Advances in Computational Mathematics

, Volume 29, Issue 3, pp 269–290 | Cite as

Improved stability estimates and a characterization of the native space for matrix-valued RBFs

Article

Abstract

In this paper we derive several new results involving matrix-valued radial basis functions (RBFs). We begin by introducing a class of matrix-valued RBFs which can be used to construct interpolants that are curl-free. Next, we offer a characterization of the native space for divergence-free and curl-free kernels based on the Fourier transform. Finally, we investigate the stability of the interpolation matrix for both the divergence-free and curl-free cases, and when the kernel has finite smoothness we obtain sharp estimates.

Keywords

Divergence-free Radial basis functions Stability Native spaces Interpolation 

Mathematics Subject Classifications (2000)

41A05 41A30 41A63 65Dxx 

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References

  1. 1.
    Devore, R.A., Sharpley, R.C.: Besov spaces on domains in \(\mathbb{R}^{d}\). Trans. Amer. Math. Soc. 335, 843–864 (1993)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Fasshauer, G.: Solving differential equations with radial basis functions: multilevel methods and smoothing. Adv. Comput. Math. 11, 139–159 (1999)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Fuselier, E.: Refined error estimates for matrix-valued radial basis functions. Ph.D. thesis, Texas A&M University (2006)Google Scholar
  4. 4.
    Kansa, E.J.: Multiquadrics – a scattered data approximation scheme with applications to computational fluid-dynamics – I. Surface approximations and partial derivative estimates. Comput. Math. Appl. 19, 127–145 (1990)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Lowitzsch, S.: Approximation and interpolation employing divergence-free radial basis functions with applications. Ph.D. thesis, Texas A&M University (2002)Google Scholar
  6. 6.
    Lowitzsch, S.: Matrix-valued radial basis functions: stability estimates and applications. Adv. Comput. Math. 23, 299–315 (2005)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Lowitzsch, S.: A density theorem for matrix-valued radial basis functions. Numer. Algorithms 39, 253–256 (2005)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Lowitzsch, S.: Error estimates for matrix-valued radial basis function interpolation. J. Approx. Theory 137, 238–249 (2005)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Narcowich, F.J., Ward, J.D.: Norms of inverses and condition numbers for matrices associated with scattered data. J. Approx. Theory 64, 69–94 (1991)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Narcowich, F.J., Ward, J.D.: Norm estimates for the inverses of a general class of scattered-data radial-function interpolation matrices. J. Approx. Theory 69, 84–109 (1992)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Narcowich, F.J., Ward, J.D.: Generalized Hermite interpolation via matrix-valued conditionally positive definite functions. Math. Comp. 63, 661–687 (1994)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Schaback, R.: Error estimates and condition number for radial basis function interpolation. Adv. Comput. Math. 3, 251–264 (1995)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, NJ (1971)Google Scholar
  14. 14.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, London (1966)MATHGoogle Scholar
  15. 15.
    Watson, G.N., Whittaker, E.T.: A Course of Modern Analysis, 4th edn. Cambridge University Press, London (1965)Google Scholar
  16. 16.
    Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2004)Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUnited States Military AcademyWest PointUSA

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