Polynomial approximation with doubling weights

Among other things, we prove that, for a doubling weight $w$, $0< p\leq\infty$, $r\in{\mathbb N}_0$, and $0<\alpha <r+1 - 1/\lambda_p$, we have $$E_n(f)_{p, w_n} = O(n^{-\alpha}) \iff \omega_\varphi^{r+1}(f, n^{-1})_{p, w_n} = O(n^{-\alpha}),$$ where $\lambda_p := p$ if $0 < p < \infty$, $\lambda_p :=1$ if $p=\infty$, $\|f\|_{p,w} := \left( \int_{-1}^1 |f(u)|^p w(u) du \right)^{1/p}$, $\|f\|_{\infty,w} := {\mathop{\rm ess\: sup}\nolimits}_{u\in [-1,1]} \left(|f(u)| w(u)\right)$, $\omega_\varphi^r(f, t)_{p, w} := \sup_{0<h\leq t} \| \Delta_{h\varphi(\cdot)}^r(f,\cdot)\|_{p, w}$, $E_n(f)_{p, w} := \inf_{P_n\in\Pi_n} \|f-P_n\|_{p,w}$, and $\Pi_n$ is the set of all algebraic polynomials of degree $\leq n-1$. Equivalence type results involving related $K$-functionals and realization type results (obtained as corollaries of our estimates) are also discussed. Finally, we mention that (*) closes a gap left in the paper by G. Mastroianni and V. Totik "Best Approximation and moduli of smoothness for doubling weights", J. Approx. Theory {\bf 110} (2001), 180-199, where ($\ast$) was established for $p=\infty$ and $\omega_\varphi^{r+2}$ instead of $\omega_\varphi^{r+1}$ (it was shown there that, in general, ($\ast$) is not valid for $p=\infty$ if $\omega_\varphi^{r+1}$ is replaced by $\omega_\varphi^{r}$).