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operators (Dell’Accio and Di Tommaso, On the extension of the Shepard-Bernoulli operators to higher dimensions, unpublished).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.
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
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
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
Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)MATHGoogle Scholar
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