On the Rate of Convergence of Discretization in Chebyshev Approximation

We compare complex or real linear Chebyshev approximation on a compact interval X with approximation on one of its closed subsets Y. We show that under suitable conditions the best approximation on X approaches the best approximation on X like $({\text{density of}}\,Y)^2 $. This indicates that we can obtain very good near best approximations on the interval by solving best approximation problems on subsets of the interval which contain only “small” numbers of points.