Peherstorfer, Franz Linear combinations of orthogonal polynomials generating positive quadrature formulas. (English) Zbl 0708.65022 Math. Comput. 55, No. 191, 231-241 (1990). Linear combinations of monic polynomials \(p_ k\) of degree k orthogonal on [-1,1] with respect to the positive measure \(d\sigma\) (x) are considered. Specifically, sufficient conditions on the real numbers \(\mu_ j\), \(j=0,...,m\) are given so that the sum \(\sum^{m}_{j=0}\mu_ jp_{n-j}\) has n simple zeros in (-1,1) and the numerical interpolatory quadrature formula for integration with respect to \(d\sigma\) (x) using these zeros as nodes has positive weights. Reviewer: W.E.Smith Cited in 20 Documents MSC: 65D32 Numerical quadrature and cubature formulas 41A55 Approximate quadratures 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) Keywords:orthogonal polynomials; positive weights; Gaussian rules; numerical interpolatory quadrature formula PDF BibTeX XML Cite \textit{F. Peherstorfer}, Math. Comput. 55, No. 191, 231--241 (1990; Zbl 0708.65022) Full Text: DOI References: [1] Richard Askey, Positive quadrature methods and positive polynomial sums, Approximation theory, V (College Station, Tex., 1986) Academic Press, Boston, MA, 1986, pp. 1 – 29. · Zbl 0613.41027 [2] T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach Science Publishers, New York-London-Paris, 1978. Mathematics and its Applications, Vol. 13. · Zbl 0389.33008 [3] Ya. L. Geronimus, Polynomials orthogonal on a circle and their applications, Zapiski Naučno-Issled. Inst. Mat. Meh. Har\(^{\prime}\)kov. Mat. Obšč. (4) 19 (1948), 35 – 120 (Russian). · Zbl 0056.10303 [4] Morris Marden, Geometry of polynomials, Second edition. Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. · Zbl 0162.37101 [5] Charles A. Micchelli, Some positive Cotes numbers for the Chebyshev weight function, Aequationes Math. 21 (1980), no. 1, 105 – 109. · Zbl 0457.41028 · doi:10.1007/BF02189344 · doi.org [6] C. A. Micchelli and T. J. Rivlin, Numerical integration rules near Gaussian quadrature, Israel J. Math. 16 (1973), 287 – 299. · Zbl 0287.65014 · doi:10.1007/BF02756708 · doi.org [7] Franz Peherstorfer, Characterization of positive quadrature formulas, SIAM J. Math. Anal. 12 (1981), no. 6, 935 – 942. · Zbl 0481.41025 · doi:10.1137/0512079 · doi.org [8] Franz Peherstorfer, Characterization of quadrature formula. II, SIAM J. Math. Anal. 15 (1984), no. 5, 1021 – 1030. · Zbl 0596.41044 · doi:10.1137/0515079 · doi.org [9] H. J. Schmid, A note on positive quadrature rules, Rocky Mountain J. Math. 19 (1989), no. 1, 395 – 404. Constructive Function Theory — 86 Conference (Edmonton, AB, 1986). · Zbl 0691.41032 · doi:10.1216/RMJ-1989-19-1-395 · doi.org [10] G. Sottas and G. Wanner, The number of positive weights of a quadrature formula, BIT 22 (1982), no. 3, 339 – 352. · Zbl 0491.41035 · doi:10.1007/BF01934447 · doi.org This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.