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The best polynomial approximation and Chebyshev approximation are bothimportant in numerical analysis. In tradition, the best approximation isregarded as more better than the Chebyshev approximation, because it is usuallyconsidered in the uniform norm. However, it not always superior to the latternoticed by Trefethen \cite{Trefethen11sixmyths,Trefethen2020} for the algebraicsingularity function. Recently Wang \cite{Wang2021best} have proved it intheory. In this paper, we find that for the functions with logarithmicregularities, the pointwise errors of Chebyshev approximation are smaller thanthe ones of the best approximations except only in the very narrow boundariesat the same degree. The pointwise error for Chebyshev series, truncated at thedegree $n$ is $O(n^{-\kappa})$ ($\kappa = \min\{2\gamma+1, 2\delta + 1\}$), butis worse by one power of $n$ in narrow boundary layer near the weak singularendpoints. Theorems are given to explain this effect.

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