Computational and Applied Mathematics

, Volume 36, Issue 2, pp 885–902 | Cite as

Weighting Shepard-type operators

  • Umberto Amato
  • Biancamaria Della Vecchia


Uniform approximation error estimates for weighted Shepard-type operators more flexible than the unweighted analogues are given. Error estimates for a linear combination of their iterates faster converging than previous ones are also showed. The results are applied in CAGD to construct Shepard-type curves useful in modeling and a weighted progressive iterative approximation technique exponentially converging.


Shepard-type operators Rate of convergence Flexibility property  Hotelling’s method Weighted progressive iterative approximation 

Mathematics Subject Classification

41A25 41A36 


  1. Allasia G (1995) A class of interpolatory positive linear operators: theoretical and computational aspects. Approx Theory Wavel Appl NATO ASI Ser C 454:1–36MathSciNetMATHGoogle Scholar
  2. Amato U, Della Vecchia B (2015a) New results on rational approximation. Res Math 67:354–364Google Scholar
  3. Amato U, Della Vecchia B (2015b) Bridging Bernstein and Lagrange Polynomials. Commun Math (in press) Google Scholar
  4. Amato U, Della Vecchia B (2016) Modelling by Shepard-type curves and surfaces. J Comput Anal Appl (in press) Google Scholar
  5. Anastassiou G, Gal S (2012) Approximation theory: moduli of continuity and global smoothness preservation. Birkhauser, BostonMATHGoogle Scholar
  6. Della Vecchia B (1996) Direct and converse results by rational operators. Constr Approx 12:271–285MathSciNetCrossRefMATHGoogle Scholar
  7. Della Vecchia B, Mastroianni G (1991) Pointwise simultaneous approximation by rational operators. J Approx Theorey 64:140–150MathSciNetCrossRefMATHGoogle Scholar
  8. Della Vecchia B, Mastroianni G, Vertesi P (1996) Direct and converse theorems for Shepard rational approximation. Numer Funct Anal Optim 17:537–562MathSciNetCrossRefMATHGoogle Scholar
  9. Ewald F, Winkler C, Zinner T (2015) Interactive comments of 3D cloud geometry using a scanner cloud radar. Atmos Meas Tech 7:C4897–C4914Google Scholar
  10. Lin H, Bao H, Wang G (2005) Totally positive bases and progressive iteration approximation. Comput Math Appl 50:576–586MathSciNetMATHGoogle Scholar
  11. Lu L (2010) Weighted progressive iteration approximation and convergence analysis. Comput Aided Geom Des 27:129–137MathSciNetCrossRefMATHGoogle Scholar
  12. Nairn D, Peters J, Lutterkort D (1999) Sharp, quantitative bound on the distance between a polynomial piece and its Bézier control polygon. CAGD 16:613–631MATHGoogle Scholar
  13. Ouassou M, Jensen ABO, Gevestad JGO, Kristiansen O (2015) Next generation network real-time kinematic interpolation segment to improve the user accuracy. Int J Navig Observ, article ID 346498Google Scholar
  14. Reif U (2000) Best bounds on the approximation of polygons and splines by their control structures. CAGD 17:579–589Google Scholar
  15. Soleymani F (2013) On a fast iterative method for approximate inverse of matrices. Commun Korean Math Soc 28:407–418MathSciNetCrossRefMATHGoogle Scholar
  16. Tachev G (2012) The rate of approximation by rational Bernstein functions in terms of second order moduli of continuity. Numer Funct Anal Optim 33:206–215MathSciNetCrossRefMATHGoogle Scholar
  17. Welsch R, Manthe U (2013) Fast Shepard interpolation on graphics processing units: potential energy surfaces and dynamics for H+CH4\(\rightarrow \)HE+CH3. J Chem Phys 138:164118CrossRefGoogle Scholar
  18. Yan F, Lv J, Feng X, Pan P (2015) A new hybrid boundary node method based on Taylor expansion and the Shepard interpolation method. Numer Meth Eng 102:1488–1506MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma ‘La Sapienza’RomeItaly
  2. 2.Istituto per le Applicazioni del Calcolo ‘Mauro Picone’ CNRNaplesItaly

Personalised recommendations