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Stieltjes-type polynomials on the unit circle


Authors: B. de la Calle Ysern, G. López Lagomasino and L. Reichel
Journal: Math. Comp. 78 (2009), 969-997
MSC (2000): Primary 65D32, 42A10, 42C05; Secondary 30E20
DOI: https://doi.org/10.1090/S0025-5718-08-02195-9
Published electronically: October 27, 2008
MathSciNet review: 2476567
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Abstract: Stieltjes-type polynomials corresponding to measures supported on the unit circle $ \mathbb{T}$ are introduced and their asymptotic properties away from $ \mathbb{T}$ are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functions to the corresponding Carathéodory function. In turn, this is used to give an estimate of the rate of convergence of certain quadrature formulae that resemble the Gauss-Kronrod rule, provided that the integrand is analytic in a neighborhood of $ \mathbb{T}$.


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  • 1. G. S. AMMAR, D. CALVETTI, AND L. REICHEL, Computation of Gauss-Kronrod quadrature rules with non-positive weights, Electron. Trans. Numer. Anal., 9 (1999), 26-38. MR 1749795 (2001c:65030)
  • 2. M. BELLO HERNáNDEZ, B. DE LA CALLE YSERN, J. J. GUADALUPE HERNáNDEZ, AND G. LóPEZ LAGOMASINO, Asymptotics for Stieltjes polynomials, Padé approximants, and Gauss-Kronrod quadrature, J. Anal. Math., 86 (2002), 1-23. MR 1894475 (2002m:41021)
  • 3. I. S. BEREZIN AND N. P. ZHIDKOV, Computing Methods, Pergamon Press, Oxford, 1965.
  • 4. A. BULTHEEL, P. GONZáLEZ-VERA, E. HENDRIKSEN, AND O. NJåSTAD, On the convergence of multipoint Padé-type approximants and quadrature formulas associated with the unit circle, Numer. Algorithms, 13 (1996), 321-344. MR 1430523 (97h:41029)
  • 5. B. DE LA CALLE YSERN AND F. PEHERSTORFER, Ultraspherical Stieltjes polynomials and Gauss-Kronrod quadrature behave nicely for $ \lambda <0$, SIAM J. Numer. Anal., 45 (2007), 770-786. MR 2300296 (2008d:33009)
  • 6. D. CALVETTI, G. H. GOLUB, W. B. GRAGG, AND L. REICHEL, Computation of Gauss-Kronrod quadrature rules, Math. Comp., 69 (2000), 1035-1052. MR 1677474 (2000j:65035)
  • 7. S. EHRICH, On product integration with Gauss-Kronrod nodes, SIAM J. Numer. Anal., 35 (1998), 78-92. MR 1618432 (99e:65044)
  • 8. S. EHRICH AND G. MASTROIANNI, Stieltjes polynomials and Lagrange interpolation, Math. Comp., 66 (1997), 311-331. MR 1388888 (97j:65013)
  • 9. W. GAUTSCHI, Gauss-Kronrod quadrature - a survey, in Numerical Methods and Approximation Theory III, G. V. Milovanović, ed., University of Niš, Niš, 1988, 39-66. MR 960329 (89k:41035)
  • 10. YA. L. GERONIMUS, On the trigonometric moment problem, Ann. of Math. (2), 47 (1946), 742-761. MR 0018265 (8:265d)
  • 11. P. GONZáLEZ-VERA, O. NJå STAD, AND J. C. SANTOS-LEóN, Some results about numerical quadrature on the unit circle, Adv. Comput. Math., 5 (1996), 297-328. MR 1414284 (98f:41028)
  • 12. CH. HERMITE AND T. J. STIELTJES, Correspondance d'Hermite et de Stieltjes, Prentice-Hall, Englewood Cliffs, NJ, 1966.
  • 13. C. JAGELS AND L. REICHEL, Szegő-Lobatto quadrature rules, J. Comput. Appl. Math., 200 (2007), 116-126. MR 2276819 (2008c:65074)
  • 14. 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 976057 (90e:42027)
  • 15. A. S. KRONROD, Nodes and weights for quadrature formulae. Sixteen-Place Tables, Nauka Moscow, 1964, English transl., Consultants Bureau, New York, 1965. MR 0183116 (32:598)
  • 16. D. P. LAURIE, Calculation of Gauss-Kronrod quadrature rules, Math. Comp., 66 (1997), 1133-1145. MR 1422788 (98m:65030)
  • 17. A. MáTé, P. NEVAI, AND V. TOTIK, Extensions of Szegő's theory of orthogonal polynomials, II, Constr. Approx., 3 (1987), 51-72. MR 892168 (88m:42044a)
  • 18. G. MONEGATO, Stieltjes polynomials and related quadrature rules, SIAM Rev., 24 (1982), 137-158. MR 652464 (83d:65067)
  • 19. G. MONEGATO, An overview of the computational aspects of Kronrod quadrature rules, Numer. Algorithms, 26 (2001), 173-196. MR 1829797 (2002a:65051)
  • 20. F. PEHERSTORFER, Szegő-Kronrod quadrature on the unit circle, manuscript.
  • 21. F. PEHERSTORFER AND K. PETRAS, Ultraspherical Gauss-Kronrod quadrature is not possible for $ \lambda > 3$, SIAM J. Numer. Anal., 37 (2000), 927-948. MR 1749243 (2001g:33010)
  • 22. R. PIESSENS, E. DE DONCKER-KAPENGA, C. W. ÜBERHUBER, AND D. K. KAHANER, QUADPACK: A subroutine package for automatic integration, Springer Ser. Comput. Math., 1, Springer, Berlin, 1983. MR 712135 (85b:65022)
  • 23. E. A. RAKHMANOV, On asymptotic properties of polynomials orthogonal on the circle with weights not satisfying Szegő's condition, Math. USSR Sbornik, 58 (1987), 149-167. MR 854969 (88b:42033)
  • 24. T. RANSFORD, Potential Theory in the Complex Plane, London Mathematical Society, Student Texts 28, Cambridge University Press, Cambridge, 1995. MR 1334766 (96e:31001)
  • 25. B. SIMON, Orthogonal Polynomials on the Unit Circle, AMS Colloquium Publications 54, Parts I, II, Providence, RI, 2005.
  • 26. H. STAHL AND V. TOTIK, General Orthogonal Polynomials, Cambridge University Press, Cambridge, 1992. MR 1163828 (93d:42029)
  • 27. G. SZEGŐ, Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören, Math. Ann., 110 (1935), 501-513. MR 1512952

