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Positive quadrature formulas III: asymptotics of weights


Author: Franz Peherstorfer
Journal: Math. Comp. 77 (2008), 2241-2259
MSC (2000): Primary 65D32; Secondary 42C05
DOI: https://doi.org/10.1090/S0025-5718-08-02119-4
Published electronically: May 1, 2008
MathSciNet review: 2429883
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Abstract | References | Similar Articles | Additional Information

Abstract: First we discuss briefly our former characterization theorem for positive interpolation quadrature formulas (abbreviated qf), provide an equivalent characterization in terms of Jacobi matrices, and give links and applications to other qf, in particular to Gauss-Kronrod quadratures and recent rediscoveries. Then for any polynomial $ t_n$ which generates a positive qf, a weight function (depending on $ n$) is given with respect to which $ t_n$ is orthogonal to $ \mathbb{P}_{n-1}$. With the help of this result an asymptotic representation of the quadrature weights is derived. In general the asymptotic behaviour is different from that of the Gaussian weights. Only under additional conditions do the quadrature weights satisfy the so-called circle law. Corresponding results are obtained for positive qf of Radau and Lobatto type.


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  • 1. H. Brass, Quadraturverfahren, Vandenhoeck u. Ruprecht, Göttingen-Zürich, 1977. MR 0443305 (56:1675)
  • 2. D. Calvetti, G.H. Golub, W.B. Gragg, and L. Reichel, Computation of Gauss-Kronrod rules, Math. Comp. 69 (2000), 1035-1052. MR 1677474 (2000j:65035)
  • 3. A. Ezzirani and A. Guessab, A fast algorithm for Gaussian type quadrature formulae with mixed boundary conditions and some lumped mass spectral approximations, Math. Comp. 68 (1999), 217-248. MR 1604332 (99c:65044)
  • 4. G. Freud, Orthogonale Polynome, Birkhäuser Verlag, Basel, 1969. MR 0481888 (58:1982)
  • 5. W. Gautschi, Orthogonal polynomials and quadrature, Elect. Trans. Num. Anal. 9 (1999), 65-76. MR 1749799 (2001b:33015)
  • 6. W. Gautschi, The circle theorem and related theorems for Gauss-type quadrature rules, Elect. Trans. Num. Anal. 25 (2006), 129-137. MR 2280368 (2007k:42067)
  • 7. G.H. Golub and H. Welsch, Calculation of Gauss quadrature rules, Math. Comp. 23(1969), 221-230. MR 0245201 (39:6513)
  • 8. A. Guessab, On a new family of Gaussian quadrature formulae of Birkhoff type and some lumped mass spectral approximations. Part II: Generalized Laguerre case on semi-infinite domains, Int. J. Pure Appl. Math. 3 (2002), 193-239. MR 1937650 (2003k:65020)
  • 9. D.P. Laurie, Anti-Gaussian quadrature formulas, Math. Comp. 65 (1996), 739-747. MR 1333318 (96m:65026)
  • 10. D.P. Laurie, Calculation of Gauss-Kronrod quadrature rules, Math. Comp. 66 (1997), 1133-1145. MR 1422788 (98m:65030)
  • 11. G. Monegato, An overview of the computational aspect of Kronrod quadratures rules, Numerical Algorithms 26 (2001), 173-196. MR 1829797 (2002a:65051)
  • 12. I.P. Natanson, Constructive Function Theory, Vol. II, F. Ungar Publ. Co., New York, 1965.
  • 13. P. Nevai, A new class of orthogonal polynomials, Proc. Amer. Math. Soc. 91 (1984), 409-415. MR 744640 (85f:42036)
  • 14. G. Nikolov, On the weights of nearly gaussian quadrature formulae, East Journal on Approximations 7 (2001), 115-120. MR 1834159 (2002g:41047)
  • 15. F. Peherstorfer, Characterization of positive quadrature formulas, SIAM J. Math. Anal. 12 (1981), 935-942. MR 635246 (82m:65021)
