Best and Chebyshev approximations play an important role in approximation theory. From the viewpoint of measuring approximation error in the maximum norm, it is evident that best approximations are better than their Chebyshev counterparts. However, the situation may be reversed if we compare the approximation quality from the viewpoint of either the rate of pointwise convergence or the accuracy of spectral differentiation. We show that when the underlying function has an algebraic singularity, the Chebyshev projection of degree n converges one power of n faster than its best counterpart at each point away from the singularity and both converge at the same rate at the singularity. This gives a complete explanation for the phenomenon that the accuracy of Chebyshev projections is much better than that of best approximations except in a small neighborhood of the singularity. Extensions to superconvergence points and spectral differentiation, Chebyshev interpolants and other orthogonal projections are also discussed.
翻译:最佳和切比谢夫近似值在近似理论中起着重要作用。 从衡量最高规范近近误的角度看, 最优近近似明显优于其恰比谢夫对等方。 但是, 如果我们从点趋同率或光谱差异的准确性的角度比较近近近近质量, 情况可能会发生逆转。 我们表明, 当基本函数具有代数奇特性时, 切比谢夫对数值 n 的预测会比其最佳对应方在每一个点上的总和快一个 n 的功率要快, 并且两者都以同一的奇异性相趋同。 这充分解释了Chebyshev 预测的准确性比最佳近近近似值的准确性要好得多的现象, 唯一性的小区除外 。 也讨论了超相趋近点和光谱差异的延伸, 切比谢夫 中间线和其他直线预测 。