Convergence of interpolation in polynomial Chebyshev approximation

The quality of a polynomial approximation on an interval to a functionf is considered as a function of its points of interpolation. Iff satisfies a Lipschitz condition of order 1, the quality depends linearly on the distance of the points of interpolation from an optimal interpolating point set: further restrictions onf still give only linear dependence. This suggests that algorithms based on interpolation are inferior to algorithms based on error extrema (such as the Remes algorithm).