# zbMATH — the first resource for mathematics

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 $$X$$.

##### MSC:
 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) 58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
##### Keywords:
convex set; Fréchet derivative; Markov inequalities
Full Text: