Skip to main content

The fundamentality of sets of ridge functions

aequationes mathematicae volume 44, pages 226–235 (1992)Cite this article

Summary

We investigate the fundamentality of the set of all continuous ridge functions in the spaceC(ℝ n) as well as inC(X) for a general Banach space,X. Both positive and negative results are obtained. Necessary and sufficient conditions for the fundamentality are given for certain sets of ridge functions inC(ℝ n).

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price includes VAT (Canada)

References

  1. Chui, C. K. andLi, Xin,Approximation by ridge functions and neural networks with one hidden layer. To appear in J. Approx. Theory.

  2. Cybenko, G.,Approximation by superpositions of a sigmoidal function. Math. Control Signals Systems2 (1989), 203–314.

    Google Scholar 

  3. Dyn, N., Light, W. A. andCheney, E. W.,Interpolation by piecewise-linear radial basis functions I. J. Approx. Theory59 (1989), 202–223.

    Article  Google Scholar 

  4. Dahmen, W. andMicchelli, C.,Some remarks on ridge functions. Approx. Theory Appl.3 (1987), no. 2-3, 139–143.

    Google Scholar 

  5. Dyn, N. andMicchelli, C.,Interpolation by sums of radial functions. Preprint.

  6. Diaconis, P. andShahshahani, M.,On nonlinear functions of linear combinations. SIAM J. Sci. Statist. Comput.5 (1984), 175–191.

    Article  Google Scholar 

  7. Hamaker, C. andSolmon, D. C.,The angles between the null spaces of x-rays. J. Math. Anal. Appl.62 (1978), 1–23.

    Article  Google Scholar 

  8. John, Fritz,Plane waves and spherical means applied to partial differential equations. Interscience, New York, 1955.

    Google Scholar 

  9. Kelly, J. L.,General topology. [Graduate Texts in Math., vol. 27]. Springer-Verlag, New York, 1955.

    Google Scholar 

  10. Natterer, F.,The mathematics of computerized tomography. John Wiley, New York, 1986.

    Google Scholar 

  11. Royden, H. L.,Real analysis. 2nd Ed., MacMillan Co., New York, 1968.

    Google Scholar 

  12. Wiener, N.,The Fourier integral and certain of its applications. Cambridge Univ. Press, Cambridge, 1933; reprint, Dover, New York, 1958.

    Google Scholar 

  13. Willard, S.,General topology. Addison-Wesley, Boston, 1970.

    Google Scholar 

  14. Zalik, R. A.,Approximation by nonfundamental sequences of translates. Proc. Amer. Math. Soc.78 (1980), 261–265.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Southwestern Missouri State University, 65804, Springfield, Missouri, USA

    Xingping Sun & E. W. Cheney

  2. Department of Mathematics, The University of Texas at Austin, 78712, Austin, Texas, USA

    Xingping Sun & E. W. Cheney

Authors
  1. Xingping Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. W. Cheney
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sun, X., Cheney, E.W. The fundamentality of sets of ridge functions. Aeq. Math. 44, 226–235 (1992). https://doi.org/10.1007/BF01830981

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01830981

AMS (1991) subject classification

  • Primary 41A63, 41A65, 41A30
  • Secondary 65D15