Rational Szego quadratures associated with Chebyshev weight functions

Bultheel, Adhemar; Cruz-Barroso, Ruymán; Deckers, Karl; González-Vera, Pablo

doi:10.1090/S0025-5718-08-02208-4

Rational Szego quadratures associated with Chebyshev weight functions
HTML articles powered by AMS MathViewer

by Adhemar Bultheel, Ruymán Cruz-Barroso, Karl Deckers and Pablo González-Vera;

Math. Comp. 78 (2009), 1031-1059

DOI: https://doi.org/10.1090/S0025-5718-08-02208-4

Published electronically: December 9, 2008

PDF | Request permission

Abstract:

In this paper we characterize rational Szegő quadrature formulas associated with Chebyshev weight functions, by giving explicit expressions for the corresponding para-orthogonal rational functions and weights in the quadratures. As an application, we give characterizations for Szegő quadrature formulas associated with rational modifications of Chebyshev weight functions. Some numerical experiments are finally presented.

References

Adhemar Bultheel, Leyla Daruis, and Pablo González-Vera, A connection between quadrature formulas on the unit circle and the interval $[-1,1]$, J. Comput. Appl. Math. 132 (2001), no. 1, 1–14. Advanced numerical methods for mathematical modelling. MR 1834799, DOI 10.1016/S0377-0427(00)00594-X
Adhemar Bultheel, Leyla Daruis, and Pablo González-Vera, Positive interpolatory quadrature formulas and para-orthogonal polynomials, J. Comput. Appl. Math. 179 (2005), no. 1-2, 97–119. MR 2134362, DOI 10.1016/j.cam.2004.09.037
Adhemar Bultheel, Pablo González-Vera, Erik Hendriksen, and Olav Njåstad, Orthogonal rational functions and quadrature on the unit circle, Numer. Algorithms 3 (1992), no. 1-4, 105–116. Extrapolation and rational approximation (Puerto de la Cruz, 1992). MR 1199359, DOI 10.1007/BF02141920
Adhemar Bultheel, Erik Hendriksen, Pablo González-Vera, and Olav Njåstad, Quadrature formulas on the unit circle based on rational functions, Proceedings of the Fifth International Congress on Computational and Applied Mathematics (Leuven, 1992), 1994, pp. 159–170. MR 1284259, DOI 10.1016/0377-0427(94)90297-6
Adhemar Bultheel, Pablo González-Vera, Erik Hendriksen, and Olav Njåstad, Orthogonal rational functions, Cambridge Monographs on Applied and Computational Mathematics, vol. 5, Cambridge University Press, Cambridge, 1999. MR 1676258, DOI 10.1017/CBO9780511530050
A. Bultheel, P. González-Vera, E. Hendriksen, and Olav Njåstad, Quadrature and orthogonal rational functions, J. Comput. Appl. Math. 127 (2001), no. 1-2, 67–91. Numerical analysis 2000, Vol. V, Quadrature and orthogonal polynomials. MR 1808569, DOI 10.1016/S0377-0427(00)00493-3
—, Rational quadrature formulas on the unit circle with prescribed nodes and maximal domain of validity, (2007), Submitted.
María José Cantero, Ruymán Cruz-Barroso, and Pablo González-Vera, A matrix approach to the computation of quadrature formulas on the unit circle, Appl. Numer. Math. 58 (2008), no. 3, 296–318. MR 2392689, DOI 10.1016/j.apnum.2006.11.009
Ruymán Cruz-Barroso, Leyla Daruis, Pablo González-Vera, and Olav Njåstad, Sequences of orthogonal Laurent polynomials, bi-orthogonality and quadrature formulas on the unit circle, J. Comput. Appl. Math. 206 (2007), no. 2, 950–966. MR 2333724, DOI 10.1016/j.cam.2006.09.003
Leyla Daruis and Pablo González-Vera, Szegő polynomials and quadrature formulas on the unit circle, Appl. Numer. Math. 36 (2001), no. 1, 79–112. MR 1808125, DOI 10.1016/S0168-9274(99)00136-1
Leyla Daruis, Pablo González-Vera, and Olav Njåstad, Szegö quadrature formulas for certain Jacobi-type weight functions, Math. Comp. 71 (2002), no. 238, 683–701. MR 1885621, DOI 10.1090/S0025-5718-01-01337-0
L. Daruis, P. González-Vera, and M. Jiménez Paiz, Quadrature formulas associated with rational modifications of the Chebyshev weight functions, Comput. Math. Appl. 51 (2006), no. 3-4, 419–430. MR 2207429, DOI 10.1016/j.camwa.2005.10.004
Philip J. Davis, Interpolation and approximation, Dover Publications, Inc., New York, 1975. Republication, with minor corrections, of the 1963 original, with a new preface and bibliography. MR 380189
