On the L ₂ Markov Inequality with Laguerre Weight

Abstract

Let w _α (t) = t ^α e ^−t, α > −1, be the Laguerre weight function, and $\Vert \cdot \Vert _{w_{\alpha }}$ denote the associated L ₂-norm, i.e.,

$$\displaystyle{ \Vert f\Vert _{w_{\alpha }}:=\Big (\int _{0}^{\infty }w_{\alpha }(t)\vert f(t)\vert ^{2}\,dt\Big)^{1/2}. }$$

Denote by $\mathcal{P}_{n}$ the set of algebraic polynomials of degree not exceeding n. We study the best constant c _n (α) in the Markov inequality in this norm,

$$\displaystyle{ \Vert p^{{\prime}}\Vert _{ w_{\alpha }} \leq c_{n}(\alpha )\,\Vert p\Vert _{w_{\alpha }}\,,\quad p \in \mathcal{P}_{n}\,, }$$

namely the constant

$$\displaystyle{ c_{n}(\alpha ) =\sup _{\mathop{}_{p\neq 0}^{p\in \mathcal{P}_{n}}}\frac{\Vert p^{{\prime}}\Vert _{w_{\alpha }}} {\Vert p\Vert _{w_{\alpha }}} \,, }$$

and we are also interested in its asymptotic value

$$\displaystyle{ c(\alpha ) =\lim _{n\rightarrow \infty }\frac{c_{n}(\alpha )} {n} \,. }$$

In this paper we obtain lower and upper bounds for both c _n (α) and c(α). Note that according to a result of P. Dörfler from 2002, c(α) = [j _{(α−1)∕2, 1}]⁻¹, with j _ν, 1 being the first positive zero of the Bessel function J _ν (z), hence our bounds for c(α) imply bounds for j _{(α−1)∕2, 1} as well.