Skip to main content
SpringerLink
Log in

Generalized Strang–Fix condition for scattered data quasi-interpolation

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

Quasi-interpolation is very useful in the study of the approximation theory and its applications, since the method can yield solutions directly and does not require solving any linear system of equations. However, quasi-interpolation is usually discussed only for gridded data in the literature. In this paper we shall introduce a generalized Strang–Fix condition, which is related to nonstationary quasi-interpolation. Based on the discussion of the generalized Strang–Fix condition we shall generalize our quasi-interpolation scheme for multivariate scattered data, too.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M.D. Buhmann, Multivariable interpolation using radial basis functions, Ph.D. dissertation, University of Cambridge (1989).

  2. M.D. Buhmann, N. Dyn and D. Levin, On quasi-interpolation radial basis functions with scattered centers, Construct. Approx. 11 (1995) 239–254.

    MATH  MathSciNet  Google Scholar 

  3. C. de Boor, Quasi-interpolants and approximation power of multivariate splines, in: Computation of Curves and Surfaces, eds. M. Gasca and C.A. Micchelli, Nato Advanced Science Institute Series C: Mathematical and Physical Sciences, Vol. 307 (Kluwer Academic, Dordrecht, 1990) pp. 313–345.

    Google Scholar 

  4. C. de Boor, Approximation order without quasi-interpolants, in: Approximation Theory, Vol. VII, eds. C. Chui, L. Schumaker and J. Ward (Academic Press, Boston, MA, 1993) pp. 1–18.

    Google Scholar 

  5. C. de Boor, R. DeVore and A. Ron, Approximation from shift-invariant subspaces of L 2(Rd), Trans. Amer. Math. Soc. 341 (1994) 787–806.

    MATH  MathSciNet  Google Scholar 

  6. C. de Boor and A. Ron, Fourier analysis of the approximation power of principal shift-invariant spaces, Construct. Approx. 8 (1992) 427–462.

    MATH  Google Scholar 

  7. N. Dyn, I.R.H. Jackson, D. Levin and A. Ron, On multivariate approximation by integer translates of a basis function, Israel J. Math. 78 (1992) 95–130.

    MATH  MathSciNet  Google Scholar 

  8. N. Dyn and A. Ron, Radial basis functions: from gridded centers to scattered centers, Proc. London Math. Soc. 71 (1995) 76–108.

    MATH  MathSciNet  Google Scholar 

  9. K. Jetter and D.X. Zhou, Seminorm and full norm order of linear approximation from shift-invariant space, in: Estratto dal “Rendiconti del Seminario Matematico e Fisico di Milano”, Vol. LXV (1995).

  10. K. Jetter and D.X. Zhou, Order of linear approximation from shift-invariant spaces, Construct. Approx. 11 (1995) 423–438.

    MATH  MathSciNet  Google Scholar 

  11. R.Q. Jia, Approximation order of translation invariant subspaces of functions, in: Approximation Theory, Vol. VI, eds. C. Chui, L. Schumaker and J. Ward (Academic Press, New York, 1989) pp. 349–352.

    Google Scholar 

  12. R.Q. Jia, Partition of unity and density: A counterexample, Construct. Approx. 13 (1997) 251–260.

    MATH  Google Scholar 

  13. R.Q. Jia, Shift-invariant spaces and linear operator equations, Israel J. Math. 103 (1998) 259–288.

    MATH  MathSciNet  Google Scholar 

  14. R.Q. Jia and J.J. Lei, A new version of the Strang–Fix conditions, J. Approx. Theory 74 (1993) 221–225.

    MATH  MathSciNet  Google Scholar 

  15. X. Li and C. Micchelli, Approximation by radial basis functions and neural networks, Numer. Algorithms (2001) to appear.

  16. M.D.J. Powell, Radial basis functions for multivariable interpolation: A review, in: Numerical Analysis, eds. D.F. Griffiths and G.A. Watson (Longman Scientific & Technical, Harlow, 1987) pp. 223–241.

    Google Scholar 

  17. C. Rabut, An introduction to Schoenberg’s approximation, Comput. Math. Appl. 24 (1992) 149–175.

    MATH  MathSciNet  Google Scholar 

  18. R. Schaback, Approximation by radial basis functions with finitely many centers, Construct. Approx. 12 (1996) 331–340.

    MATH  MathSciNet  Google Scholar 

  19. R. Schaback, Remarks on meshless local construction of surfaces, in: Proc. of the IMA Mathematics of Surfaces Conference, Cambridge (2000).

  20. R. Schaback and Z.M. Wu, Construction techniques for highly accurate quasi-interpolation operators, J. Approx. Theory 91 (1997) 320–331.

    MATH  MathSciNet  Google Scholar 

  21. R. Sibson, The Dirichlet tessellation as an aid in data-analysis, Scandinavian J. Statist. 7 (1980) 14–20.

    MATH  MathSciNet  Google Scholar 

  22. G. Strang and G. Fix, A Fourier analysis of the finite-element method, in: Constructive Aspects of Functional Analysis, ed. G. Geymonat (C.I.M.E., Rome, 1973) pp. 793–840.

    Google Scholar 

  23. Z.M. Wu, Hermite–Birkhoff interpolation for scattered data by radial basis functions, Approx. Theory Appl. 8 (1992) 1–10.

    MATH  Google Scholar 

  24. Z.M. Wu, Multivariate compactly supported positive definite radial basis functions, Adv. Comput. Math. 4 (1995) 283–292.

    Article  MATH  MathSciNet  Google Scholar 

  25. Z.M. Wu, Compactly supported positive definite radial basis functions and the Strang–Fix condition, Appl. Math. Comput. 84 (1997) 115–124.

    MATH  MathSciNet  Google Scholar 

  26. Z.M. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13 (1993) 13–27.

    MATH  MathSciNet  Google Scholar 

  27. J. Yoon, Approximation in L p(Rd) from a space spanned by the scattered shifts of radial basis functions, Construct. Approx. 17 (2001) 227–247.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Shanghai Key Laboratory for Contemporary Applied Mathematics Department of Mathematics, Fudan University, Shanghai, P.R. China

    Zong Min Wu

  2. Department of Mathematics, East China University of Science and Technology, P.R. China

    Jian Ping Liu

Authors
  1. Zong Min Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jian Ping Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zong Min Wu.

Additional information

Communicated by Z. Wu and B.Y.C. Hon

AMS subject classification

41A63, 41A25, 65D10

Zong Min Wu: Supported by NSFC No. 19971017 and NOYG No. 10125102.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wu, Z.M., Liu, J.P. Generalized Strang–Fix condition for scattered data quasi-interpolation. Adv Comput Math 23, 201–214 (2005). https://doi.org/10.1007/s10444-004-1832-6

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-004-1832-6

Keywords

Search

Navigation