2021/08/09 by Xiaolong Zhang, Zhang, Xiaolong
Mathematics · #Mathematical functions and polynomials #Iterative Methods for Nonlinear Equations #Differential Equations and Numerical Methods
paper · pdf · doi:10.48550/arxiv.2108.03836
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 \citeTrefethen11sixmyths,Trefethen2020 for the algebraic singularity function. Recently Wang \citeWang2021best 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-κ) (κ= min\2γ+1, 2δ+ 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.