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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Newton interpolation in Fejér and Chebyshev points

Authors: Bernd Fischer and Lothar Reichel
Journal: Math. Comp. 53 (1989), 265-278
MSC: Primary 65D05; Secondary 30E10, 65E05
MathSciNet review: 969487
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Abstract: Let $ \Gamma $ be a Jordan curve in the complex plane, and let $ \Omega $ be the compact set bounded by $ \Gamma $. Let f denote a function analytic on $ \Omega $. We consider the approximation of f on $ \Omega $ by a polynomial p of degree less than n that interpolates f in n points on $ \Gamma $. A convenient way to compute such a polynomial is provided by the Newton interpolation formula. This formula allows the addition of one interpolation point at a time until an interpolation polynomial p is obtained which approximates f sufficiently accurately. We choose the sets of interpolation points to be subsets of sets of Fejér points. The interpolation points are ordered using van der Corput's sequence, which ensures that p converges uniformly and maximally to f on $ \Omega $ as n increases. We show that p is fairly insensitive to perturbations of f if $ \Gamma $ is smooth and is scaled to have capacity one. If $ \Gamma $ is an interval, then the Fejér points become Chebyshev points. This special case is also considered. A further application of the interpolation scheme is the computation of an analytic continuation of f in the exterior of $ \Gamma $.

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Keywords: Polynomial interpolation, Newton form, Fejér points, Chebyshev points, van der Corput's sequence, complex approximation, analytic continuation
Article copyright: © Copyright 1989 American Mathematical Society