2023/02/12 by Xiang, Shuhuang, Kong, Desong, Liu, Guidong +1
#41A10 #41A25 #41A50 #65M70 #65N35 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2302.05824
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 Φ-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 Babuuska 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.