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

The best polynomial approximation and the Chebyshev approximation are both impor-tant 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 log-arithmic 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(-?)) (? = min{2? + 1, 2d + 1}), but increases as O(n(-(?-1))) at the endpoint singularities. Theorems are given to explain this effect.

Automated summary

Both the best polynomial approximation and the Chebyshev approximation play important roles in numerical analysis. While the best approximation was traditionally considered superior to the Chebyshev approximation in the uniform norm, recent studies have shown that this is not always the case, especially for functions with singularities. This paper presents findings regarding functions with logarithmic endpoint singularities, showing that the pointwise errors of the Chebyshev approximation are smaller than those of the best approximation of the same degree, except at the narrow boundary layer. The paper also provides theorems to explain this phenomenon.