Bernstein-type constants for approximation of |x|α by partial Fourier–Legendre and Fourier–Chebyshev sums
In this paper, we study the approximation of fα(x)=|x|α,α>0 in L∞[−1,1] by its Fourier–Legendre partial sum Sn(α)(x). We derive the upper and lower bounds of the approximation error in the L∞-norm that are valid uniformly for all n≥n0 for some n0≥1. Such an optimal L∞-estimate requires a judiciou...
| Main Authors: | , , |
|---|---|
| Other Authors: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://hdl.handle.net/10356/169935 |
| _version_ | 1811685462228598784 |
|---|---|
| author | Liu, Wenjie Wang, Li-Lian Wu, Boying |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Liu, Wenjie Wang, Li-Lian Wu, Boying |
| author_sort | Liu, Wenjie |
| collection | NTU |
| description | In this paper, we study the approximation of fα(x)=|x|α,α>0 in L∞[−1,1] by its Fourier–Legendre partial sum Sn(α)(x). We derive the upper and lower bounds of the approximation error in the L∞-norm that are valid uniformly for all n≥n0 for some n0≥1. Such an optimal L∞-estimate requires a judicious summation rule that can recover the lost half order if one uses a naive summation. Consequently, we can obtain the explicit Bernstein-type constant [Formula presented] Interestingly, using a similar argument, we can show that the Fourier–Chebyshev sum has the same Bernstein-type constant B∞(α) as the Legendre case. |
| first_indexed | 2024-10-01T04:44:54Z |
| format | Journal Article |
| id | ntu-10356/169935 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T04:44:54Z |
| publishDate | 2023 |
| record_format | dspace |
| spelling | ntu-10356/1699352023-08-15T06:29:26Z Bernstein-type constants for approximation of |x|α by partial Fourier–Legendre and Fourier–Chebyshev sums Liu, Wenjie Wang, Li-Lian Wu, Boying School of Physical and Mathematical Sciences Science::Mathematics Bernstein Constants Legendre Polynomial Expansion In this paper, we study the approximation of fα(x)=|x|α,α>0 in L∞[−1,1] by its Fourier–Legendre partial sum Sn(α)(x). We derive the upper and lower bounds of the approximation error in the L∞-norm that are valid uniformly for all n≥n0 for some n0≥1. Such an optimal L∞-estimate requires a judicious summation rule that can recover the lost half order if one uses a naive summation. Consequently, we can obtain the explicit Bernstein-type constant [Formula presented] Interestingly, using a similar argument, we can show that the Fourier–Chebyshev sum has the same Bernstein-type constant B∞(α) as the Legendre case. Ministry of Education (MOE) The research of the first author was supported by the National Natural Science Foundation of China (12271128). The research of the second author is partially supported by Singapore MOE AcRF Tier 1 Grant: RG15/21. The research of the third author was supported by the National Natural Science Foundation of China (11971131, 61873071, 51476047) and Natural Sciences Foundation of Heilongjiang Province (ZD2022A001). 2023-08-15T06:29:26Z 2023-08-15T06:29:26Z 2023 Journal Article Liu, W., Wang, L. & Wu, B. (2023). Bernstein-type constants for approximation of |x|α by partial Fourier–Legendre and Fourier–Chebyshev sums. Journal of Approximation Theory, 291, 105897-. https://dx.doi.org/10.1016/j.jat.2023.105897 0021-9045 https://hdl.handle.net/10356/169935 10.1016/j.jat.2023.105897 2-s2.0-85153045408 291 105897 en RG15/21 Journal of Approximation Theory © 2023 Elsevier Inc. All rights reserved. |
| spellingShingle | Science::Mathematics Bernstein Constants Legendre Polynomial Expansion Liu, Wenjie Wang, Li-Lian Wu, Boying Bernstein-type constants for approximation of |x|α by partial Fourier–Legendre and Fourier–Chebyshev sums |
| title | Bernstein-type constants for approximation of |x|α by partial Fourier–Legendre and Fourier–Chebyshev sums |
| title_full | Bernstein-type constants for approximation of |x|α by partial Fourier–Legendre and Fourier–Chebyshev sums |
| title_fullStr | Bernstein-type constants for approximation of |x|α by partial Fourier–Legendre and Fourier–Chebyshev sums |
| title_full_unstemmed | Bernstein-type constants for approximation of |x|α by partial Fourier–Legendre and Fourier–Chebyshev sums |
| title_short | Bernstein-type constants for approximation of |x|α by partial Fourier–Legendre and Fourier–Chebyshev sums |
| title_sort | bernstein type constants for approximation of x α by partial fourier legendre and fourier chebyshev sums |
| topic | Science::Mathematics Bernstein Constants Legendre Polynomial Expansion |
| url | https://hdl.handle.net/10356/169935 |
| work_keys_str_mv | AT liuwenjie bernsteintypeconstantsforapproximationofxabypartialfourierlegendreandfourierchebyshevsums AT wanglilian bernsteintypeconstantsforapproximationofxabypartialfourierlegendreandfourierchebyshevsums AT wuboying bernsteintypeconstantsforapproximationofxabypartialfourierlegendreandfourierchebyshevsums |