Stieltjes polynomials and related quadrature formulae for a class of weight functions, II

  • Sotirios E. NotarisEmail author


Consider a (nonnegative) measure \(d\sigma \) with support in the interval [ab] such that the respective orthogonal polynomials satisfy a three-term recurrence relation with coefficients \(\alpha _{n}=\left\{ \begin{array}{ll} \alpha _{e},&{}n~{\hbox {even,}}\\ \alpha _{o},&{}n~ {\hbox {odd,}} \end{array} \right. \beta _{n}=\beta ~{\hbox {for}}~ n\ge \ell \), where \(\alpha _{e}, \alpha _{o}, \beta \) and \(\ell \) are specific constants. We show that the corresponding Stieltjes polynomials, above the index \(2\ell -1\), have a very simple and useful representation in terms of the orthogonal polynomials. As a result of this, the Gauss–Kronrod quadrature formula for \(d\sigma \) has all the desirable properties, namely, the interlacing of nodes, their inclusion in the closed interval [ab] (under an additional assumption on \(d\sigma \)), and the positivity of all weights, while the formula enjoys an elevated degree of exactness. Furthermore, the interpolatory quadrature formula based on the zeros of the Stieltjes polynomials has positive weights and also elevated degree of exactness. It turns out that this formula is the anti-Gaussian formula for \(d\sigma \), while the resulting averaged Gaussian formula coincides with the Gauss–Kronrod formula for this measure. Moreover, we show that the only positive and even measure \(d\sigma \) on \((-a,a)\) for which the Gauss–Kronrod formula is almost of Chebyshev type, i.e., it has almost all of its weights equal, is the measure \(d\sigma (t)=(a^{2}-t^{2})^{-1/2}dt\).

Mathematics Subject Classification

33C45 65D32 



The author would like to thank the referees for their comments, which helped to improve the paper.


  1. 1.
    Allaway, W.R.: The identification of a class of orthogonal polynomial sets. Ph.D. Thesis, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton (1972)Google Scholar
  2. 2.
    Beckermann, B., Gilewicz, J., Leopold, E.: Recurrence relations with periodic coefficients and Chebyshev polynomials. Appl. Math. (Warsaw) 23, 319–323 (1995)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brezinski, C., Driver, K.A., Redivo-Zaglia, M.: Quasi-orthogonality with applications to some families of classical orthogonal polynomials. Appl. Numer. Math. 48, 157–168 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chihara, T.S.: An Introduction to Orthogonal Polynomials. Mathematics and Its Applications, vol. 13. Gordon and Breach, New York (1978)zbMATHGoogle Scholar
  5. 5.
    Damanik, D., Killip, R., Simon, B.: Perturbations of orthogonal polynomials with periodic recursion coefficients. Ann. of Math. (2) 171, 1931–2010 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gautschi, W.: Advances in Chebyshev quadrature. In: Watson, G.A. (ed.) Numerical Analysis, Proceedings Dundee Conference on Numerical Analysis 1975, Lecture Notes in Mathematics, vol. 506, pp. 100–121. Springer, Berlin (1976)Google Scholar
  7. 7.
    Gautschi, W.: A survey of Gauss–Christoffel quadrature formulae. In: Butzer, P.L., Fehér, F. (eds.) E.B. Christoffel, The Influence of His Work on Mathematics and the Physical Sciences, pp. 72–147. Birkhäuser, Basel (1981)CrossRefGoogle Scholar
  8. 8.
    Gautschi, W.: Gauss–Kronrod quadrature—a survey. In: Milovanović, G.V. (ed.) Numerical Methods and Approximation Theory III, pp. 39–66. Faculty of Electronic Engineering, University of Niš, Niš (1988)Google Scholar
  9. 9.
    Gautschi, W., Notaris, S.E.: Gauss-Kronrod quadrature formulae for weight functions of Bernstein–Szegö type. J. Comput. Appl. Math. 25, 199–224 (1989). (erratum in: J. Comput. Appl. Math. 27, 429 (1989))MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gautschi, W., Notaris, S.E.: Stieltjes polynomials and related quadrature formulae for a class of weight functions. Math. Comp. 65, 1257–1268 (1996)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gautschi, W., Rivlin, T.J.: A family of Gauss–Kronrod quadrature formulae. Math. Comp. 51, 749–754 (1988)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Geronimo, J.S., Van Assche, W.: Orthogonal polynomials with asymptotically periodic recurrence coefficients. J. Approx. Theory 46, 251–283 (1986)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Laurie, D.P.: Anti-Gaussian quadrature formulas. Math. Comp. 65, 739–747 (1996)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. Chapman & Hall/CRC, Boca Raton, FL (2002)CrossRefGoogle Scholar
  15. 15.
    Máté, A., Nevai, P., Van Assche, W.: The supports of measures associated with orthogonal polynomials and the spectra of the related self-adjoint operators. Rocky Mountain J. Math. 21, 501–527 (1991)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Monegato, G.: Stieltjes polynomials and related quadrature rules. SIAM Rev. 24, 137–158 (1982)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Notaris, S.E.: Gauss–Kronrod quadrature formulae for weight functions of Bernstein–Szegö type, II. J. Comput. Appl. Math. 29, 161–169 (1990)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Notaris, S.E.: On Gauss–Kronrod quadrature formulae of Chebyshev type. Math. Comp. 58, 745–753 (1992)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Notaris, S.E.: Gauss–Kronrod quadrature formulae—A survey of fifty years of research. Electron. Trans. Numer. Anal. 45, 371–404 (2016)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Notaris, S.E.: Anti-Gaussian quadrature formulae based on the zeros of Stieltjes polynomials. BIT 58, 179–198 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Peherstorfer, F.: Weight functions admitting repeated positive Kronrod quadrature. BIT 30, 145–151 (1990)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Peherstorfer, F.: On Bernstein–Szegö orthogonal polynomials on several intervals. II. Orthogonal polynomials with periodic recurrence coefficients. J. Approx. Theory 64, 123–161 (1991)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsNational and Kapodistrian University of AthensAthensGreece

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