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Additional Information

B. de la Calle Ysern
Affiliation: Departamento de Matemática Aplicada, E. T. S. de Ingenieros Industriales, Universidad Politécnica de Madrid, José G. Abascal 2, 28006 Madrid, Spain
Email: bcalle@etsii.upm.es

G. López Lagomasino
Affiliation: Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, Universidad 30, 28911 Leganés, Spain
Email: lago@math.uc3m.es

L. Reichel
Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
Email: reichel@math.kent.edu

DOI: https://doi.org/10.1090/S0025-5718-08-02195-9
Keywords: Quadrature rules on the unit circle, para-orthogonal polynomials, Stieltjes polynomials, Gauss-Kronrod quadrature
Received by editor(s): October 3, 2007
Received by editor(s) in revised form: April 25, 2008
Published electronically: October 27, 2008
Additional Notes: The work of B. de la Calle received support from Dirección General de Investigación (DGI), Ministerio de Educación y Ciencia, under grants MTM2006-13000-C03-02 and MTM2006-07186 and from UPM-CAM under grants CCG07-UPM/000-1652 and CCG07-UPM/ESP-1896
The work of G. López was supported by DGI under grant MTM2006-13000-C03-02 and by UC3M-CAM through CCG06-UC3M/ESP-0690
The work of L. Reichel was supported by an OBR Research Challenge Grant.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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