Advertisement

A Survey on Lagrange Interpolation Based on Equally Spaced Nodes

  • Michael Revers
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 142)

Abstract

Lagrange interpolation polynomials based on the equidistant node system have not been a popular subject in approximation theory. This is due to some famous examples published by C. Runge in 1901 and later by S.N. Bernstein in 1918 which discouraged mathematicians from considering this method of interpolation. This paper provides a brief survey of Lagrange interpolation polynomials which are based on equidistant nodes including recent results on pointwise divergence properties and certain limit relations.

Keywords

Approximation Theory Lagrange Interpolation Lebesgue Constant Interpolation Node Lagrange Interpolation Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S.N. Bernstein, Sur la meilleure approximation de 1x1 par des polynômes de degrés donnés, Acta Math. 37 (1913), 1–57.zbMATHCrossRefGoogle Scholar
  2. [2]
    S.N. Bernstein, Quelques remarques sur l’interpolation, Math. Ann. 79 (1918), 1–12.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    S.N. Bernstein, Sur la limitation des valeurs d’un pôlynome P n (x) de degré n sur tout un segment par ses valeurs en n +1 points du segment, Bull. Acad. Sci. de l’USSR, 8 (1931), 1025–1050.Google Scholar
  4. [4]
    S.N. Bernstein, Sur la meilleure approximation de IxVP par des polynômes de degrés très élevés, Bull. Acad. Sci. USSR Sér. Math. 2 (1938), 181–190.Google Scholar
  5. [5]
    L. Brutman, Lebesgue functions for polynomial interpolation–a survey, Ann. Numer. Math. 4 (1997), 111–127.MathSciNetzbMATHGoogle Scholar
  6. [6]
    L. Brutman, E. Passow, On the Divergence of Lagrange Interpolation to 1xi, J. Approx. Theory 81 (1995), 127–135.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    G. Byrne, T.M. Mills, S. Smith, On Lagrange interpolation with equidistant nodes, Bull. Austral. Math. Soc. 42 (1990), 81–89.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    S. Cobzas, I. Muntean, Condensation of singularities and divergence results in approximation theory, J. Approx. Theory 31 (1981), 138–153.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    P. Erdós, P. Vértesi, On the almost everywhere divergence of Lagrange interpolating polynomials on arbitrary system of nodes, Acta Math. Acad. Sci. Hungar. 36 (1980), 71–89.MathSciNetCrossRefGoogle Scholar
  10. [10]
    G. Faber, Über die interpolatorische Darstellung stetiger Funktionen, Jahres-ber. Deutsch. Math.-Verein. 23 (1914), 190–210.Google Scholar
  11. [11]
    M. Ganzburg, Strong Asymptotics in Lagrange Interpolation with Equidistant Nodes, Preprint 2002.Google Scholar
  12. [12]
    G. Grünwald, Uber Divergenzerscheinungen der Lagrangeschen Interpolationspolynome stetiger Funktionen, Annals of Mathematics 37 (1936), 908918.Google Scholar
  13. [13]
    G. Grünwald, P. Erdzős, Über einen Faber’schen Satz, Annals of Mathematics 39 (1938), 257–261.MathSciNetCrossRefGoogle Scholar
  14. [14]
    J. Marcinkiewicz, Sur la divergence des polynômes d’interpolation, Acta Sci. Math. (Szeged) 8 (1937), 131–135.zbMATHGoogle Scholar
  15. [15]
    Ch. Méray, Observations sur la légitimité de l’interpolation, Asens 1 (1884), 165–176.Google Scholar
  16. [16]
    I. Muntean, The Lagrange interpolation operators are densely divergent, Studia Univ. Babes-Bolyai Mat. 21 (1976), 28–30.MathSciNetGoogle Scholar
  17. [17]
    I.P. Natanson, Constructive function theory, Vol. III (1965), Frederick Ungar, New York.zbMATHGoogle Scholar
  18. [18]
    M. Revers, On the approximation of certain functions by interpolating polynomials, Bull. Austral. Math. Soc. 58 (1998), 505–512.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    M. Revers, The divergence of Lagrange interpolation for Ixr at equidistant nodes, J. Approx. Theory 103 (2000), 269–280.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    M. Revers, On Lagrange interpolation with equally spaced nodes, Bull. Austral. Math. Soc. 62 (2000), 357–368.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    M. Revers, Approximation constants in equidistant Lagrange interpolation, Periodica Mathematica Hungarica 40(2) (2000), 167–175.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    C. Runge, Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten, Zeitschrift für Mathematik und Physik 46 (1901), 224–243.zbMATHGoogle Scholar
  23. [23]
    B.L. Sendov, A. Andreev, Interpolation and Approximation, in Handbook of Numerical Analysis, Vol. III, P.G. Ciarlet, J.L. Lions, Eds., (1994) North Holland.Google Scholar
  24. [24]
    A. Schönhage, Fehlerfortpflanzung bei Interpolation, Numerische Mathematik 3 (1961), 62–71.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    G. Szegö, Orthogonal polynomials, 4th edn. Colloq. Pub. 23, Am. Math. Soc., (1975) Providence, RI.Google Scholar
  26. [26]
    R.S. Varga, A.J. Carpenter, On the Bernstein Conjecture in approximation theory, Const. Approx. 1 (1985), 333–348.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel 2002

Authors and Affiliations

  • Michael Revers
    • 1
  1. 1.Department of MathematicsUniversity of SalzburgSalzburgAustria

Personalised recommendations