Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 1258.41007
Totik, Vilmos
Bernstein-type inequalities. (English)
[J] J. Approx. Theory 164, No. 10, 1390-1401 (2012). ISSN 0021-9045

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 Szeg\H{o} variant as well as both Videnskii's inequality and its half-integer variant.
[Yuri A. Farkov (Moscow)]

MSC 2000:: *41A17 Inequalities in approximation

Keywords: Bernstein-inequalities; Szeg\H{o}-inequalities; Videnskii-inequalities; equilibrium measures

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]