Convergent interpolatory quadrature rules and orthogonal polynomials of varying measures

Original Paper

Abstract

Let (P n ) be a sequence of polynomials such that P n (x) > 0 for x ∈ [− 1, 1] and \(\lim \limits _{n\to \infty }\text {deg}(P_{n})/n = 1\). Let q n be the nth monic orthogonal polynomial with respect to \( {P}_{n}^{-1} \) d μ, where μ is a measure on [− 1, 1] that is regular in the sense of Stahl and Totik. We prove that the interpolatory quadrature rule with nodes at the zeros of q n is convergent with respect to μ provided that the zeros of P n lie outside a certain curve surrounding [− 1, 1].

Keywords

Interpolatory quadrature formulas Orthogonal polynomials Varying measures 

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References

  1. 1.
    Bloom, T., Lubinsky, D.S., Stahl, H.: What distribution of points are possible for convergent sequences of interpolatory integration rules. Constr. Appr. Theo. 9, 41–58 (1993)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bloom, T., Lubinsky, D.S., Stahl, H.: Interpolatory integration rules and orthogonal polynomials with varying weights. Numer. Algorithms. 3, 55–66 (1992)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2Nd Edn. Academic Press, San Diego (1984)MATHGoogle Scholar
  4. 4.
    Dragnev, P., Kuijlaars, A.B.J.: Equilibrium problems associated with fast decreasing polynomials. Proc. Am. Math. Soc. 127, 1065–1074 (1999)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Fidalgo, U., López, A., López Lagomasino, G., Sorokin, V.N.: Mixed type multiple orthogonal polynomials for two nikishin systems. Constr. Approx. 32, 255–306 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Series of Comprehensive Studies in Mathematics, vol. 316. Springer, New York (1997)CrossRefGoogle Scholar
  7. 7.
    Stahl, H., Totik, V.: General Orthogonal Polynomials. Cambridge University Press, Cambridge (1992)CrossRefMATHGoogle Scholar
  8. 8.
    Szegő, G.: Orthogonal Polynomials, vol. XXIII. Coll. Pub. Amer. Math. Soc, New York (1939)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Case Western Reserve UniversityClevelandUSA
  2. 2.The University of MississippiUniversityUSA

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