  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 $\varpi_\mu(x) = (1-\|x\|^2)^\mu$ for
$\mu > -1$. Our main result shows that, for $s \in \mathbb{N}$, $$ E_n(f)_\mu
\le c n^{-2s}[E_{n-2s}(\Delta^s f)_{\mu+2s} + E_{n}(\Delta_0^s f)_{\mu}], $$
where $\Delta$ and $\Delta_0$ are the Laplace and Laplace-Beltrami operators,
respectively. We also derive a bound when the right hand side contains odd
order derivatives.

Classical Analysis and ODEs