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.

我们比较了最佳多项式逼近与Legendre和Chebyshev投影的收敛行为，并针对最大范数中的解析函数和微分函数得出了Legendre投影的最优收敛速率。对于解析函数，我们表明，阶数n的最佳多项式逼近要比相同阶数的Legendre投影好一个\（n ^ {1/2} \）。但是，对于分段分析函数和分数平滑函数之类的微分函数，我们证明，仅通过某些常数因子，最佳逼近度优于勒让德投影。我们的结果为勒让德投影的逼近能力提供了一些新见解。