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arXiv · 1705.03824

Markov $L_2$-inequality with the Laguerre weight

Geno Nikolov·Alexei Shadrin
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Abstract

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

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Geno Nikolov, Alexei Shadrin. 2017-05-10. Markov $L_2$-inequality with the Laguerre weight. https://arxiv.org/abs/1705.03824

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