Newton Interpolation in Fejér and Chebyshev Points

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