V. A. Markov’s theorems in normed spaces. (English. Russian original) Zbl 1155.41004

Skalyga, V. I.

Izv. Math. 72, No. 2, 383-412 (2008); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 2, 193-222 (2008).

Let \(X\) be a normed vector space over \(\mathbb{R}\) and let \(K\) be a convex bounded closed body symmetric with respect to the zero of \(X\). By \(\operatorname{Re} _n(X,\mathbb{R})\) is denoted the space of all polynomials of degree \(\leq n\) that map \(X\) to \(\mathbb{R}\). The value of the \(k\)-th Fréchet derivative of a polynomial \(P_n\in \operatorname{Re} _n(X,\mathbb{R})\) at \(x\in X\) at the vector cortége \((h_1,\cdots ,h_k)\) will be denoted by \(P_n^{(k)}(x)[h_1,\cdots ,h_k]\). Upper bounds for \(|P_n^{(k)}(x)[h_1,\cdots ,h_k]|\) expressed in terms of some characteristics of the body \(K\) and of the value of the \(k\)-th derivative for the first kind Chebyshev polynomial of order \(n\) in the specific points are given. These estimates are multidimensional analogous of well-known Markov’s inequalities for algebraic polynomials. The special attention is paid to the study of the case \(X=\mathbb{R}^N\). For polynomials on multidimensional cubes an analogue of the Schaeffer-Duffin inequality is obtained. Moreover necessary and sufficient conditions for these inequalities to become equalities are established and the sets of extremals are described. The obtained results unify and extend some earlier results by the author and also by the others.

Reviewer: D. K. Ugulawa (Tbilisi)

MSC:

41A17	Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
41A63	Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
26D10	Inequalities involving derivatives and differential and integral operators

Keywords:

Markov’s inequality; Schaeffer-Duffin inequality; normed space; polynomial; extremal problem; derivative