Geodesic
Approximation by spherical polynomials in $L^{p}$ for $p1$
Čebyševskij sbornik, Tome 22 (2021) no. 3, pp. 453-456

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Based on recently proved estimates for the $L^{1}$-Nikolskii constants for $\mathbb{S}^{d}$ and $\mathbb{R}^{d}$, effective bounds for the constant $K$ are given in the following inequality of the type Brown–Lucier for functions $f\in L^{p}(\mathbb{S}^{d})$, $0$: $$ \|f-E_{1}f\|_{p}\le (1+2K)^{1/p}\inf_{u\in \Pi_{n}^{d}}\|f-u\|_{p}, $$ where $\Pi_{n}^{d}$ is the subspace of spherical polynomials, $E_{1}f$ is a best approximant of $f$ from $\Pi_{n}^{d}$ in the metric $L^{1}(\mathbb{S}^{d})$. The results are generalized to the case of the Dunkl weight.
Keywords: spherical polynomial, best approximant, Dunkl weight.
Mots-clés : Nikoskii constant 
D. V. Gorbachev; N. N. Dobrovolskii. Approximation by spherical polynomials in $L^{p}$ for $p<1$. Čebyševskij sbornik, Tome 22 (2021) no. 3, pp. 453-456. http://geodesic.mathdoc.fr/item/CHEB_2021_22_3_a28/
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[2] Dai F., Gorbachev D., Tikhonov S., “Estimates of the asymptotic Nikolskii constants for spherical polynomials”, Journal of Complexity, 65 (2021) | DOI | MR

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