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The Lebesgue function for Hermite-Fejér interpolation on the extended Chebyshev nodes

Published online by Cambridge University Press:  17 April 2009

Simon J. Smith
Simon J. Smith
Affiliation:
Department of Mathematics, La Trobe University, P.O. Box 199, Bendigo, Vic. 3552, Australia e-mail: [email protected]
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Abstract

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Given f ∈ C[−1, 1] and n point (nodes) in [−1, 1], the Hermite-Fejér interpolation polynomial is the polynomial of minimum degree which agrees with f and has zero derivative at each of the nodes. In 1916, L. Fejér showed that if the nodes are chosen to be zeros of Tn (x), the nth Chebyshev polynomial of the first kind, then the interpolation polynomials converge to f uniformly as n → ∞. Later, D.L. Berman demonstrated the rather surprising result that this convergence property no longer holds true if the Chebyshev nodes are extended by the inclusion of the end points −1 and 1 in the interpolation process. The aim of this paper is to discuss the Lebesgue function and Lebesgue constant for Hermite-Fejér interpolation on the extended Chebyshev nodes. In particular, it is shown that the inclusion of the two endpoints causes the Lebesgue function to change markedly, from being identically equal to 1 for the Chebyshev nodes, to having the form 2n2(1 − x2)(Tn (x))2 + O (1) for the extended Chebyshev nodes.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 66 , Issue 1 , August 2002 , pp. 151 - 162
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Berman, D.L., ‘A study of the Hermite-Fejér interpolation process’, Dokl. Akad. Nauk USSR 187 (1969), 241–244; (in Russian)(Soviet Math. Dokl. 10 (1969), 813–816).Google Scholar
[2]Bojanić, R., ‘Necessary and sufficient conditions for the convergence of the extended Hermite-Fejér interpolation process’, Acta Math. Acad. Sci. Hungar. 36 (1980), 271–279.CrossRefGoogle Scholar
[3]Bojanić, R., Varma, A.K. and Vértesi, P., ‘Necessary and sufficient conditions for uniform convergence of quasi-Hermite-Fejér and extended Hermite-Fejér interpolation’, Studia Sci. Math. Hungar. 25 (1990), 107–115.Google Scholar
[4]Brutman, L., ‘Lebesgue functions for polynomial interpolation — a survey’, Ann. Numer. Math. 4 (1997), 111–127.Google Scholar
[5]Faber, G., ‘Über die interpolatorische Darstellung stetiger Funktionen’, Jahresber. Deutsch. Math.-Verein. 23 (1914), 190–210.Google Scholar
[6]Fejér, L., ‘Über interpolation’, Göttinger Nachrichten (1916), 66–91.Google Scholar
[7]Powell, M.J.D., Approximation theory and methods (Cambridge University Press, Cambridge, 1981).CrossRefGoogle Scholar
[8]Rivlin, T.J., An introduction to the approximation of functions (Dover Publications, New York, 1981).Google Scholar
[9]Rivlin, T.J., Chebyshev polynomials, Pure and Applied Mathematics, (2nd Edition) (J. Wiley & Sons, New York, 1990).Google Scholar