Are best approximations really better than Chebyshev?

Abstract

Best and Chebyshev approximations play an important role in approximation theory. From the viewpoint of measuring approximation error in the maximum norm, it is evident that best approximations are better than their Chebyshev counterparts. However, the situation may be reversed if we compare the approximation quality from the viewpoint of either the rate of pointwise convergence or the accuracy of spectral differentiation. We show that when the underlying function has an algebraic singularity, the Chebyshev projection of degree n converges one power of n faster than its best counterpart at each point away from the singularity and both converge at the same rate at the singularity. This gives a complete explanation for the phenomenon that the accuracy of Chebyshev projections is much better than that of best approximations except in a small neighborhood of the singularity. Extensions to superconvergence points, spectral differentiation, Chebyshev interpolants and other orthogonal projections are also discussed.