arXiv subjects

Dragomir Aleksov

Publications and source records attributed to Dragomir Aleksov.

2 recordsLinked to original sources

arXiv2017

Markov $L_2$ inequality with the Gegenbauer weight

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

math.CA↗

arXiv2015

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

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

math.CA↗