On the Denseness of Weighted Incomplete Approximations

  • Peter Borwein
  • E. B. Saff
Conference paper
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 19)


For a given weight function w(x) on an interval [a, b], we study the generalized Weierstrass problem of determining the class of functions fC[a, b] that are uniform limits of weighted polynomials of the form w n (x)p n (x) 1 , where p n is a polynomial of degree at most n. For a special class of weights, we show that the problem can be solved by knowing the denseness interval of the alternation points for the associated Chebyshev polynomials.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    P.B. Borwein, Zeros of Chebyshev Polynomials in Markov Systems, J. Approx. Theory, 63(1990), 56–64.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    P.B. Borwein, Variations on Müntz’s Theme, Bull. Canadian Math. Soc, 34(1991), 1–6.Google Scholar
  3. [3]
    M. v. Golitschek, Approximation by Incomplete Polynomials, J. Approx. Theory, 28(1980), 155–160.MathSciNetCrossRefGoogle Scholar
  4. [4]
    M. v. Golitschek, G.G. Lorentz and Y. Makovoz, Asymptotics of weighted polynomials (these Proceedings).Google Scholar
  5. [5]
    X. He, Weighted Polynomial Approximation and Zeros of Faber Polynomials, Ph.D. Dissertation, University of South Florida, Tampa (1991).Google Scholar
  6. [6]
    X. He and X. Li, Uniform Convergence of Polynomials Associated with Varying Jacobi Weights, Rocky Mountain Journal, 21(1991), 281–300.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    S. Karlin and W.J. Studden, Tchebysheff Systems with Applications in Analysis and Statistics, Wiley, New York, 1966.Google Scholar
  8. [8]
    A. Kroó and F. Peherstorfer, On the Distribution of Extremal Points of General Chebyshev Polynomials, (to appear).Google Scholar
  9. [9]
    D.S. Lubinsky and E.B. Saff, Uniform and Mean Approximation by Certain Weighted Polynomials, with Applications, Const. Approx., 4(1988), 21–64.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    M.N. Mhaskar and E.B. Saff, Where Does the Sup Norm of a Weighted Polynomial Live?, Constr. Approx., 1(1985), 71–91.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    E.B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, (to appear).Google Scholar
  12. [12]
    E.B. Saff and R.S. Varga, Uniform Approximation by Incomplete Polynomials, Internat. J. Math. Soc, 1(1978), 407–420.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • Peter Borwein
    • 1
  • E. B. Saff
    • 2
  1. 1.Department of MathematicsDalhaousie UniversityHalifaxCanada
  2. 2.Institute for Constructive Math. Department of MathematicsUniversity of South FloridaTampaUSA

Personalised recommendations