Comparisons of best approximations with Chebyshev expansions for functions with logarithmic endpoint singularities

Abstract

The best polynomial approximation and the Chebyshev approximation are both important in numerical analysis. In tradition, the best approximation is regarded as better than the Chebyshev approximation, because it is usually considered in the uniform norm. However, it is not always superior to the latter, as noticed by Trefethen (Math. Today 47:184–188, 2011) for the function $f(x) = |x-0.25|$. Recently, Wang (arxiv, 2106.03456, 2021) observed a similar phenomenon for a function with an algebraic singularity and proved it in theory. In this paper, we find that for functions with logarithmic endpoint singularities, the pointwise errors of the Chebyshev approximation are smaller than those of the best approximations of the same degree, except only at the narrow boundary layer. The pointwise error for the Chebyshev series truncated at the degree n is $O(n^{-\kappa })$ ($\kappa = \min \{2\gamma +1, 2\delta + 1\}$), but increases as $\mathcal {O}(n^{-(\kappa -1)})$ at the endpoint singularities. Theorems are given to explain this effect.