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Analogues of the Markov and Bernstein inequalities on convex bodies in Banach spaces. (English. Russian original) Zbl 0926.41009
Izv. Math. 62, No. 2, 375-397 (1998); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 62, No. 2, 169-192 (1998).
Let $$X$$ be an arbitrary real Banach space and let $$\mathcal P_n(X,\mathbb R)$$ denote the set of all polynomials of degree at most $$n$$ from $$X$$ to $$\mathbb R$$. If $$K$$ is a bounded subset of $$X$$ and $$P\in\mathcal P_n(X,\mathbb R)$$, let $$\| P\| _K:=\sup \limits_{x\in X}| P(x)|$$ and $$\| P'\| _K:=\sup\limits_{x\in K}\| P'(x)\| _ {L(X,\mathbb R)}$$, where $$P'(x)$$ is the Frechét derivative of $$P$$ at the point $$x$$. In this interesting paper, the author estabishes the exact Markov type inequality $$\| P'\| _K\leq n^2/r(K)\| P\| _K$$ on the class of all centrally symmetric bounded closed convex bodies in $$X$$. In particular, this generalizes M. Baran’s [Ann. Pol. Math. 60, No. 1, 69-79 (1994; Zbl 0824.41014)] and L. Białas-Cież and P. Goetgheluck’s [EAST J. Approx. 1, 379-389 (1995; Zbl 0846.41013)] results obtained in the case of a finite-dimensional Euclidean space.
MSC:
 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) 26D05 Inequalities for trigonometric functions and polynomials
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