As one myth of polynomial interpolation and quadrature, Trefethen [30] revealed that the Chebyshev interpolation of $|x-a|$ (with $|a|<1 $) at the Clenshaw-Curtis points exhibited a much smaller error than the best polynomial approximation (in the maximum norm) in about $95\%$ range of $[-1,1]$ except for a small neighbourhood near the singular point $x=a.$ In this paper, we rigorously show that the Jacobi expansion for a more general class of $\Phi$-functions also enjoys such a local convergence behaviour. Our assertion draws on the pointwise error estimate using the reproducing kernel of Jacobi polynomials and the Hilb-type formula on the asymptotic of the Bessel transforms. We also study the local superconvergence and show the gain in order and the subregions it occurs. As a by-product of this new argument, the undesired $\log n$-factor in the pointwise error estimate for the Legendre expansion recently stated in Babu\u{s}ka and Hakula [5] can be removed. Finally, all these estimates are extended to the functions with boundary singularities. We provide ample numerical evidences to demonstrate the optimality and sharpness of the estimates.

翻译：摘要：作为多项式插值与求积中的一个未解之谜，Trefethen [30] 揭示了在Clenshaw-Curtis点上对$|x-a|$（其中$|a|<1$）进行的切比雪夫插值，在$[-1,1]$区间约$95\%$的范围内（除奇异点$x=a$附近的一小邻域外）所产生的误差远小于最佳多项式逼近（在最大范数意义下）。本文严格证明了对于更一般的$\Phi$-函数类，雅可比展开同样具有这种局部收敛行为。我们的论证基于使用雅可比多项式再生核的点态误差估计以及关于贝塞尔变换渐近性质的Hilb型公式。我们还研究了局部超收敛性，展示了阶数提升及其发生的子区域。作为这一新论证的副产品，最近Babuškova和Hakula [5]中所述的勒让德展开点态误差估计中不期望的$\log n$因子可以被消除。最后，所有这些估计被推广至具有边界奇异性的函数。我们提供了大量数值实验以证明这些估计的最优性与精确性。