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Title: Pointwise error estimates and local superconvergence of Jacobi expansions
Authors: Xiang, Shuhuang
Kong, Desong
Liu, Guidong
Wang, Li-Lian
Keywords: Science::Mathematics
Issue Date: 2023
Source: Xiang, S., Kong, D., Liu, G. & Wang, L. (2023). Pointwise error estimates and local superconvergence of Jacobi expansions. Mathematics of Computation, 92(342), 1747-1778.
Project: RG15/21
Journal: Mathematics of Computation
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 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 Babuška 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
ISSN: 0025-5718
DOI: 10.1090/mcom/3835
Schools: School of Physical and Mathematical Sciences 
Rights: © 2023 American Mathematical Society. All rights reserved.
Fulltext Permission: none
Fulltext Availability: No Fulltext
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