## Pointwise error estimates and local superconvergence of Jacobi expansions

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Shuhuang Xiang, Desong Kong, Guidong Liu and Li-Lian Wang
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## 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.## References

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## Additional Information

**Shuhuang Xiang**- Affiliation: School of Mathematics and Statistics, INP-LAMA, Central South University, Changsha 410083, People’s Republic of China
- ORCID: 0000-0002-6727-6170
- Email: xiangsh@csu.edu.cn
**Desong Kong**- Affiliation: School of Mathematics and Statistics, Central South University, Changsha 410083, People’s Republic of China
- MR Author ID: 1312014
- ORCID: 0000-0003-1995-8086
- Email: desongkong@csu.edu.cn
**Guidong Liu**- Affiliation: School of Statistics and Mathematics, Nanjing Audit University, Nanjing 211815, People’s Republic of China
- MR Author ID: 1237006
- Email: csu_guidongliu@163.com
**Li-Lian Wang**- Affiliation: Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore
- MR Author ID: 681795
- Email: lilian@ntu.edu.sg
- Received by editor(s): June 16, 2021
- Received by editor(s) in revised form: July 30, 2022, and November 16, 2022
- Published electronically: March 21, 2023
- Additional Notes: The 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.

The third author is the corresponding author - © Copyright 2023 American Mathematical Society
- Journal: Math. Comp.
**92**(2023), 1747-1778 - MSC (2020): Primary 41A10, 41A25, 41A50, 65N35, 65M70
- DOI: https://doi.org/10.1090/mcom/3835
- MathSciNet review: 4570340