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

DOI:
https://doi.org/10.1090/S0025-5718-00-01260-6

Published electronically:
June 12, 2000

MathSciNet review:
1803128

Full-text PDF

Abstract | References | Similar Articles | Additional Information

For the construction of an interpolatory integration rule on the unit circle with nodes by means of the Laurent polynomials as basis functions for the approximation, we have at our disposal two nonnegative integers and 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 are obtained. These bounds apply to analytic functions up to a finite number of isolated poles outside In addition, if the integrand function has no poles in the closed unit disc or is a rational function with poles outside , we propose a simple rule to determine the value of and hence in order to minimize the quadrature error term. Several numerical examples are given to illustrate the theoretical results.

**1.**A. Bultheel, P. González-Vera, E. Hendriksen and O. Njåstad,*Orthogonal rational functions and interpolatory product rules on the unit circle. III: Convergence of general sequences,*Preprint.**2.**A. Bultheel, P. González-Vera, E. Hendriksen and O. Njåstad,*Orthogonality and quadrature on the unit circle,*IMACS Annals on Computing and Appl. Math. 9 (1991) 205-210. CMP**94:10****3.**P. González-Vera, O. Njåstad and J.C. Santos-León,*Some results about numerical quadrature on the unit circle,*Adv. Comput. Math. 5 (1996) 297-328. MR**98f:41028****4.**P. Henrici,*Applied and computational complex analysis*(Vol. 1, John Wiley and Sons, New York, 1974). MR**51:8378****5.**W.B. Jones, O. Njåstad and W.J. Thron,*Continued fractions associated with trigonometric and other strong moment problems,*Constr. Approx. 2 (1986) 197-211. MR**88m:30087****6.**W.B. Jones, O. Njåstad and W.J. Thron,*Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle,*Bull. London Math. Soc. 21 (1989) 113-152. MR**90e:42027****7.**W.B. Jones and H. Waadeland,*Bounds for remainder term in Szegö quadrature on the unit circle,*Approximation and Computation: A. Festschrift in Honor of Walter Gautschi (Boston) (R.V.M. Zahar, ed.), International Series of Numerical Mathematics, vol. 119, Birkhäuser, Boston, 1994, pp. 325-342. MR**96i:41029****8.**J.C. Santos-León,*Product rules on the unit circle with uniformly distributed nodes. Error bounds for analytic functions,*J. Comput. Appl. Math. 108 (1999) 195-208. CMP**99:17****9.**G. Szegö,*Orthogonal polynomials*(Amer. Math. Soc. Providence, R.I., 1939).

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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

Email:
jcsantos@ull.es

DOI:
https://doi.org/10.1090/S0025-5718-00-01260-6

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