  • 16. F. Peherstorfer, Characterization of quadrature formulas II, SIAM J. Math. Anal. 15 (1984), 1021-1030. MR 755862 (86a:65025)
  • 17. F. Peherstorfer, Linear combinations of orthogonal polynomials generating positive quadrature formulas, Math. Comp. 55 (1990), 231-241. MR 1023052 (90j:65043)
  • 18. F. Peherstorfer, Finite perturbations of orthogonal polynomials, J. Comput. Appl. Math. 44 (1992), 275-302. MR 1199259 (94d:42029)
  • 19. F. Peherstorfer, On positive quadrature formulas, ISNM 112, Birkäuser, Basel, 1993, 297-313. MR 1248412 (94k:65035)
  • 20. F. Peherstorfer, Zeros of linear combinations of orthogonal polynomials, Math. Proc. Camb. Phil. Soc. 117 (1995), 533-544. MR 1317495 (96c:42055)
  • 21. F. Peherstorfer, A special class of polynomials orthogonal on the unit circle including the associated polynomials, Constr. Approx. 12 (1996), 161-185. MR 1393285 (97d:42023)
  • 22. F. Peherstorfer and K. Petras, Stieltjes polynomials and Gauss-Kronrod quadrature for Jacobi weight functions, Numer. Math. 95 (2003), 689-706. MR 2013124 (2004j:33010)
  • 23. H.J. Schmid, A note on positive quadrature rules, Rocky Mountain J. Math. 19 (1989), 395-404. MR 1016190 (90k:41041)
  • 24. J. Shohat, On mechanical quadratures, in particular, with positive coefficients, Trans. Amer. Math. Soc. 42 (1937), 461-496. MR 1501930
  • 25. B. Simon, Orthogonal polynomials on the unit circle, Part 1: Classical theory, AMS Colloquium Series, American mathematical Society, Providence, RI, 2005. MR 2105088 (2006a:42002a)
  • 26. B. Simon, Orthogonal polynomials on the unit circle, Part 2: Spectral theory AMS Colloquium Series, American mathematical Society, Providence, RI, 2005. MR 2105089 (2006a:42002b)
  • 27. M.R. Skrzipek, Generalized associated polynomials and their application in numerical differentiation and quadrature, Calcolo 40 (2003), 131-147. MR 2025599 (2005k:33008)
  • 28. G. Sottas and G. Wanner, The number of positive weights of a quadrature formula, BIT 22 (1982), 339-352. MR 675668 (84a:65020)
  • 29. M. Spalević, On generalized averaged Gaussian formula, Math. Comp. 76 (2007), 1483-1492. MR 2299784
  • 30. G. Szegő, Über den asymptotischen Ausdruck von Polynomen, die durch eine Orthogonalitätseigenschaft definiert sind, Math. Ann. 86 (1922), 114-139. MR 1512082
  • 31. G. Szegő, Orthogonal Polynomials Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc., Providence, RI, 3rd ed., 1967.
  • 32. V. Totik, Asymptotics for Christoffel functions for general measures on the real line, J. Anal. Math. 81 (2000), 283-303. MR 1785285 (2001j:42021)
  • 33. Y. Xu, Quasi-orthogonal polynomials, quadrature, and interpolation, J. Math. Anal. Appl. 182 (1994), 779-799. MR 1272153 (95a:42035)
  • 34. Y. Xu, A characterization of positive quadrature formulae, Math. Comp. 62 (1994), 703-718. MR 1223234 (94h:41067)

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Additional Information

Franz Peherstorfer
Affiliation: Abteilung für Dynamische Systeme und Approximationstheorie, Institut für Analysis, J.K. Universität Linz, Altenberger Strasse 69, 4040 Linz, Austria
Email: franz.peherstorfer@jku.at

DOI: https://doi.org/10.1090/S0025-5718-08-02119-4
Received by editor(s): June 6, 2007
Received by editor(s) in revised form: September 4, 2007
Published electronically: May 1, 2008
Additional Notes: The author was supported by the Austrian Science Fund FWF, project no. P20413-N18.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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