A discrete weighted Markov--Bernstein inequality for polynomials and sequences

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 $$ \| f \|_{c,\beta}^2 := \sum_{k=0}^{\infty}\frac{(\beta)_k}{k!}\,c^k\,[f(k)]^2<\infty\,. $$ We study the best (i.e., the smallest possible) constant $\,\gamma_n(c,\beta)\,$ in the discrete Markov-Bernstein inequality $$ \|\Delta P\|_{c,\beta}\leq \gamma_n(c,\beta)\,\|P\|_{c,\beta}\,,\quad P\in\mathcal{P}_n\,, $$ 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) $\displaystyle \gamma_n(c,1)\leq 1+\frac{1}{\sqrt{c}}\,$ for every $\,n\in \mathbb{N}\,$ and $\displaystyle \lim_{n\to\infty}\gamma_n(c,1)= 1+\frac{1}{\sqrt{c}}\,$. (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 unequality is proved for sequences in $\,\ell_{2}(c,\beta)\,$. We also establish a relation between the best Markov-Bernstein constants $\,\gamma_n(c,\beta)\,$ and the smallest eigenvalues of certain explicitly given Jacobi matrices.