On the $L_2$ Markov Inequality with Laguerre Weight

TL;DR

This paper investigates the best constants in the $L_2$ Markov inequality with Laguerre weight, providing bounds for these constants and their asymptotic behavior, which relate to zeros of Bessel functions.

Contribution

It derives new bounds for the Markov inequality constants in the Laguerre-weighted $L_2$ norm and explores their asymptotic limits, connecting to Bessel function zeros.

Findings

Bounds for $c_n( ext{alpha})$ and $c( ext{alpha})$ are established.

Asymptotic behavior of the constants is characterized.

Connections to zeros of Bessel functions are discussed.

Abstract

Let $w_{\alpha}(t)=t^{\alpha}\,e^{-t}$, $\alpha>-1$, be the Laguerre weight function, and $|\cdot|_{w_\alpha}$ denote the associated $L_2$-norm, i.e., $$ | f|_{w_\alpha}:=\Big(\int_{0}^{\infty}w_{\alpha}(t)| f(t)|^2\,dt\Big)^{1/2}. $$ Denote by ${\cal P}_n$ the set of algebraic polynomials of degree not exceeding $n$. We study the best constant $c_n(\alpha)$ in the Markov inequality in this norm, $$ | p^{\prime}|_{w_\alpha}\leq c_n(\alpha)\,| p|_{w_\alpha}\,,\quad p\in {\cal P}_n\,, $$ namely the constant $$ c_{n}(\alpha)=\sup_{\mathop{}^{p\in {\cal P}_n}_{p\ne 0}}\frac{| p^{\prime}|_{w_\alpha}}{| p|_{w_\alpha}}\,, $$ and we are also interested in its asymptotic value $$ c(\alpha)=\lim_{n\rightarrow\infty}\frac{c_{n}(\alpha)}{n}\,. $$ In this paper we obtain lower and upper bounds for both $c_{n}(\alpha)$ and $c(\alpha)$. % Note that according to a result of P. D\"{o}rfler from 2002, $c(\alpha)=[j_{(\alpha-1)/2,1}]^{-1}$, with $j_{\nu,1}$ being the first positive zero of the Bessel function $J_{\nu}(z)$, hence our bounds for $c(\alpha)$ imply bounds for $j_{(\alpha-1)/2,1}$ as well.