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On the positivity of certain Cotes numbers

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Abstract

It is shown that the Cotes numbers are nonnegative for some interpolatory quadratic rules with nodes at the zeros of the ultraspherical polynomialsC λn , when integrating with respect to the weight functions wµ(x)=(1−x2)µ−1/2.

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References

  1. Askey, R.,Orthogonal polynomials and special functions. SIAM Regional Conference Series in Applied Mathematics, vol. 21, Philadelphia, 1975.

  2. Askey, R.,Positivity of the Cotes numbers for some Jacobi abscissas II. J. Inst. Math. Appl.24 (1979), 95–98.

    Google Scholar 

  3. Askey, R. andFitch, J.,Positivity of the Cotes numbers for some ultraspherical abscissas. SIAM J. Numer. Anal.5 (1968), 199–201.

    Google Scholar 

  4. Askey, R. andSteinig, J.,Some positive trigonometric sums. Trans. Amer. Math. Soc.187 (1974), 295–307.

    Google Scholar 

  5. Brass, H.,Quadraturverfahren. Vandenhoeck und Ruprecht, Göttingen-Zürich, 1977.

    Google Scholar 

  6. Davis, Ph. andRabinowitz, Ph.,Methods of numerical integration. Academic Press, New York-San Francisco-London, 1975.

    Google Scholar 

  7. Gasper, G.,Positive sums of the classical orthogonal polynomials. SIAM J. Math. Anal.8 (1977), 423–447.

    Google Scholar 

  8. Horton, R.,Expansions using orthogonal polynomials. Ph.D. Thesis, University of Wisconsin, Madison, 1973.

    Google Scholar 

  9. Kütz, M.,Fehlerschranken und Fehlerasymptotik für eine Klasse von Interpolationsquadraturen. Doctoral Thesis, Technische Universität Braunschweig, 1982.

  10. Michelli, C. A.,Some positive Cotes numbers for the Chebyshev weight function. Aequationes Math.21 (1980), 105–109.

    Google Scholar 

  11. Pólya, G. and Szegö, G., Über den transfiniten Durchmesser (Kapazitätskonstante) von ebenen und räumlichen Punktmengen. J. Reine. Angew. Math.165 (1973), 4–49.

    Google Scholar 

  12. Riess, R. andJohnson, L.,Error estimates for Clenshaw-Curtis-quadrature, Numer. Math.18 (1972), 345–353.

    Google Scholar 

  13. Smith, W. E. andSloan, I. H., Product integration rules based on the zeros of Jacobi polynomials. SIAM J. Numer. Anal.17 (1980), 1–13.

    Google Scholar 

  14. Szegö, G.,Orthogonal polynomials. Amer. Math. Soc. Colloq. Publications, vol. 23, Amer. Math. Soc., Providence, 1978.

    Google Scholar 

  15. Tricomi, F.,Vorlesungen über Orthogonalreihen. Springer, Berlin-Heidelberg-New York, 1970.

    Google Scholar 

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Authors and Affiliations

  1. Lehrstuhl E für Mathematik, Technische Universität Braunschweig, Pockelsstr 14, D-3300, Braunschweig, West Germany

    Martin Kütz

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  1. Martin Kütz
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Kütz, M. On the positivity of certain Cotes numbers. Aeq. Math. 24, 110–118 (1982). https://doi.org/10.1007/BF02193038

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  • DOI: https://doi.org/10.1007/BF02193038

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