The mysteries of the best approximation and Chebyshev expansion for the function with logarithmic regularities

The best polynomial approximation and Chebyshev approximation are both important in numerical analysis. In tradition, the best approximation is regarded as more better than the Chebyshev approximation, because it is usually considered in the uniform norm. However, it not always superior to the latter noticed by Trefethen \cite{Trefethen11sixmyths,Trefethen2020} for the algebraic singularity function. Recently Wang \cite{Wang2021best} have proved it in theory. In this paper, we find that for the functions with logarithmic regularities, the pointwise errors of Chebyshev approximation are smaller than the ones of the best approximations except only in the very narrow boundaries at the same degree. The pointwise error for Chebyshev series, truncated at the degree $n$ is $O(n^{-\kappa})$ ($\kappa = \min\{2\gamma+1, 2\delta + 1\}$), but is worse by one power of $n$ in narrow boundary layer near the weak singular endpoints. Theorems are given to explain this effect.