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On Stieltjes polynomials and Gauss-Kronrod quadrature


Author: Franz Peherstorfer
Journal: Math. Comp. 55 (1990), 649-664
MSC: Primary 65D32; Secondary 33C45, 41A55
DOI: https://doi.org/10.1090/S0025-5718-1990-1035940-8
MathSciNet review: 1035940
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Abstract: Let D be a real function such that $ D(z)$ is analytic and $ D(z) \ne 0$ for $ \vert z\vert \leq 1$. Furthermore, put $ W(x) = \sqrt {1 - {x^2}} \vert D({e^{i\varphi }}){\vert^2}$, $ x = \cos \varphi $, $ \varphi \in [0,\pi ]$, and denote by $ {p_n}( \bullet ,W)$ the polynomial which is orthogonal on $ [ - 1, + 1]$ to $ {\mathbb{P}_{n - 1}}$ ( $ {\mathbb{P}_{n - 1}}$ denotes the set of polynomials of degree at most $ n - 1$) with respect to W. In this paper it is shown that for each sufficiently large n the polynomial $ {E_{n + 1}}( \bullet ,W)$ (called the Stieltjes polynomial) of degree $ n + 1$ which is orthogonal on $ [ - 1, + 1]$ to $ {\mathbb{P}_n}$ with respect to the sign-changing function $ {p_n}( \bullet ,W)W$ has $ n + 1$ simple zeros in $ ( - 1,1)$ and that the interpolation quadrature formula (called the Gauss-Kronrod quadrature formula) based on nodes which are the $ 2n + 1$ zeros of $ {E_{n + 1}}( \bullet ,W){p_n}( \bullet ,W)$ has all weights positive.


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  • [1] W. Gautschi, Gauss-Kronrod quadrature--a survey, Numerical Methods and Approximation Theory III (G. V. Milovanović, ed.), University of Niš, Niš, 1988, pp. 39-66. MR 960329 (89k:41035)
  • [2] W. Gautschi and S. E. Notaris, An algebraic study of Gauss-Kronrod quadrature formulae for Jacobi weight functions, Math. Comp. 51 (1988), 231-248. MR 942152 (89f:65031)
  • [3] W. Gautschi and T. J. Rivlin, A family of Gauss-Kronrod quadrature formulae, Math. Comp. 51 (1988), 749-754. MR 958640 (89m:65029)
  • [4] W. Gautschi and S. E. Notaris, Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegö type, J. Comput. Appl. Math. 25 (1989), 199-224. [Erratum: ibid. 27 (1989), 429.] MR 988057 (90d:65045)
  • [5] Ya. L. Geronimus, On asymptotic formulae for orthogonal polynomials, Izv. Akad. Nauk SSSR Ser. Mat. 12 (1948), 3-14. (Russian) MR 0023955 (9:429e)
  • [6] -, On asymptotic properties of polynomials which are orthogonal on the unit circle, and on certain properties of positive harmonic functions, Izv. Akad. Nauk SSSR Ser. Mat. 14 (1950), 123-144; English transl., Amer. Math. Soc. Transl. (1) 3 (1962), 79-106. MR 0036873 (12:177a)
  • [7] -, Orthogonal polynomials, Amer. Math. Soc. Transl. (2) 108 (1977), 37-130.
  • [8] 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)
  • [9] M. Marden, Geometry of polynomials, Math. Surveys, vol. 3, Amer. Math. Soc., Providence, R.I., 1966. MR 0225972 (37:1562)
  • [10] G. Monegato, A note on extended Gaussian quadrature rules, Math. Comp. 30 (1976), 812-817. MR 0440878 (55:13746)
  • [11] -, Positivity of the weights of extended Gauss-Legendre quadrature rules, Math. Comp. 32 (1978), 243-245. MR 0458809 (56:17009)
  • [12] -, Stieltjes polynomials and related quadrature rules, SIAM Rev. 24 (1982), 137-158. MR 652464 (83d:65067)
  • [13] S. E. Notaris, Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegö type. II, J. Comput. Appl. Math. 29 (1990), 161-170. MR 1041189 (91b:65030)
  • [14] F. Peherstorfer, Weight functions admitting repeated positive Kronrod quadrature, BIT 30 (1990), 145-151. MR 1032847 (91e:65043)
  • [15] -, Linear combinations of orthogonal polynomials generating positive quadrature formulas, Math. Comp. 55 (1990), 231-241. MR 1023052 (90j:65043)
  • [16] P. Rabinowitz, The exact degree of precision of generalized Gauss-Kronrod integration rules, Math. Comp. 35 (1980), 1275-1283. [Corrigendum: ibid. 46 (1986), 226, footnote.] MR 583504 (81j:65047)
  • [17] -, Gauss-Kronrod integration rules for Cauchy principal value integrals, Math. Comp. 41 (1983), 63-78. [Corrigenda: ibid. 45 (1985), 277.] MR 701624 (84i:65029)
  • [18] G. Szegö, Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören, Math. Ann. 110 (1935), 501-513. (Also appears in his Collected papers (R. Askey, ed.), vol. 2, Birkhäuser, Boston-Basel-Stuttgart, 1982, pp. 545-557.) MR 1512952
  • [19] -, Orthogonal polynomials, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1967.

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1990-1035940-8
Keywords: Stieltjes polynomials, orthogonal polynomials, Gauss-Kronrod quadrature formulas
Article copyright: © Copyright 1990 American Mathematical Society

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