Summary
A method is proposed for the computation of the Riesz-Herglotz transform. Numerical experiments show the effectiveness of this method. We study its application to the computation of integrals over the unit circle in the complex plane of analytic functions. This approach leads us to the integration by Taylor polynomials. On the other hand, with the goal of minimizing the quadrature error bound for analytic functions, in the set of quadrature formulas of Hermite interpolatory type, we found that this minimum is attained by the quadrature formula based on the integration of the Taylor polynomial. These two different approaches suggest the effectiveness of this formula. Numerical experiments comparing with other quadrature methods with the same domain of validity, or even greater such as Szegö formulas, (traditionally considered as the counterpart of the Gauss formulas for integrals on the unit circle) confirm the superiority of the numerical estimations.
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References
L. Brutman, On the polynomial and rational projections in the complex plane, SIAM J. Numer. Anal. 17 (1980) 366–372
L. Brutman and A. Pinkus, On the Erdös conjecture concerning minimal norm interpolation on the unit circle, SIAM J. Numer. Anal. 17 (1980) 373–375
A. Bultheel, P. González-Vera, E. Hendriksen and O. Njåstad, Orthogonal rational functions, Volume 5 of Cambridge monographs on applied and computational mathematics. Cambridge University Press, 1999
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, Analysis 20 (2000) 99–120
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
P.J. Davis, Interpolation and approximation, Dover publications, New York, (1975)
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
W.B. Jones and O. Njåstad, Applications of Szegö polynomials to digital signal processing, Rocky Mountain J. Math. 21 (1991) 387–436
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
W.B. Jones and H. Waadeland, Bounds for remainder terms 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
J.C. Santos-León, Computation of integrals over the unit circle with nearby poles, Appl. Numer. Math., to appear
J.C. Santos-León, Error bounds for interpolatory quadrature rules on the unit circle, Math. Comput., to appear
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
G. Szegö, Orthogonal polynomials, Amer. Math. Soc., Providence, R.I., 1939
H. Waadeland, A Szegö quadrature formula for the Poisson formula, in: C. Brezinski and U. Kulish (Eds.), Comp. and Appl. Math. I, 1992 IMACS, Elsevier, North-Holland, Amsterdam, pp. 479–486
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This work was supported by the ministry of education and culture of Spain under contract PB96-1029.
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Santos-León, J.C. Computation of the Riesz-Herglotz transform and its application to quadrature formulas over the unit circle. Numer. Math. 93, 153–175 (2002). https://doi.org/10.1007/BF02679441
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DOI: https://doi.org/10.1007/BF02679441