Bounds for Remainder Terms in Szegö Quadrature on the Unit Circle
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.
KeywordsReflection Coefficient Unit Circle Quadrature Formula Remainder Term Moment Problem
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