Geodesic
Sharp Bernstein Inequalities for Jacobi--Dunkl Operators
Matematičeskie zametki, Tome 112 (2022) no. 5, pp. 770-783

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We find sharp constants in the Bernstein inequality $$ \|\Lambda_{\alpha,\beta}^rf\|\le M\|f\| $$ for the Jacobi–Dunkl differential-difference operator $$ \Lambda_{\alpha,\beta}f(x) =f'(x)+\frac{A'_{\alpha,\beta}(x)}{A_{\alpha,\beta}(x)} \frac{f(x)-f(-x)}{2}\,. $$ Here $n,r\in\mathbb N$, $f$ is a trigonometric polynomial of degree $\le n$, the norm is uniform, $\alpha,\beta\ge -1/2$, and $A_{\alpha,\beta}(x)=(1-\cos x)^\alpha(1+\cos x)^\beta|{\sin x}|$ is the Jacobi weight. In the spaces $L_p$ with Jacobi weight, upper bounds are obtained.
Keywords: Bernstein inequality, Jacobi–Dunkl operator
Mots-clés : sharp constant. 
@article{MZM_2022_112_5_a9,
     author = {O. L. Vinogradov},
     title = {Sharp {Bernstein} {Inequalities} for {Jacobi--Dunkl} {Operators}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {770--783},
     publisher = {mathdoc},
     volume = {112},
     number = {5},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2022_112_5_a9/}
}
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O. L. Vinogradov. Sharp Bernstein Inequalities for Jacobi--Dunkl Operators. Matematičeskie zametki, Tome 112 (2022) no. 5, pp. 770-783. http://geodesic.mathdoc.fr/item/MZM_2022_112_5_a9/