Good Interpolation Points: Learning from Chebyshev, Fekete, Haar and Lebesgue

Abstract

The search for sets of good interpolation points is highly motivated by the fact that, due to the finite precision of digital computers, valid results can only be expected when the interpolation problem is well‐conditioned. The conditioning of polynomial interpolation and of rational interpolation with preassigned poles is measured by the respective Lebesgue constants. Here we summarize the main results with respect to the Lebesgue constant for polynomial interpolation and we present the best Lebesgue constants in existence for rational interpolation with preassigned poles. The new results are based on a fairly unknown rational analogue of the Chebyshev orthogonal polynomials. We compare with the results obtained in [1] and [2].