Exact constant in the Dzyadyk inequality for the derivative of an algebraic polynomial
For natural $k$ and $n \geq 2k$, we determine the exact constant $c(n, k)$ in the Dzyadyk inequality $$|| P^{\prime}_n\varphi^{1-k}_n ||_{C[ 1,1]} \leq c(n, k)n\| P_n\varphi^{-k}_n \|_{C[ 1,1]}$$ for the derivative $P^{\prime}_n$ of an algebraic polynomial $P_n$ of degree $\leq n$, where $$\varph...
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| Datum: | 2017 |
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| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Ukrainisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2017
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/1721 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | For natural $k$ and $n \geq 2k$, we determine the exact constant $c(n, k)$ in the Dzyadyk inequality
$$|| P^{\prime}_n\varphi^{1-k}_n ||_{C[ 1,1]} \leq c(n, k)n\| P_n\varphi^{-k}_n \|_{C[ 1,1]}$$
for the derivative $P^{\prime}_n$ of an algebraic polynomial $P_n$ of degree $\leq n$, where
$$\varphi_n(x) := \sqrt{n^{-2} + 1 - x_2,} .$$
Namely,
$$c(n, k) = \biggl( 1 + k \frac{\sqrt{ 1 + n^2} - 1}{n}
\biggr)^2 - k.$$ |
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