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A discrete weighted Markov-Bernstein inequality for sequences and polynomials. (English) Zbl 07265514
Authors’ abstract: “For parameters $$c \in(0, 1)$$ and $$\beta > 0$$, let $$\ell_2(c, \beta)$$ be the Hilbert space of real functions defined on $$\mathbb{N}$$ (i.e., real sequences), for which $\| f \|_{c , \beta}^2 : = \sum\limits_{k = 0}^\infty \frac{ ( \beta )_k}{ k !} c^k [ f ( k ) ]^2 < \infty .$ We study the best (i.e., the smallest possible) constant $$\gamma_n(c, \beta)$$ in the discrete Markov-Bernstein inequality $\| {\Delta} P \|_{c , \beta} \leq \gamma_n(c, \beta) \| P \|_{c , \beta}, \;\;\; P \in \mathcal{P}_n,$ where $$\mathcal{P}_n$$ is the set of real algebraic polynomials of degree at most $$n$$ and $${\Delta} f(x) : = f(x + 1) - f(x)$$.
We prove that
(i)
$$\gamma_n(c, 1) \leq 1 + \frac{ 1}{ \sqrt{ c}}$$ for every $$n \in \mathbb{N}$$ and $$\lim\limits_{n \to \infty} \gamma_n(c, 1) = 1 + \frac{ 1}{ \sqrt{ c}}$$;
(ii)
For every fixed $$c \in(0, 1), \gamma_n(c, \beta)$$ is a monotonically decreasing function of $$\beta$$ in $$(0, \infty)$$;
(iii)
For every fixed $$c \in(0, 1)$$ and $$\beta > 0$$, the best Markov-Bernstein constants $$\gamma_n(c, \beta)$$ are bounded uniformly with respect to $$n$$.
A similar Markov-Bernstein inequality is proved for sequences, and a relation between the best Markov-Bernstein constants $$\gamma_n(c, \beta)$$ and the smallest eigenvalues of certain explicitly given Jacobi matrices is established.”
Added by reviewer: There is an interesting section “Comments” where the authors indicate several related unsolved problems and discuss directions for further investigations.
##### MSC:
 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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