Constructive Approximation

, Volume 5, Issue 1, pp 405–414 | Cite as

On the uniqueness of canonical points in the Hobby-Rice theorem

  • András Kroó


According to the Hobby-Rice theorem for anyn-dimensional subspaceU n ofL1([a, b], ν) (ν positive, finite, nonatomic) there exist points α=s0<x1<⋯<xm+1=b, where 0≤m≤n, such that
$$\sum\limits_{j = 0}^m {( - 1)^j \int_{x_j }^{x_{j + 1} } {udv = 0,u \in U_n .} }$$
. In this paper we study under what conditions onU n the pointsx1,⋯,x m with the above property are unique for everyν. A necessary and sufficient condition for local uniqueness is found.

AMS classification


Key words and phrases

Hobby-Rice theorem Uniqueness of canonical points Weak Chebyshev spaces 


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  1. 1.
    C. R. Hobby, J. R. Rice (1965):A moment problem in L 1-approximation. Proc. Amer. Math. Soc.,16:665–670.Google Scholar
  2. 2.
    A. Kroó (1985):On an L 1-approximation problem. Proc. Amer. Math. Soc.,94:406–410.Google Scholar
  3. 3.
    A. Kroó (1986):Chebyshev rank in L 1-approximation. Trans. Amer. Math. Soc.,296:301–313.Google Scholar
  4. 4.
    C. A. Micchelli (1977):Best L 1-approximation by weak Chebyshev systems and the uniqueness of interpolating perfect splines. J. Approx. Theory,19:1–14.Google Scholar
  5. 5.
    A. Pinkus (1986):Unicity subspaces in L 1-approximation. J. Approx. Theory,48:226–250.Google Scholar
  6. 6.
    A.Pinkus (to appear): On L'-Approximation. Cambridge Tracts in Mathematics. Cambridge: Cambridge University Press.Google Scholar
  7. 7.
    L. L. Schumaker (1981): Spline Functions: Basic Theory. New York: Wiley.Google Scholar
  8. 8.
    M. Sommer (1979):L 1-approximation by weak Chebyshev spaces. In: Approximation in Theorie und Praxis (G. Meinardus, ed.). Mannheim: Bibliographisches Institut, pp. 85–102.Google Scholar
  9. 9.
    M. Sommer (1983):Some results on best L 1-approximation of continuous functions. Numer. Funct. Anal. Optim.,6:253–271.Google Scholar
  10. 10.
    H. Strauss (1984):Best L 1-approximation. J. Approx. Theory,41:297–308.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1989

Authors and Affiliations

  • András Kroó
    • 1
  1. 1.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary

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