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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: S.Smith@bendigo.latrobe.edu.au
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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

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