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

  • Simon J. Smith (a1)

Abstract

Given fC[−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.

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References

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[1]Berman, D.L., ‘A study of the Hermite-Fejér interpolation process’, Dokl. Akad. Nauk USSR 187 (1969), 241244; (in Russian)(Soviet Math. Dokl. 10 (1969), 813–816).
[2]Bojanić, R., ‘Necessary and sufficient conditions for the convergence of the extended Hermite-Fejér interpolation process’, Acta Math. Acad. Sci. Hungar. 36 (1980), 271279.
[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), 107115.
[4]Brutman, L., ‘Lebesgue functions for polynomial interpolation — a survey’, Ann. Numer. Math. 4 (1997), 111127.
[5]Faber, G., ‘Über die interpolatorische Darstellung stetiger Funktionen’, Jahresber. Deutsch. Math.-Verein. 23 (1914), 190210.
[6]Fejér, L., ‘Über interpolation’, Göttinger Nachrichten (1916), 6691.
[7]Powell, M.J.D., Approximation theory and methods (Cambridge University Press, Cambridge, 1981).
[8]Rivlin, T.J., An introduction to the approximation of functions (Dover Publications, New York, 1981).
[9]Rivlin, T.J., Chebyshev polynomials, Pure and Applied Mathematics, (2nd Edition) (J. Wiley & Sons, New York, 1990).
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The Lebesgue function for Hermite-Fejér interpolation on the extended Chebyshev nodes

  • Simon J. Smith (a1)

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