## Newton interpolation in Fejér and Chebyshev points

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- by Bernd Fischer and Lothar Reichel PDF
- Math. Comp.
**53**(1989), 265-278 Request permission

## 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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## Additional Information

- © Copyright 1989 American Mathematical Society
- 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