Refinement of Bernstein–Nikolskii constant for the sphere with Dunkl weight in the case of octahedron group
https://doi.org/10.22405/2226-8383-2021-22-5-354-358
Abstract
We continue the study of the sharp Bernstein–Nikolskii constants for spherical polynomials in the space 𝐿𝑝(S𝑑) with the Dunkl weight. We consider the model case of the octahedral
reflection group Z𝑑+1 2 and weight Π︀𝑑+1 𝑗=1 |𝑥𝑗 |2𝜅𝑗 when the explicit form of the Dunkl intertwining operator is known. We show that for min 𝜅 = 0 the multidimensional problem is reduced to the one-dimensional problem for the Gegenbauer weight, otherwise not.
About the Authors
Dmitriy Victorovich Gorbachev
Tula State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
Nikolai Nikolaevich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University; Tula State University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences, associate
professor
Ivan Anatol’evich Martyanov
Tula State University
Russian Federation
Russian Federation
postgraduate student
References
1. Gorbachev, D.V. & Dobrovol’skii, N.N. 2020. “Nikolskii–Bernstein constants in 𝐿𝑝 on the sphere with Dunkl weight”, Chebyshevskii Sbornik, vol. 21, no. 4, pp. 302–307. (In Russ.)
2. Dai, F. & Xu, Yu. 2013. “Approximation theory and harmonic analysis on spheres and balls”, Springer, N.Y.
3. Dai, F. & Xu, Yu. 2015. “Analysis on ℎ-harmonics and Dunkl transforms”, Birkhauser/Springer, Basel, CRM Barcelona.
4. Gorbachev, D.V., & Mart’yanov, I.A. 2020. “Bounds of the Nikol’skii polynomial constants in 𝐿𝑝 with Gegenbauer weight”, Trudy Inst. Mat. i Mekh. UrO RAN, vol. 26, no. 4, pp. 126–137.
5. (In Russ.)
Review
For citations:
Gorbachev D.V., Dobrovol’skii N.N., Martyanov I.A. Refinement of Bernstein–Nikolskii constant for the sphere with Dunkl weight in the case of octahedron group. Chebyshevskii Sbornik. 2021;22(5):354-358. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-5-354-358
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