How much faster does the best polynomial approximation converge than Legendre projection?

@article{Wang2021HowMF,
title={How much faster does the best polynomial approximation converge than Legendre projection?},
author={Haiyong Wang},
journal={Numerische Mathematik},
year={2021},
volume={147},
pages={481-503}
}
• Haiyong Wang
• Published 7 January 2020
• Mathematics
• Numerische Mathematik
We compare the convergence behavior of best polynomial approximations and Legendre and Chebyshev projections and derive optimal rates of convergence of Legendre projections for analytic and differentiable functions in the maximum norm. For analytic functions, we show that the best polynomial approximation of degree $n$ is better than the Legendre projection of the same degree by a factor of $n^{1/2}$. For differentiable functions such as piecewise analytic functions and functions of fractional…
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