K. Deckers and A. Bultheel, Orthogonal rational functions and rational modifications of a measure on the unit circle, J. Approx. Theory (2008), Accepted.
K. Deckers, J. Van Deun, and A. Bultheel, Computing rational Gauss-Chebyshev quadrature formulas with complex poles, Proceedings of the Fifth International Conference on Engineering Computational Technology (Kippen, Stirlingshire, United Kingdom) (B.H.V. Topping, G. Montero, and R. Montenegro, eds.), Civil-Comp Press, 2006, Paper 30.
Karl Deckers, Joris Van Deun, and Adhemar Bultheel, An extended relation between orthogonal rational functions on the unit circle and the interval $[-1,1]$, J. Math. Anal. Appl. 334 (2007), no. 2, 1260–1275. MR 2338662, DOI 10.1016/j.jmaa.2007.01.031
—, Computing rational Gauss-Chebyshev quadrature formulas with complex poles: the algorithm, Advances in Engineering Software (2008), Accepted.
Karl Deckers, Joris Van Deun, and Adhemar Bultheel, Rational Gauss-Chebyshev quadrature formulas for complex poles outside $[-1,1]$, Math. Comp. 77 (2008), no. 262, 967–983. MR 2373187, DOI 10.1090/S0025-5718-07-01982-5
Walter Gautschi, A survey of Gauss-Christoffel quadrature formulae, E. B. Christoffel (Aachen/Monschau, 1979) Birkhäuser Verlag, Basel-Boston, Mass., 1981, pp. 72–147. MR 661060
Walter Gautschi, Laura Gori, and M. Laura Lo Cascio, Quadrature rules for rational functions, Numer. Math. 86 (2000), no. 4, 617–633. MR 1794345, DOI 10.1007/PL00005412
Ja. L. Geronimus, Polynomials orthogonal on a circle and interval, International Series of Monographs on Pure and Applied Mathematics, Vol. 18, Pergamon Press, New York-Oxford-London-Paris, 1960. Translated from the Russian by D. E. Brown; edited by Ian N. Sneddon. MR 133642
William B. Gragg, Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle, J. Comput. Appl. Math. 46 (1993), no. 1-2, 183–198. Computational complex analysis. MR 1222480, DOI 10.1016/0377-0427(93)90294-L
Ulf Grenander and Gabor Szegö, Toeplitz forms and their applications, California Monographs in Mathematical Sciences, University of California Press, Berkeley-Los Angeles, 1958. MR 94840
Carl Jagels and Lothar Reichel, Szegő-Lobatto quadrature rules, J. Comput. Appl. Math. 200 (2007), no. 1, 116–126. MR 2276819, DOI 10.1016/j.cam.2005.12.009
William B. Jones, Olav Njåstad, and W. J. Thron, Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle, Bull. London Math. Soc. 21 (1989), no. 2, 113–152. MR 976057, DOI 10.1112/blms/21.2.113
Barry Simon, Orthogonal polynomials on the unit circle. Part 1, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Classical theory. MR 2105088, DOI 10.1090/coll054.1
Gábor Szegö, On bi-orthogonal systems of trigonometric polynomials, Magyar Tud. Akad. Mat. Kutató Int. Közl. 8 (1964), 255–273 (1964) (English, with Russian summary). MR 166541
—, Orthogonal polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ., vol. 33, Amer. Math. Soc., Providence, Rhode Island, 1975.
Walter Van Assche and Ingrid Vanherwegen, Quadrature formulas based on rational interpolation, Math. Comp. 61 (1993), no. 204, 765–783. MR 1195424, DOI 10.1090/S0025-5718-1993-1195424-6
Joris Van Deun, Adhemar Bultheel, and Pablo González Vera, On computing rational Gauss-Chebyshev quadrature formulas, Math. Comp. 75 (2006), no. 253, 307–326. MR 2176401, DOI 10.1090/S0025-5718-05-01774-6
J. Van Deun, K. Deckers, A. Bultheel, and J.A.C. Weideman, Algorithm 882: Near best fixed pole rational interpolation with applications in spectral methods, ACM Trans. Math. Software 32 (2008), no. 2, article no. 14, pp. 1–21.
Patrick Van Gucht and Adhemar Bultheel, A relation between orthogonal rational functions on the unit circle and the interval $[-1,1]$, Commun. Anal. Theory Contin. Fract. 8 (2000), 170–182. MR 1789681
Haakon Waadeland, A Szegő quadrature formula for the Poisson formula, Computational and applied mathematics, I (Dublin, 1991) North-Holland, Amsterdam, 1992, pp. 479–486. MR 1203369