Best polynomial approximation on the unit ball

Let $E_n(f)_μ$ be the error of best approximation by polynomials of degree at most $n$ in the space $L^2(\varpi_μ, \mathbb{B}^d)$, where $\mathbb{B}^d$ is the unit ball in $\mathbb{R}^d$ and $\varpi_μ(x) = (1-\|x\|^2)^μ$ for $μ> -1$. Our main result shows that, for $s \in \mathbb{N}$, $$ E_n(f)_μ\le c n^{-2s}[E_{n-2s}(Δ^s f)_{μ+2s} + E_{n}(Δ_0^s f)_μ], $$ where $Δ$ and $Δ_0$ are the Laplace and Laplace-Beltrami operators, respectively. We also derive a bound when the right hand side contains odd order derivatives.