On the Application of the Peano Representation of Linear Functionals in Numerical Analysis

  • Helmut Brass
  • Klaus-Jürgen Förster
Part of the Mathematics and Its Applications book series (MAIA, volume 430)


For more than 80 years, Peano kernel theory has proven to be an important tool in numerical analysis. It is one aim of this paper to elucidate the wide range of possible applications of Peano’s representation of linear functionals. In the literature, Peano kernel theory is mostly considered for restricted classes of linear functionals. In this paper, it is also our objective to give an elementary but general approach for continuous linear functionals on C[a, b].

Key words and phrases

Peano kernel theory Inequalities for linear functionals Error estimates Quadrature Interpolation Optimal formulas 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. Akrivis and K.-J. Förster On the definiteness of quadrature formulae of Clenshaw-Curtis type, Computing 33 (1984), 363–366.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Ch. T. H. Baker On the nature of certain quadrature formulas and their errors, SIAM J. Numer. Anal. 5 (1968), 783–804.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    H. Brass, Zur Theorie der definiten Funktionale, Z. Angew. Math. Mech. 55 (1975), T230–T231.MathSciNetzbMATHGoogle Scholar
  4. 4.
    H. Brass, Quadraturverfahren, Vandenhoeck und Ruprecht, Göttingen, 1977.zbMATHGoogle Scholar
  5. 5.
    H. Brass, Error estimation for the Clenshaw-Curtis quadrature method, Abhandlungen der Braunschweigischen Wissenschaftlischen Gesselschaft 43 (1992), 42–53.MathSciNetGoogle Scholar
  6. 6.
    H. Brass, Bounds for Peano kernels, Numerical Integration IV (H. Brass and G. Hämmerlin, eds.), Birkhäuser Verlag, Basel, 1993, pp. 39–55.Google Scholar
  7. 7.
    H. Brass and K.-J. Förster On the estimation of linear functionals, Analysis 7 (1987), 237–258.MathSciNetzbMATHGoogle Scholar
  8. 8.
    H. Brass and R. Giinttner Eine Fehlerabschätzung zur Interpolation stetiger Funktionen, Studia Sci. Math. Hungar. 8 (1973), 363–367.MathSciNetGoogle Scholar
  9. 9.
    H. Brass and G. Schmeisser Error estimates for interpolatory quadrature formulas, Numer. Math. 37 (1981), 371–386.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Ph. J. Davis and Ph. Rabinowitz, Methods of Numerical Integration, Academic Press, Orlando, 1984.zbMATHGoogle Scholar
  11. 11.
    R. A. DeVore and L. R. Scott Error bounds for Gaussian quadrature and weighted Lipolynomial approximation, SIAM J. Numer. Anal. 21 (1984), 400–412.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    L. Durand, Nicholson-type integrals for products of Gegenbauer functions and related topics, Theory and Application of Special Functions (R. A. Askey, ed.), Academic Press, New York, 1975, pp. 353–374.Google Scholar
  13. 13.
    S. Ehrich and K.-J. Förster On exit criteria in quadrature using Peano kernel inclusions, Z. Angew. Math. Mech. 75 (1995), 625–628.Google Scholar
  14. 14.
    H. Fiedler Das asymptotische Verhalten der Peanokerne einiger interpolatorischer Quadraturverfahren, Numer. Math. 51 (1987), 571–581.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    K.-J. Förster A survey of stopping rules in quadrature based on Peano kernel methods, Suppl. Rend. Circ. Mat. Palermo, Serie II 33 (1993), 311–330.Google Scholar
  16. 16.
    K.-J. Förster Inequalities for ultraspherical polynomials and applications to quadrature, J. Comp. Appl. Math. 49 (1993), 59–70.zbMATHCrossRefGoogle Scholar
  17. 17.
    K.-J. Förster and K. Petras On a problèm proposed by H. Brass concerning the remainder term in quadrature for convex functions, Numer. Math. 57 (1990), 763–777.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    K.-J. Förster and K. Petras Error estimates in Gaussian quadrature for functions of bounded variation, SIAM J. Numer. Anal. 28 (1991), 880–889.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    A. Ghizetti and A. Ossicini, Quadrature Formulae, Birkhäuser Verlag, Basel, 1970.Google Scholar
