Newton interpolation in Fejér and Chebyshev points

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 $\Gamma$.