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Analogues of the Markov and Bernstein inequalities for polynomials in Banach spaces. (English. Russian original) Zbl 0889.41008
Izv. Math. 61, No. 1, 143-159 (1997); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 61, No. 1, 141-156 (1997).
Let \(P_n\) be a polynomial mapping from a Banach space \(X\) into a Banach space \(Y\). The author proves for the first Fréchet derivative \(P_n'\) the following generalizations of the classical Bernstein and Markov inequalities \[ \sup_{x\in K} \bigl|P_n'(x) \bigr|_{L(x,y)} \leq{4n^2 \over r(K)} \sup_{x\in K} \bigl|P_n (x)|_y \] \[ \bigl |P_n'(x) \bigr|_{L(x,y)} \leq b_K(x)n \sup_{x\in K} \bigl|P_n(x) \bigr|_y,\;x\in K \setminus \partial K. \] Here \[ r(K): =\inf_{z\in X\setminus (\;)}\sup_{{\begin{smallmatrix} y_1,y_2 \in K \\ y_1-y_2 =cz \\ c\in\mathbb{R} \end{smallmatrix}}} |y_1- y_2|_X, \] \({1\over b_k(x)}\) is a kind of “Bernstein’s distance” (equals \(\sqrt{(b-x) (x-a)}\) for \(k=[a,b] \subset \mathbb{R})\), and \(K\) is a bounded closed convex set in \(X\).

41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
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