Chebyshev spectral methods are widely used in numerical computations. When the underlying function has a singularity, it has been observed by L. N. Trefethen in 2011 that its Chebyshev interpolants exhibit an error localization property, that is, their errors in a neighborhood of the singularity are obviously larger than elsewhere. In this paper, we first present a pointwise error analysis for Chebyshev projections of functions with a singularity and prove that the rate of convergence of Chebyshev projections of degree $n$ at each point away from the singularity is one power of $n$ faster than that of at the singularity. This gives a rigorous justification for the error localization of Chebyshev projections. We then extend the framework of our analysis to Chebyshev interpolants, Chebyshev spectral differentiations and Legendre projections and justify their error localization using similar arguments. As a result, we find that Chebyshev spectral differentiations converge faster than their best counterparts except in a neighborhood of the singularity and, in the particular case where the singularity is located in the interior of interval, they converge even faster than their best counterparts in the maximum norm.

翻译：切比雪夫谱方法广泛应用于数值计算中。L. N. Trefethen于2011年观察到，当原始函数具有奇异性时，其切比雪夫插值展现出误差局部化性质，即在奇点邻域内的误差明显大于其他区域。本文首先对具有奇异性的函数的切比雪夫投影进行逐点误差分析，证明次数$n$的切比雪夫投影在远离奇点各点的收敛速度比奇点处快一个$n$的幂次，从而为切比雪夫投影的误差局部化提供了严格的理论依据。随后我们将分析框架扩展至切比雪夫插值、切比雪夫谱微分及勒让德投影，通过类似论证验证其误差局部化。研究结果发现：除奇点邻域外，切比雪夫谱微分收敛速度优于最佳逼近；当奇点位于区间内部时，其最大范数下的收敛速度甚至快于最佳逼近。