Video library: V. Totik, Bernstein and Markov inequalities on Jordan arcs

VIDEO LIBRARY


International Conference on Complex Analysis Dedicated to the memory of Andrei Gonchar and Anatoliy Vitushkin October 8, 2018 11:00, Moscow, Steklov Mathematical Institute of RAS, Conference hall, 9th floor

Bernstein and Markov inequalities on Jordan arcs V. Totik^ab ^a Bolyai Institute, University of Szeged ^b University of South Florida, Department of Mathematics
Abstract: Let $\Gamma$ be a smooth Jordan arc and $x_0\in \Gamma$ a point that is different from the endpoints of $\Gamma$. The talk will be about the smallest constant $B_{x_0}$ for which $$ \|P_n'(x_0)\|\le B_{x_0}(1+o(1))n\\|P_n\\|_\Gamma $$ for all polynomials $P_n$ of degree $n=1,2,\dots$, where $o(1)$ tends to 0 (uniformly in $P_n$) as $n\to\infty$. Thus, this $B_{x_0}$ is the asymptotically sharp Bernstein factor at the point $x_0$. It turns out that $B_{x_0}=\max (g'_+(x_0),g'_-(x_0))$, where $g$ is the Green's function of $\overline C\setminus \Gamma$ with pole at infinity, and $g'_\pm(x_0)$ are the normal derivatives of $g$ at $x_0$ with respect to the two normals to $\Gamma$ at $x_0$. The proof uses in an essential way a result of Gonchar and Grigorian on the supremum norm of the sum of the principal parts of a meromorphic function on the boundary of the given domain in terms of the supremum norm of the function itself. The asymptotically best Markov factor $M=M_\Gamma$, i.e. the smallest $M$ for which $$ \\|P_n'\\|_\Gamma \le M(1+o(1))n^2\\|P_n\\|_\Gamma $$ is true, is also expressed in terms of the normal derivative of the associated Green's function. Similar results are established for rational functions provided the poles lie in a closed set disjoint from $\Gamma$. This is a joint work with Sergei Kalmykov and Béla Nagy. Language: English