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 | References | Similar Articles | Additional Information

Abstract: Let , , be the polynomials orthogonal on with respect to the positive measure . We give sufficient conditions on the real numbers , , such that the linear combination of orthogonal polynomials has *n* simple zeros in and that the interpolatory quadrature formula whose nodes are the zeros of has positive weights.

**[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