# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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## Newton interpolation in Fejér and Chebyshev pointsHTML articles powered by AMS MathViewer

by Bernd Fischer and Lothar Reichel
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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