Pointwise error estimates and local superconvergence of Jacobi expansions
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...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://hdl.handle.net/10356/171812 |
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| Institution: | Nanyang Technological University |
| Language: | English |
| Summary: | 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 |
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