Markov $L_2$-inequality with the Laguerre weight (1705.03824v1)

Abstract: Let $w_\alpha(t) := t^{\alpha}\,e^{-t}$, where $\alpha > -1$, be the Laguerre weight function, and let $|\cdot|{w\alpha}$ be the associated $L_2$-norm, $$ |f|{w\alpha} = \left{\int_{0}^{\infty} |f(x)|^2 w_\alpha(x)\,dx\right}^{1/2}\,. $$ By $\mathcal{P}n$ we denote the set of algebraic polynomials of degree $\le n$. We study the best constant $c_n(\alpha)$ in the Markov inequality in this norm $$ |p_n'|{w_\alpha} \le c_n(\alpha) |p_n|{w\alpha}\,,\qquad p_n \in \mathcal{P}n\,, $$ namely the constant $$ c_n(\alpha) := \sup{p_n \in \mathcal{P}n} \frac{|p_n'|{w_\alpha}}{|p_n|{w\alpha}}\,. $$ We derive explicit lower and upper bounds for the Markov constant $c_n(\alpha)$, as well as for the asymptotic Markov constant $$ c(\alpha)=\lim_{n\rightarrow\infty}\frac{c_n(\alpha)}{n}\,. $$