Best polynomial approximation on the unit ball

Abstract: 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.