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Bernstein-type inequalities. (English) Zbl 1258.41007
Let $E\subset [-\pi,\pi]$ be compact and symmetric with respect to the origin and let $\Gamma_E = \{ e^{it}\,| \ t\in E \}$. Denote by $\omega_{\Gamma_E}$ the equilibrium density of $\Gamma_E$ on the unit circle ${\Bbb T}$. 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 $$|T'_n(\theta)| \leq n2\pi\omega_{\Gamma_E}(e^{i\theta})\|T_n\|_E. \eqno (1)$$ This Bernstein-type inequality is sharp. In particular, if $E=[-\beta, -\alpha]\cap [\alpha, \beta]$ with some $0\leq \alpha <\beta \leq \pi$, then $$\omega_{\Gamma_E}(e^{i\theta})=\frac{1}{2\pi}\frac{|\sin\theta|}{\sqrt{|\cos\theta - \cos\alpha||\cos\theta - \cos\beta|}}.$$ So for $E=[-\beta, \beta]$, 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)
Full Text:
References:
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