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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)

Abstract

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.

Keywords

Reflection Coefficient Unit Circle Quadrature Formula Remainder Term Moment Problem 
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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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