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AN ELEMENTARY PROOF OF THE OPTIMAL RECOVERY OF THE THIN PLATE SPLINE RADIAL BASIS FUNCTION

  • KIM, MORAN (DEPARTMENT OF MATHEMATICS, EWHA WOMANS UNIVERSITY) ;
  • MIN, CHOHONG (DEPARTMENT OF MATHEMATICS, EWHA WOMANS UNIVERSITY)
  • Received : 2015.10.30
  • Accepted : 2015.12.15
  • Published : 2015.12.25

Abstract

In many practical applications, we face the problem of reconstruction of an unknown function sampled at some data points. Among infinitely many possible reconstructions, the thin plate spline interpolation is known to be the least oscillatory one in the Beppo-Levi semi norm, when the data points are sampled in $\mathbb{R}^2$. The traditional proofs supporting the argument are quite lengthy and complicated, keeping students and researchers off its understanding. In this article, we introduce a simple and short proof for the optimal reconstruction. Our proof is unique and reguires only elementary mathematical background.

Keywords

References

  1. J. Carr, R. Beatson, J. Cherrie, T. Mitchell, W. Fright, B. McCallum and T. R.Evans, Reconstruction and representation of 3D objects with radial basis functions, SIGGRAPH 2001 Proceedings, (2001), 67-76.
  2. E. Kansa, Multiquadrics - a scattered data approximation scheme with applications to computational fluid dynamics, I.Surface approximations and derivative estimates, in Advances in Partial Differential Equations, 1990.
  3. D. Lazzaro and L. Montefusco, Radial basis functions for the multivariate interpolation of large scattered data sets, J. Comput. Appl. Math.140 (2002), 521-536. https://doi.org/10.1016/S0377-0427(01)00485-X
  4. J. Duchon, Fonctions-spline du type plaque mince en dimension 2, Technical Report 231, Universite de Grenoble, 1975.
  5. M. Powell, The theory of radial basis function approximation, Advances in Numerical Analysis, vol. II: Wavelets, Subdivision Algorithms and Radial Functions, W. Light, ed. 105-210, Oxford University Press, 1990.
  6. H. Wendland, Scattered Data Approximation, Cambridge Monogr. Appl. Comput. Math., vol. 17, Cambridge University Press, 2005.
  7. S. Bochner, Monotone Functions, Stieltjes Integrale and harmoniche Analyse, Math. Ann. 108 (1933), 378-410. https://doi.org/10.1007/BF01452844
  8. S. Bochner, Vorlesungen uber Fouriershe Integrale, Akademische Verlagsgesellshaft, Leipzig, 1932.
  9. W.R. Madych and S.A. Nelson, Multivariate interpolation and conditionally positive definite functions , Math. Comp. 54 (1990), 211-230.
  10. W.R. Madych and S.A. Nelson, Multivariate interpolation: a variational theory, manuscipt. 1983.
  11. Z. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13 (1993), 13-27. https://doi.org/10.1093/imanum/13.1.13
  12. C. Micchelli, Interpolation of scattered data: distance matrices and conditionally positive definite functions, Contr. Approx., vol.II 11-22, Springer Verlag, New York, 1986.
  13. R. Schaback and H. Wendland, Characterization and construction of radial basis functions, Multivariate Approximation and Applications, Cambridge University Press, 1-4, 2000.
  14. C. Min, Numerial Analysis and Scientific Computing, Kyung Moon Sa, Korea, 2010.

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