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 $$\Vert f{\Vert }_{c,\beta }^2:=\sum _{k=0}^{\infty }\frac{(\beta {)}_k}{k!}{c}^k[f(k){]}^2<\infty .$$ We study the best (i.e., the smallest possible) constant ${\gamma }_n(c,\beta )$ in the discrete Markov-Bernstein inequality $$\Vert \Delta P{\Vert }_{c,\beta }\le {\gamma }_n(c,\beta )\Vert P{\Vert }_{c,\beta },P\in {\mathcal{P}}_n,$$ 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) ${\gamma }_n(c,1)\le 1+\frac{1}{\sqrt{c}}$ for every $n\in \mathbb{N}$ and $\underset{n\to \infty }{\lim }{\gamma }_n(c,1)=1+\frac{1}{\sqrt{c}}$. (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.

15 pages