Exact $L_2$ Bernstein-Markov inequalities for generalized weights (2411.16359v1)
Abstract: In this paper, we obtain some exact $L_2$ Bernstein-Markov inequalities for generalized Hermite and Gegenbauer weight. More precisely, we determine the exact values of the extremal problem $$M_n2(L_2(W_\lambda),{\rm D}):=\sup_{0\neq p\in\mathcal{P}n}\frac{\int_I\left|{\rm D} p(x)\right|2W\lambda(x){\rm d}x}{\int_I| p(x)|2W_\lambda(x){\rm d}x},\ \lambda>0,$$ where $\mathcal{P}n$ denotes the set of all algebraic polynomials of degree at most $n$, ${\rm D}$ is the differential operator given by $${\rm D}=\Bigg{\begin{aligned}&\frac {\rm d}{{\rm d}x}\ {\rm or}\ \mathcal{D}\lambda, &&{\rm if}\ W_\lambda(x)=|x|{2\lambda}e{-x2}\ {\rm and}\ I=\mathbb R, \&(1-x2){\frac12}\,\frac {\rm d}{{\rm d}x}\ {\rm or}\ (1-x2){\frac12}\,\mathcal{D}_\lambda, &&{\rm if}\ W_\lambda(x):=|x|{2\lambda}(1-x2){\mu-\frac 12},\mu>-\frac12\ {\rm and}\ I=[-1,1],\end{aligned} $$ and $\mathcal{D}\lambda$ is the univariate Dunkl operator, i.e., $\mathcal{D}\lambda f(x)=f'(x)+\lambda{(f(x)-f(-x))}/{x}$. Furthermore, the corresponding extremal polynomials are also obtained.