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

Abstract

Let $w_{\lambda }(t)=(1-t^2{)}^{\lambda -1/2}$, $\lambda >-1/2$, be the Gegenbauer weight function, and $\Vert \cdot \Vert$ denote the associated $L_2$-norm, i.e., $$\Vert f\Vert :=({\int }_{-1}^1{w}_{\lambda }(t)|f(t){|}^{2}dt{)}^{1/2}.$$ Denote by ${\mathcal{P}}_n$ the set of algebraic polynomials of degree not exceeding $n$. We study the best (i.e., the smallest) constant $c_{n,\lambda }$ in the Markov inequality $$\Vert p^{\prime }\Vert \le c_{n,\lambda }\Vert p\Vert ,p\in {\mathcal{P}}_n,$$ and prove that $$c_{n,\lambda }<\frac{(n+1)(n+2\lambda +1)}{2\sqrt{2\lambda +1}},\lambda >-1/2.$$ Moreover, we prove that the extremal polynomial in this inequality is even or odd depending on whether $n$ is even or odd.

arXiv comment10 pages

Cited by in corpus (2)

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