Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-21T13:03:17.606Z Has data issue: false hasContentIssue false
  • English
  • Français

An analogue of problem 26 of P. Turán

Published online by Cambridge University Press:  17 April 2009

Y. G. Shi
Y. G. Shi
Affiliation:
Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, PO Box 2719, Beijing, China 100080
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Explicit formulas for Cotes numbers of the Gaussian Hermite quadrature formula based on the zeros of the nth Chebyshev polynomial of the second kind and their asymptotic behaviour as n → ∞ are given. This provides an answer to an analogue of Problem 26 of Tur´n.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 53 , Issue 1 , February 1996 , pp. 1 - 12
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Abramowitz, M. and Stegun, I.A., Handbook of mathematical functions with formulas, graphs and mathematical tables (National Bureau of Standards, Applied Mathematics 55, Washington, D.C., 1965).Google Scholar
[2]Bernstein, S., ‘Sur les polynomes orthogonaux relatifs á un segment fini’, J. de Math. 9 (1930), 127–17710 (1931), pp.219–286.Google Scholar
[3]Bojanov, B.D., Braess, D. and Dyn, N., ‘Generalized Gaussian quadrature formulas’, J. Approx. Theory 46 (1986), 335–353.CrossRefGoogle Scholar
[4]Gradshteyn, I.S. and Ryzhik, I.M., Table of integrals, series and products (Academic Press, New York, 1980).Google Scholar
[5]Natanson, I.P., Constructive function theory I–III (Ungar, New York, 1964–1965).Google Scholar
[6]Shi, Y.G., ‘A solution of Problem 26 of P. Turán’, Sci. China (to appear).Google Scholar
[7]Shi, Y.G., ‘On Problem 26 of P. Turán’, Acta Sci. Math. (Szeged), (to appear).Google Scholar
[8]Shi, Y.G., ‘A class of functions having the same best approximation for all Lp norms (1 ≤ p ≤ ∞)’, Approx. Theory Appl. 5 (1989), 69–72.Google Scholar
[9]Szabados, J., ‘On the order of magnitude of fundamental polynomials of Hermite interpolation’, Acta Math. Hungar 61 (1993), 357–368.CrossRefGoogle Scholar
[10]Turán, P., ‘On some open problems of approximation theory’, J. Approx. Theory 29 (1980), 23–85.CrossRefGoogle Scholar
[11]Vértesi, P., ‘Hermite-Fejér interpolations of higher order. I’, Acta Math. Hungar. 54 (1989), 135–152.CrossRefGoogle Scholar