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Linear combinations of orthogonal polynomials generating positive quadrature formulas


Author: Franz Peherstorfer
Journal: Math. Comp. 55 (1990), 231-241
MSC: Primary 65D32; Secondary 41A55, 42C05
DOI: https://doi.org/10.1090/S0025-5718-1990-1023052-9
MathSciNet review: 1023052
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Abstract: Let $ {p_k}(x) = {x^k} + \cdots $, $ k \in {{\mathbf{N}}_0}$, be the polynomials orthogonal on $ [ - 1, + 1]$ with respect to the positive measure $ d\sigma $. We give sufficient conditions on the real numbers $ {\mu _j}$, $ j = 0, \ldots ,m$, such that the linear combination of orthogonal polynomials $ \sum _{j = 0}^m{\mu _j}{p_{n - j}}$ has n simple zeros in $ ( - 1, + 1)$ and that the interpolatory quadrature formula whose nodes are the zeros of $ \sum _{j = 0}^m{\mu _j}{p_{n - j}}$ has positive weights.


References [Enhancements On Off] (What's this?)

  • [1] R. Askey, Positive quadrature methods and positive polynomial sums, Approximation Theory. V (C. K. Chui, L. L. Schumaker, and J. D. Ward, eds.), Academic Press, New York, 1986, pp. 1-30. MR 903680 (88j:41065)
  • [2] T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach, New York, 1978. MR 0481884 (58:1979)
  • [3] Ja. L. Geronimus, Polynomials orthogonal on a circle and their applications, Zap. Naučno-Issled. Inst. Mat. Mekh. Kharkov. Mat. Obshch. 19 (1948), 35-120; English transl., Amer. Math. Soc. Transl. 3 (1962), 1-78. MR 0036872 (12:176e)
  • [4] M. Marden, Geometry of polynomials, Amer. Math. Soc., Providence. R.I., 1966. MR 0225972 (37:1562)
  • [5] C. A. Micchelli, Some positive Cotes numbers for the Chebyshev weight function, Aequationes Math. 21 (1980), 105-109. MR 594098 (81k:41021)
  • [6] C. A. Micchelli and T. J. Rivlin, Numerical integration rules near Gaussian quadrature, Israel J. Math. 16 (1973), 287-299. MR 0366003 (51:2255)
  • [7] F. Peherstorfer, Characterization of positive quadrature formulas, SIAM J. Math. Anal. 12 (1981), 935-942. MR 635246 (82m:65021)
  • [8] -, Characterizations of quadrature formulas. II, SIAM J. Math. Anal. 15 (1984), 1021-1030. MR 755862 (86a:65025)
  • [9] H. J. Schmid, A note on positive quadrature rules, Rocky Mountain J. Math. 19 (1989), 395-404. MR 1016190 (90k:41041)
  • [10] G. Sottas and G. Wanner, The number of positive weights of a quadrature formula, BIT 22 (1982), 339-352. MR 675668 (84a:65020)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1990-1023052-9
Keywords: Quadrature formula, positive weights, orthogonal polynomials, zeros
Article copyright: © Copyright 1990 American Mathematical Society

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