Bounds for Remainder Terms in Szegö Quadrature on the Unit Circle

  • William B. Jones
  • Haakon Waadeland
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 119)


This paper deals with Szegö quadrature for integration around the unit circle in the complex plane. Nodes for the quadrature formulas are the zeros ζ j (n) ( w n), j = 1,2,…, n, of para-orthogonal Szegö polynomials B n(z, w n) in z of degree n. The parameter w n is a complex constant satisfying |w n| = 1. Results are described for convergence of the quadrature formulas as n → ∞ and upper bounds for the remainder term that results when the value of the integral is replaced by an n-point quadrature approximation. The upper bounds for remainder terms apply to integrals that represent Carathéodory functions and real parts of such integrals. The latter are Poisson integrals used to represent harmonic functions determined by boundary values on the unit circle.


Reflection Coefficient Unit Circle Quadrature Formula Remainder Term Moment Problem 
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  1. [1]
    Akhiezer N. I. The Classical Moment Problem and Some Related Questions in Analysis. Hafner, New York, 1964. ( Translated by N. Kemmer. )Google Scholar
  2. [2]
    Bultheel A., González-Vera P., Hendriksen E., Njástad O. Orthogonality and quadrature on the unit circle. IMACS Annals on Comp, and Appl. Math., 9: 205–210, 1991.Google Scholar
  3. [3]
    Bultheel A., González-Vera P., Hendriksen E., Njástad O. Quadrature formulas on the unit circle and two-point Padé approximation. (Submitted).Google Scholar
  4. [4]
    Gautschi Walter. A survey of Gauss-Christoffel quadrature formulae. In E. B. Christoffel; The Influence of his Work in Mathematics and the Physical Sciences, pages 72–147. Birkhäuser Verlag, Basel, 1981. P. L. Butzer and F. Fehér, eds.Google Scholar
  5. [5]
    Hille Einar. Analytic Function Theory, Vol. II. Ginn and Company, Boston, 1962.MATHGoogle Scholar
  6. [6]
    Jones William B., Njástad Olav, Thron W. J. Orthogonal Laurent polynomials and the strong Hamburger moment problem. J. Math. Anal, and Appl., 98 (2): 528–554, 1984.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Jones William B., Njástad Olav, Thron W. J. Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle. Bull. London Math. Soc., 21: 113–152, 1989.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Jones William B., Thron W. J. A constructive proof of convergence of the even approximants of positive PC-fractions. Rocky Mountain J. of Math., 19 (1): 199–210, 1989.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Jones William B., Thron W. J. Continued Fractions: Analytic Theory and Applications. Encyclopedia of Mathematics and its Applications, 11. Addison- Wesley Publishing Company, Reading, Mass. (1980). Distributed by Cambridge University Press, New York.Google Scholar
  10. [10]
    Waadeland Haakon. A Szegö quadrature formula for the Poisson formula. Comp. and Appl. Math., I, pages 479–486. Elsevier Science Publishers B.V. (North-Holland), IMACS 1992. C. Brezinski and U. Kulish, eds.Google Scholar

Copyright information

© Birkhäuser 1994

Authors and Affiliations

  • William B. Jones
    • 1
  • Haakon Waadeland
    • 2
  1. 1.Department of MathematicsUniversity of ColoradoBoulderUSA
  2. 2.Department of Mathematics & StatisticsUniversity of Trondheim (AVH)DragvollNorway

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