Fine-grained evaluations of the best estimates for smooth functions in $C_{2\pi}$ in terms of linear combinations of modules of continuity of their derivatives
UDC 517.5 For the best approximations of $e_{n-1}(f)$ functions in $C^1_{2\pi}$ by trigonometric polynomials, Zhuk proved the exact Jackson inequality $e_{n-1}(f)\leqslant \dfrac{\pi}{4n}\omega\left(f',\dfrac{\pi}{n}\right)$. In this paper, we prove the following version of Jackson'...
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| Дата: | 2022 |
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| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Українська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2022
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/7124 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Резюме: | UDC 517.5
For the best approximations of $e_{n-1}(f)$ functions in $C^1_{2\pi}$ by trigonometric polynomials, Zhuk proved the exact Jackson inequality $e_{n-1}(f)\leqslant \dfrac{\pi}{4n}\omega\left(f',\dfrac{\pi}{n}\right)$. In this paper, we prove the following version of Jackson's exact inequality: $e_{n-1}(f)\leqslant \dfrac{\pi}{4n}\left(\dfrac{1}{2}\omega\left(f',\dfrac{\pi}{2n}\right)+\dfrac{1}{2}\omega\left(f',\dfrac{\pi}{n}\right)\right)$. |
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| DOI: | 10.37863/umzh.v74i4.7124 |