Exact $L_2$ Bernstein-Markov inequalities for generalized weights

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 $$M_n^2(L_2(W_{\hat{I}}»),\mathrm{D}):=\underset{0\ne p\in {\mathcal{P}}_n}{\sup }\frac{{\int }_I{\left|\mathrm{D}p(x)\right|}^2{W}_{\hat{I}}»(x)\mathrm{d}x}{{\int }_I|p(x){|}^2{W}_{\hat{I}}»(x)\mathrm{d}x},\hat{I}»>0,$$ where ${\mathcal{P}}_n$ denotes the set of all algebraic polynomials of degree at most $n$, $\mathrm{D}$ is the differential operator given by $$\mathrm{D}=\{\begin{array}{rlrl}& \frac{\mathrm{d}}{\mathrm{d}x}\mathrm{or}{\mathcal{D}}_{\hat{I}}»,& & \mathrm{if}{W}_{\hat{I}}»(x)=|x{|}^{2\hat{I}»}{e}^{-x^2}\mathrm{and}I=\mathbb{R},\\ & (1-x^2{)}^{\frac{1}{2}}\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{or}(1-x^2{)}^{\frac{1}{2}}{\mathcal{D}}_{\hat{I}}»,& & \mathrm{if}{W}_{\hat{I}}»(x):=|x{|}^{2\hat{I}»}(1-x^2{)}^{\hat{I}¼-\frac{1}{2}},\hat{I}¼>-\frac{1}{2}\mathrm{and}I=[-1,1],\end{array}$$ and ${\mathcal{D}}_{\hat{I}}»$ is the univariate Dunkl operator, i.e., ${\mathcal{D}}_{\hat{I}}»f(x)=f^{\prime }(x)+\hat{I}»(f(x)-f(-x))/x$. Furthermore, the corresponding extremal polynomials are also obtained.

23 pages