Markov $L_2$-inequality with the Laguerre weight

Abstract

Let $w_{\alpha }(t):=t^{\alpha }{e}^{-t}$, where $\alpha >-1$, be the Laguerre weight function, and let $\Vert \cdot {\Vert }_{w_{\alpha }}$ be the associated $L_2$-norm, $$\Vert f{\Vert }_{w_{\alpha }}={\left\{{\int }_0^{\infty }|f(x){|}^2{w}_{\alpha }(x)dx\right\}}^{1/2}.$$ 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 $$\Vert p_n^{\prime }{\Vert }_{w_{\alpha }}\le c_n(\alpha )\Vert p_n{\Vert }_{w_{\alpha }},p_n\in {\mathcal{P}}_n,$$ namely the constant $$c_n(\alpha ):=\underset{p_n\in {\mathcal{P}}_n}{\sup }\frac{\Vert p_n^{\prime }{\Vert }_{w_{\alpha }}}{\Vert p_n{\Vert }_{w_{\alpha }}}.$$ We derive explicit lower and upper bounds for the Markov constant $c_n(\alpha )$, as well as for the asymptotic Markov constant $$c(\alpha )=\underset{n\to \infty }{\lim }\frac{c_n(\alpha )}{n}.$$

12 pages

Cited by in corpus (1)

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