Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-11T23:23:17.173Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on Lagrange interpolation for |x|λ at equidistant nodes

Published online by Cambridge University Press:  17 April 2009

Michael I. Ganzburg  and
Michael Revers
Michael I. Ganzburg
Affiliation:
Department of Mathematics, Hampton University, Hampton, VA 23668, United States of America, e-mail: [email protected]
Michael Revers
Affiliation:
Department of Mathematics, University Salzburg, Hellbrunnerstrasse 34, A-5020 Salzburg, Austria e-mail: [email protected] Internet: www.michael.revers.at
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.

In this note, we discuss the exceptional set E ⊆ [−1, 1] of points x0 satisfying the inequality where λ > 0, λ ≠ 2, 4, … and Ln(fλ,.) is the Lagrange interpolation polynomial of degree at most n to fλ(x):= |x|λ on the interval [−1, 1] associated with the equidistant nodes. It is known that E has Lebesgue measure zero. Here we show that E contains infinite families of rational and irrational numbers.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 70 , Issue 3 , December 2004 , pp. 475 - 480
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Bernstein, S.N., ‘Quelques remarques sur l'interpolation’, Zap. Kharkov Mat. Ob-va (Comm. Kharkov Math. Soc.) 15 (1916), 49–61.Google Scholar
[2]Bernstein, S.N., ‘Quelques remarques sur l'interpolation’, Math. Ann. 79 (1918), 1–12.Google Scholar
[3]Byrne, G., Mills, T.M. and Smith, S.J., ‘On Lagrange interpolation with equidistant nodes’, Bull. Austral. Math. Soc. 42 (1990), 81–89.Google Scholar
[4]Feldman, N.I., Approximations of algebraic numbers, (in Russian) (Moscow State University, Moscow, 1981).Google Scholar
[5]Ganzburg, M.I., ‘Strong asymptotics in Lagrange interpolation with equidistant nodes’, J. Approx. Theory 122 (2003), 224–240.Google Scholar
[6]Li, X. and Mohapatra, R.N., ‘On the divergence of Lagrange interpolation with equidistant nodes’, Proc. Amer. Math. Soc. 118 (1993), 1205–1212.Google Scholar
[7]Natanson, I.P., Constructive function theory, III (Frederick Ungar, New York, 1965).Google Scholar
[8]Revers, M., ‘On Lagrange interpolation with equally spaced nodes’, Bull. Austral. Math. Soc. 62 (2000), 357–368.CrossRefGoogle Scholar
[9]Revers, M., On the zero-divergence of equidistant Lagrange interpolation, Monatsh. Math. 131 (2000), 215–221.Google Scholar
[10]Revers, M., ‘A survey on Lagrange interpolation based on equally spaced nodes’, in Advanced problems in Constructive Approximation, (Buhmann, M.D. and Mache, D.H., Editors) (Birkhäuser, Basel, 2002), pp. 153–163.Google Scholar