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Bernstein-type inequalities. (English) Zbl 1258.41007

Let $E\subset \left[-\pi ,\pi \right]$ be compact and symmetric with respect to the origin and let ${{\Gamma }}_{E}=\left\{{e}^{it}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{4pt}{0ex}}t\in E\right\}$. Denote by ${\omega }_{{{\Gamma }}_{E}}$ the equilibrium density of ${{\Gamma }}_{E}$ on the unit circle $𝕋$. It is proved that if $\theta \in E$ is an inner point of $E$ then for any trigonometric polynomial ${T}_{n}$ of degree at most $n$ we have

$|{T}_{n}^{\text{'}}\left(\theta \right)|\le n2\pi {\omega }_{{{\Gamma }}_{E}}\left({e}^{i\theta }\right){\parallel {T}_{n}\parallel }_{E}·\phantom{\rule{2.em}{0ex}}\left(1\right)$

This Bernstein-type inequality is sharp. In particular, if $E=\left[-\beta ,-\alpha \right]\cap \left[\alpha ,\beta \right]$ with some $0\le \alpha <\beta \le \pi$, then

${\omega }_{{{\Gamma }}_{E}}\left({e}^{i\theta }\right)=\frac{1}{2\pi }\frac{|sin\theta |}{\sqrt{|cos\theta -cos\alpha ||cos\theta -cos\beta |}}·$

So for $E=\left[-\beta ,\beta \right]$, he gets from (1) the Videnskii inequality. Also, it is shown that the original Bernstein inequality implies its Szegő variant as well as both Videnskii’s inequality and its half-integer variant.

##### MSC:
 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
##### References:
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