Best polynomial approximation on the unit ball

Let $E_n(f{)}_{\mu }$ be the error of best approximation by polynomials of degree at most $n$ in the space $L^2({\varpi }_{\mu },{\mathbb{B}}^d)$, where ${\mathbb{B}}^d$ is the unit ball in ${\mathbb{R}}^d$ and ${\varpi }_{\mu }(x)=(1-\mathrm{\parallel }x{\mathrm{\parallel }}^2{)}^{\mu }$ for $\mu >-1$. Our main result shows that, for $s\in \mathbb{N}$, $$E_n(f{)}_{\mu }\le c{n}^{-2s}[E_{n-2s}({\mathrm{\Delta }}^{s}f{)}_{\mu +2s}+E_n({\mathrm{\Delta }}_0^{s}f{)}_{\mu }],$$ where $\mathrm{\Delta }$ and ${\mathrm{\Delta }}_0$ are the Laplace and Laplace-Beltrami operators, respectively. We also derive a bound when the right hand side contains odd order derivatives.