Approximation by spherical polynomials in 𝐿𝑝 for 𝑝 < 1
https://doi.org/10.22405/2226-8383-2021-22-3-453-456
Abstract
Based on recently proved estimates for the 𝐿1-Nikolskii constants for S𝑑 and R𝑑, effective bounds for the constant 𝐾 are given in the following inequality of the type Brown–Lucier for functions 𝑓 ∈ 𝐿𝑝(S𝑑), 0 < 𝑝 < 1:
‖𝑓 − 𝐸1𝑓‖𝑝 6 (1 + 2𝐾)1/𝑝 inf 𝑢∈Π𝑑 𝑛 ‖𝑓 − 𝑢‖𝑝, where Π𝑑
𝑛 is the subspace of spherical polynomials, 𝐸1𝑓 is a best approximant of 𝑓 from Π𝑑𝑛 in the metric 𝐿1(S𝑑). The results are generalized to the case of the Dunkl weight.
About the Authors
Dmitry Viktorovich Gorbachev
Tula State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
Nikolai Nikolaevich Dobrovol’skii
Tula State University, Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
References
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5. Gorbachev, D.V. & Dobrovolskii, N.N. 2020. “Nikolskii–Bernstein constants in 𝐿𝑝 on the sphere
6. with Dunkl weight”, Chebyshevskii Sbornik, vol. 21, no. 4, pp. 302–307. (In Russ.)
7. Gorbachev, D.V. & Mart’yanov, I.A. 2020. “Bounds of the Nikol’skii polynomial constants in
8. 𝐿𝑝 with Gegenbauer weight”, Trudy Inst. Mat. i Mekh. UrO RAN, vol. 26, no. 4, pp. 126–137.
9. (In Russ.)
Review
For citations:
Gorbachev D.V., Dobrovol’skii N.N. Approximation by spherical polynomials in 𝐿𝑝 for 𝑝 < 1. Chebyshevskii Sbornik. 2021;22(3):453-456. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-3-453-456
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