Approximation by spherical polynomials in $L^{p}$ for $p<1$

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 $f\in L^{p}(\mathbb{S}^{d})$, $0<p<1$:
$$ \|f-E_{1}f\|_{p}\le (1+2K)^{1/p}\inf_{u\in \Pi_{n}^{d}}\|f-u\|_{p}, $$
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.