Approximation by spherical polynomials in 𝐿𝑝 for 𝑝 &lt; 1

Dmitry Viktorovich Gorbachev,Nikolai Nikolaevich Dobrovol’Skii

doi:10.22405/2226-8383-2021-22-3-453-456

Approximation by spherical polynomials in 𝐿𝑝 for 𝑝 < 1

Dmitry Viktorovich Gorbachev, Nikolai Nikolaevich Dobrovol’Skii

Open Access

https://doi.org/10.22405/2226-8383-2021-22-3-453-456

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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.

Highlights

effective bounds for the constant K are given in the following inequality of the type Brown
where Πdn is the subspace of spherical polynomials
The results are generalized to the case of the Dunkl weight

Summary

Introduction

2021, "Approximation by spherical polynomials in Lp for p < 1" , Chebyshevskii sbornik, vol 22, no. Установим одно интересное неравенство для приближений функций сферическими полиномами в пространстве Lp при 0 < p < 1, возникшее в связи с работами [1, 3, 4, 2]. Из результатов работы [1] вытекает следующее утверждение. Пусть f ∈ Lp(Sd), E1f — элемент наилучшего приближения функции f полиномами из Πdn в метрике L1(Sd) (он существует). ‖f − E1f ‖p (1 + 2K)1/p inf ‖f − u‖p, (1)

Results

Conclusion