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Error bounds for interpolatory quadrature rules on the unit circle

Author: J. C. Santos-León
Journal: Math. Comp. 70 (2001), 281-296
MSC (2000): Primary 41A55, 65D30
Published electronically: June 12, 2000
MathSciNet review: 1803128
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For the construction of an interpolatory integration rule on the unit circle $T$ with $n$ nodes by means of the Laurent polynomials as basis functions for the approximation, we have at our disposal two nonnegative integers $p_n$ and $q_n,$ $p_n+q_n=n-1,$ which determine the subspace of basis functions. The quadrature rule will integrate correctly any function from this subspace. In this paper upper bounds for the remainder term of interpolatory integration rules on $T$ are obtained. These bounds apply to analytic functions up to a finite number of isolated poles outside $T.$ In addition, if the integrand function has no poles in the closed unit disc or is a rational function with poles outside $T$, we propose a simple rule to determine the value of $p_n$ and hence $q_n$ in order to minimize the quadrature error term. Several numerical examples are given to illustrate the theoretical results.

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Additional Information

J. C. Santos-León
Affiliation: Department of Mathematical Analysis, La Laguna University, 38271-La Laguna, Tenerife, Canary Islands, Spain

Keywords: Error bounds, quadrature formulas, singular integrands, Szeg\"{o} polynomials
Received by editor(s): February 2, 1999
Published electronically: June 12, 2000
Additional Notes: This work was supported by the Ministry of Education and Culture of Spain under contract PB96-1029.
Article copyright: © Copyright 2000 American Mathematical Society

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