How to Deal With Interpolation Points That Are Not Well-Spaced in Polynomial and Rational Interpolation

We discuss the growth rate of the Lebesgue constant of polynomial Lagrange interpolation and barycentric rational interpolation with pre-assigned poles. Its magnitude is of importance when evaluating the conditioning of the interpolation problems under investigation or answering the question whether the extra work to calculate the best polynomial or rational approximant isworthwhile. Since the growth rate of the Lebesgue constant has such important numerical consequences, the problem remains of course what to do if the interpolation points are not at the ideal positions. In this paper we tackle this problem and propose a possible alternative approach in that situation. The technique is illustrated on several numerical examples.