Geodesic
Refinement of Bernstein--Nikolskii constant for the sphere with Dunkl weight in the case of octahedron group
Čebyševskij sbornik, Tome 22 (2021) no. 5, pp. 354-358

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We continue the study of the sharp Bernstein–Nikolskii constants for spherical polynomials in the space $L^{p}(\mathbb{S}^{d})$ with the Dunkl weight. We consider the model case of the octahedral reflection group $\mathbb{Z}_{2}^{d+1}$ and weight $\prod_{j=1}^{d+1}|x_{j}|^{2\kappa_{j}} $ when the explicit form of the Dunkl intertwining operator is known. We show that for $\min \kappa=0$ the multidimensional problem is reduced to the one-dimensional problem for the Gegenbauer weight, otherwise not.
Keywords: spherical polynomial, reproducing kernel, Dunkl weight, Bernstein–Nikoskii constant. 
@article{CHEB_2021_22_5_a25,
     author = {D. V. Gorbachev and N. N. Dobrovol'skii and I. A. Martyanov},
     title = {Refinement of {Bernstein--Nikolskii} constant for the sphere with {Dunkl} weight in the case of octahedron group},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {354--358},
     publisher = {mathdoc},
     volume = {22},
     number = {5},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2021_22_5_a25/}
}
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D. V. Gorbachev; N. N. Dobrovol'skii; I. A. Martyanov. Refinement of Bernstein--Nikolskii constant for the sphere with Dunkl weight in the case of octahedron group. Čebyševskij sbornik, Tome 22 (2021) no. 5, pp. 354-358. http://geodesic.mathdoc.fr/item/CHEB_2021_22_5_a25/