  20. 20.
    H. H. Gonska and J. Meier On approximation by Bernstein-type operators: best constants, Studia Sci. Math. Hung 22 (1987), 287–297.MathSciNetzbMATHGoogle Scholar
  21. 21.
    P. Köhler, Estimates for linear remainder functionals by the modulus of continuity, Open Problems in Approximation Theory (B. Bojanov, ed.), Science Culture Technology Publishing, Singapore, 1994, pp. 109–124.Google Scholar
  22. 22.
    P. Köhler, Error estimates for polynomial and spline interpolation by the modulus of continuity, Approximation Theory (Proc. IDoMAT 95) (M. M. Müller, M. Feiten and D. H. Macke, eds.), Akademie Verlag, Berlin, 1995, pp. 141–150.Google Scholar
  23. 23.
    M. Levin and J. Girshovich, Optimal Quadrature formulas, Teubner, Leipzig, 1979.zbMATHGoogle Scholar
  24. 24.
    A. A. Ligun Inequalities for upper bounds of functionals, Analysis Mathematica 2 (1976), 11–40.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    G. G. Lorentz, Approximation of Functions, Holt, Rinehart and Winston, New York, 1966.zbMATHGoogle Scholar
  26. 26.
    K. Petras, Asymptotic behaviour of Peano kernels of fixed order, Numerical Integration III (H. Brass and G. Hämmerlin, eds.), Birkhäuser Verlag, Basel, 1988, pp. 186–198.Google Scholar
  27. 27.
    K. Petras, Normabschät-zung für die ersten Peanokerne der Gauβ-Formeln, Z. Angew. Math. Mech. 69 (1989), T81–T83.MathSciNetzbMATHGoogle Scholar
  28. 28.
    K. Petras, Error bounds of Gaussian and related quadrature and applications to r-convex functions, SIAM J. Numer. Anal. 29 (1992), 578–585.MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    K. Petras, One sided L 1-approximation and bounds for Peano kernels, Numerical Integration (T. O. Espelid and A. Genz, eds.), Kluwer Academic Publisher, Dordrecht, 1992, pp. 165–174.CrossRefGoogle Scholar
  30. 30.
    K. Petras, Gaussian quadrature formulae — second Peano kernels, nodes, weights and Bessel functions, Calcolo 30 (1993), 1–27.MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    K. Petras, On the integration of functions having singularities, Z. Angew. Math. Mech. 75 (1995), 655–656.Google Scholar
  32. 32.
    R. Piessens, E. Doncker-Kapenga, C. W. Überhuber and D. K. Kahaner, QUADPACK — a Subroutine Package for Automatic Integration, Springer Series in Comp. Math. 1, Springer Verlag, Berlin, 1982.Google Scholar
  33. 33.
    A. Ponomarenko, Estimation of the error functional for quadrature formulas with Chebyshev weights, Metody Vychisl. 13 (1983), 116–121. (Russian)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Ph. Rabinowitz On the definiteness of Gauss-Kronrod integration rules, Math. Comp. 46 (1986), 225–227.MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    J. Radon Restausdrücke bei Interpolations-und Quadratur formein durch bestimmte Integrale, Monatsh. Math. Phys. 42 (1935), 389–396.MathSciNetCrossRefGoogle Scholar
  36. 36.
    F. Riesz and B. Sz.-Nagy, Vorlestungen über Funktionanalysis, Deutscher Verlag der Wissenschaften, Berlin, 1956.Google Scholar
  37. 37.
    A. Sard Integral representation of remainders, Duke Math. J. 15 (1948), 333–345.MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    D. D. Stancu Evaluation of the remainder term in approximation formulas by Bernstein polynomials, Math. Comp. 17 (1963), 270–278.MathSciNetzbMATHGoogle Scholar
  39. 39.
    A. H. Stroud and D. H. Secrest, Gaussian Quadrature Formulas, Prentice Hall, Englewood Cliffs, N.J., 1966.zbMATHGoogle Scholar
  40. 40.
    G. Szegö, Orthogonal Polynomials, 4th edition, Amer. Math. Soc, Providence, R.I., 1975.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Helmut Brass
    • 1
  • Klaus-Jürgen Förster
    • 2
  1. 1.Institut für Angewandte MathematikTechnische Universität BraunschweigBraunschweigGermany
  2. 2.Institut für MathematikUniversität HildesheimHildesheimGermany

Personalised recommendations