Pointwise error estimates and local superconvergence of Jacobi expansions

Xiang, Shuhuang; Kong, Desong; Liu, Guidong; Wang, Li-Lian

doi:10.1090/mcom/3835

Pointwise error estimates and local superconvergence of Jacobi expansions
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by Shuhuang Xiang, Desong Kong, Guidong Liu and Li-Lian Wang HTML | PDF

Math. Comp. 92 (2023), 1747-1778 Request permission

Abstract:

As one myth of polynomial interpolation and quadrature, Trefethen [Math. Today (Southend-on-Sea) 47 (2011), pp. 184–188] 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 Babus̆ka and Hakula [Comput. Methods Appl. Mech Engrg. 345 (2019), pp. 748–773] 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.

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