321 (2007)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Caira, R., Dell’Accio, F., Di Tommaso, F.: On the bivariate Shepard–Lidstone operators. J. Comput. Appl. Math. 236, 1691–1707 (2012)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Cătinaş, T.: The combined Shepard–Lidstone bivariate operator. Trends and applications in constructive approximation. Internat. Ser. Numer. Math. (Birkhäuser, Basel) 151, 77–89 (2005)CrossRefGoogle Scholar
  12. 12.
    Cheney, W., Light, W.: A Course in Approximation Theory. Brooks/Cole Publishing Company, Pacific Grove (1999)Google Scholar
  13. 13.
    Coman, Gh., Trîmbiţaş, R.T.: Combined Shepard univariate operators. East J. Approx. 7, 471–483 (2001)MathSciNetMATHGoogle Scholar
  14. 14.
    Coman, Gh.: Shepard operators of Birkhoff-type. Calcolo 35, 197–203 (1998)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Costabile, F., Dell’Accio, F.: Expansion over a simplex of real functions by means of Bernoulli polynomials. Numer. Algorithms 28, 63–86 (2001)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Costabile, F.A., Dell’Accio, F.: Lidstone approximation on the triangle. Appl. Numer. Math. 52, 339–361 (2005)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Costabile, F.A., Dell’Accio, F., Guzzardi, L.: New bivariate polynomial expansion with boundary data on the simplex. Calcolo 45, 177–192 (2008)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Costabile, F.A., Dell’Accio, F., Luceri, R.: Explicit polynomial expansions of regular real functions by means of even order Bernoulli polynomials and boundary values. J. Comput. Appl. Math. 176, 77–90 (2005)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Farwig, R.: Rate of convergence of Shepard’s global interpolation formula. Math. Comput. 46, 577–590 (1986)MathSciNetMATHGoogle Scholar
  20. 20.
    Golightly, G.O.: Absolutely convergent Lidstone series. J. Math. Anal. Appl. 125, 72–80 (1987)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Gordon, W.J., Wixom, J.A.: Shepard’s method of “metric interpolation” to bivariate and multivariate interpolation. Math. Comput. 32, 253–264 (1978)MathSciNetMATHGoogle Scholar
  22. 22.
    Lidstone, G.J.: Notes on the extension of Aitken’s theorem (for polynomial interpolation) to the Everett types. Proc. Edinb. Math. Soc. 2, 16–19 (1929)CrossRefGoogle Scholar
  23. 23.
    Lorentz, G.G., Jetter, K., Riemenschneider, S.D.: Birkhoff Interpolation. Encyclopedia of Mathematics and its Applications, vol. 19. Addison-Wesley, Reading (1982)Google Scholar
  24. 24.
    Poritsky, H.: On certain polynomial and other approximations to analytic functions. Trans. Am. Math. Soc. 34, 274–331 (1932)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Renka, R.J., Cline, A.K.: A triangle-based C 1 interpolation method. Rocky Mt. J. Math. 14, 223–237 (1984)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Renka, R.J., Brown, R.: Algorithm 792: accuracy tests of ACM algorithms for interpolation of scattered data in the plane. ACM Trans. Math. Softw. 25, 78–94 (1999)MATHCrossRefGoogle Scholar
  27. 27.
    Shepard, D.: A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings of the 1968 23rd ACM National Conference, pp. 517–524. ACM Press, New York (1968)CrossRefGoogle Scholar
  28. 28.
    Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)MATHGoogle Scholar
  29. 29.
    Whittaker, J.M.: On Lidstone’s series and two-point expansions of analytic functions. Proc. Lond. Math. Soc. 36, 451–459 (1933–34)MathSciNetGoogle Scholar
  30. 30.
    Widder, D.V.: Completely convex functions and Lidstone series. Trans. Am. Math. Soc. 51, 387–398 (1942)MathSciNetGoogle Scholar
  31. 31.
    Wu, Z.: Hermite-Birkhoff interpolation of scattered data by radial basis functions. Approx. Theory Appl. 8, 1–10 (1992)MATHGoogle Scholar
  32. 32.
    Zuppa, C.: Error estimates for modified local Shepard’s interpolation formula. Appl. Numer. Math. 49, 245–259 (2004)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Zuppa, C.: Modified Taylor reproducing formulas and h-p clouds. Math. Comput. 77, 243–264 (2008)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • F. A. Costabile
    • 1
  • F. Dell’Accio
    • 1
  • F. Di Tommaso
    • 1
  1. 1.Dipartimento di MatematicaUniversità della CalabriaRende (Cs)Italy

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