Some Properties of Polynomials Bounded at Equally Spaced Points

  • Theodore J. Rivlin
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 119)


Suppose the absolute value of a real polynomial, p(x), of degree d is bounded by 1 at k equally spaced points of the real line. For pairs (d, k) we present some results about how large are: (i) the absolute value of p(x) for a given real x; (ii) the maximum norm of p(x) on the span of the k points; (iii) the absolute values of the coefficients.


Real Line Chebyshev Polynomial Lagrange Interpolation American Math Real Polynomial 
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  1. [1]
    Cantor D. G. Solution of Problem 6084. American Math. Monthly, 84: 832–833, 1977.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Coppersmith D., Rivlin T. J. The growth of polynomials bounded at equally spaced points. SIAM J. Math. Anal., 23: 970–983, 1992.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Ehlich H., Zeller K. Schwankung von Polynomen zwischen Gitterpunkten. Math. Zeitschr., 86: 41–44, 1964.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Ehlich H., Zeller K. Numerische Abschätzung von Polynomen. ZAMM, 45: T20–T22, 1965.MathSciNetMATHGoogle Scholar
  5. [5]
    Ehlich H. Polynome zwischen Gitterpunkten. Math. Zeit., 93: 144–153, 1966.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Faber G. Uber die interpolatorische Darstellung stetiger Funktionen. Jahresber. der deutschen Math. Verein, 23: 190–210, 1914.Google Scholar
  7. [7]
    Landau E. Abschätzung der Koeffizientsumme einer Potenzreihe (Zweite Abhandlung). Archiv, der Math. u. Phys, (3) 21: 250–255, 1913.Google Scholar
  8. [8]
    Micchelli C. A., Rivlin T. J. Optimal recovery of best approximations. Resultate Math., 3: 25–32, 1980.MathSciNetMATHGoogle Scholar
  9. [9]
    Mills T. M., Smith S. J. On the Lebesgue function for Lagrange interpolation with equidistant nodes. J. Austral Math. Soc., (Series A) 52: 111–118, 1992.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Mills T. M., Smith S. J. The Lebesgue constant for Lagrange interpolation on equidistant nodes. Numer. Math., 61: 111–115, 1992.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Rivlin T. J. Proposal of Problem 6084. American Math. Monthly, 83: 292, 1976.MathSciNetCrossRefGoogle Scholar
  12. [12]
    Rivlin T. J. Chebyshev Polynomials, 2nd Edition. John Wiley h Sons, New York, 1990.MATHGoogle Scholar
  13. [13]
    Runck P. O. Uber Konvergenzfragen bei Polynominterpolation mit äquidistanten Knoten. I. J. Reine Angew. Math., 208: 51–69, 1961.MathSciNetMATHGoogle Scholar
  14. [14]
    Runge C. Uber empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten. Z. f. Math. u. Phys., 46: 224–243, 1901.MATHGoogle Scholar
  15. [15]
    Schönhage A. Fehlerfortpflantzung bei Interpolation. Numer. Math., 3: 62–71, 1961.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Tietze H. Eine Bemerkung zur Interpolation. Z. angew. Math. u. Phys., 64: 74–90, 1914.Google Scholar
  17. [17]
    Turetskii A. H. The bounding of polynomials prescribed at equally distributed points (Russian). Proc. Pedag. Inst. Vitebsk, 3: 117–127, 1940.Google Scholar
  18. [18]
    Vértesi P. Remark on a theorem about polynomials. To appear.Google Scholar

Copyright information

© Birkhäuser 1994

Authors and Affiliations

  • Theodore J. Rivlin
    • 1
  1. 1.T.J. Watson Research CenterIBM ResearchYorktown HeightsUSA

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