×

zbMATH — the first resource for mathematics

Linear combinations of orthogonal polynomials generating positive quadrature formulas. (English) Zbl 0708.65022
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

MSC:
65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
PDF BibTeX XML Cite
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.