论文标题
最好的多项式近似比Legendre投影要快多少？
How much faster does the best polynomial approximation converge than Legendre projection?
论文作者
论文摘要
我们比较了最佳多项式近似值和Legendre和Chebyshev投影的收敛行为，并得出了最大规范中的分析和可区分功能的Legendre投影的最佳收敛速率。对于分析功能，我们表明，$ n $的最佳多项式近似比$ n^{1/2} $的legendre投影要好。但是，对于分数平滑度的分段分析函数和功能，我们表明，最佳近似值仅比某些恒定因素更好。我们的结果为Legendre预测的近似能力提供了一些新的见解。
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 smoothness, however, we show that the best approximation is better than the Legendre projection by only some constant factors. Our results provide some new insights into the approximation power of Legendre projections.
