Bounds for Remainder Terms in Szegö Quadrature on the Unit Circle

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 (n)j ( 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.KeywordsReflection CoefficientUnit CircleQuadrature FormulaRemainder TermMoment ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.