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dc.contributor.authorXiang, Shuhuangen_US
dc.contributor.authorKong, Desongen_US
dc.contributor.authorLiu, Guidongen_US
dc.contributor.authorWang, Li-Lianen_US
dc.identifier.citationXiang, S., Kong, D., Liu, G. & Wang, L. (2023). Pointwise error estimates and local superconvergence of Jacobi expansions. Mathematics of Computation, 92(342), 1747-1778.
dc.description.abstractAs 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 estimatesen_US
dc.description.sponsorshipMinistry of Education (MOE)en_US
dc.relation.ispartofMathematics of Computationen_US
dc.rights© 2023 American Mathematical Society. All rights reserved.en_US
dc.titlePointwise error estimates and local superconvergence of Jacobi expansionsen_US
dc.typeJournal Articleen
dc.contributor.schoolSchool of Physical and Mathematical Sciencesen_US
dc.subject.keywordsPointwise Error Analysisen_US
dc.description.acknowledgementThe research of the first three authors was supported in part by the National Natural Foundation of China (No. 12271528 and No. 12001280). The research of the second author was supported in part by the Fundamental Research Funds for the Central Universities of Central South University (No. 2020zzts031). The research of the third author was supported in part by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 20KJB110012), and the Natural Science Foundation of Jiangsu Province (No. BK20211293). The research of the fourth author was supported in part by the Singapore MOE AcRF Tier 1 Grant: RG15/21.en_US
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