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Advances in Computational Mathematics

, Volume 40, Issue 1, pp 185–200 | Cite as

Wendland functions with increasing smoothness converge to a Gaussian

  • A. Chernih
  • I. H. Sloan
  • R. S. Womersley
Article

Abstract

The Wendland functions are a class of compactly supported radial basis functions with a user-specified smoothness parameter. We prove that with an appropriate rescaling of the variables, both the original and the “missing” Wendland functions converge uniformly to a Gaussian as the smoothness parameter approaches infinity. We also explore the convergence numerically with Wendland functions of different smoothness.

Keywords

Radial basis functions Compact support Smoothness Wendland functions Gaussian 

Mathematics Subject Classification (2010)

33C90 41A05 41A15 41A30 41A63 65D07 65D10 

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References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, vol. 65 of National Bureau of Standards Applied Mathematics Series. Dover Publications, New York (1972)Google Scholar
  2. 2.
    Andrews, G.E., Askey, R., Roy, R.: Special Functions, vol. 71 of Encylopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2000)Google Scholar
  3. 3.
    Buhmann, M.D.: Radial Basis Functions, vol. 12 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  4. 4.
    DLMF: Digital Library of Mathematical Functions, National Institute of Standards and Technology (2011)Google Scholar
  5. 5.
    Fasshauer, G.E.: Meshfree Approximation Methods with MATLAB. World Scientific Publishing Co., Singapore (2007)MATHGoogle Scholar
  6. 6.
    Fasshauer, G.E., Zhang, J.G.: On choosing ‘optimal’ shape parameters for RBF approximation. Numer. Algorithms 45, 345–368 (2007)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Fornefett, M., Rohr, K., Stiehl, H.S.: Radial basis functions with compact support for elastic registration of medical images. Image Vis. Comput. 19, 87–96 (2001)CrossRefGoogle Scholar
  8. 8.
    Franke, R.: A Critical Comparison of some Methods for Interpolation of Scattered Data, Tech. Report NPS-53-79-003, Naval Postgraduate School, March (1979)Google Scholar
  9. 9.
    Gneiting, T.: Radial positive definite functions generated by Euclid’s hat. J. Multivar. Anal. 69, 88–119 (1999)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Hubbert, S.: Closed form representations for a class of compactly supported radial basis functions. Adv. Comput. Math. 36, 115–136 (2012)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Moreaux, G.: Compactly supported radial covariance functions. J. Geod. 82, 431–443 (2008)CrossRefGoogle Scholar
  12. 12.
    Rippa, S.: An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Adv. Comp. Math. 11, 193–210 (1999)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Schaback, R.: Creating surfaces from scattered data using radial basis functions. In: Daehlen, M., Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods for Curves and Surfaces, pp. 477–496. Vanderbilt University Press, Nashville (1995)Google Scholar
  14. 14.
    Schaback, R.: The missing Wendland functions. Adv. Comput. Math. 34, 67–81 (2011)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces, vol. 32 of Princeton Mathematical Series. Princeton University Press, Princeton (1971)Google Scholar
  16. 16.
    Wendel, J.G.: Note on the gamma function. Am. Math. Monthly 55, 563–564 (1948)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4, 389–396 (1995)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Wendland, H.: Scattered Data Approximation, vol. 17 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2005)Google Scholar
  19. 19.
    Wu, Z.: Compactly supported positive definite radial basis functions. Adv. Comput. Math. 4, 283–292 (1995)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Zastavnyi, V.P.: On some properties of Buhmann functions. Ukr. Math. J. 58, 1045–1067 (2006)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia

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