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

DOI:
https://doi.org/10.1090/S0025-5718-1989-0969487-3

MathSciNet review:
969487

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a Jordan curve in the complex plane, and let be the compact set bounded by . Let *f* denote a function analytic on . We consider the approximation of *f* on by a polynomial *p* of degree less than *n* that interpolates *f* in *n* points on . 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 as *n* increases. We show that *p* is fairly insensitive to perturbations of *f* if is smooth and is scaled to have capacity one. If 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 .

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1989-0969487-3

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