A weighted estimate of best approximations in $L_2(\Omega)$

The best approximation $\widetilde f$ [in the space $L_2(\Omega)$] of a function $f$, satisfying a Lipschitz condition with exponent $\alpha$, $0\le\alpha\le1$, with the aid of certain spaces of local functions, dependent on a parameter $h$, is discussed. We obtain the estimate
$$ \|f-\widetilde f\|_\beta\le\widetilde C(f)h^{\min\{\alpha,\beta\}}, $$
where
$$ \|u\|_\beta=\max_{x\in\overline\Omega}|r^\beta u(x)|,\quad\beta\ge0\quad u\in C(\overline\Omega) $$
and $r=r(x)$ is the distance of the point $x$ from the boundary of the domain $\Omega$.