Sharp error bounds for Jacobi expansions and Gegenbauer--Gauss quadrature of analytic functions
This paper provides a rigorous and delicate analysis for exponential decay of Jacobi polynomial expansions of analytic functions associated with the Bernstein ellipse. Using an argument that can recover the best estimate for the Chebyshev expansion, we derive various new and sharp bounds of the expa...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Journal Article |
| Language: | English |
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2013
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| Online Access: | https://hdl.handle.net/10356/101170 http://hdl.handle.net/10220/18308 |
| _version_ | 1811688565071937536 |
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| author | Zhao, Xiaodan Wang, Li-Lian Xie, Ziqing |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Zhao, Xiaodan Wang, Li-Lian Xie, Ziqing |
| author_sort | Zhao, Xiaodan |
| collection | NTU |
| description | This paper provides a rigorous and delicate analysis for exponential decay of Jacobi polynomial expansions of analytic functions associated with the Bernstein ellipse. Using an argument that can recover the best estimate for the Chebyshev expansion, we derive various new and sharp bounds of the expansion coefficients, which are featured with explicit dependence of all related parameters and valid for degree $n\ge 1$. We demonstrate the sharpness of the estimates by comparing with existing ones, in particular, the very recent results in SIAM J. Numer. Anal., 50 (2012), pp. 1240--1263. We also extend this argument to estimate the Gegenbauer--Gauss quadrature remainder of analytic functions, which leads to some new tight bounds for quadrature errors. |
| first_indexed | 2024-10-01T05:34:13Z |
| format | Journal Article |
| id | ntu-10356/101170 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T05:34:13Z |
| publishDate | 2013 |
| record_format | dspace |
| spelling | ntu-10356/1011702023-02-28T19:30:51Z Sharp error bounds for Jacobi expansions and Gegenbauer--Gauss quadrature of analytic functions Zhao, Xiaodan Wang, Li-Lian Xie, Ziqing School of Physical and Mathematical Sciences DRNTU::Science::Mathematics::Applied mathematics::Numerical analysis This paper provides a rigorous and delicate analysis for exponential decay of Jacobi polynomial expansions of analytic functions associated with the Bernstein ellipse. Using an argument that can recover the best estimate for the Chebyshev expansion, we derive various new and sharp bounds of the expansion coefficients, which are featured with explicit dependence of all related parameters and valid for degree $n\ge 1$. We demonstrate the sharpness of the estimates by comparing with existing ones, in particular, the very recent results in SIAM J. Numer. Anal., 50 (2012), pp. 1240--1263. We also extend this argument to estimate the Gegenbauer--Gauss quadrature remainder of analytic functions, which leads to some new tight bounds for quadrature errors. Published version 2013-12-18T04:19:58Z 2019-12-06T20:34:34Z 2013-12-18T04:19:58Z 2019-12-06T20:34:34Z 2013 2013 Journal Article Zhao, X., Wang, L.-L., & Xie, Z. (2013). Sharp error bounds for Jacobi expansions and Gegenbauer--Gauss quadrature of analytic functions. SIAM journal on numerical analysis, 51(3), 1443-1469. 0036-1429 https://hdl.handle.net/10356/101170 http://hdl.handle.net/10220/18308 10.1137/12089421X en SIAM Journal on Numerical Analysis © 2013 Society for Industrial and Applied Mathematics. This paper was published in SIAM Journal on Numerical Analysis and is made available as an electronic reprint (preprint) with permission of Society for Industrial and Applied Mathematics. The paper can be found at the following official DOI: http://dx.doi.org/10.1137/12089421X. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law. application/pdf |
| spellingShingle | DRNTU::Science::Mathematics::Applied mathematics::Numerical analysis Zhao, Xiaodan Wang, Li-Lian Xie, Ziqing Sharp error bounds for Jacobi expansions and Gegenbauer--Gauss quadrature of analytic functions |
| title | Sharp error bounds for Jacobi expansions and Gegenbauer--Gauss quadrature of analytic functions |
| title_full | Sharp error bounds for Jacobi expansions and Gegenbauer--Gauss quadrature of analytic functions |
| title_fullStr | Sharp error bounds for Jacobi expansions and Gegenbauer--Gauss quadrature of analytic functions |
| title_full_unstemmed | Sharp error bounds for Jacobi expansions and Gegenbauer--Gauss quadrature of analytic functions |
| title_short | Sharp error bounds for Jacobi expansions and Gegenbauer--Gauss quadrature of analytic functions |
| title_sort | sharp error bounds for jacobi expansions and gegenbauer gauss quadrature of analytic functions |
| topic | DRNTU::Science::Mathematics::Applied mathematics::Numerical analysis |
| url | https://hdl.handle.net/10356/101170 http://hdl.handle.net/10220/18308 |
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