Markov $L_2$ inequality with the Gegenbauer weight

Abstract

For the Gegenbauer weight function $w_{\lambda }(t)=(1-t^2{)}^{\lambda -1/2}$, $\lambda >-1/2$, we denote by $\Vert \cdot {\Vert }_{w_{\lambda }}$ the associated $L_2$-norm, $$\Vert f{\Vert }_{w_{\lambda }}:=({\int }_{-1}^1{w}_{\lambda }(t)f^2(t)dt{)}^{1/2}.$$ We study the Markov inequality $$\Vert p^{\prime }{\Vert }_{w_{\lambda }}\le c_n(\lambda )\Vert p{\Vert }_{w_{\lambda }},p\in {\mathcal{P}}_n,$$ where ${\mathcal{P}}_n$ is the class of algebraic polynomials of degree not exceeding $n$. Upper and lower bounds for the best Markov constant $c_n(\lambda )$ are obtained, which are valid for all $n\in \mathbb{N}$ and $\lambda >-\frac{1}{2}$.

arXiv comment14 pages