On the Markov inequality in the $L_2$-norm with the Gegenbauer weight

Abstract

Let $w_{\lambda }(t):=(1-t^2{)}^{\lambda -1/2}$, where $\lambda >-\frac{1}{2}$, be the Gegenbauer weight function, let $\Vert \cdot {\Vert }_{w_{\lambda }}$ be the associated $L_2$-norm, $$\Vert f{\Vert }_{w_{\lambda }}={\left\{{\int }_{-1}^1|f(x){|}^2{w}_{\lambda }(x)dx\right\}}^{1/2},$$ and denote by ${\mathcal{P}}_n$ the space of algebraic polynomials of degree $\le n$. We study the best constant $c_n(\lambda )$ in the Markov inequality in this norm $$\Vert p_n^{\prime }{\Vert }_{w_{\lambda }}\le c_n(\lambda )\Vert p_n{\Vert }_{w_{\lambda }},p_n\in {\mathcal{P}}_n,$$ namely the constant $$c_n(\lambda ):=\underset{p_n\in {\mathcal{P}}_n}{\sup }\frac{\Vert p_n^{\prime }{\Vert }_{w_{\lambda }}}{\Vert p_n{\Vert }_{w_{\lambda }}}.$$ We derive explicit lower and upper bounds for the Markov constant $c_n(\lambda )$, which are valid for all $n$ and $\lambda$.

arXiv comment20 pages